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THE

BUILDERS Complete Assistant,

O R, A

LIBRA

O F

ARTS and SCIENCES,

Absolutely Neceffary to be Underftood by
BUILDERS and WORKMEN in General.

V I Z.

IV. MENSURATION.

V. PLAIN TRIGONOMETRY.

VI. SURVEYING of Land, i$t,

VII. MECHANICK POWERS.

VIII. HYDROSTATICKS.

I.ARITHMETICK.VnlgarandDemal ia whole N umbers and Fraflio

II. GEOMETRY, Lineal, Superficand Solid.

Hi. ARCHITECTURE, Umverfa

Illustrated by above Thirteen Hundred Examples of Lines,
Superficies, Solids, Mouldings, Pedejtals, Columns, Pilaflers,
Entablatures, Pediments, Impefts, Block Cornices, Ruji�k �hioins_}Fruiilifpieccs, Arcades, Portico's, &c.

PROPORTIONED

By MODULES and MINUTES,

According to ANDREA PALLADIO,

AND BY

E O^U A L PARTS.

Likewife great Varieties of Tniffed Roofs, Timber Bridges, Centerings,
Arches, Groins, Twifted Rails, Compartments, Obelifques, Vafs, Pedejtals
for Bit/lo's, Sun-Dials, Fonts,Sic. and Methods for railing heavy Bodies by
llie Force of Levers, Pulleys, Axes in Pcritrochio, Scre-ws, and Wedges ;
As alfo Water, by the common Pump, Crane, &c,
W H E R E l N
The Properties and Pressure of Air on Water, &c. are explained.
The Whole exemplified by 77 large Quwto Copper-Plates.

By 1IT"T"n~g""l- e y7~

T H � F O U R T H E D ITION.

LONDON:

Printed for C. and R. Ware at th; Bible and Sun on Ludgrtte-bilL

M.JB�C.LXV1.

(Pi.#f -\o s 6 it. the Two Volumes.)

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TABLE

OF THE

PLATES, and PAGES wherein they
are explained.

Plates ] Pages where explained. Plates.] Pages where explained.

�. P Age 6_3> 63,64,6c, 66.
�l JL 66, 67,68, 6q, 70.71.

tu' ⁷³'�⁴' ⁷ⁱ'⁷⁶' 77- 78.

IV. 79, 80, Si, 82,83, 24. 85.

V. 84, 85, 86, 87.

VI. 87,88, 89,90, 91, 92, 93.
VII ₉, 9Z.93,94,95,96.

V11I. 95,97,98, 99, IOO, 101.

IX 101, ir-, 103.

X. 106, 107, 114, 1 ij, 159.

xr .₃₉,140.

Ail. 120, 121.
XII [. 120, 121.

XIV. 129, 14,-.

XV. 110, 124.

XVI. 142, 143,

XVII. 142, 141.

XVIII. 104, i_4?.

XIX. 105,106,107.

XX. 103, 107, 108, 156.

XXI. 1C9, 110, 1 n.

XXII. 109, in, 112.

XXIII. 113, 114.

XXIV. 113,,14,115-, ,16,137,

AAV, 117.

XXVI. 1 j 6, 117.

XXVII. n₇.

AX1X. II_9j _I22j i2₃

XXX. 122, I2₃₎ ,34, ,_4I.,

XXXI, 122, 123. 124.

XXXIr. 105, r:8. 125. 1.-6, ir:

XXXIII. ,�_Sj _la6) ,_z8j ,_a9

XXXIV. IOC.I2S.

XXXV. 127.

XXXVI. ,₃o.

XXXVII. 1 jo.

XXXVIII. 117, 150, i$i;

XLI. 105, 132, 133, 134, 138.

XLII. ice, in, 124, 131, 134, 135.

XLIIF. 131, 136, 141, 146.

XL1V. 115, 117, 130,141.

XLV. 138, 144.

XLVI. 145, 146.

XLVII. 138, «44, 146.

XLVIIF. 146.

XLIX. 146, 147. ' '

L. 147.

LI. 150.

LIL 149, 150, 1 ��.

LIU. 149, 151, ici.

LIV. 152.

LV. 152, 153,

LVI. 152, 156, 157.

LV!I. 153, 1J4.

I.VIII. .155.

LIX. 154, 156.

I'X. i>5> ^!57> ^!5^S-

LXI. 159.

LXII. 159, 160.

I,XIII. 1C9. 110.161.

LXIV. 161, 162, 163.

LXV. 164.

LXVI. 16c, 166.

I,XVII. 163,166.167.

LXVIII 111, 166.

I.X1X. 168..

LXX. 163.

LXXI. 163,169.

II. 169, 173, 17c.
LXXIII. 173,174.175,1-6.
LXXiV. .73,174, 175,(76.
LXXV. 177, 178, 180, i8i,18i, i8i-

18.1.
L'XXVf. 185,186", 187,18S, 189, -.90,

101, �Q2.

LXXViJ, 193, 194, 195, ;c�, ;�.;-.

�9§, l₉0, 2CO.'

A TAP L �

XXXIX. 10c, 132, n;

J �7-

XL. ,«,

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T A B L E

O F T H E

CONTENTS.

Page

XX. Of Pediments 146

XXI. Of trufi'ed Partitions 147

XXII. Of naked Flooring 148

XXIII. Of Roofs ' 1 �o

XXIV. Of Angle Biackets 156
XXV. Of Niches ibid.

XXVI. Of Timber Bridges 159

XXVII. Of Brick and Stone Arches

161

XXVIII. Of centering to Arches and
Groins 163

XXIX. Of-Stair-cafes 165

XXX. Of Ornaments for Buildings-

and Gardens it>8

PART IV. Of Mensuration.
Lect. I. Rules lor meafuriitg Super-
ficies 169
I'. Rules formcafuringSolids 172

PART V. Plain Trigonometry.
Lect. I. Of the Solution of plain Tri-
angles 177

II. Of Heights and Difiances 179

PART VI. Of Surveying Lands, fcV.

1S2
PART VII. Of Mechanicks.
Lect. 1. Dcfinitionsof Matter, Gra-
vity, and Motion 18,
II. Of the Laws of Nature 187

III. Of mechanical Powers in ge-
neral 190

IV. Of the Balance 192
V. Of the Lever 193

VI. Of the Pulley I93

VII. Ofthe AxisinPeritrochio 196

VIII. Ofthe Wedge ibid.

IX. Of the Screw 197

X. Ofthe Velocities with which,

Bodies are railed 198

PARTVili; Of I-Ivqro*t.\tscs 199

T H E

PART I. Of Arithmetics.


	i	age
Lect. T.	/AF Numeration \J Of Addition	i
II	/AF Numeration \J Of Addition	6
IIF.	Of Subtraction	24
IV.	Of Multiplication	33
V.	Of Divifion	43
VI.	Or Reduction	47
VI�.	Of the Golden Rule	49
VIII.	Of Fractions	5'
IX.	Of fquare and cube Root	s 56

PART II. O/Geometry.

Introduction 61
Lect. I. Of Definitions 62
II. Of Angles 7«
Hi; Of Line» 73
IV'. Of plain Figures 79
V. Of inscribing geometri-
cal Figures 90

VI. Of proportional Lines 93

PART III. O/Architecturi.
Lect. I. Of Mouldings �7

IT. Of making Scales �cz

III. Of the principal Parts of

an Order 103

JV. Of the Tufcan Order 105

V. Oi'i ulcanFrontifpieces,

fjff. log

VI- Or the Dorick Order 113

VII. Of the lonick Order 119
VIII. Ofthe Corinthian Order 125

IX. Of the CorhpofiteOrder 1 yz

X. Queries on the Orders 130

Xf. Of the GrotcfqueOrder 138

XII. Of the Attlck Order ibid,

XIT.L Of wreathed Columns 139

XIV. Of Flutes and Fillets ibid.

XV. Of placing Columns 141

XVI. Of Ornaments fortheEnrich-

rnents of the 5 Orders 142

X'vil. Of rufticating Columns 144

XVIIL Of-Block Cornices ibU,

;ilX. Of Doors aadWaidsws 1*5

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		THE Builders Complete Assistant, Q R A Library of Arts and Sciences^ &c.

			PART I. Of A R I T II M E T I C K. Sect. i. Of the feveral Parts of Arithmetick, and the Notation or Art of expreffing Numbers by Characters, and to read their Values. P'What is Arithmetick ? _f * M. Arithmetick is a G reek Word, and .imports an Art or Science, that teaches the Ufes and Properties of Figures, or right Art of numbering. P. What doth right numbiring confifl of ? M: To denote any given Quantity with proper Characters, and to exprefs them by Wotd?, whichis called Notation. P. Hozv many arc the Kind! of Notation ? M. There are manv Kinds of Notation by which Quantity is expreffed, but the moil ufual are Literal and Figurai. P. What is Literal Notation? M. The expreffing Numbers by Letters, and is therefore called Literal, and which was anciently made Ufe of by the Hebrews or Jetus, Chaldeans, Syrians, �Arabians, Perjians, and others of the Eailern Nations. 'The Greeks aUo expreffed Nurnbers by divers of their alphabetical Letters, and initial Capital Letters of fome of their numeral Words, as n ni7e, Five, A �Suttt,* Ten, E Efcajic, an Hundred, XX'tMoi, a Thoufand, HUo'otot, Ten Thoufand. P. Pray ivhat Kind of Letters are ufed norm for Notation ? M. Divers of the Roman Capitals, which Method it is very reafonable to be- lieve the Latin; firil took from the Greeks, as is very evident from the initial Letters of feveral of their num�ral Words, as follows ; 'viz, The Capital C which is the initial Letter of Centum, the Latin Word for an Hundred, is aow ufed of itfelf to fignify an Hundred. P. But pray hwu is half an hundredexprefs'd? M. By the Capital L. P. Pray ivhy is half an Hundred'exprefs''d by an L ? M. You muff, understand, that the ancient Form of the Capital C, was thus written E ; and as it then fignified an Hundred, therefore the Ancients fignified half a Hundred by one half Part of it, as thus l, which being like unto the Ca- pital L ; therefore Printers take the Liberty to denote half a Hundred by chat Letter. B P. I

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		2 Of N U MERATIO N. P. I thank you, Sir ; pray proceed. M. i will : The Capital Letter D, which is the initial Letter of Decern {the Z�f�» for Ten) was anciently ufed by the Latins to denote Ten, and one half thereof, as thus u, did alio denote five. Now as this half Letter hath more of the Likenris of the Capital V than of any other Capital, therefore Printer: and others have uled the V (inllead of the half Letter v) for Five ; and to denote i en, inflead of uiing the Capita! D, as the Ancients did, they join together two V's at their narrow Ends, the one upright, the other downright, in manner of the Capital l etter X, which now is ufed to denote Tert. Again, as Miile'tt Latin for a Thoufand, therefore the Ancients ufed the Capi- tal M to denote a thoufand, as it is now ufed at this Day; and as the old Cha- racter of the Capital M was this ®, whofe Right-hand Side being like unto the Capital D, therefore Printers, hfc. denote Five hundred by the Capital D. You are.alfo to note, that as this ancient M CD had fome Refemblar.ee of the Letter ] p'ae'd between two C's, of which one is turned the wrong Way, as thus CI3, therefore thofe Letters are now ufed by fome to denote a "1 houfand, jultead of the Letter M, and I3 to denote Five hundred, inrlead of the Letter D: P. Pray by tvhat CbaraSt-.r did the Ancients vfe to denote One? M. Both Greekt and Latins denoted One by one fingle Stroke, as being the na- tural and molt fimpre, Character of one fingle Thing ; and theiefore One is repre- fented by the Letter I. Now from thefe l�verai Characters the following Num. bers are expiefled by the Romans or Latins, viz. I One, II Two. Ill fhiee, IV or IIU Four, V'Five, VI Six, VII Seven, ViII Eight, IX Nine, XTen, XI Eleven, XII Twelve. XV Fifteen, XX Twenty, XXX Thirty, XL Forty, L Fifty, LX Sixty, LXX Seventy, LXXX Eighty, XC Ninety, C a Hundred, CC Two Hundred, CCCThree Hundred, CCCC Four Hundred, Dor [3, or la Five Hundred, DC Six Hundred, DCC Seven Hundred, M or CI3, or do, a Thoufand, I33 Five Thoufand, CCIq;) Ten Thoufand, I333 Fifty Thoufand, CCCI33-) an Hundred Thoufand, �)D>D Five Hundred Thoufand, CCCfclpnoO a Million, and fo MDCCXXXVIIl or CI3DCLXXXVTU de- notes the Date of the Year One thoufand Seven hundred and Thirty-eight. P. But, pray Sir, ivhy is Nine and Eleven denoted by tbr Jams Letters* ? M. As the I, being fet after the X, adds One to th� X and makes it Eleven, fo on the contray, when the I is fet before the X. as in Nine, it leflens its \ alue one, and therefore fignifies but Nine. For the lame Reafon the I placed before the V, Five, leflens its Value one, and fignifies but Four. The fame is alfo to be obferved of Forty, and Ninety, where the X, being fet before the L Fifty, k-fiens its Value T'en, andfignif.es but Fortv, and being placed before the C, a Hun- dred, leflens its Value Ten, and fignifies but Ninety. And it is further to be ob- ferved, that fome ufe I:X for to denote Eight, and XXC to denote Eighty, as being more concife The V and L are never repeated, nor are any of the other Characters repeated more than four times ; the I repeated four times, thus IIII, fignifies Four, but the Vis Five, notlllll. So likewife 4. C's, thus CCCC, fig- nifies Four Hundred, but Five Hundred is denoted by D or I 3, as aforefaid, and ftyt by CCCCC. Now as by this Method the Notation of Numbers by Letters is verv tedious, the Figura! .Notation was invented, as being more (expedite, P. What is Figurai Notatim ? M. The Manner of espreffing Quantities by the Ten Arabick Charaflers, 7 :- 1234567890, which fignify as follows, <viz. I one, 2 two, 3 three, 4 four, 5 five, 6 fix, 7 feven, 3 eight, g nine, o nought, Cypher, or nothing. P. Pray bo<wlong may tbeje C/jaraclers have been ufedin England ? M. Df. Wallis'. ;n his Treatife of Algebra, Page IZ, fays, they were intro- duced about the Year One thoufand One hundred and Thirty, which is Six hun- dred and Thirty Years fince. P. Htiv many diflinS Parts is Arithmetic', dividedinto ? M. Three; Two of which are properly called Natural, and the Third Art» f P. What are thvfe which you call Natural ? M. Th-r-

			»

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Of N U M ERATION. 3

M. The firft Part is that Kind of Arithrnetick which is called Vujgar, and
Which is the Doftrine of whole Numbers, and the mort plain and eafy, because
every Unit or One (which is called Integer) represents or fignifie* one entire
Thing or Quantity of fome Kind of Species, as a Nail, Lath, Brick, i3c The
f�cond Part is the Doftrine of broken Quantities, or Parts of Units or Integers,
wmch is called Vulgar Fractions, and wherein the Unit or integer is dWjded Into
a certain Number of even or uneven Parts. As for Example, if a Foot be the
given or propofed Unit or Integer, and be divided into twelve Inches; then one
�Inch becomes a Fraftion, or twelfth Part thereof, two Inches one fixth Part, three
Inches one fourth Part, four Inches one third Part thereof, CjV. This Part of
-Arithrnetick may be confidered either as pure, confuting of fractional Parts only,
each lefsthan a Unit ; as Quarters, Halves, is'c. or of Integers and fractional
Parts intermix'd, as one and a half, two and one third Part of one, f5>. The
third Part, which I call Artificial, is alio called Deamai Arithmetic!^ which
is an Artificial Method of working Fractions or broken Numbers in a mach eafier
Manner than that of vulgar Fractions, and which differs very little from vulgar
Arithrnetick. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^�

P._ Pray ivhy is this artificial Kind of Jri/hmetici called Decimal Arithmetic!: ?
From, the Latin Decern, Ten, into which every Integer is fupoofed

lubdivided. and indeed, in many Cafes, every Subdivision is fubdivided again
into iolefferParts, �V. Suppofe one Foot in Length be an Integer or Unit
given, and let it be divided into io equal Parts; then, we fay, the loot is de-
cimally divided, and if every tenth Part be decimal!', divided again i:i the like
Manner, then the Foot wi.l be divided into one hundred Parts, and is then faid
to be centefimally divided.

P. I und,_:rjtand you, Sir, anddefi.re to know in'the next Place, 'what Vfie is the
Cyfbervf, firr.ee that of itfelfi it fignifies nothing ?

M. 'Io augmentor increafe other Figures ; thus if next after the Figure I, I
P'ace an o, as thus io, they together fignify Ten, and 20 fgnifies Twenty 30
Fhirty, 40 Forty, &c. whersby the Value of every Figure is increal'ed ten times.

00 alfo if to 10 you add another Cypher, as thus loo, it will increafe the 10
ten times, and together fignify one Hundred. So in like Manner 200 fiunifies
^X"aa ^Hundred' 3°° ^t0'ee Hundred, 400 four Hundred, �V. And if to 100 you
add another Cypher, as 1000, it will increafe the 100 ten times, and make it
one Thoufand.

So in like Manner, 2000 lignifies two ThouVand, 3000 three Thoufand, 4000
four Thoufand, &c. Again, if to 1000, you add another Cypher, as thus looco,
the 1000 will be made ten Thoufand; and in like Manner if a Cypher be added
JP 2C0O, as thus 20000, they will fignify twenty Thoufand, and 30000 thirty

1 noufand, (s~e.

�il f^ery �^f/^' ^'^r' ^andf^uPpoJ^e that to 10000/ add one, tivo, or more Cyphers,

Jj^r ^C'^^waJ' '«e'eafie the Value oj'the former ten times?

M. 1 es ; for if to 10000 you add another Cypher, as thus ; 100,000. the Va-
M ^1S ^InCreafed,^from ^tcn Thoufand to one hundred Thoufand : and fo in like
Manner, the Addition of another Cypher to 100,000, as thus 1,000,000, will
increafe them into ten hundred Thoufand, which is called a Million.

Now if you confider the Increafe that has been made by the Addition of the
Cyphers, it will be very eafy to read or exprefs the true Value of any Number
of Cyphers, when written, or to write down any given Number propofed. But
to make this more plain, I will give you a Table of the Increafe of Unity by the
Addition of Cyphers, unto cne thoufand Millions, as follows.

2 2 1 Ur

'�

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4 O/ NU M E R A T I O N.

i, Unit.

to, Ten.

�co, one Hundred, or ten tinnes ten.

lcoo, one Thoufand, or ten times one hundred.

loooo, ten Thoufand.

100,000 one hundred Thoufand, or ten tim s ten Thoufand.

1,000,000 one Million, or ten times one hundred Fhoufand.

I0,OCO,0:O t. R Million.

Ioo,coo,ooo one hand red Million, or ten times ten Million.

10.0,000,oco one thoufand Million, or ten times one hundred Million.

P. I perfettly under/land the Increafe that is made by adding of a Cypher or Cy-
phers is any of the nine Figures ; but hoiv are Numbers to be underjiood nuhen di~
�vers of them are placed together, either ixiitb ornuitbout Cyptftrs, as 12, or IZ3, or

1234, &c.

¹ M. This I will make very eafy to you, and which increafe each otheVs Value,
juft iii the very fame Manner, as is done by tiie Addition of Cyphers; as for Ex-
ample, if to 1 I place 2, as thus, 12, they together fignify twelve, which 15
no more than thel'alue of the 2, placed in the Cypher's Place, added to 10 ; and
fo in like Manner 13 figriines thirteen. 14 fourteen, 15 fifteen, oV. fo likewife
23 lignifies twenty -three, 25 twenty.five, csY. So it is plain, th.it the firft Fi-
gures to the right fignify fo many Units, and the other fo many times Ten, as
their Characters exprefs. And therefore the firft Place is called the Place of Units,
and the f�cond thePlace of Tens. And as the Figures in the f�cond Place are
Tens, and fignify ten times, their Number of Units, fo Figures in the tliird Place
are Hundreds, and fignify ten times their Number of 'lens; as 123, wherein
the 1 lignifies one Hundred, the 2 twenty, and the 3 three, and the ^ Whole
one Hundred twenty and three.

To make this plain, oblerve the following Range of Figures, where every one
lignifies ten times the Figures it precedes, and-where their, Places are not only
expriiVd in Words at 1 ength, bin aifn divided into the feveral diftintt Co-
l�mes of Periods, by which they are to be numbered or exprcfled.

Tti'da L .1 >. ;

Unit*.

524.
HTU,

Hun-
dreds'.

Thds. Units.

444. 444.
HTU, HTU,

Thds,

HT�,

'i'i'.oo.-

faiWs.

Thds. Units.

444. 444-
HTU, HTU,
Thds. Billions.

of
Bill. E.

/ 77»

HTU,
Tril-
lions.
G.


	HTU,	*3 3 3'* HTU
	Ttids.	Qua
	of	drill
	Qtia-

	driil.	I.
	K.

HTU.

Thds.

Tril-
lions,
If.

Mill.
C.

Thds,
of
Mill.
D.

New it is tobeobferved, Firft, that the'Places of Numbers are always #hH
letkoi ed or numbered, from the right Hand to the left, and then read or ex-
prefied in Words from the left to the right. Soin the firft Column A to reckon
the Number 524, 1 begin at the 4, calling that Units, then proceed to the 2. call-
ing that Ten?, and hilly to the 5, calling that Hundreds. Saying, Units, Tens,
! Hundreds, which then read from the left to the right, faying, Five hundred
twenty and four Units. Again, if to the Column of Units I join 3 ■�; 1 the Column
of Thoijfiunds, 1 begin t� numerate them as before, faying Units as 4. Tens at 2,
Hundreds Et5, Thoufand: at 1, Ten6 of Thouiands a-t 7, and Hundr.d,s of Thou-

fc.nds.

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Of N U M E R A T I O N. 5

lands at 3 ; which I exprefs or read, Three Hundred Seventy and one Thoufand
five Hundred twenty and four, and fo in like manner any other Number.

Secondly, by the Capital Letters HTU, placed under the Figures of every
Column, you are to underftand the repeating of the Denominations of Units,
Tens, and Hundreds of the Units and f houfands of each Period.

". Pray, nvbat do you mean by a Period?

M. A Period is a Quantity expreffed by fix Figures, and are Units, Millions,
Billions, Trillions, Quadrillions, Quintillions, Sextiliions, &c. So here, the
Period of Units is the Columns A B, which are three Hundred feventy and
one Thoufand, five hundred twenty and four Units. The Period of Millions, is
the Columns C D, which are four Hundred forty and twoThoufand, four Hun-
dred forty and four Millions. The Period of Billions, is the Columns E F,
which are four Hundred forty and four thoufand, four Hundred and forty four
Billions, and fo the like of Trillions, Quadrillions.

P. Pray, ivbat do you mean by a Billion, Trillion, CSV. ?

M. A Billion is a Million of Millions, a Trillion is a Million of Millions cf
Millions, cSV. and therefore as you fee that every Column confifts but of three
places of Figures, njiz. of Units, Tens, and Hundreds, which in general begin
with Hundreds, altho' the Units may be Units, as in Column A, or Thoufands
as in Column B, or Millions as in Column C, &c. and as every Period con-
tains two Columns, or fix Figures, 'tis very eafy to read any range of Figures,
that can be propofed, as is evident from the aforefaid, which are thus expreffed in
Words, 'viz. three Hundred thirty and three Thoufand, three Hundred and
thirty three Quadrillions j feven Hundred feventy and feven Thoufand, feven
Hundred feventy and feven Trillions ; four Hundred forty and four Thoufand,
four Hundred forty and four Billions ; four Hundred forty and two Thoufand,
five Hundred forty and four Millions, three hundred feventy and one Thoufand,
five Hundred twenty and four.

But that you may perfectly underftand how to reckon or numerate any Range
of Figures propofed, and to truly underftand the value of their refpeftive Places,
I will therefore give you the following Table.

"2 �
a .« -a

o A a ^c

I' . i?*l*z

^rs �■» ^ j=s e >- c �>

H ^s-J ^oSjj 12

'2 8- U^°�l In 1234

W 5 3 �� ^H-Sl|�§ '234�O

�T

-S'S-S.S -3 °°K 1234567
b«3ms «s� 12345678

^S-o^�gh 123456789

^l^^m ,234567898

t%l:M ^Sm 1 2 3 4 5 6 7 8 9 8 7 v,

or^B^ 1234567898765

�g^ojffi «2345678987654

�-5 S H '23456789876543

.2 g H 123456789876543Z

51 i2 3 456789876 5 4.321

h 123456789876543ZIZ

,J> ² 3 4'5 6 7.8 9

2 3
B ft*

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		6 Of A D D IT I O N. In this Table, you fee a Demonftration of all that I have been informing you, with regard to the Places of Figures, exceeding each other ten times. P. Tit -very trite, Sir, fray is there any thing further to be known, relating to the Numeration and Exprejpon of Figures ? M. Yes, 'tis neceffary, and indeed a very ready way, in long Numbers, to place a Comma before every third Figure, thereby diltinguidling the Units, Tens, and Hundreds in every Column as aforefaid, and the Millions, Billions, Trillions, &e. by one, two. three, &c. Dots or Points placed under them, as is done in the loweimoii Line of the preceding fable. LF.CT. H. Of Addition-. PWhat is to be underfiood by Addition ? _a M. To collect, or gather into one Sum or Total, all fuch Sums or Quan- tities, as may be given or proposed, which is performed by the two following Rules. Rule I. Place all the Numbers given, to be added together; fo as that each Figure may Hand directly under thofe Figures of the (�iTw Value, �, Units under 7012 Units ; Tens under Tens ; Hundreds under Hundreds, &c. Which be- 540 ing done, (always) draw a Line under the iowermofr Number, tpVfeparate iz their Sum when found. As forExampte . Suppofe the Numbers 7012, 90 540, 12, and 90, were to be added together, they inult be placed as in ------- the Margin. Rule II. Always begin to add the given Quantities together, at the Place of Units; add- ing together all the Figures that Hand in that Column ; and if their Sum be lefs than Ten, fet it down underneath the laid Column; and if their Sum be more than Ten, fet down only the. Overplus, or odd Figure more than Ten or Tens ; and as many Tensas are contained in the Column of Units, fo many Ones you rauft carry and add unto the iecond Column of 'I'ens ; adding them, and all the Figures that (land in the Column of Tens together, in the fame Manner as thofe of the Column of Units were added : and io in like manner proceed to the Co- lumn of Hundreds, Thoufands, &c. until every Column is done ; and placing the whole Amount of the laft Column underneath the lame, the Sum aniiijg from thofe Additions, will be the total Amount required. Example 1. To 7543, add 234s' which place as in the Margin. Praclice. Begin at the Place of Units, and fay �, and 3 is 8, 7543 which being lefs than Ten, f t i: underneath that Column. Then 2345 proceed to the f�cond Column of Tens, and (ay 4 and 4 is 8, ------- which being lefs than ten. place it alfo underneath that Column. Sum. 9888 Again, in the third Column of Hundreds fay, 3 and 5 is 8, which being alfo lefs than ten, place it alio underneath that Co- lumn. Laflly. in the Column of Thoufands, fay 2 and 7 make 9, which place underneath that Column j then will the Product be equal to 9888, the true Sum required. Example II. To 9999999, add 8888, which place as in the Margin. Prailice. Beginning at the Column of Units, fay, 8 and 9 9990990 is I-; now, as 17 is 7 more than 10, therefore fet the 7 8888 underneath, and carry the ten unto the f�cond Column or �:�.�- Place of Tens, calling it one, and then fa>ing, one that I Sum 1000�S87 carry and 8 is 9, and gis 18; then place the 8 under the place of Tens, and carry the Ten unto the nest Column of Hundreds,

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Of ADDITION.

Hundreds, (becaufe 10 times 10 is one Hundred) faying, one that I carry and 8
is �, and 9 is 18 ; place the S under the Column of Hundreds, and carry one
for the Ten. to the next Column of Thousands, (becaufe to. Hundred is equal
to one Thoufand). Proceed in like manner, to the Column of tens of Thoufands,,
&c. and the true Sum required, will be 10008887.

Example III.

It is required to find the true Sum of 1430, more 234, more 456, more 78g,
more 91, which place as in the Margin.

Begin as before, at the Column of Units, faying, 1 and 9 is 10,

and6 is 16, and4is 20. Now as zx> contains ten twice, and none
remains, therefore under the Column of Uaks place a Cypher o,
and carry the two Tens to the Column of Tens, faying, z that I
carry, and 9 is n, and 8 is 19. and � is 24. and 3 is 27, and

1430
234
456

789

3 is 30. Now as 30 contains ten three times, and nothing remains,
therefore under the Column of Tens place an o, and carry 3 to the

9^!

Sum 20C�

three that 1 carry, and 7 is 10, and
ow as 20 contains 10 twice,

4 ^ls ^!4, and 2 is 16, and 4 is ?.o. j..^.......------------------- - .

and nothing remains, therefore place o under the Column of Hundreds ; ar.d
carrying the two Tens'to the place of Thoufands, fay, two that I carry and 1
make 3, whjcji being placid under the Pbce of Thoufands, the true Sum wil}
be 30C0, as required.

P. � underpaid your Method of cafting up every Column by itfelf and to carry the

■ _______________

proceed any further, pray dtmsnjlrate the Reafo.

M. I will with the )a!l Example, as followeth.

Add together each fingle Column of Figures by
itfelf, as if there were no ether Columns or figures
to be added, and underneath each Coiomn place
the Product.

Thus the Produft of the fir ft Column of Units,
is 20; the Product of the Column of Tens, is 28;
the Produft of the Column of Hundreds is 17 ;
and the Produft of the Column of Thoufands,
is 1.

' Now thefe four f�veral Produits bting added
together, in like manner, the Product .will be
3000, as following.

Sum required. But before you
eof?


	4	3
	z	3
	4	5
	7	8
		9


	2	z 8	°i
I	'7		s
		*A°l*
2	.9	1 *
I 2	0	0	1

The particular Produ&s of the
above four Columns.

By this continual Addition
of the Prod lifts, they atlength
terminate in theTotal, which
was to be demonftrated.

Their Sums added as above, a
f�cond Time.

Their Sums added as above, a
third Time.

J 3 I o I o [ o The Total or Product, as above.

P. � thank you, Sir, for this Demonftration, 'which has 'well informed me of the
renfon of carrying on the Tens, as they arife, to the next Column. Pray Sir, tc

B z pleafed,

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	8 Of A D DITIO N. f leafed, in the next-place, to -proceed to other Examples, for "'tis a P leaf ure to ivori, (when I know the Reafon of my Operations. M I am glad to find that you are fo pleafed with Demonfirations, which very few Youths care to trouble themfelves with. P. Such there are, then s no doubt of; hut did they knovu the Snueetnefs of De- tnonftration, they vjould friclly purfue it; for by this fingle Demonjiration only it is proved, That the whole is equal to all its Parts taken together, that is, I am taught to know that the Numbers which are propofed to be added together, are the federal Parts, and their total Sum found by Addition to be the whole. M. 'Tis true, you rightly conceive it, and you will as eafily conceive the reafon of the Proof of Addition. P. Pray ho<w do you prove the Truth of Addition ? M- By parting or feparating the given Quantities or Numbers into two (or more) Parcels, according to the Largenefs of the feveral Numbers contained therein ; and then adding up each Parcel by itfelf, their particular Sums being added together, the Sum total thereof will be equal to the other Sum total firit found, if the Work be truly performed; if otherwife, 'tis falfe, and care mull be taken to difcover and correct the Error, by going over the whole again. Example. IZ3456 123456 _c4²3^l65 21436c B 21436; 432615 A241356 241356 ----------- 423165 '---------- 855780 432615 579*77

M34957 (1) In this Example, the given Quantities are 123456, more 214365, more 141356, more 423165, more 432615, whofe Sum total is equal to 1434957. (2) Dividing thefe five given Quantities into two Parts, as the firft three by themfelves, as B, and the laft two by themfelves, as C ; their two Sums 01* Totals, added together, will be equal to the Total of the whole five Numbers taken together at A. The Sum Total of B, is 579177 The Sum Total of C, is 855780 The grand Total is - - 1434957 which is equal to the Total of the five given Numbers at A, as required. And foin like manner any other Sum or Quantities given, may be proved. P. � underfiandyou perfectly well, and can now prove the truth of any Total re- quired. Pray proceed to my further Information in other things neceffary to my Pur- pof e ? M. I will : and firft, with refpefl to Meafures of Length. P. What Meafures of Length are moft generally ufed in Bufenefs ? M. 1 he Foot, the Yard, and the Pole or Perch. P. How is the Foot commonly divided ? M. Generally into twelve equal Parts called Inches, and every of thofe Inches into eight, and fonietimes ten equal Parts, which laft is called a Decimal Diviiion of the Inch, and then the whole Foot is divided into 120 equal Parts. P. Is the Foot divided into any other Sorts of Parts or Divifons ? M. Yes, 'tis fometimes divided into one hundred Parts; which is called, the pentefimal DiviftOn of the P'oot, as has been already obferved ; by which the Di- p;t n liens of Glafs, Marble, &c. are taken. .p. Pray give kc fine Examples in thefe Kinds of Feet Meafure ? jV. I will ; and firft, of the Foot divided into \z Inches, and each Inch into fighj Farts,

		II, 4ddition

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Of ADDITION.

II. Addition of Feet, Inches, and 8ths.

Example I.-

Feet. Inch. Sth Parts.

11.

7
10

«3

14
18

J?»/*.
For every 8 Parts, carry 1
to the Inches ; for every 12
Inches, carry 1 to the Feet,
which add as Integers.

Collefl: into one Sum thefe
l�verai Lengths, tri».

Anfvver 115 8 3
Take the following Examples for Pra&ice.
Example II.
Feet. Inch. 8ths.

Example III.
Feet. Inch. 8ths.
123 IX 7

27
4

2
10
11

2'
11

3^ 7

4
o

172 11
75 10

Sum 66 2 6 Sum 411 2 4

I will now proceed to Examples of Foot Meafures^ centefimally divided ; that
is, the Foot divided into 100 equal Parts.

III. Addition of Feet and Parts.
Feet. Hund. Parts.

! 2 \ ,09

456 ,75

Collett into one Sum thefe
fevcral Lengths, tri».

7⁸9 »99
101 ,82

f071 ,29

172 ,25

2 ,50

Sum total 1937 ,69
_ Now, as the Foot is herefuppofed to be divided into 100 equal Parts, which.
is a Centefimal Pivifion ; therefore the Manner of adding thefe Sums together is
the very fanae as in whole Numbers ; the Tens of every Column being carried on
to the next, and the Remainders placed underneath : this is fo very plain, needs
no farther Examples hereof. But obferve, that as the Foot contains an 100 Parts,
75 Parts thereof are equal to f of a Foot ; 50 Parts thereof are equal to * a

¹ and pleafant, ana I am mach
Pray no<w proceed to the other Meafures you before mentioned ;
�which, 'f I remember right, you faid, lucre the Tard, and the Pole, or Perch,

IV. Addition of Tards, Quarters, and Nails,
M. The Yard is a Meafure of Length, containing three Feet precifely ; of
which other Meafures of Length are compofed, as the Pole, or Perch, Furlongs,
Miles and Leagues.

P. In tvhat Manner is the Tard vfually divided f

M. Into four equal Parts or Quarters, each (containing nine Inches) fubdivided
into four equal Parts, called Nails ; therefore the Divifions of a Yard, are Nails
and Quarters, and the Manner of theit Addition is performed by this Rule.

Example

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lo O/ ADDITION.

Example.
The following Lengths are to be added into one Sura.
Yds. Quar. Nails.
123 3 3 For every 4 Nails, carry i
456 1 2 to the Quarters; for every
789 214 Quarters carry I to the
«587 ' o 3 Yards, which add as Ia-
966 3 2 tegers.

Sum total required 3323 3 3
Take the following Examples for Practice.

Yds. Qu. Nails. Yds. Qu. Nails.

765 3 2 1456 3 3

834 2 1 325 1 3

799 * ° 444 2 3
888 2 2 . ------------------

------- Total 2227 o 1

Total 3288 1 1

You muft alfo underltand, that there are three other fmall Meafures of Length
proceeding from the Yard, namely, the Flemifo and Englijh EI!s, and the,Fathom.
The Flemijh Ell is equal to three Quarters of a Yard ; the Englijh Ell is equal to
one Yard and Quarter ; and the Fathom is equal to two Yards, or fix Feet.

P. Thank you, Sir; 1Jhallremember their Quantities : pray proceed unto the larger
Meafures, as Poles, Furlongs, &c.

M. I will ; but firft, 'tis neceflary that you fliould have, at lead, one Example
in each of the preceding Meafures : for always remember, that the pradtice of
one fingle Example ingrafts a fhonger Impreffion on the Mind, than the bare
hearing or reading of twenty.

V. Addition of Cloth-Meafure, Flemijh.
Fl, Ells. Inch.

f"2l3 26 Rule.

/mi n' ... .c _.l r I²?¹ ^ll For every 27 Inches carry 1
Colled into one Sum thefe)_I2' �' _t0 _the _E1f_8f �_hich _add _as f_n.
l�verai Quantities, *&. V₂J _2Q ^^

(.222 15

Sum total 1J53 8

VI. Addition of Cloth-Mea/ure, Englijh.
El. 4thsofYds. Na.of Yd.

r 12 4 3

e } 123 3 2

] 71 4 2

i 72 4 1

Rule.

Collect into one Sum thefefeveral Quantities, -viz.

For every 4 Nails carry I
to the Quarters, for every 5

Quarters carry 1 to the Ells,

which add as Integers.

281

VII. Addition of Fathoms,
Fath. Feet.

5123 2 Rule.

173 1 For every 6 Feet carry 1 to
275 5 the fathoms, and add them as
222 4 Integers.

'794 5
Now I will proceed to Poles, Furlongs, &t

P. Pray,

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Of ADDITION. ii

P. Pray, wohat Number of Feet are equal to one Pole or Perch ?
M. 'There are three different Poles or Perches, by which Lands are meafured.
The firftis called the Statute Pole, containing 16 Feet and f. The f�cond, the
Woodland Pole, containing 18 Feet; and the third, the Foreft Pole or Perch,
containing 21 Feet.

The Statute Pole is ufually ufed in the Menfuration of meadow, arable, and
pafture Lands, and Brick-works, &c. the Woodland Pole in the Menfuration of
copious Woods, tiff, and the Forell-Pole in the Menfuration of large Chaces,
Foiefts, &c.

VIII. Addition of Statute Poles.
Poles. Feet.
f*999 13 Rule.

Colleft into one Sam thefe J¹²? '5 ^For ^eVery ^l6 ^Feet ^ *

l�verai Lengths,*/*,

⁰ '

W ^II carry r or. every 33 Feet

888 2 carry z to the Poles, and add

777 4 them as Integers.

Sum 3522 12
IX. Addition of Woodland Poles.
Poles. Feet.

5796 17 Rule.

127 i� For 18 Feet carry 1 to the
493 11 Poles, which add as Inte-
101 16 eers.

Sum 174.2 14
X. Addition of Foreft Poles.
'Tis required to collect into one Sum, the following Lengths.

Poles. Feet.
f9999 20 Rule.

Colled into one Sum thefe ) �# »? f'YT, ^2I ^L^f ^lfeveral Lengths, «fe. )⁸⁸⁸ '⁵ » the Poles, which «Id as

⁰ ' I 201 20 Integers.

i S5S 9

Sum 12423 20
Thefe are the various Kinds of Poles, of which the Statute Pole is the molt in
ufe, and it is by the Statute Pole, that Chains, Furlongs, Miles, and Leagues,
^ate compofed.

P- Pray what Meafure is a Chain ?

"+. A Chain is a Meafure of Length, containing four Statute Poles, precifely
equal to 66 Feet, and is divided into 100 equal Parts, called Links : it is by this
Meafure, that Land is ufually meafured ; and was firft invented by that late emi-
nent Mathematician, Mr. Edmund Gunter; and as the whole Length is divided
into 100 Links, and contains 4 Poles, therefore 25 Links is equal to one Pole;
50 Links equai to two Poles, and 75 Links equal to 3 Poles.
XI. Addition of Chains and Links.
Cha. Links.

f 10 7J Rule.

\ 5 95 For every 100 Links carry I
Colled into one Sum, thefe } 2 99 to the Chains, which add as

Quantities, «was.

"\ 27 _2l1 28 96
V 00 18

Integers.

Sum 76 04

XII. Ad-

-ocr page 16-



12 Of ADDITION. XII. Addition of Furlongs, Chains, and Poles. P. Pray <what is a Furlong ? M. A Furlong is a Length, containing 10 Chains, or 40 Statute Poles or Perches, and is one eighth Part of a Mile. It is alfo called, an Acre's Length ; and one Chain's Length, is called an Acre's Breadth ; becaufe a Piece of Land, whofe Length is 10 Chains, and Breadth one Chain, is equal to 160 fquare Poles, the Quantity of one Statute Acre. The Addition of thefe Meafures, is made by this Rule : For every 4 Poles, carry 1 to the Chains, for every 10 Chains, carry 1 to the Furlongs, which add as Integers. Example. Fur. Ch. Po. rzi2 33 Colleft into one Sum thefe} '" f ^zl�verai Lengths, viz. � 777 $ * �ooo 7 o Sum 1335 9 o P. Sir, Inonu under/land thefe Additions very well, and'therefore defire you to pro- ceed unto Miles, Leagues, &C Pray, hoi» many Furlongs are equal to one Mile ? XIII. Addition of Degrees, Leagues, Miles, and Furlongs. M. Eight Furlongs are equal to one Mile, and three Miles are equal to one League. P. And is a League the great eft Meafure of Length ? M. No ; a Degree is the greateft Meafure of Length. P. What is a Degree ? M. A Degree is ftated at 60 Miles, of which, 360 is faid to be the Circum- ference of the Earth. P. Prav gi<ve me an Example hereof? M. I will. Example. Colleft into one Sum the following Meafures. Rule. For every 8 Furlongs carry 1 to the Miles, for every 3 Miles, carry 1 to the Leagues, for every 20 Leagues, carry 1 to the Degrees, aBd add them as Integers. Degr. Lea. Mi. Fur. 70 18 2 7 25 15 1 6 18 18 2 5 '25 04 o 2

	Sum. 140 17 1 4 Thefe are the feveral Meafures of Length ufed in England, whofe Proportions to each other are exhibited in the following Table.

		A

-ocr page 17-

Of ADDITION.

. A Table of Englifh Meafures of Length.

Barley-corns, taken out of the Middle of an Ear of Barley;

3 Inch.


36	I!	Foot.
81	27	2\|	Flemil	hE�.
ic8	36	3	'1	Yard
^T3>	45	3*		11 ¹ 4.	Englifh Eil.
216 S 94 648	72 198	6 i6\|	2\|	2	if	Fathom
	72 198	6 i6\|	7*	Si	4f	^Z"3=F	Statute Pole.
	216	18	8	6	4iT	3	'�T	Woodland Pole.
756 2376	252	21	9}	7		31	Wi	1*	Foreft Pole,
756 2376	792	66	²9t	22	1-1 1 ^X/TT	10	4	3f	5*	Chain.
23760 '90080	7920 63360	660	99 3 �	220	«76	no	40	36I	31!	10	Furlong
23760 '90080	7920 63360	5280	79465	1760	1408 880		320	²931	^zS^l ?	80	8	Mile

I 2 3 4 5 6709101112

P- Pray explain unto me the Nature and Vf e of this Table ?
f> p You fee that it contains 12 Columns, as numbered, I, 2, 3, 4, y,'

» 7> 5, 9, _IOj _Il; _I2) _eacj₁ _rep_ref_entj_ng the Number of Times that they are
contained in the next greater Mcafure." Thus in'a Mile, ftie e is contained
190080 Barley-corns Length; or 63360 Inches ; or 5280 Feet; or 7946-j F/e-
**/» Jills ; or X760 Yards; or 1408 Englifo Ells ; �r 880 Fathoms; or 320
orTir ^P0!eS ^; ^or ²93t ^w°odland Poles; or 2 5 if Foreft Poles ; or 80 Chains*
^⁰ Furlongs; as exhibited in the lowermoft Line of the Table. Again, ad-
' ^u was required to know what Number of Inches is in a Furlong, &c. pro-
ved as follows.

b_ri ^lr.^ft' ^filK* out the Word Furlong on the right hand Side of the Table, and
of) /'"? ■^y0Ur ^^e '^eve' ^^ncrefr°^m> until you come under the Title (or Column
tain i ^the ^ieconc* Column, there ftands 7920, the Number of Inches con-

ti N ^ln ^,0tle -^"^o^g- ^as required. Likewiie under the Title Foot, Hands 660,

__. ^x "^nU)er of Feet in a Furlong ; and fo in like manner, any other Meafure^

Or 1

^S %,"^rts °* which 'tis compofed, may moll readily be found by Infpeclion.
� """» / am <very much obliged to you for your painful Information of�'ong Mea^J

es, pray be pleafed to infrucl me in like manner, of fuck fqu�re Meafures as are
'ujea in B if nef ?

P «S ! j ^^arf> the Foot, the Square, apd the Rod, or Pole.

erformed and

treat doyon mean by the Foot? You have already informed me, that a F

B�t l S

« Length containing 1 ₂ I_KCh_es, «which I already know

M. Tis \try true a Foot in Length is 12 Inches as you fay, buta fqaare Foot,-
s a iquare Space, each Side thereof equal to 12 Inches; that is, as wei' int
length as m Breadth, and contains 144 fquare Inches.

P: Pray

-ocr page 18-



H Of ADDITION. P. Pray explain this to me in fuch a manner as I may rightly un�erfiand it ; fof at pre/,MI cannot comprehend your Meaning ? M I will, 'tis very eafily underftood ; Suppofe that the Square ADC D, fg. IX. PL LV1I. have each of its Sides equal to one Foot in Length. And each Side divided into 12 equal Parts; that is, the Indies in a Foot. Then I fay that if from the feveral Divifions of the Inches at the Points i, 2, 3, 4, 5, 6, 7, 8, 9, �o, 11, and 12, in the Sides A B and A C, right Lines be drawn from Side to Side, refpedlively oppofite, they will form 144 little Squares or fquare Inches : For every ore thereof will be an Inch fquare precifely. Hence it is, that a fquare Foot contains 144 fquare Inches. P. Sir, 1under/land you perfeil'.y ivt11, and upon the fame Principle Ifuppsfe that a fquare Yard contain) 9 fquare Feet. M. 'Tis true. For if each of the Sides of the Square ABC D, fg. I. PL LVIL contain cne Yard, divided into 3 equal Parts or Feet, as at the Points 1, 2, 3, 4, &c. and the Lines 3, 7 ; 4, 8 ; and 1, 5 ; 2, 6 ; be drawn, they will divide the fquare Yard into 9 little Square?, each containing one fquare Foot. Therefore \is evident, that one iquare Yard contains 9 fquare Feet, as you have before cb- ferved. P. I f e plainly that it doth, hut 'what do you mean by the Meafure tabicb you call a Square ? M. A Square of Work is a Space containing 100 fquare Feet, or it is � fquare Figure wliofe Sides are each equal to 10 Feet, divided into Feet, as the Square ABCD.^.II. PL LVII. P. I underftand you, Sir, and fee that if from the feveral refpeuive Dii)ifons of Feet, there he right Lines draivn, in the jame Manner as before in the fquare Fiot andlard, they ivit'l generate ICO little Squares, each equal to one fquare Foot. Pray nvherein is this Kind of Ivlecfurc tfed? M. In the Menfuration of Flooring, Tyling, Slating, �fe. which you will be acquainted with, when you come to learn Menfuration. P. 'Ibankyou, Sir, pray be pleafed to proceed? M. I will. The next fquare Meafure is a Rod, or Pole, and is a Space con. taining 272 J fquare Feet. P. Pray Jhew me its Fioure ? M. I will ; Suppofe each Side of the Square ABC D, fig. III. PL LVIL to contain i5 Feet J-, divided into 16 Feet and \| as at the Numbers, 1, 2, 3, isc. in the Sides A B and A C\ Then I fay, that if the fight Lines la, 2 b, 3 c, \d, 5 e, c5V. be drawn, as before in the preceding fquare Figures, they will ge- nerate 256 complete little Squares, each containing one fquare Foot, as in the Scheme. P. Very well, Sir, hut I thought that you faid, that a fquare Rod contain'd 272 J fquare Feet, and beninyou produce but 2j6, M. Within the Square of 16 Feet \ A B C D, there are 32 little long Squares* or Oblongs, marked with Dots; now as each of thefe oblongs are 16 Inches in breadth, and cne Foot in Length, therefore one of them is equal to but 4 of one of the whole fquare Feet. And confequently the 32 boing taken together, are equal to but iG whole Feet. Nov/if unto 256 You add 16

	TheSumis 272 The Number of Feet in oneRod. And laftly the little Square r, at the Corner D, having each of its Sides equal to but \ a Foot or fix Inches, therefore it contains but J of a Foot ; that is 36 Inches, which is buc ^of I44, the Number of fquare Inches {as before proved) in one fquare Foot. Therefore the Sum of the whole Square is equal to 272� Feet. Having thus defined unto you thefe feveral fquare Meafures, 1 wiil in the next Place proceed ^!� force Examples of the Addition of fuch Quantities.

		7 *X\Y.*

-ocr page 19-

Of ADDITION. 15

XIV. Addition offq aure Feel. ^^^^^^^^^^_m

Mote, That as the fquare Foot is divided inco Quarters, therefore one Quarter
)nrainc �< r-------r_-i_ _ ^* >� -^T

contains 36 fquare inclus.

Sq.Feet.Qrs.Sq.In.

run- f ^I23 3 31 Rule.

^ouect into one Sum thefe J 729 2 29 For every 36 Inches carry

'everal Quantities, -viz. j 80 1 25 1 to the Quarters,for every

L 7¹ ° 35 4 Quarters carry 1 tothefq.

b eet, which add as Integers.

_T IOOJ I 12

I muft alfo inform you, that the fquareFoot is by fome divided into 12 equal

ans, each being 12 Inches long, and one Inch in breadth, as a � c d e f V h i

* lm mf_s. vill. PI. LVII Which Parts are called long Inches, of which

you a fee more at large in crofs Multiplication hereafter. By this Manner of

ending the fquare Foot, its Faits are moll readily added together, as following.

Example.
Sq. Feet. Inches,

'999 11. Rule.

Colka into one Sum thefe \ '° ¹⁰/ For every .12 Inches carry

Quantities, _w*. "\ 7 ⁶- 1 to the Feet, and add then»

2 3 as Integers.
99 8

Sum iizo

XV. Addition of fquare Yard Meafure.
Example.
Yds. Feet.
C-₇ 8

Collect into one Sum thefe \^lZ ⁷ ^Ruie-

Quantities, -viz. ^ 9 4 For every 9 Feet carry. 1 to

58 the Yards, and add them as In-

6 2 teeers.

Sum 62

XVI. Addition of fquare Meafure, as Flooring, &c.
Sq. Feet.

�°^I,e& into one Sum thefe \ 70 .83' Add up the Feet'as Integers
wsta.l Quantities of Floor --i 70 .96. and for every ico carry j
°' *^/a:' � 10 25 to the Squares.

Sum 396 2/

..j �. . XVIL Addition of fquere Pole Meafure.

omitted d 'h ^lnefs ^{thefraaionalPart} or one Quarter of a Feot is general!/

The Rod is taken at 272 Feet.

The 3 Quarters 204

The Half j₃6

The Quarter 68

C ? To

-ocr page 20-

itf Of ADDITION.

Rod. Qr. Ft.

To add thefe Quantities together, this is (he Rule. For every 68 27 3. 30

Feet carry one to the Quarters, and for every 4 Quarters carry 1 29 1 a8

to the.Rods. 16 3

The Quantities in the Margin, are given to he added into one 8 1

3
9

Sum ----------

Sum 8:

XVIir. Addition of Land Meafure.
Note, That an Acre of Land contains 160 Poles or 4 Roods, and each Rood
4c fquare Poles or Perches.


Acre.	Rd.	P.
f-27	3-	39.
*\z6*	�y	2 1.
'< 18	1	�S-
1^:0	3-	₃«.
(. 21	1.	30
n 1 15	2	03

Rule.
For every 40 Poles carry
I to the Roods, for every
4 Roods carry 1 to the
Acres, which add as In-
tegers.

Collcfl thefe federal Qnan
titles into Sum, yitc.

A Table of /quart Meafure.

Thus have I delivered unto you,
all the ufeful fquare Meafures, by
which all manner of fuperhcial
Works are meafured. I fhall
now exhibit them together in his
Table, which by Tnlpectior. will
fliew their refpective Quantities,
in any of the lefler Meafures.

Sq. Inches.

�4'

9; Yards.

1296

1.29�0:

loo

oquares

Statute Pole.
___1

40;Roods.
(60I /(Acre.

39204

^z5

3°y,

1568160

10890
435�C

I 2iC

6272640

43 51

5 40

P. Pray f civ me the Ufe of this Table ?

M. I will. Suppofe it was required to know how many fquare Feet were
contain'd in one Acre of Land, Statute Meafure ; looking in the f�cond Column,
under the Title Feet, and againft the word Acre, Hands 43560, the Number of
fquare Feet in an Acre of Land, as required ; and fo in like manner any other
Meafuifi in the Tabic.

P. / thank tou, Sir, I und'.rfta.nd it, and Jo in like manner an Acre of Land is
eaualto 6:'.72640 fquare Inches, or 484c fquare Yards, or 43 J�It Squares of too
'Feet ; or 16b fquare Statute Poles ; or 4 Riods. And a Rood is equal to 1568160
fquare Inches, or to 10890 fquare Feet, or to li 1 q fquwe Tards ; or to loSj^-
Squares of (CO Feet ; or to AG Statute Poles.

M. "Fis very well, I find yea have a right Underftanding of its Ufe. I fhall
iri the nexc Place proceed to inform you of the feveral Weights ufed in this King-
dom, from which the feveral Meafures of'Capacity were taken.

V. Ijhaxkyou, S:r, bu.'if there -veere any filid Meafures necejfary to folioiv the
jupe>f:ial or fquare ones koz'j taught me, 1'Jhouldgladly knoiu their..

M. 1 here are folid Meafures which you are to be informed of, as the folid

Foot, which contains i r 2 3 sbiid or cubicle Inches ; and the folid Yard, which

cciitains 27 folid, Fest_; a Tun, of Tiiribcr .10 folid Feet, and a Load �o iblid Feet.

" ' . But

-ocr page 21-

Of ADDITION. 17

But before lean inform you thereof regularly, I muft teach you Multiplication,
or otherwife you cannot fo readily, or fo well underftand them.

P. Iejk pardon for my Forwardnefs. Pray proceed to the Account of the Weights
you nvas mentioning ?

M. I will. The Original of all Weights ufed in this Kingdom was a Grain of
wheat, taken out of the Middle of a well-grown Ear, and being well dry'd, 32
of them were called and made a Penny Weight, 20 Penny Weights one Ounce and
12 Ounces one Pound. See the Statutes of 5 I Hen. 3. 31 Ed. 1. 12 Hen. 7. But
the Moderns, fmcethe making of thefe Statutes, have divided the aforefaid Penny
Weight into 24 equal Parts, which are called Grains, and is the leaft Weight now
in common Ufe.

P. What do you call this original Weight ?

M. It is called Trey Weight, becaufe 'tis fuppofed to be the fame that was
ufed by the Trojans. By this Weight, Ojbright a Saxon King of England 200 Years
before the Conqueft caufed an Ounce Troy of Silver to be divided into twenty-
Pieces, which were at that Time called Pence, and at that Time an Ounce of
Silver was worth but 20 Pence.

This Value of Silver continued unto the Reign of Hen. VI. who to preventthe
enhancing of Money in foreign Parts, valued the Ounce at thirty Pence, and ac-
cordingly divided the fame into thirty Pieces, each beingthen a Penny. And the
old Pennies made in OJbrigbfs Time went then for three Pence half penny
each, and which continued unto the Time of Ed. IV. who valued the Ounce of
Silver at 40 Pence, and divided it into 40 Pieces each a Penny, and then the
old Penny of OJbrighfs went for Two-pence.

This continued until the Reign of Hen. VIII. who valued the Ounce of Silver
at 45 Pence, which was not altered until the Reign of Queen E/iz.who valued
the old Penny oiOfbright at three Pence; fo that at that Time, all Three-pences
coin'd by Qseen BJiz. weigh'd but one Penny Weight, every Six-pence two Penny
Weight, and the like Proportion in Shillings and other Pieces then coin'd.

This kill Alteration was the Caufe of the Ounce oi Troy Stiver to be valued
at 60 Pence, or five Shillings, as it now is at this Time.

By this Weight Jewels, Gold, Silver, Corn, Bread, and all Liquids are
weighed.

XIX. Addition of Troy Weight.
Thefe Weights are added together by the following Rule.

For every 24 Grains carry 1 to the Penny Weights, for every 20 Penny
Weights carry 1 to the Ounces, and for every 12 Ounces carry 1 to the Pounds.


	EXAi	IPLE.
ft	Oz.	Pw.	Gr.
22	II	'9	20
l6	9	11	'7
20	8	3	4
l6	11	7	8
Sum 77	5	2	1

But befides thefe common Divifipns of the Troy Pound, I find in the Prefent
State of England, for the Year 1699, ^tnat ^cne Grain is fubdivided as following,
'viz. I Grain is divided into 20 Mites, 1 Mite into 24 Droits, I Droit into 20
Periots, and 1 Periot into 24 Blanks, from which the following Table of Troy
Weight is made.

Blanks

-ocr page 22-

Of ADDITION,

Blanks


24	Pcriot
480	20	Droit
11520	480	24	Mite
230400	9600	480	20	Grair	1
532960c	230400	11 520	480	24 480 5760	Penny Weight
102892000	4608000 230400		9600 115200		20 240	Ounce
1,234,704,000	55,296,000 2764800		9600 115200		20 240	12 Pound.

Thefe Weights are added together by the following Rule.

For every 24 Blanks carry one to the Periots, for every 20 Pericts carry 1 to
the Droits, for every 24 Droits carry 1 to the Mites, for every 20 Mites carry
1 10 the Grains, for every 24 Grains carry one to the Penny Weights, for every
20 Penny Weights carry one to the Ounces, and for every 12 Ounces carry 1 to
the Pounds.

Example.


	12	20	²4	20	24	20	24
Pounds.	Oun.	Pwts.	Gr.	Mites	Droit.	Per.	Blanks
To f6	7	9	iS	J S	«7	'9	23
Addj 20	5	7	13	16	'4	is;	16
I 02	] 1	'9	19	iS	16	IJ	11
Total 40	0	'7	4	11	1	14	2

Now feeing that by this Table a Grain contains two Hundred and thirty
Thoufand, four Hundred Parts, or Blanks, furely the Commodities that have
been fold by thefe Weights rnuft have been of great Value, as that they them-
lolves noil be real Atoms, or at leaft as frna'l as one Particle of the fineft Kind
of Sand. But this Example 1 give you more for Curiofity than real Ufe.

By Avoirdupoife Weight all Kind of heavy Commodities are fold, as Iron,
Lead, Brafs, Copper, Grocery Wares, �V. vvhofe fmailelf. Part is called a
Dram, of which 16 make one Ounce, 16 Ounces one Pound, and 112 Pounds,
one Hundred Weight, 56 fe. half a Hundred and 28 a Quarter of a Hundred.

P. Pray is /he Pound Troy, and Pound Avoirdupoife equal to each other ?

III. No. The Pound Avoirdupoife, is equal to one Pound two Ounces and
12 Penny Weights, of Troy Weight, and the Pound Troy, is but nearly 13
Ounces 2 Drams and a half of A voirdupofe ; fo that the Pound Avoirdupoife is
about two Ounces, 13 Drams and a half, Avoirdupoife, greater than the Troy
Pound, which is very near a lixth Part of a Pound Avoirdupoife, Jefs than a
Pound Avoirdupoife. And therefore fix Pound of Bread, which is fold by Troy
Weight, is very little heavier than five Pound of Butter or Cheefe, which is fold
by Avoirdupoife Weight. So that thofe who believe the Pound Troy and Pound
Avoirdupoife to be equal, are much miftaken ; but, however, though the Pound
Troy is lefs than the Pound Avoirdupoife, yet the Ounce Troy is heavier than
the Ounce Avoirdupoife, for 292 which are the Number of Penny Weights in 14
Ounces 12 Penny Weights, which are equal to one Pound Avoirdnpoiie, being
divided into 16 equal Parts, each Part will be found to be but 18, and five fix-
teeathsj, which are the Number of Penny Weights in one Ounce Avoirdupoife,
of which the Ounce Troy contains zo.

N. B,

-ocr page 23-

Of ADDITION. 19

N. B. The Hundred Weight Troy, is 100 ifc. the half Hundred 50 ft. and
the Quarter of a Hundred 25 ft.

The following is a Table of Avoirdupoife Weights.
Drams


16 256	Ounce 16 Poun 1	i
7168	448' 28	Q��	arter of a Hundred
I433⁶	8₉6\| 56	2	Hi	ilf a Hundred.
28672	1792; 112	4	2	A Hundred
573440	358402240	80	40	zo A Ton Weight

XX. Addition of A-voirdupoife Weight.

Thefe Weights are added together by the following Rule.

For every 16 Drams carry 1 to the Ounces ; for every 16 Ounces carry 1 to
the Pounds; for every 28 Pounds carry 1 to the Quarters; for every 4 Quarters
carry 1 to the Hundreds; and for every 20 Hundred carry 1 to the Tons,

Example.
A Smith made five Parcels of Iron-works j


To.	H.	Q- p.	Oz.	Dr.
7 2	'5 11	3 27 2 14	'3 10	14 11	I demand the to-
9	19	¹ 9	1	^l5	tal Weight of the
27	J 5	2 25	12	9	whole.
18	>7	1 11	'5	15

The firft weighed
The Second
The Third
The Fourth
The Fifth

5 13 °°

P. Pray tahy is this Kind rf U'eight called A-voirdupoife ?

M. From the French, Have your Weight ; that is, you fhall h.a.vc full H'tight,
and therefore 12 Pounds over and above 100 are added.

P. Pray is the Troy Pound diwded in any other Manner than the preceding ?

M. No : but the Troy Ounce is, by Apothecaries, as follows, 'viz. Firft into 8
Parts, called Drams, a Dram into 3, called Scruples, and aScruple into 20, called

Grains; therefore f 20 Grains
J 3 Scruples

�8 Drams
12 Ounces
Note, That by thefe Weights

r;g;_a^UP^le'7whofeMarks,7 9

is equal to*» .-. ' >or Characters^H J < Ounce, f

L 1 Pound, J

^
5

Medicines are compounded, but Druo-s are

bought and fold by Avoirdupoife Weight.

From the'Pound Troy all the Meafures of Capacity were taken; a Pound of
Wheat filling that which was called a Pint : but in regard to the Difference that
was found in Wheats, which were fome of more material Subftance and Space
than others, and thereby filled more or lefs Space, as forne but 286, and others
288 folid Inches ; it was therefore Hated by Parliament, that 282 folid Inches,
fliould be equal to one Gallon of Beer Meafure, and 23I folid Inches, to one
Gallon of Wine Meafure ; and from hence it follows, firft in Beer Meafure,
that 2 Pints make 1 Quart ; a Quarts one Pottle ; 2 Poules 1 Gallon ; S Gallons

1 Bum.l}

-ocr page 24-

2� Of ADDITION.

t Bufhel ; g Gallons 1 Firkin ; 2 Firkins 1 Kilderkin ; 2 Kilderkins f Barrel ;-
63 Gallons 1 Hogfhead ; and 2 Hogfheads one Pipe or Butt ; and therefore

Pint

Quart

Pottle

Gallon

Bufhel

35 and 1 Quarter '

70 and 1 half

141

282

2156

■ folid Inches*

One

■contains -

! Firkin
Kilderkin
Barrel
Hogfhead
Butt

2438
4876

9752
17762

35>3²II. In Wine Meafure, that 18 Gallons and half make 1 Runlet of Wine ; 4&
Gallons 1 Tierce, or third Part of a Pipe ; 84 Gallons I Tertian, or third Part
of a Tun ; 63 Gallons one Hogfhead ; z Hogfheads 1 Pipe ; 2 Pipes one Tun 3
and therefore _;

'Pint

Quart

Gallon

Runlet

Tierce
¹ Tertian

Hogfhead

Pipe
.Tun

28 and 7 eighths -j
57 and 3 Quarters 1

²3^l4273 and half

One

contains

9702

folid Inche

19404

�45S3

zgio�
58212

Example I.
XX'I. Addition of Beer Meafure.
Ba K. F. G.

{

3 8
3

Rule.
For every 9 Gallons carry 1 to'
the Firkins; for every 2 Fir-
kins carry I to the Kilderkins ;
for every 2 Kilderkins carry 1
to the Barrels, which add as
Integers.

Four Veffels contain th

feveral Quantities, I de-
mand the total Sum of the
tohole.

Total 18 o i 1

Note, That although 4 Fiikins of 9 Gallons each, which are equal to 36
Gallons, make 1 Barrel of Beer; yet a Barrel of Ale contains but 3.-:
Gallons.

Example I.
B. Hhs.Gal.

Hide.

For every 63 Gallons carry 1 to
the Hogfheads; for every 2'
Hogfheads carry 1 to the Butts,
and add the Butts as Integers.

("3 « S3

J 7 o 61

I9 ! ,39

Four Veffels contain the fe-
veral Quantities^ I demand
the Total

Total 26 1 54

XXII. Addition of TFine Meafure.

Tu. Pip. Ter. T�ir. Run.Gal. Qa.

Four Veffels con-
tain thefe Quan-
tities, I demand
the Total.

Rui;.
For every 4 Quf.
carry 1 to the Gal-
lons; for every 42
- Gailoris carry 1
to the Runlets; for
Tierces carry 1 to �i�
Tertians,

1 1 1 1 37 3
01 1 o 40 2
1 o o 1 41 3
1 1 1 1 39 2

Total 24 1 000 33
etery 2 Runlets carry one to the Tierces ; for.every

-ocr page 25-

Of ADDITION. at

Tertians ; for every i and half Tertians, carry I to the Pipes ; he eve<y 2 Pipes,
carry 1 to the Tons, and add the Tons as Integers.

� XXIII. addition of Dry Meafurei.

Note, That 4 Bufhels make one Sack or Comb ; z Combs t Quarter; 4 Quar-
ters one Chaldron of Corn ; 5 Quarters 1 Wey; 2 Weys I Laft.
Example I.
. Chal. Quar. Comb. Burn. Gall.
Co!!eclthefefeveralf9 ^r' " 3 7; Rule.

Quantities into one^ S o o ₂ 6 For every 8 Gallons
Sim, �&, /73*35 carry . to the Bufhels;

{.z 2 1 5 7 tor every 4 Bufhels

-----------------------------------carry 1 to the Combs ;

Total 25 1 02 1 for every 2 Combs

carry 1 to the Quarters ; for every 4 Quarters carry I to the Chaldrons, and
'add them as Integers.

Example II.
Lafts. Weys. Quar. Bufh. Gall.

Colleathefefeveralf? ^l 4 7 7 Rale.

Quantities into one 1$ ^l 3 ⁶ S For every 8 Gallons
Sin,�*. /² ° 4 7 7 carry 1 to tne Bufhels ;

1,7 o 3 5 o for every a Bufhels

--------- �-----------.-�■ carry 1 to the Quar-

Total 23 1 2 4 1 ters ; for every 5 Quar-

ters carry 1 to the Weys ; ior every 2 Weys carry 1 to the Lafis, and add the
Lafts as Integers.

Note, A Chaldron of Coals is 36 Bufhels, and one Hundred of Scotch Coals,
112 Pound, Avcirdupoife.

XX[V. Addition of Decimals.
Note here, the Integer is divided into ten equal parts,

Inte ioths.

71 9
41 7 Ruk

tities together, viz.

32 g For every join the ioths carry I

11 8 to the Integers, which add as be*

I 6 4 fore taught.

I 7 9

Total S72 6

Note, Decimals are ufually expreffed by having Fractional Parts 271,9

feparated from the Integers by a Comma, which is called a 541,7

Separatrix, as in the Margin; where the aforelaid Example is 3²>9

expreffed in that manner. J t ,8:

6,4
7>9

872,6

XXV. Addition of Duodecimals.

Note, As in Decimals, the Integer is divided into 10 equal Parts; fo here in
Duodecimals the Integer is divided into twelve equal Parts (as the Inches
in a Foot, or Pence in a Shilling). It is alfo to be noted, that in many
Cafes not only the 12ths are divided again into 12 Parts called Primes,
but each Prime into 12 again, called Seconds, and every Second in like
manner, into i2, which are called Thirds, is'c. which are denoted by
Dafhes over them, according to their Place or Value. As for Example,
10 Primes are expreffed thus, 10', 10 Seconds thus, 10", loThirds thus,
|p", fcV.

D Colled

-ocr page 26-

22 O/ A D D I T I O N.

Rule.

�I

o io io ii For every 12 Thirds carry
9 11 7 10 1 to the' Seconds; and the

CollecT: into one Sum th

following Quantities, <^'z.

5 9 4 7 fame from the Seconds to

z 7 11 11 the Prime; ; and from the

' ------------ Primes to the Integers, which,

Total 29 3 H 3 add as before taught.

XXVI. Addition of Degrees and Minnies.
Note, A Degree is divided into 60 equal Parts, called Minutes.

Deg.Min.


n	59 47 59	Rule. For every 60 Minu the Degrees, and	tes carry 1 add them	to as
b	42 55	Integers.

CollecT; into one Sum thefe
feveral Degrees and Mi-
nutes, iiiz.

Total 57 22

XXVII. Addition of Time.

Note, A Year is fuppofed to be divided into 13 equal Months ;
4 equal Weeks ; a Week into 7 Days, of 24 Hours each ;

a Month into
an Hour into

60 Minutes, and a Minute into 60 Seconds.

Years. Mon. Weeks. Days. Hours. Min. Sec.
CollecT: into one Sum thefe C17 II 3 6 17 57 jo

feveral Quantities of Time, s ij 10 2 5 23 5} 59

�VIZ. C 20 9 34 2 2 40 30

Total J4 6 2 3 16 34 19

XXVIII, jWiW of Sand and Lime.
Example I. Of Sand.
Note, A Load of Sand is 18 heaped Buihels.

Loads. Bulh.


27	11		Rule.
18	17	For every	18 Buftels	carry	1
is	'3	to the Loads, and add		them	as
.6	l�	Integers.
.12	9

CollecT; into one Sum thefe
feveral Quantities of Sand,
njiz.

Total 91 12

E x a M P L � 11. Of Lime.

Note, 25 Bags, which ought to be one Bufhel, is accounted one Hundred of
Lime; and in many Countries, 30 Bafiiels is called a Load.
Hund. Bag.

J 2 21 Rule.

3 17 For every 2j Bags carry 1 to

) 4 24
15 ²²

Total 17 9

Loads. Bum.
Again, collect into c� Sum [2 27 Rule.

thefe feveral Quantities of J 3 29 For every 30 Buihels, carry 1 to
Lime, -viz. | 4 26 the Place of Loads, and add

/2 18 them as Integers.

Total 14 lo

xxix.

-ocr page 27-

�f A D D I T I O N. 23

_v XXIX. Addition of Bricks.

Note, 500 Bricks make i Load.

Loads. Bricks.
Collect thefe four Quanti- f'z 480 Rule.

ties of Bricks into one Sum, I 3 472 For every 50b Bricks carry 1 to
*�. I2 137 the Loads, and add them as Iri-

CS 493 tegers.

Total 15 87

XXX. Addition of Timber and Planks.
Note, That 50 folid Feet make one Load.

Loads. Feet.

f"2 45 Rule.

Collect into one Sum thefe \ 3 42 For every 50 Feet carry 1 to the
feveral Quantities of Tim- -^ z 28 Loads, and add them as In-
ter, <uiz. I] 37 tegers.

Total 14 1
Note, That in the Addition of Planks, 1 Inch in Thicknefs, every 600 Feet
is 1 Load ; of 1 Inch and half Thicknefs, 400 Feet ; of 2 Indies Thick-
nefs, 300 Feet j of 3 Inches Thicknefs, 200 Feet ; and of 4 Inches Thick-
nefs, 150 Feet.

XXXI. Addition of folid Tards.
Note, That in I folid Yard there are 27 folid Feet.

Yards. Feet.
f 3 26 Rule.

Collect into one Sum thefe J 2 17 For every 27 Feet carry � to'th�
feveral Quantities, viz. j 4 25 Yards, and add the Yards as In-

I5 26 tegers.

Total 17 13

XXXII. Addition of Money.
Note, That /. ftands for Pounds ; s. for Shillings ; d. for Pence ; and qr. for
Farthings ; with refpedt to Libra, which lignifies a Pound, Solidus a Shil.
Hng, Denarius a Penny, and ��uadrans a Farthing.


/.	S.	d.	*qr,*
12	H	11	4 3
IO	ij	9	Z
� 2	7	8	3
^J'i	'9	11	2
²i	16	7	3

Rule.

For every 4 Farthings carry⁴..

^--^ _v I to the Pence ; for ever/

!2 7 83 12 Pence carry I to the Shil-

ij 19 11 2 lings; for every zo Shil-

23 16 7 3 lings carry 1 to the Pc�ndsjj

^^^�^^�^^^

Collect into cne Sum thefe <

ievera.1 Sums, viz. »

(,1

-----------------~------ which add as Integers.

total 175 �8 1 1
As � have thus gone through the Addition of all that is neceflary, Ifhal! there-
fore Conclude this Lecture with cbferving,
r. That 1 Load of Earth is one folid Yard.

2. A Hundred Weigh: of Nails, Iron, Brafs, CSV. is 112 Pound.

3. A Hundred of Deals or Nails, fix Score or X20.

4' A Bundle of � Feet Laths, 100, and of 4 Feet in Length,' 17.0, which
ihould be 1 Inch and half in Breadth, and half an Inch in Thicknefs.

5- A Fodder of Lead is 19 Hundred and a half, or 2184 Pounds Avoir�u
poife. _

6. A Bale of Paper is ten Reams j a perfect Rears, 2% Quires, or j c'o Sheets j
i perfect Qgirc, 25- Shetts-

13 2 7, k

-ocr page 28-



�4 Of A D D I T I O N. lb. ioths. 7. A folid or Cubick Foot of fine Gold, weighs ----- ■ ■ ■ 13ja 4 Ditto of Standard Gold ----------■--------------� 1180 4 Ditto of Qnickfilver -------■----------■-------;-------------------- 874 9 Ditto of Lead----------------------------------------------■ 707 7 Ditto of fine Silver------------------------------------------------ 693 I Ditto of Standard Silver ■------------------------------------------ 658 3 Ditto of Copper---------------------------------_:--------------------- 562 4 Ditto of Brafs--------------------------------------------------------- 521 8 Ditto of Caft Brafs------------------------■--------------'---------- 500 o Ditto of Steel------------------------------------------------------■ 490 7 Ditto of Iron----------------------------------------------■------------ 477 5 Ditto of Tin ■-------:-------------------■----------■�-----'------- 457 4. Ditto of Marble-----------------�----------------------------------- 196 3 Ditto of Glafs--------------------------■------------------ 161 2 Ditto of Alabafter---------------------------------------�■---------- 117 ;o Ditto of Ivory ■-------■------------------------------------------------- 113 9 Ditto of Clay moderately moift ■---------------------■ 112 o 3}itto of fandy Gravel of common Moillure� ■ 96 o Ditto of Sea Water-----------------'~~■�"'~~--------------------------- 64 ,1 Ditto of River Water~~----------------------------------------------~~ 62 3 Ditto of Dry Oak ■-------------------------------------------�� 57 8 8. A circular Foot contains 11 3 fquare Inches, and one Seventh of an Inch ; that is, there are Co many fquare Inches in a Circle of one Foot in Di- ameter, which I call a circular Foot, for the fame Reafon as a fquare Foot, which makes a fquare Figare, is called a fquare Foot. '9. A folid or Cube Fpot, is j 728 folid Inches, that is 12 Times 144, the_vfquare Inches in a fquare Foot. 10. A Cylindrical Foot is 1573 folid Inches, and two Sevenths of an Inch ; that is, 12 Times 113 and one Seventh, the fquare Inches in a circular Toot. ' S1. A Cylindrical Foot of Sea Water, is about ;o Pound and half, and of frefii Water, about 49 Pound and one Tenth. L E C T. IH. Of Subtraction. M Subtraction is a Rule for finding the Diff�rence of any two Numbers, by _# taking or drawing the ieffer from the greater, whereby the Difference or Excei's (which is called the Remainder) will appear. P. Pray nvhat is particularly to be obfernied herein ? M. To take care that you always place the lefier Number under the greater, and that the Units, Tens, �fc of the Subtrahend, be placed under the Units, Tens, Hundreds, 'Off. of the given Number. P. Pray which of ibe tnxo Numbers are the Subtrahend, and which the given Number ? M. The greateft is the given Number, and the lefier the Subtrahend, as this Example makes plain. I. Subtraction of Integers, Example I.

Place your Numbers as in the Margin, and be- ginning at tie right Hand, fay, 1 from 7, there remains 6, and 2 from 8, remains 6. Note, if in fubtrafting any want fhould hap- pen, then oorrow 10 from the next Place, end for every 10 fo borrowed, carry 1 to the w%i Place
		From S7 the given Number, take 21 the Subtrahend, rem. 66 the Difference Or Excels.


			Example.
	?		Example.
	?

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Of SUBTRACTION.

Example II.

Operation. Fiift, 3 from 4 remains

I bought 7 524 Bricks,
and have fold 5643, what are re-
maining ? ------

Anfwer 1881 remain.

t ; fecondly, 4 from 2 I cannot, but

4 from 12 (for borrowing 10, makes
the 2, J2) and there remains 8 : third-

ly, 1 I borrowed, and 6 is 7, from

5 I cannot, but (borrowing to as be-
fore) 7 from 15, reft 8. Laftly, 1 that I borrowed, and 5 is 6, from 7, reft I,
fo the remains is i88t.

P. Pray ho-iu Jhall I know when SubtraBion is truly performed?

M. All kinds of Subtraction are proved by adding theSubtrahend and Remains
together, which will be equal to the given Number, if the Subtraction be truly
performed. As for Example, if 5643

7524 given Number,
5643 Subtrahend.

the Subtrahend, be added to 1881,
the Remains, their Sum will be 7524,
as in the Margin, which being equal
to the given Number, the Subtraction
is therefore truly performed.

si remains:

7524 Sum of given Num. and Subtra.
Other Examples for PraBice.
From 547213 From 772543279

take 439197 take 619987654

remains 108016

remains 152555625

Proof 547213 Proof 772543279

II. SuhtraBion of Money.

Example I. Example II.

s. d. q. '� s. d.

From 19 11 3 From 272 19 10

take 17 92 take 229 15 9

rem. 43

Proof 19 11 3

Example III.

/. s. d. q.

From 275 5 1 2

take 199 19 3 3

Procf 272 19 10

Example IV.

/. s. d. q.

From 927 571

take 832 19 8 3

rem.

75 5 9 3

Proof 275 512 Proof 927 571

In thefe !aft two Examples, at the Farthings you borrovv 4, and carry I to
the Pence, becaufe 4 Farthings make one Penny ; at the Pence you borrovv 12
and carry t to the Shillings, becaufe 12 Pence make 1 Shilling; and at the Shil-
lings you borrovv 20 from the Pounds and carry 1 to the Pounds, becaufe 20
Shillings make 1 Pound. The Pounds you fubtraft as Integers.


	Example. Inch. From 372 take 245	lit. 1. ioths. eg 09
	rem. 127	DO
	Proof 572	cq

SuhtraBion of Inches and loths.

Example II. Example Ilf.

Inch. �oths, Inch, ioths.

From 342 5 From 971 z

take 213 9 take 725 9

rem. 128 6
Proof 342 5

rem. 245 3
Proof 971 2

Here

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Of SUBTRACTION.

Here, at the loths, you borrow 10 from the Inches, and carry I to th<3
Inches, becaufe 10 Parts make i Inch.

IV. Subtraclion of Feet and Inches.
Example I. Example II. Example III.

Feet. Inch. Feet. Inch. Feet. Inch.

■ From 279 5 From 972 3 From 999 8

take 217 11 take 165 7 take 777 11

rem. 806 8

rem. 61 6

rem. 221

Proof 279 5 Proof 972 3 Proof 099 S

Here, at the Inches, you borrow iz Inches, or t Foot, from the Feet, and
carry 1 to the Feet, becaufe 12 Inches make 1 Foot.
V. Subtraclion of Decimals.

Example I. Example II. Example JII.

From 217,9 From 2754,8 From 729,02

take 206,5 take 1234,9 take 561,97

rem. 167,05

rem. 1519,9

rem. 11,4

Proof 217,9 Proof 2754,8 Proof 729,02

Here you fubtradt the,whole as Integers.

VI. Subtraction of Duodecimals.
P. Pray <wha( are Duodecimals ?

M. Duodecimals fignify twelfths, and as thefe Examples are of Feet, Inches,
and Parts, you are to obferve, that the Inches are each divided into 12 Parts, the
fame as the Feet are divided into 12 Inches.

Example I. Example II.

Feet.Inch. Parts. Feet. Inch.Parts.

' From 12 7 3 From 92 9 9

take 07 11 11 take 73 11 11

Example III.
Feet.lnch.Parts.
From 67 2 9
take 27 10 10

rem, 18

rem.

Proof 12 7 3 Proof 92 9 9 Proof 67 - 9

Here, at the Parts and at the Inches, you borrw 12, and carry 1 to the Inches,
and to the Feet, becaufe 1 z Parts make 1 Inch, and 12 Inches J Foot.

VII. Subtraction of Yards, Feet, and Incur:*.
Example I. Example II. Example III.

Yds. Feet. Inch. Yds. Feet. Inch. Yds. Feet. Inch.-

Prom 127 2 7 From 72 1 3 From 172 o 5


take 43	2	9	take 99	2	10
rem. 28	1	6	rem. 72	0	7

take 97 211

rem. 29

Proof 127 2 7 Proof 72 1 3 Proof 172 o 5

Here you borrow iz at the Inches, and carry I to the Feet; and borrow 3 at

tlie Feet, and carry 1 to the Yards; becaufe t?. Inches make a Foot, and 3 Feet

J Yard.

V�I�.

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			Of SUBTRACTION. 27 VII I. Subtraaion of Cloth Mea/ure. Example I. Example II. Example III. Yds. Qurs.Nails. Yds. Qurs.Nails. Yds. Qurs.Nails. From 527 1 ■ 2 From 270 2 1 From 127 3 2 take 399 3 3 take 211 3 2 take 96 3 3

				rem. 127 1 3 rem. 58 2 3 rem. 30

	Proof 527 j ₂ Proof 270 2 I Proof 127 3 2 Here, at the Nails, and at the Quarters, you borrow 4, and carry 1 to the Quarters and to the Yards, becaufe 4 Nails make 1 Quarter, and 4 Qurs. I Yard. IX. SubtraBion of flemijh Meafure. Example I. Example IL Example III. Ells. Inch. Ells. Inch. Ells. Inch. From 2794 22 From 37255 18 From 32594 22 take 1372 26 take 27532 20 take 12345 23 rem. 1421 23 rem. 9722 25 rem. 20248 26 Proof 2794 22 Proof 37255 18 Proof 32594 22 Here, at the Inches, you borrow 27, and carry 1 to theElls ; becauie 27 Inches roake one Flemijb Ell. X. SubtraBion of Englifh Ells. Example I. Example II. Example III. Ells.Qurs.Nails. Ells.Qurs.Nails. Ells.Qurs.Nails. From 772 2 1 From 987 2 3 From 888 3 2 take 666 4 3 take 912 4 4 take 699 4 3 rem. 105 2 2 rem. 74 2 3 rem. 188 3 3

		Proof 772 2 1 Proof 987 2 3 Proof 888 3 2 Here, at the Nails, you borrow 4 and carry I to the Quarters, becaufe 4 Nails make 1 Yard. And at the Quarters you borrow 5, and carry 1 to the Ells, be- Paufe 5 Quarters make one Englijh Ell. XI. Subftra�ion of Fathoms and Feet. Example I. Example II. Example III. Fath. Feet. Fath. Feet. Fath. Feet. From 729 4 From 999 3 From 3279 4 take 499 5 take 777 4 take 1999 5 rem. 229 5 rem. 221 5 rem. 1279 5

Proof 729 4 Proof 999 3 Proof 3279 4 Here, at the Feet, you borrow 6, and carry I to the Fathoms, becaufe 6 Feet P>ake 1 Fathom. XII. Subtraaion of Statute Poles. Example I. Example II. Example III. Poles. Feet. Poles. Feet. Poles. Feet. From 729 14 From 987 13 From 3729 12 take 666 15 take 599 16 take 1999 15 rem. 62 151 rem. 387 13a rem. 1729 13J- Proof729 14 Proof 987 13 Proof 3729 iz Here you borrow 16 Feet and I from the Poles, and carry 1 ; becaufe 16 Feet *»d I make a Statute Pole. XIIL

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28 Of SUBTRACTION.

XIII. Subtraelion of Woodland Pales.

Example I. Example II. Example III.

Poles. Feet. Poles. Feet. Poles. Feet

From 972 10 From 275 n From 299

take 699 17 take 196 15 take 199 16

rem. 272 11 � _rem. ₇8 ,4 rem. 99 15

Proof 972 10 Proof 275 n Proof 299 13

Here you borrow 18 from the Poles, and carry 1, becaufe 18 Feet make r
Woodland Pole.

XIV. SubtraSion of Foreft Poles.

Example I. Example II. Example III.

Poles. Feet. Poles. Feet. Poles. Peet.

From 1234 15 From 222 19 From 777 13

take 7S8 2C take 211 20 take 237 .19

rem. 445 16 rem. 10 20 rem. 539 15

Proof 1234 15 Proof 222 19 Proof 777 13
Here you borrow 21, and carry 1, becaufe 21 Feet make 1 Foreft Pole/.

XV. SubtraSion of Chains and Links.

Example I. Example II. Example I IT.

Chains. Links. Chains, Links. Chains. Links.

From 72 6j From 27 8j From 279 88

take 37 98 take 19 99 take 176 94

rem. 34 67 rem. 7 86 rem. 102 94

Proof 72 65 Proof 27 8y Proof 279 88

Here you borrow 100, and cany i, as in Integers, becaufe 100 Links make
1 Chain.

XVI. Subtraction of Miles, Furlongs, Chains, and Poles.
Example I. Example II, Example III

Mi.Fur.Ch. Po. Mi.Fnr.Ch. Po. Mi.Fur.Ch. Po.

From 7252 From 29 4 7 1 From 127 6 5 2

take 5793 take 12 7' 8 3 take 99 7 9 3

rem. 1253 rem. 16 4 8 2 rem. 27 6 5 3

Proof 7252 Proof 29 4 7 1 Proof 127 6 5 2
Here, at the Poles you borrow 4, at the Chains you borrow 10, at the Furlong
you borrow 8, becaufe 4 Poles is 1 Chain, 10 Chains is 1 Furlong, and 8. Fur?
longs is i Mile.

XVII. Subtra�lion of Degrees, Leagues, Miles, andFurhnvs.
Example I. Example II. Example III.

Deg.LeaMi.Fur. Deg.Lea.Mi.Fur. Deg.Lea.Mi.Fur.

From 27 15 1 4 From 127 12 1.5 From 29 1$ 2 5

take 14 19 2 7 take 99 18 2 6 ' take 21 19 a 7

rem. 12 15 1 5 rem. 27 13 1 7 rem. 7 15

Proof 27 15 1 4 Proof 127 12 I 5 Proof 29 15 2 5

Here you borrow 8 at the Furlongs. 3 at the Miles, and 20 at the Leagues,
becaufe 8 furlongs make 1 J4'fe; 3 Miles \ League, and 20 Leagues 1 Degree.

XVIII,

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Of SUBTRACTION. 2^

XVII1. Suit ration of Degrees, Miaules, and Seconds.

Example I. Example II. Example III.

Deg. Min. Sec. Deg. Min. Sec. Deg. Min. Sec.

Erom 102 40 49 From 221 47 23 From 28 47 4g -■

take 97 57 54 take 127 55 47 take 19 49 53

rem. 4 42 55 rem. 93 51 36 rem. 8 57 56

Proof 102 40 49 Proof 221 47 23 Proof 28 47

Here at the Seconds, and at the Minutes you borrow 60, and carry one to thi
Minutes and Degrees, beeaufe 60 Seconds make 1 Minute, and 60 Minutes i
Hour.

XlX. Subtraction of fquare Feel and fquare Inches.
Example I. Example II. Example III.

Feet. Inch. Feet. Inch. Feet. Inch.

From 729 19 From 927 75 From 5^5- 139

take 672 141 take 526 13; take 274 141

rem. 56 22 rem. 400 84 rem. 280 142

Proof 729 19 pfoof 927 75 Proof 555 139

Here at the Inches you borrow 144, and carry 1 to the Feet, beeaufe that 14^
fquare Inches make I fquare Foot;

XX. Subtraclion rf fquare Feet and long Inches.
Example I. Example II. Example III.

Feet. Inch Feet. Inch. Feet. Inch,

From 127 7 From 271 5 From 555 4

. take 93 11 take 136 10 take 449 10

rem. 33 8 rem. 134 7 rem. ioc

Proof 127 7 Proof 271 5 Proof 55; 4

Here at the Inches you borow 12 and carry 1, beeaufe 12 long Inches (whic4
are each �z Inches long and � wide) make 1 fquare Foot.

XXI. SultraBion of fquare Yard Meafure. ,
Example I. Example. II. Example III.

Yds. Feet. Yds. Feet. Yds. FeeS.

From 73 7 From 92 3 From 27 5

take 51 8' take 57 8 take 18 8

rem. 21 8 rem. 34 4 rem. 8

Proof 73 7 Proof 92 3 Proof 27 5

Here at the Feet yoi� borrow 9 and carry i, beeaufe 9 fquare Feet make %
fquare Yaid.

XXII. Subtraclion of folid Yards,

Example I. Example II. . Example III.

From 4� 2i From 72 20 From 97 19

. take ;6 26 take 49 25 take 9� 24

rem. 8 22 - , rem. 22 22 rem. o zt

� Proof 45 , 21 Proof 72 -20 Proof 97 19- .

Here at the Feet you borrow 27 and carry t, beeaufe 2 ? folid Fe�t make i
folid Yejd.

g: 5»v I f 1 ■

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�f SUBTRACTION.

XX.ni. Subtraction of Squares, as of Flooring, &c.
Example I. Example II. Example IT.

Squ. Feet. Squ. Feet. Squ. Feet.

From zj 98 From 29 11 From 127 86

take 1S 99 take 21 75 take 97 99

rem. 29 87

rem. 9 99

rem. 7 3

Proof 25 98 Proof i9 it Proof 127 86

Here at the Feet you borrow 100 and carry I, becaufe loO fquare Feet make
Square of Work, as of Flooring, Roofing, Tyling, bfc.

XXIV. SuhiraSion of Land Mcojures. I. Of fquare Statute Poles.

Example II.
Poles. Feet.
From 275 $i
take 223 127

Example III.
Poles. Feet.
From 123 270
take 99 27X

Example I.

Polo Feet.

From 192 120

take 72 \zi

196

J iq 24�

rem. 23 271

rem. O

Proof 192 120 Proof 275 51 Proof 123 270

Kcte. That altho' a Statute fquare Pole contains 272 fquare Feet, and one
Quarter, yet in thefe Examples the Quarter of a Foot is rejected, as it
ufually is in Bufinefs, and the fquare Rod or Pole is allowed at 272 fquare
Feet only,* therefore at the Feet, borrow 272 and carry 5.

II. Of Woodland Poles.

Example III.

Poles. Feet.

From 279 138

take 172 219

Example I.

Poles. Feet.

From 76 311

Example II.
Poles. Feet.
From 217 19g

take %6 320

take 120 220

rem. 96 303

rem. 106 243

rem. 39 315

Proof 76 311 Proof 217 199 Proof 279 138

Here at the Poles you borrow 324 and carry 1, beeaufe 324fquareFeet make

i Woodland Pole.

III. Of Foreft Poles.

Example III.
Foles. Feet.
From 123 138
take 75 375

Example I. . Example II.

Poles. Feet. Poles. Feet.

From 8: 399 From 594 322

take 71 439 take 437 440

rem. 156 323

f em. 47 204

rem. 10 401

Proof 82 399 Proof 594 322 Proof 123 138

He.e you borrow 441 and carry i, becaufe 441 fquare Feet, make 1 fquare

Foreft Pule.

XXV. SitltraHion of Acres, Roods; andfoles.
Examplk J. Example II. Example III.

Ac 1 es. Rds. Poles. Acres. Rds.Poles. Acres.Rcis.Poles.

From 127 2 31 From 27 1 27 From 120 1 sg

take 93 3 39 take 1S 3 38 take in 3 3j

rem. 8

rem. 8 1 29

Proof 127 2 31 Proof 27 1 27 Proof 120 1 19

Mere, at the Pole you borrow 40 and carry i, and at the Roods bonow 4 and

carry

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Of SUBTRACTION. 31

-Carry i to the Acres, which fubtraft as Integers, becaufe 40 Poles make 1 Rood»
and 4 Roods 1 Acre.

XXVI. SubtraBion of Troy Weight.
Example I. Example II. Example. III.

fc.Oun.Fwt.Gr. ft) Oun.Pwt.Gr. ft>. Oun.Pwt.Gr.

From 25 9 14 17 From zi 8 17 12 From 127 5 5 5
take 17 11 19 ig take 17 to 19 14 take 83 10 17 12


rem. 7	9	H	23
roof 25	9	14	17

9 18 22 rem. 43 6 17 17

Proof 21 8 17 12 Proof 127 5

Here at the Grains you borrow 24; at the Penny Weights 20, and. 12 at the
Ounces, becaufe 24 Grains make I Penny Weight, ana 20 Penny Weights I
Ounce, and 12 Ounces 1 Pound.

XXVJI. Subtraction of Jpoihecaries Weight.

Example I. Example II.

ft. Quo. Dr. Scr. Gr. ft. Oun. Dr. Scr. Gr.

From 12 941 15 From 127 5 3 1 17

take 9 II 7 2 19 take 99 10 7 2 iS

rem. 2 9 4 1 16 rem. 27 6 3 1 19

Proof 1* 9 4 1 15 Proof 127 S 3 ' '7

'.------------ '-------------*---------

Here at the Grains you borrow 20, at the Scruples 3, at the Drams 8, and
12 at the Ounces, becaufe zo Grains make 1 Scruple, 3 Scruples I Dram, 8
Drams I Ounce, and 12 Ounces 1 Pound.

XXVIII. Subiragion of 4-voirdvpoife Weight.
Exf.MPi.E I. Example IT.

Hun. Qiirs. ft). Oun. Dr. Hun.Qurs.Jfj. Oun. Dr.

From 27 2 21 13 ro From 25 1 18 7 11

take 21 3 27 15 1 j take 17 3 24 14 u

■■~~......~~��■��-<�*-*---------------------------------------------------------"-----------------------------------------*

rem. 5 2 21 13 it rem. 7121 8 15

Proof 27 2 21 13 10 Proof 25 1 18 71?

Here, at the Drams and at the Ounces you borrow 16. at the Pounds 28, and
4 ^at the Quarters, becaufe 16 Drams make 1 Ounce, 16 Ounces i Pound, zS
Pounds 1 Quarter of a Hundred, and 4 Quarters one Hundred.

XXIX. SublraSiion rf Beer Meafure.
. Example I. Example II.

Bar. Kilder. J-irk. Gall. Quarts. Hog. Gail.

From 27 o o 2 i From zz 57

take 18 1 j 3 3 take 18 62


rem. 8	0	0	7	2
Proof 27	0	0	2	I

rem. 3 jS
Proof 22 V7

In Example I. borrow 4 at the Quarts, 9 at the Gallons, 2 at the Firkins and
at the Kilderkins, becaufe 4 Quarts make 1 Gallon, 9 Gallons 1 Firkin, 2 Fir-
kins 1 Kilderkin, and 2 Kilderkins 1 Barrel.

'n Example II. at die Gallons borrow 63, becaufe 63 Gallons mal»e 1 Hog-
ftiead,

E 2 XXX. S«i-

-ocr page 36-

3* Of SUBTRACTION.

XXX. Subtrai�ion of Wine Meafure.
Example I. Example II.

Tuns Pipes Tier. Gall. Tuns Pipes Tier. Gall,

from 57 o i 35 From 20 01 27

take 52 1 2 4P fake 15 1 1 4;

rem. 4 o 1 37

Proof ^1 o 1 35 Proof 20

Here at the Gallons you borrow 42, at the Tierces 3, and 2 at the Pipes,
becaufe 42 Gallons make 1 Tierce, 3 Tierces 1 Pipe, 2 Pipes 1 Tun,

XXXI. SubtraBion of Dry Meafure.
Example.
Qrs. Sacks. B(ifh. Pecks, Gall. Quarts. �

From 50 p 2 2 o

take 39 1 3 3 1 3

rem. 10

Proof 50

Here you borrow 4 at the Quarts, 2 at Gallons, 4 at the Pecks and Bufhels,
and 2 at the Sacks; becaufe 4 Quarts make 1 Gallon, z Gallons 1 Peck, 4
Pecks 1 Bufhel, 4 Bufhels 1 Sack, and 2 Sacks 1 Quarter.

XXXII. Suhtraflion of Timber.

Example I. Example IT. Example Uf.

Loads Feet Loads Feet Loads Feet

From 123 44 From 57 38 From 75 38

take 117 49 take 26 39 take 25 47 '

rem. 5 45 rem. 30 49 rem. 49 41

Proof 123 44 Proof 57 38 Proof 75; 38

Here at the Feet you boirow 50, becaufe 1 Load of Timber contains 50 folid
-F�et.

XXXIIT. Subtraction of Plant I Jnch thick.
Note, 'jco Square Feet at one Inch thick, make 1 Load
Exampe I. Example II. Example III.

Loads Feet Loads Feet; Loads Feet


FiOm 127 take 38	42 5 599 426	From 372 take 263 rem. 108	472 525 547	From 725 take 632	500 584
rem. 88	42 5 599 426	From 372 take 263 rem. 108	472 525 547	rem. 92	j.6
Proof s 27	425	Proof 372	472	Proof 725	500

Here at the Feet you borrow 6co, becaufe 600 Feet make 1 Load, asaforefaid.
Note, If the Thicknefs of Plank be 1 Inch and half thick, then borrow 400 _;

if two Inches "thick, borrow 300 ; if three Inches thick, borrow 200 ; and laftly

if four Inches borrow ic�, becaufe

4⁰⁰1 f¹ ^^ncn ^and I") 'I'icknefs

³⁰⁰ I Peet at)² ^Inches (,^m;lks °ne
200 f * ) ] Inches (Load of

'�°J �4 Inches J Plank

XXXIV.

-ocr page 37-

QfSUBTR-ACTION. 33

XXXIV. Sukraaion of Bricks.

Note, 500 make I Load.

Example I. Example II. Example III.

Loads Bricks Loads Bricks Loads Bricks

From 27 491 From 14 57 From 23 372

take 13 499 take 12 451 take 14 428

rem. 13 492 rem. 1 106 rem. 8 444

Proof 27 491 Proof 14 57 Proof 23 372

Here at the Place of Bricks you borrow 500, and carry 1, becaufe 500 Bricks
."make 1 Load.

XXXV. Sultraaion of Lime.

Example I. Example II. Example III.

Hund. Eags Hund. Bags Hund. Bags

From 27 19 From 22 19 From 18 15

take 14 24 take 17 21 take 11 21

rem. 12 10 rem. 4 23 rem. 6 19

Proof 27 19 Proof 22 19 Proof 18 ij

Here at the Bags you borrow 25 and carry 1, becaufe 25 Bags (which ought to
be a Bufhel each) make a Load of Lime.

XXXVI. Sultraaion of Sand.

Example I. Example II. Example III.

Loads Bum. Loads Bufli. Loads Bu(h.

From 18 16 From 21 11 From 29 iz

take 15 17 take 20 16 take 25 15

rem. z 17 rem. o 13 rem.

Proof 18 16 Proof 21 11 Proof 29 11

XXXVII. Sultraaion of Time.

Example
Months Weeks Days Hours Min. Seconds.
From 11 2 1 20 41 53

take to 3 6 23 59 59

Proof 11 2 1 20 41 53

M. As I have now given you a fuflicient Number of Examples, of all the va-
rious Kinds of Bulinefs in general, and which I think are much more copious
than has been yet taught by ail the Mailers that have wrote on Arithmetic, I
iliall now proceed to Multiplication.

L E C T. Of Multiplication.
What is Multiplication ?
o M. By Multiplication is meant an Increafe, and therefore to multiply is
to increafe from a fmall Number to a greater; and which being confidered, is
ao more than the adding cf di«crs Numbers together.

7 ^°-^r

-ocr page 38-



			34 Of MULTIPLICATION. 7 For if 3 times 7 be added together the Sum is 21, as in the Margin : And 7 if 3 be multiplied into 7, the Product is 21 alfo. Hence 'tis plain that 7 Multiplication is nothing more than a compendious Manner of adding � Numbers together, and therefore may be called fhort Addition. 21 P. Pray, tuhat is principally to be obferiied herein ? <� M. Three Numbers or Members, which are called the Multiplicand, the Multiplicator or Multiplier, sud the Product. P. Pray, what is the Multiplicand, Multiplier, and Produit ? M. In every Multiplication, there are always two Numbers given to be mul- tiplied into each other, which are called the Multiplicand and the Multiplier, or Multiplicator, either of which being placed uppermoft is called the Multiplicand, and the lower the Multiplier; as for Example, if 8 be multi- A 8 Multiplicand plied into 9, as at A, then 8 is the Multiplicand and 9 the 9 Multiplier Multiplier j or if 9 be multiplied into 8, as at B, than 9 is � the Multiplicand, and 8 the Multiplier, and the Number 72, 72 Produce arifing by 9 times 8, and by 8 times 9, is called the Product. � But however as it is beft to make the greateft Number'of the B 9 Multiplicand two the Multiplicand, therefore it is moll ufuallydone, ob- 8 Multiplier ferving to place the Units, Tens, &c. of the Multiplier, un- ■� der the Units, Tens, �jfr. of the Multiplicand, 72 Product I. Multiplication of Integers, P. tlotv is Multiplication perforned? M. The Multiplication of Integers is performed by the following Rules. Rule I. Write down the Multiplicand and Multiplier under each other as aforefaid.and draw a Line under the Multiplier to feparate it from the Product, that arifcs from its firft Figure. Rule 'II. Multiply every Figure of the Multiplier into the Multiplicand, obferving as you proceed to carry one for every Ten, to the nexc Place, and fet the Remains under it, and the Products arifing from the feveral Figores of the Multiplier being added together, their Sum is the general Product of the whole Multiplication. Rule III. When the Multiplier confifls of many Figures, as in the following Example, the Product arifing from each Figure is to be placed by itfelf in fuch manner, that thefirftor righ:-hand Figure thereof may ftand under that Figure of theMul- tiplicator from which the faid Product arifes. Thefe will be made familiar by the following Example. Example. Multiply 7254, by 734,9, ivhicb place as in the Margin. Begin with 9 the firft Figure of the Multiplier, and thereby 7254 multiply all the Figures in the Multiplicand as follows. Firil fay 7349 9 ^ilmes 4 '^s 36» fet down 6 and carry 3, for the three Tens ; �------then fay 9 times � is 45, and 3 1 carry is 48, fet down 8 and 6�286A carry 4; then 9 times z is 18, and 4 I carry is 22, fet down 2 zgoj�B and carry 2 ; then 9 times 7 js 63, and 2 I carry is 65, which 21.762C being the lalt in the Multiplication therefore fet down 65,. and 50778D that Product will be 65286, as at A. Proceed in the fame man. -----------r- ner to multiply the remaining three Figures of the Multiplier, 53,309,646 4, 3, and 7, into the Multiplicand, and their Products will be as at B C and D, and which with that of A, being added to-

		gether, will be 53,309,646, the Product required. Rule IV. When Numbers given have one or more Cyphers a; the right Hand, the Mul- tiplication may be performed, without Regard being had to the Cyphers, until the Prod�� cf the other Figures be found, to which they are then to be annexed.

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Of MULTIPLICATION.

------As for Example, multiply 17 by 60, as at

A ; 2790 by 500, as at B ; 237000 by 25, as at
C ; which being placed as in the Margin, and
Multiplication of the fignificant Figures being
made, without any Regard being had to the Cy-
phers ; unto the Sum ef their Products, annex or

102 O

279J0 s
5|oo

1395(000

add thereto as many Cyphers, as belong to both
Multiplicand and Multiplier: foto 102, in Ex-
ample A, you add one Cypher, making the Pro-
duet 1020 : and in Example B, to 1395, the Pro-

5925I000

duce of 279, multiplied by 5, you add 3 Cy-
phers which makes the whole 1395000 ; and fo in like manner 5925, in Ex-
ample C, by the Addition of 3 Cyphers, belonging to the Multiplicand, the
Product, is made 5925000.

Rule V.

When Multiplication has any Cyphers intermixt with its
other Figures, the Cyphers need not be regarded ; as for In-

9274�
20017

fiance, the Product 1856476665, is produced by the Products
at A B C, which arifes by the 7, 1, and 2 of the Multiplier,
multiplied into the Multiplicand, without Regard being had
the Cyphers in the Multiplier.

A649215
B 92745
C 185490

1856476665
great Ufe to know readily the Produ� of any
; for which Purpofe this Table muft be learned

Multiplicatiom Table.

In Multiplication it is of very
two of the nine Digits or Figures
perfectly by Heart.


	�	�		�		~�
1	2	3	4	S	6	7	8	9	12
2	4	5	8	10	12	H	to	18	24
3	6	9	12	15	18	Zl	24	27	36
4 5 6	8 10 12	12 ^!5 18	16 20 H	20	24	28	3²	36	48
				²S 30	30 36	35 42	40 48	45	60
				²S 30	30 36	35 42	40 48	54	72
7 8 9	16 18	21 24 *7 3~	28 32 36 48	35 40 45 60	42 48 54 72	49 S6 63 84	56 64 7²96	63	84
								72	96
								81 108	108 H4 \
rz	24							81 108	108 H4 \
rz	24								" ■

The Ufe of this Table is
eafy, Suppofe the Product of
8 times ^is required; Look for
Son the Side andgontheTop,
and againlt thofe Numbers in
the Angle of" Meeting is 72,
the Product required. Sp 7
times 9 is 63, and 5 times 12
is 66, as in the Angles of
Meeting you will find, and fo
Of all other Numbers,

S»AMPLE8

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%6 Of MULTIPLICATION.


Example I. Mult. '27960 by 200	E	a amples for Practice. Example If. Mule. 972403 by 30007	Example II�. Mult. 7235 by 1000
5592,000 Prod.		6806S21 2917209	7235000 Prod
		6806S21 2917209
		29178896821 Prod.

In the firft Example T contracted my Work, by placing the 2 of the Multiplier
under the Units of the Multiplicand, which fhould always be done, when the
other Figures of the Multiplier to the right Hand are all Cyphers. In the f�cond
Example, I contracted my Work, by omitting of the Cyphers in the Multiplier,
and multiplying only by the 7 and the 3. In the third Example, I add three
Cyphers to the Multiplicand, becaufe one neither multiplies or divides.

Multiplication of Integers may be performed without giving any Trouble to
the Mind, in carrying on the Tens, according to the Rule I. as follows.

Example I.
Multiply 8342 by 7, as in the Margin.

Operation. Firit, 7 times 2 is 14, which fet down ; then 7 times 4

S342 is 28, which fet down, 2 before the 1, and 8 under the 1 ; then f

7 times 3 is 21, fet 2 before the 2, and 1 under ; then 7 times 8 is 56,

*�j� fet 5 before the laft 2, and 6 under; laftly, add the two Numbers 52214,1

52214 and 618 together, as they Hand, their Sum will be the true Product

618 required.

58394-

-��. Example II. Multiply 98254, by 3729, as in the Margin.

98254 The Operation of this Example is the

3729 fame as Che laft,only it is4 times repeat-

- ed ; and when the Produit of any Figure

871436 ? _D j n _r.T i is lefs than 10, place a Cypher in the

n ? Product of the 9 _Di , -r-_iL , , ^r

1285 J ⁷ Place, where if it had made 10, or more

iioioS '� p_rcd_ua of the ■ than 10, the Figure for 10, or above 10,

8�40 $ " "" mull have flood, as you will fee in the

6c 1 328 ? n j n. r 1 Produit that arifes by z, the f�cond Fi-

3645
220112

74⁶>

5 Produft of the 7. _gure _of _the M^ipl�er.

l Produit of the 3.

366389166 Produit of the whole. For a Proof of this Manner of work-
ing, I have fubjoined the fame Example,
worked after the comnloii Method, 39'
at A.'

4286 Produit of the 9.
Product of the 2,
6S7778 Produftof the 7.
294762 Produit of the 3.

366389166 Produit of the whole as before.

A� I have thus explained the Multiplication of Integers, you are to obfervc,
that therein is th,is Analogy, <w'k, As, an Unit is to the Multiplier, fo is the Mul-
tiplicand t$ the Product.
* ?.Prty

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		Of M ULTIP.LICATIO N. 37 P. Fray explain this, for atprefent 1don't conceive niobat you mean ? M. I will : by this Example. Suppofing one Lead of Timber coil 50 Shil- lings, how much will 12 Loads coft f If 12 Loads be multiplied by 50 Shillings, as in the Margin, 12 the Product, coo Shillings, is the Anfwer : and therefore one Load 50 being confidered as an Unit, bears the fame Proportion to coShil-------- lings, the Multiplier, as 12 Loads, the Multiplicand, doth to 600 600 Shillings, the Produft. , ------. P. 'Tis t'ery true, Sir, pray proceed, for you make Multiplication a Pleafure to me. M. The next in order, is to (hew you, how in many Cafes you may contract your Multiplications, as follows. Contraction!. To multiply any <rj~o^en Number [fuppofe 5.17) by 11. Rule. Set down the Multiplicand twice, the lower one beinp- removed one Place, either towards the right or left Hand, as at A 547 547 B AandB, where at A 'tis placed one Place 'towards the left 547 547 Hand, and at S, one Place towards the fight Hand. �------ ■ 6017 6017

			Contraction II. To ifmltiply afiy gitten Number [fuppofe 7925 ) by 12, j 3,14, �v. Rule. Multiply the Figures in the Multiplicand, by the Units in the Multiplier, obferving, as you proceed, to add that Figure of the Multiplicand, which. Hands next on the right Hand to the Produft oi' the Figure you multiply by. As for Example, mu+tiplj 792;, by 14, as in the Margin. Firft, 4 times 5 is 20, fet down o, and carry 2; then 4 times 2 is' deba 8, and 2 1 carry is 10 ; and 5 at a, being the next Figure on the right 79-> Hand of 2, which you are then multiplying, make 17, iet down J 14. and carry 1 ; then 4 times 9 is 36, and i 1 carry is 37 ; and 2, the-------~� next Figure on the Right at /�, is 59, fet down 9 and curry 3 ; then 1 10970 4 times 7 is 28, and 3 1 easy, is y. ; and 9, the next Figure to the �------ Right at c, is 40, fet down c and carry 4. Now, as there are no more Figure; in the Multiplicand, to add ^rhe 4 carried unto, therefore adding the 4 to the lait Figure 7 at d, makes xi, which fet down, and the Produft is 110970, as re- quired. Contraction III. To multiply any given Number [fuppofe 09--27) by !ti, 1 it, if J, 114. 1 � .-.ci',;. Rule Multiply the Figures ijj the Multiplicand, by the Units in the Multiplier, and as you proceed, add the two Figures of the Multiplicand, which ftand netws on the r'ght Hand, to the Produft cf the figure you multiply* by : a: ior Fx ample, multiply 99725, by 1 15, as in th.. Ma>gin. Fir It, 5 times � is 25, let down 5 and carry :: ; 'hen 7 times 2 is ilc'ba 10, and 2 I carry i; 12, and 5 at a is 17, fet down 7 and carry 1; 7 , . then 7 times 7 is 57, and 1 1 Carry is 36, and 2 at b is 38, and 7 at ; 1 � a is 43 ; fet down 3 and cany 4 ; then 7 times 9 is 4;, and 4 I cany -------...... . is 49, and 7 at c is 56, and % at b is 58, fet down 3 and carry 5 ; �146837� then'') time. 9 is 4;. and 7 I carry is 70, and 9 a: d is o. : nd 7 at ---------_ c is 60, fet down b and carry 6 ; then 6 I carry, and 7; at eh 15, 2nd o at d is 24, fet down 4and carry 2, which fjeing. added to e/a� e, make's ti, v rico fee down, and which makes the Contraction IV. To muJtip,.....; 7 ■■ o, ;.■.>.. [juffofi 7?>4J2j by f'�tj IC2, -03. 10 1. #»&. Multiply the Figures in the I" by the �nib cf ■' ' ■-.' and as you proceed, add that Figur�tof your Mi f h ai d, that ftands ne;;:, ti : �jghtHa-nd, except one onto the Pr�do'ft, �f tfatFij ire yam Itipb -, ■ .o:::-.~ Exanrpiy. multiply 7r3.;-3^7 ico, as in the :: an ^: -■-■<*

-ocr page 42-

S8 O/ MULTIPLICATION.

fedcba Ffrft, 9 times 2 is 18, fetdown 8 and carry t ; then 9 times 3 h

725432 27, and 1 I carry is 28, fetdown 8 and carry 2 ; then 9 times 4 is

109 38, and 2 I cairy is 58, and 2 at «is 40, fee down o and carry 4 ;

then 9 times 5 is 4;, and 4 I carry is 49, and 3 at b is 52, fetdown :

79072088 and carry 5 ; then 9 times 2 is 18, and 5 I carry is 23, and 4 at ris
Z7, fetdown 7 and carry z ; then 9 times 7 is 63, and 2 I carry is

65, and 5 at a'is 70, fetdown o and carry 7; now 7 I carry, and 2 at � is 9, fet
down 9 ; and becaufe you have nothing to carry to the 7 at_/", therefore fet down
7, and the Produce will be 79072088, the Produft required.
IT. Multiplication of Decimals.
M. Multiplication of Decimals, both in placing the Multiplicand and Multi-
plier, is the fame as the Multiplication of Integers, only when your Work is
completed, you mull obferve, that with a Dafh of your Pen, you cutoffas many
Places of Decimals in your Produft, as there are Places of Decimals both in your;
Multiplicand and Multiplier, and in cafe of want in your Produft, prefix Cyphers
to the left Hand.

It is alio to be obferved, firfl, That it will be convenient to make that Num-
ber the Multiplicand, which contains the moll Places, though fometimes it may
be lefs in Quantity : Secondly, that if the Multiplicand and Multiplier be both
.Decimals, that is, both Parts of Integers, the Produft will be a Decimal.
Thirdly, if the Multiplicand and Multiplier be mixed, that is, Integers and De-
cimal.Parts of Integers, the Produft will be mixed. Laftly, if the Multiplicand
and Multiplier be mixed, and the other aDccimal, the Produft will be fometimes
mixed, and fometimes a Decimal.


	Ex A Ml'LE I. Of Decimals alone. >743²■7�.3	Example II. Cf Integ. and Decimal 7.234S 1,25
	22296 ?43²52024	36172; 144�9Q 72345
	facit f299016	Facit 9104212c

Example III.
Where the Multiplicand
is mixed, and Multi-
plier a Decimal.
72,4072
>3S7

5068494
3640350

2172216

25(8693594

In Example I. of Decimals alone, the Produft is 5299016, that is, it is
'^299016 Parts of an Integer, or 1 -divided into 10,000,000 Parts, becaufe the
Denominator of every Decimal confifbof as many Places of Cyphers annexed
to I, as there are Places in the Decimal.

In Example II.' there being 7 Places of Decimals in the Multiplicand, Ithere-
i ire have cut off 7 Places of Figures from the Produft, and the Produft is 9 Inte-
gers, and, 042125 Parts of an Integer, divided into 10,000,000 Parts.

In Example III. I have alfo cut off 7 Places of Decimals, becaufe there arc
4 Places in the Multiplicand, and 3 in the Multiplier, and the Produft is 25 In-
tegers, and 8693594 Parts of an Integer divided into 10,000,000 Parts.

III. Multiplication of Duodecimals, 'vulgarly called' Crofs Multiplication.

As in Decimal Multiplication the Integer is divided into 10, fo here it is di-
i'iciad into 12 Parts, as a Shilling into 12 Pence, or a Foot into 12 Inches.�1�>
In'the following Examples, 1 fuppofe the Integers to be Feet, and the Duodeci-
mals Inches. As this kind of Multiplication may be performed, as well by tak-
ing the Aliquot, or even Parts of 12, out of the Multiplicand, as will be imme-
diately fhewn, as by multiplying the Multiplier into the Multiplicand : Before
I proceed any farther, you are to obferve, that the Aliquot (which are the even)
Parts of a Foot, are as follows; vis. In 12 there is twice 6, three times,4,
fiwr times ?.. fix titr.j. 2, waht tintes 1 snd |, aud 12 times 1 ; and therefore.

-ocr page 43-

Of MULTIPLICATION. 39

6 r� a half, 4 is one third, 3 is one quarter, 2 is one fixth, I and half one eighth,

and 1 one twelfth.-----In this kind of Multiplication there is a great Variety, as

iollo

3WS.

I. Tq multiply Feet, Inches, and Parts, into Inches, hy aliquot Parts.

Rule. Place under the Multiplicand the Number of Times that the aliquot
Multipl ier can be had in the Feet, Inches, and Parts, obferving to begin at the
left Hand, and for every one that remains at the Feet, more than the Times
that the aliquot Multiplier can be had in them, to add 12 to the Inches, and (0
the like to the Parts, C3Y.

In Example I. 6 being contained twice in

12, I therefore fay the two's in 20 is 10, the Example I.

two's in 8 is 4, and the two's in 6 is 3 ; fo Feet. Inch. Parts,

that the Product is 10 Feet 4 Inches 3 Multiply 20 8 6

Parts. By 00 6 Inches

In Example II. 4 being contained 3 times -----------------

in i2, therefore I fay the three's in 16 is 5 Produce 10 4 3

time?, and 1 remains, fet down � under the 16; �--------------

then the 1 remaining, being a Foot, equal to

12 Inches, I add it to the 8 Inches, which makes Example II.

20, and then fay, the three's in 20 is 6 times, Feet.Inch Paxts.

fet down 6 under the Inches, and carry the 2 Multiply 16 8 7

Inches remaining to the Parts, which 2 being By 4 Inches

equal to 24 Seconds, and added to the- 7, makes -------------------------

31 Seconds, wherein I find three 10 times, and Produit 5 6 10 4

1 remains, therefore I fet down 10 under the-------------------------

Seconds, and the 1, being one third of 3, the

Bliquot Part, is equal to 4 Seconds, and the Example III.

Produit to 5 Feet, 6 Inches, 10 Parts, 4 Se- Feet.Inch.Parts,

conds. Multiply 27 II 9

In Example III. 3 Inches being contained By 3 Inches

4 times in 12, I therefore lay tie fours in 27 -------------------------

is 6 times, fet 6 under z;, and 3 remains, e- 6 11 11 3
qua! to 36, and 11 is 47, which contains 4

11 times, fet 11 under Inches, and remains 3, equal to 36, and 9 is 45, which
contains 4 11 times; fet 11 under parts, and the remaining 1, being one Quarter
of 4, the aliquot Part is equal to 3 Seconds, and the Product to 6 Feet, 11
Inches, 11 Parts, 3 Seconds.

II. To multiply Feet, Inches, and Parts, into Inches, by multiplying the Multiplier

into the Multiplicand.
Rule. Firll-, Place a Cypher ir.ftend of an Integer, under the Parts of the Mul-
tiplicand, and the Inches of the Multiplier, one Place farther to the right Hand.
Secondly, multiply the Inches of the Multiplier into the Parts, Inches, and Feet
of the Multiplicand, as if they were Integers or whole Numbers, carrying 1 for
every 12, and fetting down the fint Remains, when any, under the Figure you
multiply by, rjfr.

To illuttrate the preceding Rule by aliquot Parts, I have here made Ufe of
the foregoing Examples.

Example I. Example II. Example III.

Peet. Inch. Parts. Feet. Inch. Parts. Feet. Inch. Parti.

20 8 6 to S 7

06 04


27	11	9
		0	3
6	1 e	11	3.

In Example I. 6 times 6 is 36, fet down o, and carry 3, then 6 times 3 is 4U

F 2 and

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₄o Of M U L T I P L I C A T I O N.

and 3 I carry is ji, fet down 3 and carry 4 ; then 6 times 20 is 120, and 4 I
carry is 124,whiereiethere is iOtioa.es 12 and 4remains. fet4 under the Inches,
"and :o under the Feet, and the Product is 10 Feet, 4 Inches, 3 Parts.

By either of' thefe Rules, any Number may be readily multiplied, when the
Multiplier is an aliquot Part of a Foot : But when the Multiplier is nor an a
Part, then the Operation r.uil be done by the'alt Rule, which indeed is o-eneral,
Note, For the ready fmdiiig the Tiu�kves i_;, any PrednG, 'tis h jl to make a Table
of Twelves, et-jd U get it prj'Ully by Heart, as /cliches.

■

61 r 7zit"t f132 I-6-J-' r 1

7 � \ 84 12 / � 144 17/ V l

S S tirn. 12 is ^ 96 13 > tira, izis ^ 156 18 5. tan. 1 z is < 2

.04

tirn. 12 is>

9 1 / ^Io8 HI � '63 19 1 J 22J

10 J 1120 iv J (.1 So 20 J C.2.1C

III.

... �..dtiply Fret, Inches, end Parts, by Pm ts.

, Place a Cypher under the lalt place of the Multiplicand, inflead «of
and alfo another Cypher in ac,e of Inches, and then the Baits.

Rale. FirSt

I Inteeer ;

following to the right Hand. Secondly, Multiply the parts of the Muiti-

ia the Mattipii

: for ever)' : z, as before.
Operation. 9 times 7 is 63, fe! down 3 and

carry m

I i x a m P L E.
F. 1. P.

2� i I 7

then g times 1 1 is 99, and j J carry

in I have 12 8 times and
: down S and carrv 0 ; thee 9times
wherein I

is 10

I ^

25 is 2.2J

9 Part
9

have 12 ,9 times, and ; remains; let down ;
ai d carry 19. Jvfow as the'wfrole Multiplica-

/ 5

tion is ended, and 19 r< mains take 12 out of
it, and there remains 7,letdown under Inches,

and � for the 12. under the Feet,

^^^^^^^^^^^^^^^^^^_m.- ^ⁿ^ ^�ne Producl will be I Foot, 7 Inches, c
parts, § Sec nds, 3 Thirds.

IV'. To multiply Fs-t, Inches, and Parts, by Inches and Parts.
Rule. Fifil, Place a Cypher under the lafl Place of the .Multiplicand, inflead
of an Integer, aad the Inches and Parts in their Places, towards the right, Hand,
idly, Multiply fhelnches in the Parts, Inches, and Feet, carrying 1 for every 12.
jdly,.Multiply the Parrs into the farts, Inches, and Feet, in the ian.e Manner,
iud .he two Produits added together is til required.

Operation. Firit, 8 titr.es 9 is "2,
Example. fet down o, and carry 6 ; then 8 times

;6, and 6 i carry is 62, fet down

9 by 8 Ii ch

o 8 7

2 and carry � ; then 8 times 32 is 2 :6

Multiply

is 261, wherein I find iz
11ns, let down g
Place of Fee:.

and 5 1 carry

2 I tim.es, and

and cairy 2!


21 1	9 7	2. 0 ■	1
23	4	2 6	3

2d!y, 7 time; 9 is 63, fet down 3 end

carry 3 ; then 7 times 7 is 49. and 5 i

carry is 54, fet down 6 and carry. ■

then 7 times 32 is 224, and 4 I carry is

eed, wherein 1 find 12 iq times end

<5remain-, fet down o and carry 19, out or which
i Irche , and 1 for the 12: under t!

taking 12, 7 remains, which

V. To tniFFjdy Feet, Inezes, '<::d Fart;, into Feet, Inches, and Parts, <zxcbc;z the

F e?; of toe \h ... ayid and'Multiplier doth nut exceed 20.
Rule. Fir;:, Place the Feet of the Multiplier vr.dcc the !aft Place of the Mel.
�and the Inches and Parts towards the right Hand in their Places.
the; Feet, [riches, and Pants of m -, iP_pa[

trp

rVt-.-lv into the Parts, Inches, and Feet of the Multiplicand, as before in the pre-

ceding Rules ; and their feveral pjcuda�s beifig added, will bj tire true Pre ■
feqer.id. "' ' ..... '-" ' OpeniF::F

-ocr page 45-

Of MULTIPLI C A- T ION. ₄r

Operation. Firft, 7 times 5 is 3c, fet
down 11 and carry 2 ; then 7 time:. 6 is Example.

42, and 2 I carry is 44, fet down 8 and T'. i. P. F. I, P.

carry 3; then 7 times M is 77, and 3 Multiply 11 6 5 by 1 1 9 7

I carry is 80, fet down 8 and carry 11 97

6, which put one Place to the Left.
Secondly, 9 times 5' is 45, fet do.vn 9
and carry 3 ; then 9 times 6 is 54, and
3 I carry is 57; letdown 9 and carry
4; then 9 times II is 99, and 4 I carry
is 103, fet down 7 and cany 8. Third-


	6	8	8	11
8	7	9	9
126	10	/
I 56	i	1	S	11

ly, 1 1 times 5 is 55, fet down 7 and
carry 4 ; then 11 times 6 is 66, and 4

is 70, iet down 10 and carry � ; then 11 times 11 is 121, and � [ carry is 126,
xvhich fet down, and the Produisis 136 Feet, 1 Inch, 1 Part, 5 Seconds, and 11
Is,

Note I. It matters not whether the Feet, Inches, or Parts hefirfi multiplied, fa that
their rejpcc�i-ve Produits are but duly placed

V. Ta multiply any Number of Feet and'Inches into any Number of Feet and Inches.

Ru/e. Firft, Multiply the Feet into themselves as Integers. Secondly, Ihftead
of multiplying the Feet into the Inches, take the aliquot Par's of a Foot, as
often as they can be found in the Feet, that ftand diagonally againft the n ( >y
Rule I. hereof ) and halve them when required. Thirdly, The Inches multi-
plied into themfeives, every 12 is an Inch, the Remains are Parts.

In Example I. the Feet being iiift multiplied into the
Feet, proceed to the Feet into the Inches as following : Example I.

Firft, as 1 Inches is the 4th of 12, therefore by Rule I. Feet. Inch,

find the fours in 218, faying the 4's in 21 is 5 times, and Multiply 272 3
1 remains, fet down c as at A ; and then fay, them's in by 218 6

18 !s 4 times, and z remains, fet down 4, and the 2 �-------�----------.

remaining being the half of 4, therefore fet down half 2176

pee for it, <viz. 6 Inches ; then will 54 Feet 6 Inches, 272

which is equal to a quarter,Part of 218 Feet, be the 544

Product of" 21S Feet, multiplied into 3 Inches. Se- A 54 6

�ondly, As 6 is contained twice in 12, therefore to mul- B 136

tiply 276 Feet into 6 Inches, is no more than to take its I 6

half, or fay, the 2's in 2 is 1, (et down 1 at B, and fay,

the 2's in 7 is thrice, fet do.vn 3 next after the 1. and 59486 7 �

carrying the 1 to the 2, which makes 12, ,fay, the 2's ■----------------------------

lis 12 is 6 times, fet down 6, and then the Produit of 272

Feet, into � Inches, will be 136 Feet. Thirdly, Multiply the 6 Inches into 3
Inches, is equal to 1 Inch, 6 Parts; and the whole Produit to 594S6 Feet, 7
Inches, and 6 Parts.

In Example II. Firft, as 9 Inches is threeQuarters of Example II.

12, therefore to multiply 531 Feet into 9 Inches, firft F. f.

take, the half of 531, which is 26c � 6 as at A, and the Multiply 752 9
half of 265� 6, which is 132�9 as at B. by 531 2

Secondly, As 2 is the fixth of 12, therefore take the ----------�

(fs in 752, which is 125, as at C. Thirdly, The Inches 752

into themfeives, make 1 Inch, 6 Parts, and the Whole 2256

t-eir.gadded as in Example I. is 39983,- Feet, 4 Inches, 3760

X Parts. A 265 6

B 132 9
«2j o
1 6

, ., 339⁸35' 4 ⁶

J»

-ocr page 46-

₄2 Of MULTIPLICATION.

In Example III. Firft, As 1 Inch is the twelfth Part
of tz, therefore to multiply 325 Feet into I Inch, take
the i2's in 325, which axe 27 I, as at A. Secondly,
To multiply 392 Feet into 11 Inche's, firft take the
half of 392, which is 196, as at B, whofe half is 98, as
at C ; and which two Produ&s are equal to 392 Feet
multiplied into 9. Now as the Remains to 11 is 2,
which is a fixth Part of 12, therefore by Rple I. take
the 6's in 392, which is 6� Feet 4 Inches. Laftly,
The Inches multiplied into themfelves make 11 Parts,
and the feveral Produces added, are 127786 Feet, j
Inches, and 11 Parts.


Example	ru.
F.	i.
Multiply 392	1
J«	n
i960
784
1176
A 27	1
B 196
C 98
D65	4
E	0	11
I277»6	�	11
Example	[V.
F.	1.
Multiply 524	4
372	5
1048
36�8
1�72
� 124
B 131
C 87	4
	1	8
^J:"⁰-	_5_	8
Example	V.
f.	I.
Multiply 723	1
'by �12	8
1446
7^Z3
,. � r -
3015
A 256
B 43
C361	6
D 120	6
	4	8
37©9<7	r	S
Example VI
F.	1.
2,-9	10
1 72	10
_8~
1813
259
A 86
B 57	4
C 129	6
JD86	4
.	8	4
44907	10	4

In-ExampleIV. Firft, As 4 Inches is the third pf 12,
therefore to multiply 372 Feet into 4 Inches, tak'e the
3's in 372, which, are 124, as at A, Secondly, As
in c there are 2 aliquot Parts of 12, <vi%. 3, which
is a 4th, and 2, which is a 6th, therefore firft take the
4's in 524, which are 131, as at B, and then tke 6's in
524, which are 87 4. Thirdly, The Inches into them-
felves, are 1 Inch S Parts, and the whole Product
195 2 70, 3 Inches, 8 Parts.

In Example V. Firft, As in 7 Inches there are two
siiquot Parts of 1 2, tvix. 6 which is a half, and 1 which
is a 1 2th, therefore to multiply 5 12 Feet into 7 Inches,
firft take the halves or 2's in 512 Feet, which are 256,
as at A, then the i 2's that are in 43, as at B. Secondly,
As in 8 there are alio 2 aliquot Parts of 12, <viz. 6 and
2, therefore to multiply 723 Feet into 8 Inches, firft
take the halves or 2's in 723, which are 361 6, as at
C, and then the 6's, which are 120 6, as at D.
Thirdly, the Inches into themfelves are 36, equal to
4 Inches, 8 Parts, and the whole Product 3709^7 Feet,
4 Inches, 8 Pa;ts.

In Example VI, Firft, As in 10 there are two aliquot
Parts of 12, viz. 6 which is a half, and 4 which is a
third ; therefore to multiply 172 Feet into io Inches,
firft take the halves or 2's in 172 Feet, which are 86,
as at A, and then the 3's, which are 37 4. Secondly,
There being the fame aliquot Parts in the other 10
Inches, therefore firft take the halves or z's in 239 Feet,
which are 129 6, as at C, and then the 3's, which are
86 4, as at D. Thirdly, The Inches 10 into 10 equal
to 100, are equal to 8 Inches, 4 Parts, and the whole
Product to 44907 Feet, 10 Inches, and 4 Parts.

Thus

-ocr page 47-

O/ MULTIPLICATION. 43

Thus have I given you a Number of Examples in all the Variety of odd Inches
that can happen, which, being well underftood, will make the Menfuration of
Superficies and Solids very eafy and delightful to every Capacity. And in con-
fid er ation thatfome Kinds of Works are performed by Yard iVIeafure, I fhall
therefore, before I proceed to Divifion, fhew the Multiplication of Yards andFeet.
IV. Multiplication of Tards and Feet.

Note, ift, That Yards multiplied into Yards produce Yards. 2dly, That
Yards multiplied into Feet every 3 is a Yard, the Remains more than 3 are long
Feet, a long Foot is one Foot in Breadth and 3 Feet in Length. 3dly, Feet
multiplied into Feet produce Parts, which are fquare Feet, 3 of which make 1
long Footaforefaid.


	Feet.
	273 i
	251 z
	273
	I3⁶5
	546
	A 83 2
	B 91
	C 91
	0	2
	68788 2	2

Operation. Firft, The Yards being multiplied as Integers, to
multiply 251 Yards into 1 Foot, as I is the third Part of 3, the
Feet in a Yard, therefore take the Thirds of 251, which are
83 2, as at A. Secondly, As 2 is two Thirds of 3, therefore to
multiply 272 Feet into 2 Feet, take the Thirds, twice in 273,
which are 91, and 91 as at B and C. Thirdly, The Feet multi-
plied into themfelves are two Parts, and the whole Product is
equal to 68788 Yards, z Feet, and 2 Parts.

The next Thing in Order, to conclude this Lecture, is to
fhew,

H01V to prove Multiplication.
Rule. Make that which was your Multiplier your Multiplicand, and then
multiplying as ufual, if the Produft be the fame, your Work is true; if not
'tis falfe.

L E C T. V. Of D 1 v 1 s 1 o n.

Ivifion is nothing more than a compendious Subtraction ; for as many times
as the Divifor can be fubtracied out of the Dividend, fo many Units is the
Quotient. In Divifion there are foar principal Parts to be obferved ; -viz. 1. The
given Number which is to be divided, called the Dividend. 2. The given
Number by which the Dividend is to be divided, called the Divifor. 3. The
Number arifino- from the Number of Times that the Divifor is contained in the
Dividend, which is called the Quotient ; and laftly, a Number that fometimes
happens to remain when the Divifion is ended, lefs than the Divifor, which is
called the Remains.

Divilion in general is per- D F EG TableofDivifors.

formed by this Analogy, viz. 3725)99725432(26664§°|f
As the Divifor is to 1, lb is the /7450 �.'��': abed e

Dividend to the Quotient; _____■::::

which I (hall illuftrate by the ff2C22,*5:: :

following Examples. �223c'o":;:

Example.----------:::

It is required to divide 22475,4:-.

997²5432, by 3725; firft place ^2235 o : :■


H	3725	1
A	745°	»
K	'i «75	3
C	14900	4
L	18625	5
B	22350	6
M	26075	1
N	29800	&
O	335 � 5	9

'theDividend and Divifor as at 1----------: :

DE, feparated by a Crotchet as /2404,3 :

F. Alfomake another Crotchet m 2235,0 :

as G to fe parate the Dividend ----------:

from the Quotient. Secondly, »16932

make aTableof Divifors as in ^14900

the Margin, thus ift, place ■

3725 and againft it feti; 2dly, q 2032 remains.

double 3725, asatA74;o,and .<��

M&inffc

-ocr page 48-

43- Of D I ■ V I S I O N.

againft it fet 2, figriifyihg, that' 7450 is the Divifor 2 times. Thirdly, Add
3725 and 7450 together, which make 11175; as at K, againil which fet 3.
Fourthly, To 1 ! 175, add 372c, which make 14500, as a: C, and againft it fet 4.
Fifthly, To 14900, add 3725, which make 18625, as at L, and againil it fet 5.
Proceed in hke Manner, to add the firft and lail together, until you have repeated
the Operations 9 times, placing the Number of 'F�mes againil each. Or other-
wife, multiply the D�v�for 3725, by 2, 3, 4, 5, 6, 7, 3, 9, and their Producls
will be againil A, K, C, I,, ti, M, N, O. This being-done, the Work is very
eafy, ana is thus performed. Firff, As 3725 cannot be had in the firft 3 Figures
�f the Dividend 997, therefore under ;he fourth Figure 2, make a Point; then
i'ny, how often 3725 in 9972 : Look in the Table of Divifors for the leis near-
eft Number to 9972, « Bien is 7450, againft which viands 2, as a: A.

Place 2 in the Quotient, as at a, and 7450 under 9972, as at/, and fubtracl
7450 from.9972, the Remains is 2522, as the firft 4 Figures towards the left
Hand ktg. Secondly, Make a Point under 5 in the Dividend, which bringdown
and place againft 2522, as thus, 25225 for a new Dividend. Then fay, how
often 3725 in 25225 ; look in the Table of Divifors for the neareft lefs Number,
which is 2^350, againft which ibnds 6 ; place 6 in the Quotient, as at b, and
22350 under 25225, as at b, and fubtracl 22350 from 25225, the Remains is
2475, as the fnft 4 Figures to the left at i. Thirdly, Point the next Figure 4,
jn the Dividend', and bring it down to 2457, as thus, 24754 at /, for a f�cond
new Dividend. Then fay, how often 3725 in 24754 ; look in the Table of
Divifor?,' and the neareft leis Number is 2Z350, againft which ilands 6, as at B ;
place 6 in the Quotient, as at c, and 22550 under 24754, as at /-, and fubtracl
22350^0.01 24754, the Remains is 2404, as the firft 4 Figures to the left at/.
Fourthly, Point the next Figure 3, in the Dividend, and bring it down to 2404,
as thus. 24043, as at /, for a third new Divifor. Then fay, how often 3725 in
24043 ? Look in the Table of Divifors for the neareft lefs Number, which is
22350 (as before) againft; which ftar.ds 6 ; place 6 in the Quotient, «id 22350,
under 2404.5, and the Remains is 1693, as the Iirft 4 Figures to the left at ».
Fifthly, Feint the (lest arid la'ft Figure 2 of the Dividend, and bring it dWwh
to 1693, as thus, 16932, as at p, for a fourth new Divifor. Then fay, how
often 3725 in 16932 ; look in the' Table of Divifors for the neareft lefs Number,
which is 149de, againil which Hands 4; place 4 in the Quotient, as at e, and
14900 under 16932, and fubtracting 14900 from 16932, the Remains is 235.:,
and which being the 'nil Remains, is 2052 Parts of 5725, and which together
inake a Fraction thus, j?|j, which mull be fet in the Quotient, next after
.26664, as in the Margin.

Nate, That as many Points as are placed under the Figures of the Dividend,
Co many Figures will be in the Quotient.


The Value of	this Fracuoi	i, or any othe
found as fbliowi	lg, Admit	lite Integers
ling.

in the Parts of the Integer may be
n this Example to be Pounds S

Firft, Multiply 2052, the Remains by
Shillings in a Pound, nsne A, and divide the Pro.
duct 4064b) 'by 3725, the former Divifor, as at


i"	- -7 ■ ) ! " 'j 1	4064	0(1
		572	)
	c	IV	.10 1
		D	^:-

o bhiJiin

Li', arid the Quotient 10 are Shilii-ings, and
remains, as at C. Secondly, M

ins, by 12, the Pence in a Shilling, a 1
and'.divide the Produce 40680 by 37.. ', '
Divifor, as at E, and the Quotient ro ar'e

and 3450 remains. Thirdly, Multiply' 3430. ;;
� 3725)406^0(10 Pence. main's, by 4, the Farthings in one Penny, as at F,
372; a:.d divide the Pro-duel 13720, by 3725, as fc,

��1 and theQviotient 3 are Farthings, and 2 54 5 remains,

3-430' which are 2545 Parts of 3725 of a Farthing, the

B '4 . Farthing being divided into 37 ^ Farts. T-heMan-

7' - �-*

-ocr page 49-



		Of DIVIS I O N. 45 ner of reducing this and other Fractions, into the 3725) 13720 (3 Farth. leaft equivalent Parts, is taught in LeiSure VIII. ^llll$ If this Example be well underflood, it is fully '---------■ fufficient for performing all Varieties of Cafes in ²545 rem, whole Numbers that can happen, and more efpe- ■ cially when you have alfo learned the following Contractions in �i<vifait. I. When theDiviforis 10, 100, 1000, {fff. cut from the Di- A i�] 732JO vidend the fame Number of Figures to the right Hand as are B 100; 27JI43 Cyphers in the Divifor, and the Figures remaining to the Left C loco) T2\|'3�4, are the Quotient required. So 7320, divided by 10, I cut off the lalt Figure o, and 732 remaining to the Left, is the Quotient required, as at A. In like manner, 27543, divided by 100, the Quotient is 275-^5-; and 72354, divided by 1000, the Quotient is "]2_r\y-_s, as at B and C, as the Figures cut off to the right Hand are fo many Parts of the Divifor. And as in every of thela Cafes, the Divifor is decimally divided, therefore thefe Remains are Decimal Fractions; and tho¹ I have here fez their Denominators under each for Plainneis fake, yet in Practice they are to be omitted, and the Fractions annexed to the whole Numbers, as following, viz. 10,732, not 1o_tV�V> ^an<^ 275,43, not 27c _T⁴_S%; and 72,354, not 7��« of which I have already advertiled you in the preceding Leftures. II. When your Dividend and Divifor conflits of Cy- 63I000) 7735jooo[l22 phers to the right Hand, cut off an equal Number of Cyphers in both, and then proceed as before taught: So to divide 7735000 by 63000, cut off three Cyphers in each, and divide 7735, by 63, as in the Margin. HI. If your Divifor have Cyphers annexed, and I20o)73254\|79(6io4₁�j§ your Dividend none, cut off as many Figures in 72 . . . your Dividend, as there are Cyphers in your Divi- -� for, and then proceed as before. So to divide 12 7325479 by 1 zoo, cut off 79, the laft two Figures , 054 in the Dividend, and dividing 73254 by 12, the ■ Quotient will be 6104, and 6 remains as in the 48 Margin. The 6 remaining, is to be placed before � 70, cut from the Dividend, making it 679, and 6 rem. which is the true remains, and the Numerator of � the Fraction _T��, as anr.exed to the Quotient. To prove Divifion.^* Multiply the Quotient by the Divifor, and to the Proluft add the Remains, when any, and if the Work be true, their Sum will be equal to the Dividend. Division of Decimals. Divifion of Decimals is performed in every Refpeft as whole Numbers, and for difcovering the true Value of the Quotient, this ia the general Rule : Rule. The Places of Decimal Parts in the Divifor and Quotient, being accounted together, nmjl always be equal in 'Number ivitb thofe in the Dividend ; and therefore as many Figures as are cut off in the Dividend, fo many mufl be cut off in the Divifor and Quotient : or thus ; cut off as many Figures in the Quotient, as will make thofe cut off in the Divifor equal to thofe in the Quotient ; always oblerving, that if there be not fo many in the Quotient, to add Cyphers to the left Hand. And alio, that if your Dividend be an Integer, or have lefs cut off than is in the Diviior, to add Cyphers to the Dividend, till they are equal. ' This general Rule admits of four Cafes.

			G Gsfi

-ocr page 50-

Of DIVISION.

Example.

Z5>⁶35) 4^72.565 (^l8*
²�>⁶3�"

Cafe 1. When the Places of Decimal Parts in the
Divifor and Dividend are equal in Number, as in
this Example in the Margin, where both Divifor and'
Dividend are mixed Numbers, then the Quotient will
be all whole Numbers.

210906
205080

5826c
51270

6995 rem.

427) 7264,271 (.17,012:
427/

Example TI.
Divide 7264,271, by 427, as in the Margin*
Here the Dividend is a mixed Number, and the
Divifor is Integers, and as here are three Decimals
in the Dividend, and none in the Divifor, there-
fore cut offo 12, the fall 3 Figures in the Quotient'
and the Quotient will be 17,012.

147 rem.

Example TIT. Divide 7 5 by ,012;, as in the Margin.
7500 (60 Here the Dividend is Integers, and the Divifor a Decimal:

750 and feeing that 75, the Dividend, confifts but of two Places,

�� 1 therefore add two Cyphers to it. making it 7500, that

00 thereby both Divilor and Dividend may be made Fractions,

� and by their being both of equal Number of Places, there-

fore by Cafe I, the Quotient is Integers..
Cafe 2. When there are not fo many Places of Decimal Parts in the Dividend,
as there are in the Divifor, then annex Cyphers to the Dividend, to make them
equal, and the Quotient will be all whole Numbers, as in Cafe 1

Example IV.
Divide 3.425, by ,725,83
in the Margin. Now here
the Dividend being In ■
tegers, and the Divifor 3
Decimal, to bring out In-
tegers in the Quotient, I
add 3 Cyphers to 3425,
the Dividend, and the
Quotient is 4724, and 100
remains. But if 'tis re-
quired to have the Quo-
tient to a greater Exact-
nefs, then 1 add a compe-
tent Number of Cyphers
more, to the Dividend.
In the following Ex-
ample, at A, in the Mar-
gin, 'tis required to have
two Places of Decimals,
alter the Integral Part of
the

,725) 34²i'^00OOO(4724-^I5

2900

3425,000(4724
2900 � ��

?²S

,100 rem.

57S rem.

-ocr page 51-

Of D I VI S I O N. 47

the Quotient, where the Quotient is 4724,13, and 57c remains ; for by adding
two Cyphers more to the Dividend, than was required before to make the Di-
vifor and Dividend equal ; and cutting off the fame Number of Places from the
Quotient, leave 13 for the fractional Part required, and 575 remains.

In this manner, by annexing of a greater Number of Cyphers, you may come
nearer to the Truth ; but in all Cafes like this, where the Diviior is not contained
an exadt Number of Times in the Dividend, there will always be a Remainder.

Cafe 3. When the Number of Places of Decimal Parts
in the Dividend exceed thofe in the Divifor, cut off the 7,54) 71,4038 (9,47
Excefs of Decimal Parts in the Quotient. As for Exam- 6786

pie, divide 71,4038, by 7,54, as in the Margin; where -■

the Number of Decimal Parts in the Dividend is 4, and 3543

but 2 in the Divifor ; therefore, as the Excefs is 2, cut 3016

off 47, the laft two Places in the Quotient .

5278
5278

Cafe 4. If after Divifion is finifhed, there are not fo
many Figures in the Quotient, as there ought to be Places 43) 5* 397 5 G0032J

of Decimal Parts by the general Rule, then fupply their 129

Defect by prefixing Cyphers before the Figures produced 1-----

in the Quotient. As for Example, divide ,13975 by 43. 107

Now here the Dividend is a Decimal, and the Divifor is 86

Integers, whofe Quotient is 325. But as in the Dividend ■

there is 5 Places, therefore, according to the general 215

Rule, I prefix 2 Cyphers before the Quotient 325, mak- 215

«ng it ,00325, which is the true Quotient required. ��

Note, When any Decimal Fraflion, or mixed Number, is to be divided by ars
Unit, with any Number of Cyphers annexed, remove the Separatrix, as
many Places towards the left Hand, as there are Cyphers annexed to the
Unit; fo if 57,27 were given to be divided

� the Quotient wi

Now, from the preceding Examples, it may be obferved, firft, That when the
Dividend is fuperior to the Divifor, the Quotient is either Integers, or Integers
and Decimals : and laftly, That when the Divifor is fuperior to the Dividend,
the Quotient is a Decimal, and which in both Cafes holds good in all other Ex-
amples.

LECT. VI. Of Reduction.

REDUCTION is nothing more than Multiplication or Divifion, or both,
and its Ufe in whole Numbers is for changing Quantity out of one Deno-
mination into another, as greater into lefs by Multiplication, or lefs into
greater by Divifion.

G %

Example,

-ocr page 52-



	₄3 Of REDUCTION. Exam'Ple I. In � 287 fuperjicial Feet, hciv many fuperfcialInches ? 5378 Here becaufe 1 fuperficial Foot contains 144 fuperficial Inches, 144 therefore multiply �278 by 144, and the P.oduft 76,1032, as in the ------- Margin, is the Anfwer required. 21112 2 11 1 ? 760032. ExAMPlrE II. In 760032 fupeifui'al Inches, hows tnavy fuferfi�ial Feet ? I44) 760032 (5278 Here you divide 760032 the Number given by 144, the

		720:': : 400 : : 2�8:: '1123 : 1008 : 11 J? 1152 9 rem.
			fquare Inches in a fouare Foot, and the Quotient is 5278. Now thefe two Examples, which are converfe to each other, iiluftrates all that can be done in Reductions, and; therefore I reed only add the following Rules, by Vhicl} Reductions in general may be performed.


JLuk 1. To reduce Pounds into Shilling?, multiply the Pounds by 20, the Shillings in a Pound, the Product will be Shillings ; and to reduce Shillings into Pounds, divide the Shillings by 20, the Quotient will be Pounds. Rule 2. To reduce Shillings in<o Pence, multiply the Shillings by 12, the Pence; in a Shilling, the Product will be Pence ; and to reduce Pence into Shillings, «livide 'he Pence by 12, the Quotient will be Shillings. Rule 3. To reduce fquare Yards into Feet, multiply the Yards by 9 the fquare f Set in a Yard, and the Produit will be Feet ; and to reduce fquare Etet into Yards, divide the Feel by 9, the Quotient will be Yards. Rule 4. To reduce folid Ym�s into foiid Feet, muEiply the Yards by 27 the folid Feet in a folid Yard, and the Produit will be folid Feet ; and to reduce fo'iJ Feet into folid Yards, divide the Feet by 27, and the Quotient will be foiid Yards. fiufe c. To reduce fquare Statute Rods into fquare Feet, multiply the Rods by 272.!, the fquare Feet in a fquare Rod, and the Product will be fquare Feet, and to reduce fquare Feet into fquare Rods, divide the Feet by 272^, and the Quotient will be fquare Rods. Ruled. To reduce Squares of Roofing, Tyling, &c', into fquare Feet, multiply �the Squares by ioo, the fquare Feet in a Square of Work, and the Product will be fquare Feet. And to 1 educe fquare F'eet into fquare Rods, divide the Feet b.v 272-J, and the Quotient will be fquare Rods. R;il« 7. To reduce folid Feet into folid Inches, multiply the Feet by 1728, the Number of folid Inches in one folid Foot, and the Product will be iblid Inches^ and 10 reduce folid Inches into folid Feet, divide the folid Inches by 1 728, and the Quotient will be folid Feet. ;\.:'e 3. To reduce Loads of Timber to folid Feet, multiply the Loads by 50, the Number of folid Feet in a Load of Timber, and the Product will be folid Teer. And to reduce folid Feet into Loads, divide the folid Feet by 50, and the Quotient will be Loads. ■ Thei'e Ruls, which are very plain, being underftood, will render the Reafon of . U othej Kinds of Rtduciion eafy to the meaneft Capacity ; and as the Re- ,w.uc,^r; hi pe�imals will be b/jft undit��od when Vulgar Fractions have beea explained,

-ocr page 53-

The Golden Rule, or Rule of Three. 49

explained, I fhall therefore proceed to the Golden Rule, or Rule of Three in
whole Numbers.

LECT. VII. The Golden Rule, or Rule of Three.

THIS Rule for its excellent Ufe is called the Golden Rule, and teaches to
find a fourth Number, which fhall have the fame Proportion to one of
three Numbers given, as they have to one another, and therefore is alfo called
the Rule of Proportion. This Rule is DireS, Indirect, and Compound.

I. The fingle Rule of Three Direct, finds a fourth Number in fuch Proportion
to the third, as the f�cond is to the firft ; or as the f�cond is to the firft, fo is the
third to the fourth.

Example I. If the Diem? ter e f one Circle be 7, and its Circumference22, txihat
Zj" the Circumference of another Circle <vjhefc Diameter is 14 Feet ?


	C. D. C. 22-14:44 i 22t
	2,8 28
	7) 308(44 2.3:
	28 28

Rule. Firft place your Numbers as in the Margin, fecondly D.
multiply 14. rhe third Number by 22 the f�cond Number, and 7:
divide their Produit 308, by 7 the firft Number, the Quotient a
44 is the fourth Number and Aufwer required.

Now you rnuft obferve that as the firft and third Numbers
are always of i,ke Kinds, <cix. both Diameters, fo likewife are
the second and fourth Numbers of like Kind?, being both Cir-
cumferences, of which the firft is always given, and the laft is
the Anfwer required.

Note, When the fourth Number is thus found, place it next
after the third Number, with two Dots of Separation between
them as is cone at c. The fame Kind of Separation muft be al-

fo ahvays placed between the firft and f�cond Numbers, as at ��
(f. But between t.ie f�cond and third, always place four Dots o

or Points, as at b. Thefe Points of Separation, fq placed, fig- -----

nify the following Words, <viz. the two Points at a thus:, fignify the Words,
« t'a, the four Points at b thus : :, fignify the Words, 'fo is, and the two Points as
e thus ■.. fignify the Word to; and therefore the four Vumbers, 7 : 22 : : 14 : 44,
arc thus to be read, viz. as 7 is to 22, fo is 14 to 44. And fo in like Manner,
all other Numbers having the fame Analogy.

Example IT.
If the Circumference of a Circle be 22, vvhofe Diameter is 7, what is the Di-
amerer of another Circle whole Circumference is 44 ?

Here the Nature of the Queftion require: the two firft Num- Analogy,
bets to be placed theReverfe to thofeof the foregoing Example; C, D. C. D.
foras there the 4th Number required was the Circumference of 22 : 7 1:44 : 14
a Circle, fp hereon the contrary the Diameter of a Circle is
required. Eut the Manner of working by multiplying the third
Nui iber by the f�cond, and dividing by the firft, is the fame
here as before, as is feen in the Margin, where the Qnotient 14,

is the Diameter required. Now as in both thefe and all other-------.

Examples in the Rule of Three Direct, the fourth Number is 88

always equal to, or more, than the f�cond : So in the Rule of 88

Three Indirect the fourth Number is always lefs than the fe- _

cond ; and as the 4th Number in the Direct Rule is found by o rem.

multiplying the f�cond and third Numbers together, and di- _,.

viding oi their Product by the firft Number ; fo on the contrary in the Indirect
Rule you multiply the firft and f�cond into one another, and divide their Produit
by the third, as following.

II. The Rule of Three IndireB,
Example.
If 20 Men can perform a certain Quantity of Work in 50 Days, how long 3
Time will 40 Men be employed to perform the fame .'

Rule.

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			The Golden Rule, or Rule of Three.
		*°	The Golden Rule, or Rule of Three.
		*°

Men. Days. Men. Days. Rule. Multiply 50 the f�cond Number by 20 20 50 40 25 the firft, and their Produit 1000, divide by 40 the 20 third Number, and the Quotient 25 is the Anfwer ». ■ ■» required. 40) 1000 (25 III. The GoMen Rule Confound. In the Golden Rule Compound, there are five Numbers given to find a fixth -in Proportion thereto, which Numbers muft be �0 placed, as that the three firft may contain a Suppofition, and the two laft a Demand. And that you may place your Numbers truly, always obferve. that the firft Number be of the fame Denomination with the fourth ; the f�cond of the fame Denomination with the fifth; and the third with the fixth required. Example t. If 20 Bricklayers, in 136 Days, perform 680 Rods of Biick-work, how many Rods can 12 Bricklayers perform in 28 Days? Rule. Firft, ftate your Numbers as in the Mar- M. D. R. M. D. gin; fecondly, multiply the two firft Numbers to- 20 136 680 12 28 gether, �vix. 136 into 20. whofe Produft is 2720, 20 12 as alfo the two laft, 12 and 28, whofe Produit is ------- �� 336. Now the Anfwer to this Queftion is found 2720 336 by the Rule of Three Direit, for making 2720, ,---------,�-------------------------(the Produit of the firft two Terms) the firft Num- 2720 680 336 ber; the third given Number, 680 Rods, your 680 f�cond, and 336 (the Produit of the two laft) your �---------- third Number j then 228480, the Produit of 680, 26880 multiplied into 336, the two firft Numbers, being 2106 divided by 2720 the Quotient is 84, as in the Mar- ■---------- gin at A, which is the fixth Number, and the An- 2720) 228480 (84 A fwer required. 91760 10880 10880

				o rem.

	To prove the Golden Rule. As the four Numbers are Proportionals, that is, the 4th is to the 2d, as the 3d is to the ill ; therefore the Square of the two Means (which are the f�cond and third) are always equal to the Square of the two Extremes (which are the firft and laft:) that is to fay, if the Produit of the firft and laft Numbers, multi- plied into each other, be equal to the Product of the two middle Numbers multi- plied together, the Work is right, elfe not. So 228480, the Produit of 336, multiplied into 680, 336 2720 which are the two Means of the laft Example, as in the A 6.80 B 84 Margin at A, is equal to 228480, the Produit of 84, mul- ----------:---------tipiied at 2720, the two Extremes of the fame Example, 26880 12880 asatB. Hence'tis plain, that when the given Numbers, 2016 21 760 in the foregoing three Varieties of the Rule of Three are ■ » truly ftated (and which indeed is the only Difficulty in the 2:8480 228480 whole) the Manner of performing the Operations is very . ��------- ' eafy.

					LECT.

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		Of Vulgar and Decimal Fractions. 51 LECT. Vin. Of Vulgar and Decimal FraBions. I. Notation of Fraclions. A Fraction is a broken Number, fignifying one or more Parts, proportionally of any thing divided, and therefore is always lefs thanUnity. It conflits of two Numbers fet one over another, with a Line between them, as -J, which ligni- fies one fourth, or Quarter of an Integer or Unit ; and fo in like manner, \| lig- nifies one half; -J three fourths, or three Quarters ; \| two thirds ; \ one third ; \ three eighths ; f five eighths, &c. The upper Number is called the Numerator, and the lower the Demoninator. In all Fraclions, as the Numerator is to the De- nominator, fois the Fraction itfelf to that Whole.of which it is aFraclion. Hence 'tis plain, that there may be infinite Fractions of the fame Value one with an- other, for there may be infinite Numbers found, which fhall have the fame Pro- portion one to another. So f, j⁴_?, _T⁸_�( are each of the fame Value as, J ; and I, f, j\, \|\|, are each of the fame Value with-J. When the Numerator is Jefs than the Denominator, the Fraclion is lefs than an Unit, and therefore is called a Proper Fraclion ; but when the Numerator is either equal to, cr greater than its Denominator, the Fraclion is called Improper, becaufe 'tis equal to, or greater than an Unit. So \| is equal to i, as alfo $, and J-, csV. and f is equal to I -f-, and \| to l �. Fractions are Single or compound : Single Frac- tions are fuch as have but one Numerator, and one Denominator, as f two thirds, f three fifths, /_T nine elevenths, _T^s_s five twelfths, &c. Compound Fractions are Fraclions of Fractions, and are fuch as confift of more than one Numerator, and one Denominator, a of _�'_f of -J^, that is to fay, one Farthing, which is i of a Penny, which is j'_s of a Shilling, which is ₅g- of a Pound Ster- ling. All Fraclions, whofe Numerators and Denominators are proportional to one another, are equal to one another, as before oblerved. So f- is equal to �, and-\| to f, �fc. When Integers and Fractions are joined together, a if, ^or 7�³T' ^or �J-J-) ^tney are called.mixed Numbers. Things commonly expreffed by Fraclions, are the Parts of Coin, Weight, Meafure, &c. So Inches are Frac- tions, in refpecl of Feet, and Feet are Fraclions in refpeel of Yards, Rods, He. As Addition and Subtraction of Fraclions cannot well be performed without the Knowledge of the Reduction, I fhall therefore firft teach you the Reduction. II. Reduclion of Vulgar Fraclions. By Reduction you are taught, firft, how to bring fraclions into their leaft equivalent Parts, and their various Denominators into common Denominators, or into one Denominator. Secondly, to find the Value of any Fraclion, in the known Parts of the Integer. And laftly, to reduce whole or mixed Numbers into improper Fraclions, and improper Fraclions into mixed Numbers. I. To bring Fraclions into their leaft equivalent Parts. Rule. Firft, Divide the Denominator by the Numerator, and the Divifor by the Remainder, if any be: thus continue to divide the laft Divifor, by the laft Remains, 'till nothing remain, and the laft Divifor is your greateft common Meafure ; by which dividing the Numerator and Denominator, and their Quo- tients being placed in a fractional' Manner, will be a new Fraclion equal to the given Fraction, and in the leaft Parts. Example. Let f\|f, be a Fraction given, to be reduced into its leaft Terms. Firft, the Denominator 819, divided by 637, the Numerator, the Remains is 182, as at A. Secondly, the Divifor 637, divided by ^^^^^^^^^^^ 182 the Remains, as at B, the Remains is 91. A 182 rem. Thirdly, the laft Divifor i8z, being divided by the laft Remains 91, as at C, and o remains ; 182) 637 (3 therefore 91, the laft Divifor, is the greateft 546 common Meafure required. Fourthly, divide ------ 637, the Numerator of the given Fraclion, by B 91

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52 Of Vulgar and Decimal Fractions.

C91} 182(2
182

91, as at D, and the Quotient 7 is a new
Numerator. Fifthly, divide 819, the De-

nominator of the given Fraction, by 91,
as at E, and the Quotient 9 is new Deno-
minator Laftly, the laft two Quotients,
7 and 9. being placed as at F, will be the
new Fraction required; and equal to �f�,
the given Fraction.

o rem.

D 91) 637 (7 r.ew Numerator.
⁶37

o rem.

E 91) 81 q (9 new Denominator,
819

o rem.

F J new Frafiion equal to f f �.

Note, When it happens that your lad Divifor is an Unit, the Fraction is in its
leall Terms already, becaufe 1 neither multiplies nor divides.

It is alfo to be obferved, that fome Fractions may be abbreviated, by halving
Doth your Numerator and your Denominator as often as ycu can, and which
may always be done, when both Numerator and Denominator end with aC>pher.

II. To reduce federal Fractions, ixihefe Denominators are different, into other

FraSions having a common Denominator.
Rule. Firft, multiply the Denominators into themfelves, and their Product is
a new Denominator common',o every Fraction. Secondly, multiply every Nu-
merator into each Denominator continually, except its own, which lhall be new
Numerators.

Example. Let%,%, #, beFrafli'ons given, to he reducedinto other FraSions,
which /ball have one common Denominator.
i- 1 k Operation. Firft, to find the common Denominator, I fay, the

² * * Fenominator 2, into the Denominator 4, is 8 ; and 8 into the Deno-
�t It $T m'oator 6, is 48, the new Denominator required, which place under
abc each Fraction, as at a h c. Secondly, to find the new Numera-
tors, I fay, the Numerator 1 into the Denominator 4, is 4 ; and
4 into the Denominator 6, is 24, which I fet over 24 at a. Then the Nu-
merator 3, into the Denominator 2, is 6, and 6 into the Denominator 6 is 36,
which I place over 48 at b. Thirdly, the Numerator y, into the Denominator 2
is 10, and 10 into the Denominator 4 is 40, which I place over 48 at c. Then
will �|, ||, and |f, which have one common Denominator, be equal to the given
Fractions §, f, f, as required.

III. To find the Value of any vulgar Frailion in the known Parts of the Integer.

Rule. Multiply the Numerator of the Fraction, by the known Parts of the next
Jefler Denominator, and that Produtt being divided by the Denominator, the
Quotient is the Parts of that Denominator required.

Example. How many Inches are contained in -j"^- of a Foot,

75 as the next leffer denominative Parts of a Foot are Inches? I

12 therefore multiply 75, the Numerator, by 12, the Inches in a Foot,

�� and the Product 900, being divided by too, the Denominator,

ico) 9|oo ( the Quotient 9, is the Number of Inches, which are equal to ^n

�� as required. This may alfo be found by the Rule of Three

Direft : For ice : 12 : : 7; : 9.'

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		Of Vulgar and Decimal Fractions. J3 If the given Fraction _T\|4 be Parts of a i^rard, 7 c and it is required to know how many Feet and In- A3 ches are equal thereto, multiply the Numerator 75, � by 3, the Feet in a Yard, as at A, and theProduct 100) 2)25 (zFeet. 225 being divided by the Denominator too, the 25 rem. Quotient is 2 Feet, and 25 remains. Now in all 12 InchesinaFoot kind of Cafes, when a Remainder happens, multi- ------- ply the Remainder by the Parts of the next lefs 100) 3\|oo (3 Inches. Denomination, and divide by 100, as before. So ■ here, as Inches are the next lefs Denomination, therefore the Remainder 2 j, being multiplied by 12, the Inches in a Foot, and the Product 300, divided by 100, as before, the Quotient is 3 Inches. Thefe two Quotients, 2 Feet, and 3 Inches, are the Feet and Inches which are equal ^t0 ��- °f ^a Yard, as required. After the fame' manner, the Value of ^\|§ of a Pound Sterling, will be found to be 5 j. 6 d. 2 q, which to find after having multiplied the Numerator into 20, the Shillings in a Pound, which are the next lefs Denomination, and divided the Product by 480 the Denominator ; multiply the Remains by 12, the Pence in 3" Shilling ; and the Remains of that Product, after dividing it by 480, multiply by 4, the Farthings in a Peny, the next lefs Denomination, isfc. IV. To reduce ivbole or mixed Numbers into improper Fractions, and improper Fractions into mixed Numbers. Firft, If any Number be an Integer, and the given Denominator be 12, it is done by making an Unit the Denominator, and 12 theNumerator, as thus '/w Secondly, If the given Number be mixed, as _T2, then making 12 the Deno- minator, add 7 to 12, equal to I9, is the Numerator, and the Fraction is thus expreffed J-f. Thirdly,. To reduce an improper Fraction to a proper Fraction, divide the Numerator by the Denominator, the Quotient will be Integers, and the Remains, if any, will be a Numerator to the former Denominator. So -f-f is 4 i\|, for 59 divided by 12, the Quotient is 4, and 11 remains. V, To reduce a compound FraRion into a fingle Fraction. Rule. Multiply all the Numerators one into another for a new Numerator, and tne Denominators one into another for a nevv Denominator, which being placed in a Fraction, will be^the Fraction required. So \\ of _s%, is ₂\|^-, that'is 11 Pence, which is \\ of a Shilling, which is -V of a Pound, is j'/s, that is, it is yet 11 Pence, becaufe the new Denominator 240, is equal to the Pence in a Pound Sterling. III. Addition of Fractions. Before the Addition of Fractions can be well performed, you mufl firft obferve to reduce every given Fraction to be added, into its leaft Terms, and then the Work is very eafy, as appears by the following Rules. I. To add Fractions of the fame Denomination. Rule. Add all the Numerators into one Sum for a new Numerator, keeping, the fame Denominator ; and when the new Numerator is greater than the De^ nominator, divide the Numerator by the Denominator, and the Quotient will be the Integers and Parts. So if j\, _T\, ₇\, ,\, ,_t, be given Fractions to be added, the Sum of the Nu- merators added together, is equal to 32, and the Fraction is \ \ ■�* and as t-ha Numerator 32, is greater than 12 the Denominator, therefore divide 32 by 12, and the Quotient is 2 -,\, equal to 2 J, or 2-f, which is the Sum of the Fractions as required. II. To add FraBions of divers Denominations. ' Rule. Firft, Reduce the Fractions to be added into one Denomination. Z'dlv'j ■Add all the Numerators into one Sum. 3dly, If the Sum of the Numerators be' greater than the Denominator, divide the Sum of the Numerators by the Deno- minators, as before taught, and the Quotient is the Sum required. Eut when the Sum of all the Numerators is lefs than the Denominator, then the Sum of the Fractions is the new Numerator reqiiicsd-. H IV,

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		54 Of Vulgar and Decimal Fractions, IV. Subtraction of Fractions. Rule. Firft, Reduce the two Fraft'ons into one Denomination. Secondly, Subtract the ltfl'er Numerator from the greater, and the Difference is the Re- mains required. V. Multiplication of Fractions. Before Fractions can be multiplied, if there beany mixed Numbers, they muft be reduced'into improper Fraftions, and if any are compound Fractions, they muft be reduced to fingle Fraftions ; and then the Fractions being all reduced to the loweft Terms, this is the Rule. Firft, Multiply the Numerators into each other, their Produft is a new Nume- rator. Secondly, Multiply the Denominators into each other, and their Produft is a new Denominator. So �, multiplied by f, the Produft is if, equal to � ; and fo in like manner \|, -j% f, fy, multiplied into each other, their Produft is TS⁴i°p ' V^'^c'¹' reduced into the lead Terms, is _s^lj%. Now from hence it is plain, that the Multiplication of Fraftions is the very fame thing as to reduce a com- pound Fraction into a fingle Fraftion, as was but row taught in the Reduction I Of.Fractipns. And {o in the fame manner, ten thoufand Fraftions placed before cne another in a right Line, may be multiplied into each other. VI. Division of Fractions. Before any Proceeding can be made in the Divilion of Fractions, that are mixed or compound, and not in their leaft Terms, they muft be prepared as be- fore was taught in Multiplication, and then proceed by the following Rule. Rule. Multiply the Denominator of the Divifor, by the Numerator of the Di- vidend, and thsir Sum is the Numerator of the Quotient ; and the Numerator of ' the Divifor, being multiplied into the Denominator of the Dividend, the Pro- duft is the Denominator of the Quotient. Suppofef be to be divided by \|, as in the Margin at A, then A 6, the Denominator of the Divifor, multiplied into 3. the N.ii- !') "4 [le ^or Vlr meratorof the Dividend, the Produft is 18.for the Numerator B of the Quotient, and �, the Numerator of the Divifor, m�lti- ■5%) � (s?^or K plied into 4, the Denominator of the Dividend, the Product 20 is the Denominator of the Quotient required. So,\|, divided be^l, asatB, the Quotient is f�> equal to J. A general Rule for all Sorts of' compound Divifions. I. When there is a Fraclion in the Di-vifor or Di-viJend. Rule. Multiply the Divifor and the Dividend by the Denominator of the Frac- tion, adding the Numerator to that, to which it belongs, and their Produfts be- ing divided as Integers, the Quotient will be the true Quotient required. So 271, divided by 7 �, the Divifor 7, multiplied by 9 the Denominator of the Fraftion, whofe Product is 63, being added to 8 the Numerator of the Fraftion, their Sum 71 is a new Divifor. And then 271, multiplied by the Denominator 9, the Produft 2439 is a new Dividend, which being divided by 71, the Quo-' tient is 34ft ; and io in like manner, if 295! be to be divided by 27, then 27 multiplied by 8, the Denominator of the Fraction, the Produft 216 is the new Divifor, and 29c, the Integers of the Dividend, multiplied by 8, and the Nu- merator 7, added totheProduft, the Sum 2367 is a new Dividend. Now 2367, divided by 216, the Quotient is ioJ?J, equal to -J^. II. When there are Fractions in both Divifor and Dividend. Rule. Firft, Reduce the two Fraftions into oneDei.omination ; fecondly, Multi- ply the Divifor and Dividend by the Denominator common to both Fraftions, and to their respective Produfts add their Numerators ; and then, their Sums being divided as Integers, the Quotient will be the Anfwer required. So if 275! be to be divided by 39^ the two Fraftions reduced into the fame Denomination will be 4° and \|\|.. Now 39, the Integers of the Divisor, being multiplied by 56, and 40, theNumeratorof its Fraction added toit, is equal to 2224, which is a new Divifor, and 275, the [ntegersof the Dividend, multiplied into 56, with 21, its new- Numerator,

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Of Vulgar and Decimal Fractions. 55 Numerator, added to the Produit, is equal to 15421, which being divided by 2224, the Quotient is 6J°-\|^, which Fraction is in its Ieaft Terms. VII. Reduction, or rather the changing of'Vulgar Fraclions into Decimal Frac- tions, and Decimal Fraclions into Vulgar Fraclions. Rule. Annex as many Cyphers to the Numerators of the given Fraction, as you would have Places in the Decimal, which being divided by the Denominator, the Quotient will be the Decimal required. So co reduce f into a Decimal of two Places, I add two Cyphers to 3, the Numerator, making it 300, which being divided by 4, the Denominator, the Quo- tient 7j is the Decimal required. In like manner, if it was required to have had the Decimal of 3 Places, then I Ihould have added 3 Cyphers to the Numerator 3> making it 3000, which being divided by 4, as before, the Quotient would be 750, which is equal to ,75. For -j^ is equal to _T^\|\|, becaufe cutting off the jail Cyphers in both Numerator and Denominator, thus ^J §, the Remains _TJJ- is then the fame as the other Fraction. Vulgar Fractions may be changed into decimal Fractions by Uns Analogy, vn, as the Denominator of the vulgar Fraction is to its Numerator, fo is the given, Denominator of the decimal Fraction to its Numerator required. So if _T?~ jje a vulgar Fraction given to bechangedinto a decimal, whofe Denominator is ioo; then as 120 : 96 : : 100 : 80, fo that 80 is the Decimal required ; and on the con- trary, decimal Fractions may be changed into vulgar Fractions by this Analogy, fiz. as the d�cimal Denominator is to its Numerator, fo is the given vulgar De- nominator to its Numerator required. Let fg§ be changed into a vulgar Fraction, whofe Denominator is 120, then, as ico : 80 : : 120 : 96, io that _T\|f is the vulgar Fraction required. Note, It will happen in many Cafes of changing vulgar Fractions into deci- mals, that there will be ftill a Remainder altho' you ihould annex tenthoufand Cyphers to theNumerator of the given Fraction ; and therefore it is to be noted, that if yon make the Decimal to conflit of 5 or 6 Places, it will be near enough Jn almoli every Cafe of Bufinefi, and the Remainder may be rejected as of no Value Now there only remains to Ihevv how to find the Value of any given decimal Parts of a Foot, Pounds Sterling, &c. which is done by this Rule. Multiply the given Decimal into the Units that are contained in the Integer \os in decimal Multiplication) and the Producl �will be the Value of the Decimal. E X A M P L E I. Suppofe ,7852 be a given Decimal, whofe Integer is a Foot. Here the Decimal ,7852, multiplied by 12, the Inches or Units ,78,"2 that ave contained in a Foot, which is the Integer, the Product is 12 9,4124, which is 9 Inches, and ,4124 Parts of an Inch. And if we ■ iuprole an Inch to be divided into 100Parts, then multiplying 4124, 9,4124 theReaiains, by 100, the Product is 41,2400, which is 41 hundred 100 Par;s of an 1. ch. and the Remains 2400, is 2400 Parts of one Iran- ■-------------. dr�dth Part of an Inch divided into ten thoufand Parts, So thatre- 41,2400 jeeiing this laft Remains 2400, the Value of the given Decimal is 9 Inches and 41 hundred Parts of an Inch. Example II. Suppofe the aforefaid Decimal fignify a decimal p.artof a Pound Sterling.

	II ?
	II ?	Then
		Then

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jj6 Of Vulgar and Decimal Fractions.

,7852 Then, ,7852, multiplied into zo, the Units, or

20 the Shillings in I /. Shillings in the Integer or Pound, the Product:

. 15,7040 is 15 Shilling*, and 7070 remains, which

15,7040 being multiplied by iz, the Units in the next lefs

iz the Pence in l s. Integer, <vix, the Pence in a Shilling, the Product

. 8,4480 is 8 Pence, and 4,480 remains ; and which

8,4480 being multiplied by 4, the Farthings in a Peny,

4 the Farthings in id. the Product is {,7320, which is one Farthing, and

7920 Parts of � Farthing, the Farthing being di-

1,7020 vided into ten tbotifand Parts. So the Value of the

_________. Decimal ,7852 Part of one Pound Sterling, is 15

Shillings, 8 Pence, and ; Farthing, rejecting the

laft Remains 79Z0. Thus, a due Regard being

had to the Number of Units, which are contained in the Denomination of the

Integer, to which the Decimal Parts belong, any propofed Number of a Decimal

may be reduced or changed into the known Parts of what they reprefe.nt.

L E C T. IX. The Extra f�o'i of the Square and Cube Roots.

TO extraft the fquare Root, is nothing more than to find the Side of a geo-
metrical Square, whofe Area is equal to a given Number of Units, which
are generally called a fquare Number. A fquare Number is that which is pro-
duced by any Number multiplied intoitfelf: As for Example, 16 is a fquare
Number, which is produced by 4 multiplied into 4. So in like manner 9 is a
fquare Number, produced by 3 multiplied into 3. The Side of a geometrical
Square, equal to any given Number, is called its Root.

In the Margin is a Table of fquare Numbers, whpfe Roots are the
Ro. Sq. nine Digits, and which being nothing more than a Part of the Multi-
i 1 plication Table, it is fuppofed you have it already by Heart.

2 4

3 9

4 16

5 ²5

6 36

7 49

8 64
Q Si

Let 67 2 be a Root given to find its fquare 'Number.

I &l^z Rule. Multiply 672 into itfelf, as at/, m,

v.
1

°~^z whofe Produft is 451584, the fquare Number

requited, and whole Root is thus extrafted,

344 aiiz. Firft, Place a Point under the firft Figure

47°+ to the right Hand, as at c, and at every other

4°3² Figure towards the Left, as at b and a; and ob-

' ' ^nlieJ ferve, that as many Points as the fquare Num-

451^84 (672 ber contains, fo many Places of Figures the

a b c p Root will conflit of. Secondly, Make a Crochet,

;6 as at n and p, on the right Hand Side of the

g i�' fquare Number, as is done in Divifion ; and

12.7J91 .5 firft Refolvend. note, that every two Figures fo pointed, are

889 called a Punftation. Thirdly, Find in the

h i------- Table the r.eareit fquare Number that is con-

134.2)268.4 f�cond Refovend. tained in the firft Punftation to the left Hand,
2684 viz. in 45, which is 36, whofe Root is 6.

s-*-------< Place 36 under 45, and its Root 6 in the Quo-

0 rem, tient, as at d, and fubtrafting 36 from 45, the

Remains is 9, which place under 36, This is

your

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		Of Vulgar and Decimal Fractions. $j your firfl Work, and is no more to be repeated. Fourthly, Bring down the next Pun&ation 1$, and join it to the Remains 9, making it 915, which is your firfl; Refolvend, and on its left Side make a Crochet, as is done in Divifion to fe- parate the Divifor from the Dividend. Fifthly, Double the Root 6, it makes 12, which place on the left of the Refolvend, as at g. Then rejecting the lad Figure 5 in the Refolvend (which is always to be done) fee how often the Di- vifor 12 is contained in the remaining Figures 91, which being 7 times, there- fore put 7 in the Quotient at, and alfo on the right Hand of the Divifor at»", and muitiply 127, the Divifor increafed by 7, whofe Produ� is 889, which place under 935, and being fubtrafted from it, the Remains is 26. This being done, bring down the next Punftation 84, and join it to the Remains 26, making it 2684, which is a f�cond Refolvend, and then proceed as before, as follows, <visz.* Firft, Double 67, the Root fo far found, makes 134, which place on the left of the f�cond Refolvend, as at b, and fee how often 134 is contained in the Re- folvend, the laft Figure excepted, viz. in 268, which is two times. Set 2 in the Quotient at/, and on the right Hand of that laft Divifor 134, making it 1342, which being multiplied by 2, the laft Figure in the Quotient, its Product is 2684, which being placed under the f�cond Refolvend, and fubtracled from it, as be- fore, o remains ; which fliews that 451584 is a fquare Number, whofe fquare Root is 672, as required. Note, Firft, When the fquare Number contains 4 or more Punctations, as the Remains are produced, the next Pimftation is to be brought down, and joined to the Remains for a third, �V. Refolvend ; with which you are to proceed in every refpeft, as before with the firft and f�cond Refolvend. Secondly, That if at any time, when you have multiplied the Numoer ftanding in the Place of the Divifor, by the Figure laft found in the Quotient or Root, the Product, be greater than the Refolvend, then in fuch a Cafe, you are to put a Figure lefs by one, than the former, in the Quotient, and multiply by it as before : and when the Re- mainder be greater than the Divifor, put a Figure greater by one in your Quo- tient, and multiply it as before. Thirdly, If at any time the Divifor cannot be had in the Refolvend, then place a Cypher in the Quotient, and alfo on the right Hand of the Divifor, and to the Refolvend annex the next Punftation for a new Refolvend, with which proceed as before. When it happens, that after Extraction is made, there is a Remainder, the Number given to be extracted is called an irrational or furd Number, and its Root cannot be exactly obtained, although by adding Cyphers you may come as near the Truth as is required, but never can come at the Truth itfelf. As for Example, it is required to extract the fquare Root of 160

			Firft,

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		;8 Of Vulgar and Decimal Fractions.

16.9 (12,64911 I ■ lm ---------' 22 (c6o fi.-ft Refolvend. 44 n p � ab 246) 1600 fcccnd Refolvend. 1476 t r-------cd 252,4) 12400 third Refolvend, 10096 j t----------e f �528,9) 230^00 fourth Refolvend 227601 u tv gb 25298,1) 279900 fifth Refolvend. 252981 x y----------■ i It 252982,1) 2691900 fixthRefoIv. 2529821 162070 rem.			Firft, The firft Pumfdation being 1, the Square of I is 1, which place under», and fubtracling I from 1, remains o, fet 1 in the Quotient, and to o bring down the next Punclarioa 60, making the Remains 0,060. Secondly, Double the Quotient 1 makes 2, which place for your Divifor at 7. Now as 2 is contained 3 times in 6, if you was to place 3 in the Quotient, and 3 on the right Hand of the Divifor 2, as before taught, to make the Divifor 23, then 23 multiplied by 3 would be equal to 69, which is greater than 60, the firft Refolvend, and therefore cannot be fab- tracled from ic : Therefore in this Cafe, as was before noted, place a Figure in the Quotient lefs by t than the 3. 11'z. 2, and the fame on the right Hand of the Di- vifor 2, as at vi, and then multiplying the Divifor 22, by 2 in the Quotient, the Product is 44, which being placed under the firft Refolvend '60, and fub- tracted from it, the Remains is 16. Third-


	ly, to the Remains 16, annex two Cy- phers, as at ab, making it 1600 for a f�cond Refolvend; and then proceeding as before, the next Figure in the tjuo.- ti�nt w:ll be 6, and 124 remains, to which annex two Cyphers more, as at c 'd, making the Remains 124, 12400. which is your third Refolvend. Proceed in like manner, by continually adding two Cyphers to each Remainder, until you have ericreafed the Figuies in the Quotient to as many Places as may be required. In this Example 1 have er.creaied them to 5 Places, which I apprehend to be near enough foi" any Bufinefs, for if Unity was divided into a hundred tboufand Parts, there would not be t� o Parts wanted ; for 1264911, being multiplied �nt� itfelf, its Produit is 159,9999837921, which is very near equal ro 160, the given Number to be extracted, and as the Fraction ,9999837921, is U's than the Fraction ,00002, therefore the Root is not two Parts of one hundred thouland Parts o'f an Unit lefs than the Truth. To extraS the /qu'are Root of' avulgarFratlion, ivhieh is eommmfurableto its R'iOt; that is, a Fraction ivbich, afttr that Extraction is ended, haih no Remains. Rate. Extrait the fquare Root of the Numerator, for the Numerator of the Root, and alio the fquare Root of the Denominator, for the Denominator of tire faid Root. To extra� the fquare Root of a vulgar Trailion, "juhich is ineommenfuralle to its Root ; that is, a FraBion ivhieh, after that Extraction is ended, hath a Remain. Rale. Reduce the given Fra&ion into a Decimal, and then extract: its Root as before taught ; or find the integral Part 1 f the Root, to its Quadruple, and then adding Unity for the Denominator of the fractional Part, the Remainder, being doubled, is the Numerator. So the Root of 160, in the foregoing Example, is The Extraction of the Cube Root. A Cube Number, is that Number which is produced by multiplying any Number into itfelf, and its Product again by the fame Number. So 64 is a Cube Number, produced by 4 multiplied in 4, equal to 16, and 16 into 4, equal to 64. A Cube Number is a fuppofed Quantity ot Matter, put together in trie Form of aDhe, as Figure Y, Plate If, and the Length orMeafureof one Side offuch a Body, is called it: Root; therefore to extract the Cube Roo; of any given cubi- 7 cal

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Of Vulgar and Decimal Fractions.

Cal Number, is nothing more than to find the Length of the Side of
which contains a Quantity equal to the Numer given.

As in the Square Root, a Table of me Squares of the 9 Digits, is
of Ufe for the ready finding the neareft iefs Square in a Punccauon, fo

a Cube
Ro. Cu.

here a Table of the cubick Numbers of the nine Digits, is of very
great Ufa for the immediate finding the nearelt lefs cubick Number in
a Pundation, and is therefore placed in the Margin, and which is thus
made.

Let 8 be a Root given, to find its cuhed Number.

Multiply 8 into 8, its Produel equal to 64 is the Cube Number re-
quired. This is alio called the cubing of a Number, as fuppofing
� had been a Number given to be cubed.


	I	1
	2	8
	3	Z7
	4	64
	?	I2�
	6	21�
	7	343
	y	yia
	9	729

To extraS the Cuhe Root.

Let 146363183 be a cubed Number given to find its Root,

Firft, Point the firft Figure towards the
right Hand, and then every third Figure
towards the left, as at fed. Secondly,
Look in your Table of cubed Numbers,
and find the neareft lefs Cube Number to
146, the firft Punclation, which is 125,
whofe Root is 5. Place 5 in the Quo-

d e � abc
146363183 (527

i -3

_g-----.

75) 21.3.63 firft Refolvend.
b 1 ;o T

tient at a, and 125 under 146, and fub-

Sabducends.

60
k

trading 125 from i46,theRemains is 21.
This is your firft Work, and no more to

be done. Thirdly, To 21, the Remains,
annex 363, the next Punclation, making
21,21363, which is your firft Refolvend.
Now to find a Divifor, by which you are
to divide this Refolvend, its two laft Fi-
gures excepted, which are always to be
rejected, proceed as follows, <viz. Firft,
Square the Quotient 5, makes 25, which

15608. Subtrahend.-

5755,1,83. f�cond Refolvend.'
56784 I

7z 7044. > Subdueends.
P 31-3 3

5755183 Subtrahend.

811

Triple make 75, which is the Divifor re-
quired, asat,�-. Then fay, the 75's in 3 rem.

213 (the Figures remaining in the Re- ---------�.

foivend, excluiive of the two laft rejected

as aforefaid) is 2 times, equal to 1 50, which place under 213, as at h, and fet
² in the Quotient at b. Secondly, Treble 5, the firft Figure of the Root, equal
to 15, which multiplied by 4, the Square of 2, the laft Figure in the Quotient'
makes 60, which place under 150, onePiace forward to the right Hand, as atz'j
alio Cube 3, the laft Figureof the Quotient, equal to 8, whicti place under 60,
one Place more to the right, as at*. Then the 3 Subdueends, 150, 60, and 8,
being added as they Hand, their Sum make a Subtrahend 15608, which being
fubtracted from the firft Refolvend, there remains 5755 ; to which bring down
and annex the next Punclation 183, making 5755183, for a f�cond Refolvend,
with which you are to proceed, as before,; but ;o make the Performance quite
eafy, I will explain this Repetition alfo, as follows

Firft, Find cheDivifor as follows, viz. Square 52, the Quotient already found,
makes 2704, which trebled makes 8112, the Divifor required. Then fay, how
often 8] 12 in 57551 (for here, as before, the two laft Figures S3, of the Rqfol-
vend, are to be rejected) anfw.er 7 times, equal to 56784, winch place under
57551, of the Refolvend, and fet 7 in the Quotient at c. Secondly, Treble 52,
ft and f�cond Figures of the Root, equal to 156, which multiply by 49, the
Square cf 7, the kit Figure in the Quotient, makes 7644, which place under

�O784.,

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6o Of Vulgar and Decimal Fractions.

56784, one Place more to the right Hand, as at n; alfo Cube 7, the lail Figure irf
the Quotient, equal to 343, which place under 7644, one Place more to the right,
as ?Xp. Then the threeSubducends 5�784 at?,?, 7644 at n, and 343 at p, being
added as they Hand, their Sum make a Subtrahend, 5755 1 83, which being fub-
tracled from 5755183, the fecond Refolvend, nothing remains; which fhews that
the given Number 146363183 is a CubeNumber, whofeRootis 527, as required.

Note I. As many Punciations as any given Number contains, except the firft,fo many
times is the Work to be repeated.

II. 'That in all Extradions, when a Divifir cannot be found fo often as once in its
Dividend, orifitcanheJound,andyetther*jhailarife a Subtrahend greater than the
Rcfolljen�, in both thefe Cajes a Cipher muft be put in the Quotient and annexed to the
iafl DinjifoT a/Jo, for a ne.iv Divifor ; and the next Punilation being brought doivn and
added to the laft Refii-vend, makes a nenu Refol-vend, with tuSicb proceed in every
Rejpei? as before.

ill. When Numbers remain after the laft Subtrahend is fubtracled from the
Jaft Refolvend, which very often happen, fuch are called irrational or furdNum-
bers, becaufe their Roots cannot be exaftly difcovered. But if no fuch Remain-
der, you annex three Cyphers continually, as you did two Cyphers inthefquare-
Root, you may come very near to the Truth, as was there fhewn.

To ex tra ft the Cube Root of a Vulgar FraSlion, ivhich is commcnfurable to its Root.

Rule. Extract the Cube Root of the Numerator for the Numerator of the
Root ; and the Cube Root of the Denominator for the Denominator of the faid
Root,

To tfctra� the Cube Root nearly, of a vulgar Fraction remaining, incommenfurable
to its Root.

Rule.. The integral Part of your Root being firft found, as before taught, to
the Treble thereof add one, and that Sum added to the Square of the (aid Root
tripled, is a Denominator ; to which the laft Remainder, after Extraclion is
fuiifhed, is the Numerator.

A Table of the Roots cf all fquarc and cubed whole Numbers, from I f050, caU
culatedby Thomas Langley.


R.	Sq.	Cube.
4? 4^!	1600	64000
4? 4^!	1681	68921
42	1764	74088
43	1849	79507
44	1936	85184
45	2025	91125"
46	2116	973 3⁶
47 48	2209 2304	103825
47 48	2209 2304	110592
49 5°	24OI 2500	117649-
49 5°	24OI 2500	125000'

R. Sq. Cube. R. Sq. Cube. R. Sq. Cube.


27 28	729 784	.9683
27 28	729 784	21952 24389 27000
3° 1' V- 33	841 900	21952 24389 27000
	961	29791
	1024 1089	32668 35937 393°4 42875
34	1156
3 5 3~6 37 38	1225
	1296 1369	46656 50653
	1444	54652
39	1521 5931-9�


'4	196	²744
'5	225	3375
16J256		4096
		-----
�?	28c	49^T3
,8	324	5832
rgU�i		6859
2040c		80C0
2l!₄41		9262
22484 ... 1		10648
1 ■ ²3529		12167
24.(576 i		13824
25J�25		15625
2-61676		17576


I	I 4	j
I	I 4	8
3 4	9	27
3 4	16	64
5	25	125
6 7 8	36 49 7*	216 243
6 7 8	36 49 7*	5 1²
9J ⁸' i io^:ieo		7"2� ;occ
1 1112]		'33'
� 2 I * .i		172S
1 ~> >.1	log	2197

Thus

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			INTRODUCTION. 6i Thus have I given all the ufeful Rules in Vulgar and Decimal Arithmetic!? both in whole Numbers and in Fractions, which if well confidered will be, not only very foon and eafily understood but vaftly advantageous to every Work- man, in the Execution of his Imploys. And as a perfect Knowledge herein majr be foon acquired by employing the leifure Hours of Evenings when the Labour of the Day is over, I humbly conceive that every one who wdl fo employ him* felf will find, not only a very agreeable Amufement, but very great Helps in the Performance of his feveral Works, exch five of the Reputation that will attend him alfo. But fuch P�rfons who will be fo remifs as to lay by this Work in their Chefts, &c. without taking either Pains or Pleafure herein, cannot expect tha� Advantage, which others will enjoy.

		"■ ~~-■■-■..... - ■ '........... i. .■..........~~ i ~~i..... » -i i ■ =~~3* PART II. Of Geometry, INTRODUCTION. THE next Science in order after Arithmetick is Geometry, the moft ex- cellent Knowledge in the World, as being the Bafis or Foundation of all Trade, and on which all Arts depend. Geometry is fpe�ulative and practical ; the former demonftrates the Proper- ties of Lines, Angles, and Figures ; the latter teaches how to apply them to Prac- tice in Architecture, Trigonometry, Menfuration, Surveying, Mechanicks, Per/peilive, Dialling, AJlronomy, Navigation, Fortification, &c. This Art was firft invented by Jabal the Son of Lamech and Ad ah, by whom the full Houfe with Stones and Trees was built. Jab al was alfo the firft that wrote on this Subject, and which he performed, with his Brethren, Jubal, Tubal Cain, and Naamah, who together wrote On two Columns the Arts of Geometry, Mufick, working in Bra/s and Weaving, which were found (after the Flood of Noah) by Hermarines, a Defcendarjt from Noah, who was afterwards called Hermes the Father of Wifdom, and who taught thofe Sciences to othermen. So that in a (hort Time the Science of G�omtry became known to many, and even to thofe of the highelt Rank, for the mighty Nimrod King of Babylon underltood Geometry, and was not only a Mafon himfelf, but caufed others to be taught Mafonry, many of whom he fent to build the City of Nineve and other Cities ia the Eajfl. Abraham was alfo a Geometer, and when he went into Egypt, he taught Euclid, the then mc� Worthy Geometrician in the World, the Science of Geometry, to whom the whole World is now largely indebted for his unparalleled Elements of Geome- try. Hiram, the chief Conductor of the Temple of Solomon, was alfo an exr cellent Geometer, as wasGREcus, a curious Mafon who worked at the Tenjple, and who afterwards taught the Science of Mafonry in France. England was entirely unacquainted with this noble Science, until theTime of St. Alban, when Mafonry was then eftablifhed, and Geometry was taught to moii Workmen concerned in Building ; but as foon after, this Kingdom was frequently invaded, and nothing bu: Troubles and Confufion reign'd all the Land over, this noble Science was difregarded until Athelstan a worthy King of Englan4 fupprefs'd thpfe Tumults, and brought the Land into Peace; when Geometry antj Mafonry were r*'-eflablifhed, and great Numbers of Abbeys and other ftately Build, ings were erected in this Kingdom. Edwin the Son of Athelstan was alio a great Loyer of Geometry, and ufed to read Lectures thereof to Mafons. Hs I alfn

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62 Qf GEOMETRY.

alfo obtained from his Father a Charter to hold an Affembly, where they would,
within the Realm, once in every Year, and hiinfelf held the firft. at York, where
he made Mafons; fo from hence it is, that Mafons to this Day have a grand
Meeting and Feail, once in every Year. Thus much by way of Introduction, to
Ihew the Ufe, and how much the Science of Geometry has been efteemed by
fome of the greateft. Men in the World, and which with regard to the publiek
Good of my Country, I have here explained, in the moil plain and eafy Manner
that I am able to do, and to which I proceed.

LECTURE I. Geometrical Definitions. Plate I.

THE Principles of Geometry are Definitions, Axioms and Poftnlates. D�fi-
nitions are the Explication of fuch Words and Terms which concern aPro-
poicion towards rendering it intelligible and eafy to the Underftanding, avoiding
in Densonilraiion all Difficulties and Objections. Axioms are fuch evidentTruths,
as are not to be denied, as one and one are two, two and two are four, &c. Pojiu �
lutes are Demands, or Suppofuions of things practicable, and the Manner of doing
them to eafy, plain, and evident, that no Man of Senfe and Judgment can deny
o^r cont ft them, fuch as to draw a Line by the Side of a Ruler, from one given
Po,nt to another.

Qi'ANi iTvis confidered in three different Manners, viz. Firft, Length with-
out Breadth, as an Interval or Diftance between two Points. Secondly, Length
with Breadth only, as a Shadow, &c. Thirdly, Length with Breadth and
Thicknefs, or Depth, as a Brick, �jfc. The Bounds or Limits of Quantity are
Points, Lines and Superficies.

T) f , of A Point, in the Pra&ice cf Geometry, is-the fmalleft Object of

p ■'f ' Sight, that can be made, and which is fuppofed to have no geo-

metrical Magnitude, capable of being divided to our Sight, and is
made by the Point of a Pin, Pen, Pencil, is'c. as the Point A. Plate I.

The Varieties of Points, and their particular Denominations are many ; as

7) f 2 Of ^^or Example, if a Point be affigned, in any certain Place, as the

. � � . . Point b, in the Line a d, 'tis called a give» Paint, from whence
given Point. . _T . ' , . ' ,.,-■■ ^s r � » - j

° the Line b c proceeds, or to which the Line b c is drawn from

n f i Of the End or Point f. Secondly, when the two Lines cut a-erofs eacj»

p ■'_t f j Other, zsxc,yy, orel,if, trie Points % and g, are called Po^:r.ts of

r_ar Interjection; and when fuch a Point happens to be in the Middle

- ' of a fuperikial Figure, as g, 'tis caiUd its Centre, or central

De/. 4. Of Point. ThirMy, when tao Lines meet together, and flop in one

an angular Point, as i m, and ml, in the Point m, fuch a Point is called an

Point. angular Point. Fourthh, if two Lines, touch one another, but do

D f Of ^not ^cut ^a ^Cl0^ ^c'^lc'¹ °"^i�1'> ^as ^at B> ^⁶ P°i^nt of touch B, is called

� %'� � J e ^tne Point of Contact.

a foint of � ' ( w � 1 > r. � 1 r in c

Contait ^R ^E ^are ^rnany otr.er Kinds of Points, in the l�verai Parts or

Mathematicks, which at prefent do not concern us ; as for Ex-
ample, in Perfpective there are Points of Sight, Points of Diltance, vifual Points,
ts'c- which will be better underftood hereafter, when I come to explain the Prin-
ciples and Practice of that Arc.

D f 6 Of When Quantities are confidered asLength^ only, they are called
I s Sutie Lines ; thole of Lengths with Breadths only, are called Superficies -,
f'^-s'ai 'So- ^anc' ^¹⁰^^c °f Lengths, Breadths, and Depths, are called Solids, or
i-'l ' Bodies.

Kinds of ^The K'ⁿ^^s °f Limes are three, viz, a right Line, a curved Line,

and a mixta Line.

mes.

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		Of G E O M E T R Y. 63 Aright Line, is a Length without Breadth, as the neareft t\ e _n r\f Diftance between two Points ; but in Praftice, 'tis a ftreight Line, ■'.' f\. * defcribed by the Motion of a Pen, Pencil, i�c. drawn by the Side * '^ne' of a ftreight Rule, wherein its vifible Breadth is not confidered, as a d. A Curved Line, is any Line that is not a right Line, and r\ f o m therefore all crooked, arched, or bended Lines, are curved Lines. -^' ,'.. f There are many Kinds of curved Lines, namely a circular or arched Line, as E, Fig. II. an elliptical or oval Line, as h I, or i m l, a parabolical Line, as iv zy, a hyperbolical Line, as 123, a ferpentine Line, as B, a rampant arched Curve, asF, and an irregular curved Line, as �. 1 here are alfo many other Kinds of Curves, as the Epicycloid, Cycloid, Algebraick Garve,^xLogarithmttical Curve, C'ftbid, Catenaria, Evolute Curve, Catacaufiick anti �ia- caujiick Cut ves, Helicoid Parabola, or Parabolick Spiral, &c. But as rhey have no Relation to the Bufinefs of Builders, for whom this Work is only defigned, I ihall forbear to fay any thing of their Generation and Uie. ACircular or arched Line, is that whofe Curvature or Bend- Def. 9. Of ing is the fanae in every Part, asfc e, Fig. II. a circular or ■rtN oval or elliptical Line, is fo called, as being a Part of the arched Line. Boundary of an Oval or Ellipfis, as i hi, and the Lines <u> z �/, and Def. 10. Of 123, are called parabolical and hyperbolical Lines, as being the an elliptical. Boundaries of a Parabola, and of a Hyperbola. paralogical, A Serpentine Line, as A, is fo called, from its being like and hypcrbo- the Form of a Snake when 'tis travelling along; and the Spiral Heal Line. Line B, may be alfo called a ferpentine Line, as reprefenting a Snake when coyl'd up. The Artinatural Line C, is fo called Reafons lufy ■ i-'Om its being an artificial Reprelentation of the natural Turnings the Serpentine- and Windings of Brooks, Rivers, is'c The rampart Curve F, is Spiral, arti- called fo from its rifing higher on the one Side than on the other, natural,ram- And lalfly, the Curve D, is called irregular, as not having any of pant, andir. its cppolite Parts equal. The circular Lines ufed in Architecture, regular Lines are either fmgle or compound, as in Fig. III. The Mouldings are/0 called. compofed of iingle Curves, are the Ovolo A, the Cavetto B, the ApophygesE, the fingle Aftragal G, the double'Aftragal H, the Flute M, the Fillet N, and the Bead i. The compound Curves are the Cima Recla C, the Ci- rca Inverfa D, the Scotia F, and the Volute K. A mixed Line, is both right and curved, safe d c b a, Fig. Def. Ii. Of IV. being compounded of the right Lines f e, d c, a b, and of the a mixedLine. two curved Lines d e and b c Lines are diilinguilhed into finite and infinite, a'fo into apparent and occult. A finite Line, is a known Length, bounded by two known Def. 12. Of Points, as the Line g h, Fig. IV. and therefore all Lines of known a finite Line. Lengths, are finite Lines. ' Def. 13. Of An infinite Line, is that whof� Length is undetermined, or an infinite cannot be known, as the Diameterof the Univerle, &c. Line. An apparent Line, is a Line, defciibed by the Point of a Pen, Def. 14. Of Pencil, (jlc, &s g b, Fig. IV. a a apparent An occult Line, is drawn or defcribed with the Point of a Pair Line. �f Compaffes, and in Praftice is always exprefled by Points, as Def. 15. Of i k, and therefore is made generally a dotted or pricked Line. av. occult Lines have their particular Denominations, according to their Line. different Pofitior.s and Properties, as following. Firft, If a right Def 16. Of Line as », Fig. IV. ftand on a Line, as on b 0, fo as not to incline a perpendic-U'- . either to the xight Hand or to the Left, it is then called a perpen- larLittyr.- dicuiar Line, and that the Line bo being firft made, is called a Def. 17. Of given Line. Stcondly, If a Line be level with equal Inclination oa a given Line.. I » bot»

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		Of GEOMETRY

Bef. l8. Of a horizontal Line, Def.ig. Of on oblijue Line. Dr/20. Of parallel Lines. Def. 21. Of concentrich drches. T>ef 2 2, Of excentriek Arches. Def. 23. Of the Circum- ference of Circles and �> lipjes. Def 24. Of the Sides of the right lined Figures. Def. 2^ Of a bafe Line. Def 26. Of a Diameter, Radius, and Semi-diame- ter. Def. 27. 0/ a diagonal Line. Def. 28. Of tranfivsrfe and conjugate Diameters. Def.iq. Of a Chord Line. Def 10. Of a taiigent Lint. Def. J 1. Of the Kinds of Superficies- Superfici Def 32. Of a Circle. t)ef.y_?. Of a Semhirc/e. Df.yi- Of Si Quadrant.	both Sides, as p q, 'tis called a horizontal Line. Thirdly, Tf* right Line be fo fuuated. as to be neither perpendicular, or hori- zontal, as the Line % z, fuch a Line is called an obliaue Line* And here note, that one Line may be perendicuiar to another Line, altho' it may not be perpendicular to a horizontal Line : So K I is a Perpendicular to the oblique Line F E A Plumb Line is a direct downright Line, as G H. whkh is always perpen- dicular to a horizontal Line. Fourthly, If two right Lines are at an equal Diftance from each other, as r r and s s, they are called parallel Lines, and which being infinitely continued, would rtver meet. Fifthly, If two circular Lines are at equal Durances from each other, as t and u, they are called concentrick Arches, as be- ing both defcribed on the fame Center. Sixthly, If two circular Lines have two different Centers, as the circular Lines vu x, they are called excentriek Arches, as being defcribed on different Cen- ters. Seventhly, The curved Line that bounds a Circle, Eflipfis^ or Oval, is called the Circumference; and by fome, the Perime- ter, or Periphery, b c d g. Fig. V. But the boundary Lines of all right-lined Figures, as of A B C, aie ca�ert .Mces, excepting when at any Time, fuch Figures are placed upright, fo as 10 Hand on their Sides, and then the lower aide o( every fuch Figure is called its Laie : therefore that Line on which a Figure frauds, is a bafe Line. Eighthly, A right Line drawn throli; h the Center of a Circle, as b d, Fig- V. is called a Diameter ; and one half of fuch a Diameter, asba, ox ad, is called the Radius, or Semi-diameter < Ninthly, If fquare Figures, as A or C, Fig. V. have right Lines' dra\vn through their Ceiters, and are parallel to their Sides or Ends, as A k, in A, and tn m in C, they are allo cal ed the Diame- ters of thofe Figures : But ail right Lines drawn from one oj-pc-* ,fite Angle to the other, as 0 0 in A, and n n in C, are called diago- nal Lines. The like is alio to be obferved in regular Figures, con- fifting of more Sides than four, as B, where/ p is the Diameter, and 2 q the Diagonal. In all Figures that are not fquare, as C, the longed Diameter, as /1, is called the traefverfe, and the mort- elt as m m, the conjugate Diameter ; and which is alfo to be ob-, ferved in the Diameters of an Oval, and of an Ellipfis, as in D. Every right Line drawn th ough any Part of a Circle, as ef, Fig. V. is call.d a Si btenfe, Ordinate, or chord Line ; as alfo is a Line which joins the two Extremes of an Arch, as x x ; and if a right Line be drawn foas to touch a Figure, without cutting into it, the Point of Contact either at a Side or at an Angle, as h i, in g, and 2, 'tis called a tangent Line. The f�cond Kind of Quantity, namely Superficies, is a Surface of whatever has Length and Breadth, wi:hout Depth orThick- nefs (as by Def. 6.) and is pf three Kinds, viz. Firlt, exactly flat, as the Surface of a Table. Secondly, Convex, as the Outfide of a Ball. Thirdly, Concave, as the Infide of a Bowl es are bounded by one or more Lines, and from thence it is, that they receive their various Names, by which they are known ; as, firlt. if a Superficies be bounded by one curved Line that is regular in all its Parts as A, Ftp V!. 'tis called a Circle Every half Part of a Circle as D, is called a Semi-circle which is bounded by tbe Diameter and one half of the Circumference of a whole Circle. A Quadrant as H, is a Figure bounded by two Semi-diameters (called the Sides) and one quarter P^rt of the Circumference, called the Limb, If

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Of GEOMETRY.

�F a Circle be cut into two unequal Parts by a right Line, usai,
fig. VI. each Partis called a Portion or Segment, and which are
diftinguifhed the one from the other, by a greater and leffer j fo
a c b is the lefler Segment, and a d b the greater; and

If two Lines as h k, hi, in C, are drawn from the Center of
any Circle unto its Circumference, and thereby divide the Whole

Def_3S. Of

the Segment
of a Circle.

Def. 36. Of

a Seclor.

into two unequal Parts, the Partlefs than a Semicircle, as k i h, is
is called a Seclor, and the remaining Part, h k I i, is called the Complement of
the Seclor, and by fome the great Seclor.

Now fince, by this Definition, a Seclor is a Part of a Circle which is lefs than a
Serai circle, therefore a Quadrant is a Seclor alio, as being but half a Semi circle.

The Parts of an Oval or Ellipfi", are denominated in the fame Manner as

the Parts of a Circle. So the Figures B and C, Fig. VII. are both
Semi-Ellipfes, equal to each other; that o i B being on thetranf-
verfe, and thar of C, on the conjugate Diameter. And as every
righ; Line drawn through the Center of a Circle, doth divide the
Superficies tht-reof into two equal Parts, fo likewife every right
Line dra.vn through theCenter of an El!ipfis,doesthefame. So ce,
divides the Eihpfis c m e n in two equal Parts, as alio doth either of
the Lines m.n, or a i. The Segments of an Eilipfis a- e either re-
gular as d c b, and & m i, or rampant as a k m i ; and the L-nes b d,
or k i are called Ordinates, that of k i being an Ordinate on the
tranl'verfe Diameter, and that of b d on the conjugate Diameter.
The Seclor of an Ellipfis or of an Oval, as in A, Fig. VII. is the
fame as in the Circle, as likewife is tne Complement thereof

The Farts of
an Fliipfis
have the fame
Deitamitratfoit
as the Parts
of a Circle.

Def 17- Of

the Ordinate:
of an Ellipfis-.

SeSlor of an
Ellipfis.

Now from hence you fee, that Circles, Ovals, andEll.pfes are
the only regular Superficies that are bounded by one Line, and that all regular
Superficies bounded by two Lines only, are no other than their Segments, either
fingleas the Segment a c b, in B, Fig. VI. or compound, as a b czndadc, in H,
Pig, VII. which laft is no more than two Segments, applied together (the Line
a c being common to both) and is called an Ox Eye.

Triangles have their different Denominations, as being of dif-

ferent Forms, -vise. [1) If a Triangle have all its Sides equal as
G, Fig. VI. 'tis called an equilateral Triangle. (21 If two Sides
are equal, and the third unequal as E, 'tiscalledai<d Iibfceles Tri-

D�fit. Of

an Equilate-
ral, IfofceleS)
and Scale-
nous Tri-
angle.

angle. (3) If all the Sides are unequal as P, 'tis called a Scalene
Triangle. Triangles are alfo diftinguifhed by the Quantity of
their Angles; but this I fliall refer, until 1 have intlrucled you

in the Nature and Kinds of Angles.

All Triangles, whole Sides are Arches of Circles, are called Def. 39. Of
fpherical Triangles, as N P Q^Fig. VII. And when Triangles a Spherical
are compofed both of right Lines, and circular Lines, as O R S, Triangle.
and V, they are calied mixt Triangles, with one or two convex Def. 40 Of
or concave Sides ; as for Example. (1) The Triangle O, hath mixt Tri-
two Sides that are right Lines, and the third that is a concave angles.
Arch. (2) The Triangles R and S, have each but one Side that
is a right Line, and the others are Arches of Circles, of which, thofeofthe
Triangle R are convex, as being fwelling outward, and thofe of S, are con.
Cave, as being holiow outward. (3) The Triangle V, hath alfo but one Side
that is a right Line, but the other two which are circular are one convex, and
the other concave.

Every Triangle contained under three equal Sides, be they

right-lined, circular, or mixt, is called an equilateral Triangle,
and fo the like of Ifofceles and fcalenous Triangles ; and to dif-
tinguifh right lined Triangles from fpherical and mixt Triangles,

D�fini. Of

plain Tri-
angles.

they are in general called plain Triangles.

2 Superficies

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Of G E O M E T R Y.

Four-ftded
Figures.

Superficies bounded by four right Lines are the geometrical
Square Fj Fig. VII. the Parallelogram G, the Rhombus f, the

Rhombo�des K, the Trapezoid L, and the Trapezium M.
Def.4.2. Of The Geometrical Square F, is fo called, becaufe all its�
a geometrical Sides are equal and fquare to each other, and the Parallelogram
Square and receives its Name from irs oppofite Sides and Ends beirg parallel'
Parallels- to each other. The Parallelogram is alfo called a long Square or

gram. Oblong, with regard to its being longer than wide.

The Rhombus f, is nothing more than a geometrical Square
Def. 43. Of pufiYd out of its natural fquare Form into any other : for fuppof-
a Rhombus ing the Angles da e foi the geometrical Square da e f, in the
and Rhom- Rhombus I, have each a moveable Joint at the feveral Angles.
boides. If the Angle d be pufhed to c, the Angle a will be moved to b_t

and the Side de will be removed to f e, the Side a f to if, and
the Side d a to c b. The fame is alfo to be underftood of the Rhomboid, which
is nothing more than a fquare Parallelogram, vvhofe Ends are pufhed out of
their iquare Pofitions into oblique Pofitions.

V ,- Of
a ! zoid.

Def. 45. Of
a Irafezium.

A Trapezoid is a Figure containing four Sides, of which two
are parallel, and the other two are not, as Figure L.

A Trapezium is a Figure containing four unequal Sides, of
which no two of them are parallel.

Regular Superficies bounded by five or more Sides are called
Polygons, or Polygonals, or Multilateral (that is, many Sides)

Def. 4;

Polygons

as 5, 6, 7, 8, 9, 10, 11, 12, &c. and which take their Name*

from the Number uf their Sides,
'Five

(Pentagon
Hexagon
Septagon or Heptagon

Sides, :

called

resular

So a

Plain, Figuie
confifting; of

as are ex-
hibited in
Plate 11.

Octagon
"j Nonagon

� Decagon
I Undecagon
( Duodecagon

Def. 4.?. Of Figures which have the fame Number of Sides and are un-
an irregular equal are called irregular plain Figures, confifiing of 5, 6, 7, 8,
f lain Figure. &c. Sides, as the irregular figure onde- th; Octagonin Plate II.
Def. 48. Of All Figures bounded with right Lines and curved or mixt
an irregular Lines are called mixtilineal Figures ; which are either irregular or
compound Fi- regular; that is to fay, if an irregular Figure have fome of its
gure. Sides curved, and fome that are right Lines unequal, it is called

Def. 49. Of a'compound irregular mixtilineal Figure; but when a Figure is
a regular composed of equal right lined Sides and cf equal arched Sides,

compound Fl- they are called compound regular Figures.

gure. Wh kn Figures have Voids or Imperfe�iions in their Superficies,

Def 50.' Of they are called imperfect Figures, fucli as A B, Plate II. wherein
Imperfe&.Fi- the dark or (haded Parts reprefent the Superficies, and the light
gares, Con- Parts the Deficiencies, Voids, or f m perfusions thereof, and which
centrick and are differently diftingtiifhed, as thole of A and B ; having their
Excentriek. Voids, or defective Parts bounded by Fines defcribed on the
fame Centers, are called concentrick Figures or Superficies; and'
that of the Lunula, whofe Void is bounded with Circles defcribed upon differ-
ent Centers is called an excentriek Figure or Superficies ; vide Definitions XXI.
and XXli. The imperfect .figures B and the Square on the Left of the Lu-
nula are alfo to be considered in the fame Manner, as A and the Lunula, not-
withlianding that their Voids are bounded with parallel, right Lines. For
as the Center of the Void in B is the faane as that of the Superficies whirls
bounds it, th^ whole is therefore a concentrick Figure, for the fame Reafon a*

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O/ GEOMETRY. 67

is Figure A. And fo in like manner, as the Center of the Voids, in the Square,
is not in the fame Points as the Centers of the fhaded Superficies; that is alfo
an excentriek Figure, as the Lunula.

To thefe imperfeel Figures I inuft add Fig. C, which is a Parallelogram di-
vided into four Parallelograms, that meet all together on the Diagonal Line in
the Point ».

Now if any three of thofe four Parallelograms, as n d, n b, j. . _n

and n a, be taken together, and confidered as one Figure, 'tis "h
called a Gnomon ; but if the four Parallelograms are confidered ^a ⁿ ^mon'
Separately, then the Parallelograms n b, and n c, are called Parallelograms de-
scribed about the Diagonal b c, and the other two Parallelograms a n, and » d,
are the two Supplements thereof, and which are always equal to one another,
as will Ije hereafter demonirrated.

As fuperficia! Figures are bounded by one or more Lines, fo Def. 52. Of
Solids or Bodies are bounded by one or more Superficies ; as for the Bounds of
Example, a Brick is a Solid, bounded with fix Surfaces, that are Solids or
all Parallelograms, <viz. the upper and the under, the two Sides, Bodies.
and both Ends.

The Number of entire Solids are principally twenty, <viz. a The Number
Sphere, a Spheroid, a Cylinder, a Cone, a Conoid, a Spindle, a and Names of
Tetrahedon, a Pyramid, a Pyramis, aPyramidoid, Conedoid, a Solids.
Cylindroid, a Prifm, a Hexajiedron or Cube, a Parallelopipedon,
an Oftahidron, a Dodecahedron, an Icofahedron, the twelve and the thirty
Rhombus's.

An entire geometrical Solid is a Body from which no Part has Def. 53. Of
been taken, and therefore the Remains of a Body, when a Part an entne
thereof is taken away, is called a Fruftum, as the Fruftum of Solid.
a Sphere, or of a Cone; &c. Def. 54. Of

A Sphere is a round Body, bounded by one convex Superficies, the Fruftum
whofe Parts are all at the fame Diftance from the central Point of of a Sphere.
the folid ; and is commonly called a Ball, as R, Plate II. Def 5c. Of

A Spheroid, is a round folid Body, bounded by one convex a Sphere.
Superficies alfo, but its Curvature is not the fame in every Part Def. 56. Of
over its Center, as. the Curvature of the Sphere; becaufe its a Spheroid,
Length is greater than its greateft Thicknefs, and therefore it is
what may be properly called an ovallar Solid, if we cenfider the Sphere as a
circular folid ; as S, Plate II.

A Cylinder is a long and round Body of equal Thicknefs, Def. 57. Of
as a Garden rolling Stone, or the lowermoft third Part of the a Cylinder.
Shaft of a Column, as X, Plate II. and is bounded by three Su-
perficies, of which one is convex, and two are plane or flat, and whofe Figures
depend upon the Manner of the Cylinder being cut at each End ; that is to fay,
(1) if the Ends of the Cylinder be both cut iquare to its Length, as X, then
the Superficies of the two Ends are both Circles (which are equal to each other,
becaufe the Cylinder is of equal Thicknefs, and the convex Superficies is no
more than a Parallelogram whofe Length is equal to the Length of the Cylin-
der, and Breadth to its Circumference, being bended about the fame. (a-) If»
Cylinder as D, (on the right Hand Side of the Plate) have its _r._f a _nc

Ends cut obliauely and parallel to each other, the fuDerfkial Fi-

J' 5 , ^>

'ye 'various

gure of each End will be an Ellipfis, and the Convex Superficies
Will be a double Rhombo�des. (3) If a Cylinder, as E, have ^Klnf:V ««*
its End cut obliquely, and not parallel to each other, they will T^irJ^ulei ^that.
be both Ellipfes, but unequal (as not being parallel, which caufes ^bou�JJ�f''^ar
the tranfverfe Diameter to be longer in the one than in the other) �J,~r'?^ueand the convex Superficies will be an irregular Hexagon ; a De- yl'naers,
monflration or which you will fee in the Menfuratioa of Solids and their Su*
perfides, ." .

A Con-e

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			68 O/ G E 0 M E T R Y. _ ~, A Cone is a round Solid, which rifes either from � Circle or ls'f- 59- I an Ellipfis, with a gradual and equal Diminution until it termi-

		^Cane' nates or ends in a Point, as Fig. T, on the left Side of Plate II, and therefore is bounded by two Superficies, of which that ofth� Outfide is convex, and that of its End or Bottom is a circular or elliptical Plane. In every , _ . Cone there is an imaginary Line fuppofed to be drawn from its u v' ^°P ^or ^vert'^ca' P°'^nt> ^unt0 Ae centrical Point of its Bafe, which is the Vertex _caU_ed the Axis of the Cone, and which is fo called becaufe it paries and Axis of a _dire�lv through the Middle of the Solid, and on which the Body Lone. _may he made to revolve or turn about, as that every oppofite Part is equidiftant therefrom. The fame is alfo to be underftood of a b, the Axis of the Sphere R, alfo of e c, in the Spheriod S, and of all other regular Solids. Now when a Cone hath its Bottom cut fquare to its Axis, as T, 'tis called a regular Cone, and its Bottom, which is called its Bafe, will be a Circle. But if its Bot- tom be cut obliquely to its Axis, as G, on the right Hand Side of the Plate, it is then called an oblique Cone, and its Bafe will be an Ellipfis. . _ A Conoip is a Solid, diminiihing in its upper Parts nearly r J ^l^^e ^^ame ^as ^a Cone, and takes its Rife from a Ciicle alfo ; but as * Conoid, _tlie g_ide ₀f _a _Con� j_s ft_rejght frorp its Bafe to its Vertex, this of a Conoid is either the Semi-curve of a Parabola or of a Hyperbola, or the Seg- ment of a Circle, or an Ellipfis ; and therefore terminates at its Vertex either in a Point, as the Cone doth when the outward Curve is of a Circle or an Ellipfis, as B L, or with a curved Top, like unto a Sugar-Loaf, as A, when a Semi- parabola, or Semi-hyperbola. _ , A Spindle is a Solid, thus to be conceived ; fuppofe a g in B, p i /■ 4 ^t0 k^e ^tne Diameter of a Circle, on which a Semi-fpindle is to be t f H I ^raifed' ^Wh0fe ^Axis ^is ^d' ^a,f° ^fuPP^ofe *^lhe ^Curve ^a ^dto* ^be ^the ^Semi'** and Hyperbo,- _curve ₀c _a Parabola ; now if from every Part of the Circumference luk Spindle, _of _a _circ]_�) ₀f _whi_ch _a _g is the Diameter, a Solid be railed with a Curvature equal to the Semi-parabola a d, that Solid will be a Semi-fpindle, and therefore two fuch, being equal and applied together, as B, will form that folid which is called a Spindle. And as the outward Curve may be either a Hyper- bola, or a Parabola, therefore a Spindle may be Hyperbolical or Parabolical. , A Tetrahedron is a triangular Solid, which rifes from an <r uj ^ecl«'l^ater^ triangular Bafe, with a gradual and equal Diminu. letranedron. _{j_ofr. _un(;i ^ terminates in a Point, as a Cone doth, which Point is alfo called its Vertex. This Solid is terminated by four equilateral Triangles, as B F, on the left-hand Side of the Plate. Ttff, nr A Pyramid is a Solid, which rifes from a geometrical Square, i' "., ■ with a gradual Diminution (as the Tetrahedron rifes fromanequi- aryra/md. lateral Triangle) and terminates in a vertical Point alfo. This Solid hath its Height at Pleafure, and is bounded by four Equ�aterals or Ifofccles Triangles on its Sides, and a geometrical Square at its Bale, as Fig. V. T\ f fi- nf ^ P^yRAMis '^s ^tne fame Solid as a Pyramid, only with this ■%' *. Difference, that whereas a Pyramid ftands on a geometrical a yrama. Square, and has but four Sides, which are ail equilateral, or Ifofceles Triangles, a Pyramis has fome regular Polygon, as a Pentagon, Hexa- gon, csv. for its Bafe, with five, fix, i$c. Sides, which are all Triangles, as in a Pyramid, and meet in a vertical Point alfp. T) ■<■ /■,(■ of A Pyramidoid is a pyrarnental Solid whofe Bottom if a tri- p' '. S angule geometrical Square, or fome regular Polygon, and Sides a yrarni- _arg ^ (j_urve ₀f _a Circle, Ellipfis, Parabola, or Hyperbola, as "« Fig.* IV. T) f lm Of ^A Cyukproip is a Solid, fomething like B I, the Fruflum of C ')' J 'A ^a ^one, but ^w'^tn ^tni» Difference, that as the Fruflum of a Cone is > y i ' "i* . terminated at its Ends either with two Circles, if cut fquare to its Axis, or w.Uh two Ellipfes, if cut oblique, or with a Circle and an Ellipfis, if one

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Of GEOMETRY

one End be bat fquare, and the other oblique, the Ends of the Cylindroid are
both cut fquare to its Axis ; but the one is an Ellipfis, and the other a Circle,
as Fig. C ar the Top on the right Hand.

Tut next Kind of Solids, in Order, are Prifras.

A Prism is a folid Body of equal Thicknefs, as a Cylinder ;
but as a Cylinder is round, and its Length is thereby bounded by
one Superficies only ; fo a Prilm is bounded by three, five, fix, or
more Parallelogram-, and its Ends are either Triangles, geome-

Def.68. Of
the 'various

Kinds of
Prifens.

trical Squares, Trapezranw, or fome Kind of Polygon, as a Pen-
tagon Hexagon, tjfc. So B C is a triangular Prifm bounded by two Triangles
at its Ends, an i three Parallelograms on i s Sides. B A is a Trapezium Prifsi,
bounaeu by two Trapeziums at its Ends, and tour Parallelograms on its Sides.
B E i' a pentangular Prifm, bounded by two Pentagons at its Ends, and five Pa'.
raiielo^ams on its Sides. And laftly, BD is a hexangular Prifm, bounded by
two Hexagons at its Ends, and fix Parallelograms on its Sices.

It is alio to be noted, that if theafbrefaid Prifms have their Ends cut obliqus
to their Sides, that then their Sides will be either Trapezoids or Rhomboids, and
their Ends will be changed into different Kinds of Triangles, Parallelograms,
and unequal fided Polygons.

A Cube, or Hexahedron, is anexaft fquare regular Solid (asa
Dice) and is bounded by fix equal geometrical Squares, as Fig. Y.

A Par ALLELOPiPEDON is alfo called a long Cube, and by
fome a Prifm ; but as its Ends, as well as its Sides, are bounded
by Parallelograms, which are never more nor lefs than fix in
Number, as Fig. Z, it is therefore with refpect co its Surfaces, be-
ing all Parallelograms, properly aParallelopipedon.

An Octahedron is a regular Solid, bounded by eight equi-
lateral Triangles, and is compofed of ttvo equal Pyramids, hav-
ing their Bottoms applied together, fo as to make but one Solid in
the whole, as Fig. P. Plate II.

A Dodecahedron is a regular Solid, bounded by twelve
Pentagons, as Fig. O. Plate II.

An Icosahedt on is a regular Solid alfo, and is bounded by
twenty equilateral Triangles, as Fig. Q. Plate II. �The twelve
Rhombs, and the thirty Rhombs, are Solids, bounded by as many
Rhombus's, but though they have a Uniformity in themfelves,
yet they are not regular Solids.

The regular Bodies are the Tetrahedron, the Hexahedron or
Cube, the Octahedron, the Dodecahedron, and the icofahedron,
which being the only Bodies that can be infcribed within a Sphere,
are therefore called regular Bodies.

A Body is faid to be infcribed, when being inclofed within an-
other Body, every of its folid Angles terminate at the Superficies
thereof; and that Body which contains the infcribed Body is called
the circumfcribing Body.

A solid Angle is theMeeting together of three or more right-
lined Superficies.

A Frustum, as in Def. 54. is the Remains of a Body when a
Part is taken away ; fo if from the Sphere B G, the Part A be
taken away, the Part B G remaining is the Frultum of a Sphere ;
and if from the Spheroid B N, the Part A be taken away, the
Part N B is the Fruftum of a Spheroid ; and fo the fame of B I,
and B K, which are the Fruitums of a Cone, and of a Pyramid,
when the top Pars D and A are taken from them. Fruitums of

Def. 69. Of

a Hexahedron
or Cube.
Def. 70. Of
a Parallela-
fifedon.

Def. ₇|, Of
an O&ahe-
dfon,

Def. 72.Of
a Dodecahe-
dron.

Def. 73. Of
an Ic'fake-
dron.

The 12 and
30 Rhombs.
What Solids
aref.riBly
regular Bo-
dies ?

The Reafon.
Def. 74. Of
infcribed and
cirumfcribed
Figures and
Bodies.

Def. 7;. Of
afolidAngk.
Fruftums of a
Sphere, Sphe-
roid, Cone,
(Jc. ex-
plained.

Bodies are cut obliquely, and that not only at their upper, bnt alfo at their under
Parts, as Hi KLM, and are then called oblique Fruftums. When a Part
is taken from dit Bottom of a Pyramid, or of a Cone, as the Paris a and x, in

K F

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Of G E O M E T R Y.

7°

F and G, then the remaining upper Parts being confidered feparately, become
entire Bodies with oblique Bafes ; but it they are confidered with
Def. 76. Of the Parts a and x, then they are no more than the greater Seg-

the Segments
of a Cone,
Pyramid,&c.

De/. ₇₇. 0;

the Segment
of a Frujl�m.
.The Frujium
of a Cube.
The Frufium
of a Tetraht*

ments, and the Pans a and x are the leffer Segments, which to-
gether do but complete the two Solids ; and when the upper Parts
are confidered as entire oblique Bodies, and the Parts a and x aie
confidered by themfelves, the Parts a and x are called Segments
of Fruftums, whofe Axis is equal to their perpendicular Height.

If all the folid Angles of a Cube be fo taken away as to make
every fquare Face of the Cube an Oftagon, then the Remains
will be the Frullum of a Cube, contained under fourteen Super-
ficies or Faces, of which eight will be equilateral Triangles, and
fix will be Odagons. If the folid Angles of a Tetrahedron be fo

taken off, as to make each of its equilateral triangular Faces a
Hexagon, the Remains will be the Fruftum of & Tetrahedron,
bounded by eight Superficies ; of which four will be equilateral Triangles, and
four will be Hexagons.

I mention thefe Fruftums, only to give a Hint, that by ;his Method of cutting
off the folid Angles of Bodies, there may be a very great Variety of uncommon
Bodies produced.

The Shaft of
a Column is
a Cylinder,
and Fruftum
of a Conoid.
Def. 78. Of
the SeBion of
a Solid.

The Body or Shaft of a Column is compofed of two'Kinds of
Solids, that is to fay, the lower one Third part of its whole
Height, up to S B, is a Cylinder, and R, the Remainder, is the
Frullum of a Conoid.

A Section of a Solid is a fuperficial Figure, produced by
cutting off a Solid, direftly through, in any Part ; fo if from a
Sphere, a Segment was to be cut, the flat Surface or Superficies
of that Cut, which is a Circle, is called its Seftion. And in like

manner, if any Cone be cut quite through its Axis, from the Top
to its Bottom, the fiat Superficies of that Section will be a Triangle.
The Bafe of an upright Line is a Point.

Def.₁₉.Of
the Bafe of a
Line.

Def. 80. Of
the Bafe of a
Circle and
Ellipfis.

The Bafe of a Circle is a Point alfo, as the Pointy, of the Cir-
cle E [Fig* V. Plate I.) Handing on the tangent Line hi, which
by its Curvature can touch the Line h i, but in the Point g ; for
as every Point in the Circle's Circumference, is at the fame Di-
ftance from the Center, and as the very next Point to g, in the
Line h i, is at a greater Diftance from the Center a than the
Point �, therefore the Circle cannot touch the tangent Line in

two Points, and confequently the Bafe of the Circle is the Pointy.

The fame is to be underftood of the Bafe of an Ellipfis. Right-lined Figures
may have a Point for their Bafe alfo, by being fet on angular Points, as the Hex-
agon B, Plate I. which refis on its Angle 2, on the tangent Line h i.

As Points and Lines are the Bafes of Lines and Superficies ; fo Points, Lines,
and Superficies are the Bafes of Solids ; as for Example : Firft, the Bafe of a
Sphere is a Point, for the fame Reafon, as it is the Bafe of a Circle; the fame is
alfo to be underllood of the Bafe of a Spheroid. Secondly, If we conceive the
curved Superficies of aCylinder, to be an infinite Number of Circles, like Hoops
fet clofe together, it is very eafy to conceive, that the Bafe of a Cylinder lying
down is a right Line, becaufe every Circle can touch the Plane it lies on, but
in one Point only ; and therefore all thofe Points in the feveral Circles of the
Cylinder's Length, will form a right Line.�The fame is alfo to he underftood
of a Cone laid on its Side. Thirdly, If a Cylinder be fet upright, then the
End it fiands on is its Bafe ; as indeed is every Surface on which any Body
Hands. Fourthly. The Bafe of a Cone, Conoid, Pyramid, Pyramis, Pyratnidoid,
faff, is that Superficies which is oppofite to the Vertex, and on which they com-
monly (land ; but in their Fruftums, the Superficies of both Ends are called
Bafes, as the leffer Bafe and the greater Bafe : But tho' Cuftora lias thus diftin-

Kuiihed

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		O/GEOMETRY. 71 Suiftied the fmali End from the greater, I mull own, I think it a very improper Manner of Diftin�tion, becaufe one Body cannot Hand on two oppofite Ends at the fame time, and therefore cannot beconfidered as two Bafes, but as two Ends, as they really are, and which may be diitinguimed by the Names of Greater and Lefier. by only making Ufe of the Word End, inftead of the Word Bafe ; for ftridtiy fpeaking, except the Pruftum of a Cone Hands on one of its Ends, neither of the Ends is a Bafe ; for when a Fruftum is laid on its Side, its Bafe is a right Line contained between the two loweft Points of the Superficies of its Ends. LECTURE II. On the Formation, Names, Kinds, end Menfuratien of Angles. TH E Angles I am now going to explain, are Angles on Superficies, or ra^ ther fuperficial Angles. ASuPERi-iciAtAngleisa Space contained between two Lines, of which one mud be oblique, and which meet each other in the Def. 81. Of famePoint; as for Example, Fig. I. Plate J I. If the oblique Line a Superficial de be continued forward, fo as to meet the Linegf, in the Point Angle. f, the Space that is contained between them is called an Angie. There are three Kinds of fuperficial Angles, that is to fay; (t) Def 8z. Of Right-;ined, as o n p, Fig. II. Plate II. (2) Curvilineal, as xy z, the Kinds of and 1 2 3, of which xyz is a. convexAngle, and 1 2 3, is a con- Angles. cave Angle. (3) Compound, or mixtilineal, as qrt, or / iv <u. Right lined Angles have three Denominations, which they re- Def. 83. Of ceive according as their Openings are greater or leffer, Right, the Kinds of Acute, and Obtufe. Angles. A Right Angle is that, when two right Lines meet, and are fquare to each other, as h k and m h, Fig. II. Plate If. or when a Def. 84 Of perpendicular Line Hands on a given Line, as h k on m I; then a Right An- the Angles on each Side of the Perpendicular h i, are both right gle. Angles. An Acute Angle is an Angle whofe Opening is lefs than a Def $$-Of right Angle, as the Angle made by the Lines i k and k I, or by an Acute the Lines i k and h k. Angle. An Obtufe Angle is an Angle whofe Opening is greater than Def 86. Of a Right Angle, as the Angle made by the Lines i i and m k. an Obtufe An Angle is meafured by the Arcii of a Circle defcribed on Angle. its angular Point ; and therefore the Meafure of an Angle is the Def. 8 'J Quantity of that Arch which is contained be�vveen its Sides. The the Meafure Quantity of an Arch is the Number of Degrees that are con- of an Angle. tained therein. A Begr�B is the 360th Part of the Circumference of any Cir- Def. 88. Of cle, as appears by the following Example. Suppofethe Circ'ee, a Degree. 90, b d. Fig. I. Plate II be divided into four Quadrants, by the two Diameters c b, and 90 d, and that the Limb of each Quadrant be divided into 90 equal Parts, then the whole Circumference of the Circle will be divided into 60 equal Parts, which are called Degrees, and confequentlv any one of them, which is the 360th Part of the whole, is a Degree. And from hence it is very plain, that the Limb of a Quadrant Degrees in ' contains 90 Degrees; that the Limb of a Semicircle contains 180 the Limb of» Degrees ; that a Right Angle contains 90 Degrees ; that an Acute Quadrant Angle contains lefs than 90 Degrees ; and that an Obtufe Angle and Semi- eontains more than 90 Degrees. circle. In every Circle there are 36oDegrees; for if from the Center 360 Degree* A, you draw right Lines through every Degree, in the Circle c 90, in every b d, unto the Circle h g fi, they wil! divid-e the Circumference of Circle. tha.t Circle, into the fame Number of Decrees, as the Circle c 90., K 2 l�

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		72 0/ GEOMETRY. b d ; and in like Manner the fame Lines will divide thefmall Circle m I in, for the Arches q k, p b, and o f, do each contain the fame Number of Degrees. Hniv to fndtke Quantity or Mcafure of an Angle. Jjfl&cf Before the Quantity of an Angle can be found, a Scale of Ceoidi.ha-M Chords mud be made, as following, via. Firft, Draw a right much. Line at Pleafure, as c�, Fig. III. PJat� II, and ailign a Point therein, as d, whereon with any Radiuj, 01 Opening of your Compaffes, defcribe a Semi circle, is tab. Secondly, With any Opening of jour Compaffes, greater than dc, on the Points c and b, de�r,be the Arches, as e e and ff, and from d, through the Point of intersection h, draw the Line da. Thirdly, Set the Radius dc, from c to 60, alio from « to 30, :n the Arch c a, which will then be divided imo three equal Parts, at the l'oints 30 and 60. fourthly, Divide each third Part of the Arch a c, into three equal Parts, and then the whole Arch a c will be divided into 9 Parts. Fifthly, Divide each Par: into Halves, and each Half into five equal Parts, and then the whole Arch a c will be divided into 93 Degrees. � This being done, fet one Foot of your Compaffes on the Point c, and the other being opened to 10 Degrees, turn down that Opening, on the h\ue b b, from I� to 10. In the fame Manner, on the Pointe, take the Di�lances c 20, c 30, c 40, c yo, c 60, c 70, c 80, and e go, on the Arch ac, and turn them down to the Line c b, as before, and thus you will have transferred even teeth Degree from the Limb c a, unto the right Line c b. In the fame Manner transfer every intermediate Degree, and then will the Scale, or Line of Chords, be com- pleted and made fit for Ufe. � f f j To find the Quantity of an Angle, you muff proceed as fol» wo 0 fin lowing;, hst da h, Fig. II. Platell. be an Angle given, to find the Quantity �,->.�. ■ j 1 "^s Quantity. "■? 9 Ta re 60 Degrees in your Compaffes, from the Scale of Chords, and on the angular Point a, defcribe an Arch, as e c ; take the Extent of the Arch e c in your CompafTes, and apply one Foot to your Line of Chords, at the Beginning c, and the other Foot will fall on the Number of Degrees that is con- tained in the Angle. The Reafon why you muft take exactly 60 Degrees in ycur D/f. 89. Of Compaffes for to defcribe the Arch e c, is necaufe that theRadius, the Degrees in or Semi-diameter of every Circle, is equal to die Chord Line of the Radius of 60 Degrees of its Circumference. And note, that if in the mea- ■ every Circle, furing of Angles, it fhould happen, that the Sides of an Angle fliould be fliorter than to Degrees, the Radius of your Line of Chords, you muft, in fuch a C�fe, continue out the Sides of the Angle, unto a fnrfrcierjt Length. tr r j ' To lay down an Angle equal to any Number of Degrees given, � if- ° is a very eaiy Work, and very little different from the lafl ; as -, I "'''' for Example, fuppofeit is required to lay down an Angle equal " ^:* " to 30 Degrees : I-'irft, Draw a right Line, as h a, Fig. II. Plate II. Secondly, Take 60 Degrees in your CompafTes, from your Line of Chords, and on a', the End of the Line, defcribe an Arch at Pleafurc, asab. Thirdly, Take 30 Degrees, the Angle given, from your Line of Chords, and fet them on the Arch, from e toc. Laiily, from a, through the Pointe, draw the Line a d; then will the Lines da and a h make an Angle equal to 30 Degrees, as /�'quired. V f co Of ^^s Qi^,3m"i^es ^or" Angles are femetimes whole Degrees, and M'Z�tr ' fometimes Degrees and Parts of Degrees, it is therefore to be oh- ",,'"' '" ferved, that every Degree is fuppofed to be fubdivided into fixty _;* ' equal Parts, which are called Minutes, and therefore-J of a De- gree is ic Minutes, § a Degree is 30 Minutes, -\| t/f a Degree is 45 Minutes,» i is 10 Minutes, ^t is 5 Minutes, �fc.

			Ds-

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		0/GEOMETRY. j$ Be.os.ses and Minutes are thus written or expreffcd, viz. ten Degrees and Degrees, forty Minntes, and twenty-five Seconds, is thus written, Minutes, ho=w io<\ 40', 25", and 40 Degrees, ^Minutes, thus, 40⁰ I5'. written. Anglfs are expreffed by three Letters, of which it is to be Hew an remembered, that the middle Letter always denotes the angular Angle is Point.�As for Example, to write or exprefs the Angle made by -written and the Lines da h a, Fig. II. Plate II. 1 write thus, the Angle denoted. d ah, or h a d ; in both which Cafes you fee that a, which ftands at the angular Point, is kept in the Middle, and fo the like of all other Angles. The Complement of an Angle is to be confidered in two different Manners, that is to fay, when it is to a Quadrant, and Def. qi. Of when to a Semi-circle. But be it which it will, the Complement the Cample- of an Arch, or ol an Angle, is fo many Degrees as will make ment of An- the given Angle, or given Arch, equalt.090, or to 180 Degrees, giesand So 70 Degrees is the Complement of an Angle of zo Degrees, to Arches. a Quadrant, and 160 Degrees is the Complement to a Semi-circle. ' Angles are external, internal, and oppofite. An external Def.gz.Of Angle of a Figure is an outward Angle, as the Angle � or g, in external Fig.O. PlateTV. whofe angular Point points outward ; and an and internal internal Angle, is an inward Angle, that points inward, as the Angles. Arr^le h, in Fig. P. Plats IV. but an external Angle, fitigly con- fidered, without RefpeiSt being had to a Figure, is the Complement of am (in- ternal) Angle, to a Circle, or 360 Degrees. So the Angle a x m, Fig. M. Plate VII is an internal Angle, whofe Meafure is the Arch 2' in, and the external Angle is all the Space that is without the Lines a x and* m, and whofe Meafure is the Arch i k I m, which, with the Arch i m, is a complete Circle, and therefore is the Complement of the Arch i m, to ^6oDegrees. Opposite Angles are fuch, that are againft, or oppofite to one Def. 94. Of another ; as for Example, if two right Lines, as a c and b e, oppofite dn~ Fig.G. Plate VII. crofs each other, the oppofite Angles which gles. they make are b d c, and a d e ; that is, the Angle b dc is oppo- fite to the Angle ade. So likewife the Angle a b d is oppofite to the Angle crV, and which are always equal to one another, becaufe the Arches ab and e c,* which are their Meafures, are equal, and fo the like of all others. L E C T. III. Of the Defcription of Lines. AS the feveral Works of this and the following Le&ures are very often de- pendant on one another (like the Links of a Chain) I ihall therefore de- liver the whole by way of Problem or Propvfition. A Problem is a Proportion for lomething to be done or made, as following. Prob.I. Plate V. FigA. To drain a Right Line from the gi<ven Point e to the given Point X, and to continue it infinitely from X towards f. Operation. Firft, Apply the Edge of a (freight Ruler to the Points e X, and with a Pencil draw the Line required. Secondly, Lay the Edge of a Ruler ta the Line f X, and applying the Point of a Pencil, &c. tothePointX, continue the Line e X, from the Point X, towards/. Prob.II. Fig. II. Two Points being given {as g g) tofindaPointoflnterfeSion, as i. Operation. Open you; Compaffes to any Di fiance, greater than half the Diftance of the Points propofed, and upon the Points g g deferibe Arches, asgX>, g'k ; then will the Point i be the Point of Intersection required, the Ufe of which will be prefently fhewn ; and it is to be obferved, that it is no matter what the Opening of your Compaffes is. fo that they are more than half the Diftance of the 2 given

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		74 O/ GEOMETRY. given Points ; and the Reafon thereof is, that if the Opening is lefs than half the Diftance, as g 4 and g6, the Arches defcribed on that Opening, cannot meet to interfe� each other, io as to make a Point of Interfe&ion, as is alio the Cafe if the Opening be exaftly half the Diftance, as g 3, as is evident by the Figure. Hence it is plain, that unlefs the Opening be more than half the Diftance of the given Points, there cannot be any Point of Interfcclion made. The Points /' and jr are both Points of Interferon ; that of /, being found by an Opening equal to the vvholeDillar.ee of the given Points, and that of g, by an Opening that is leis. Pros. III. Fig. Ill, IV, V, VI, and VII. To ereel Perpendiculars from given Points, in or near the Middle, and at or nef the Ends of given Right Lines. Op-ration. Firft. In Fig. III. let m be a given Point, in or near the Middle of he given Line 0 n. Set any equal pittances on each Side the given Point m, as 0 and », whereon, by the 1 aft Problem, find the Point of Interfe&ion. as q. From m '.o q draw the Line m q, which will be the Perpendicular required : for as m n and m 0 are at equal Diftances from m, therefore (by De/. 15.} the Line m q is a Perpendicular ; becaufe the Diftances n q and 0 q are equal. Secondly, 'Io ered a Perpendicular from the given Point r, Fig IV. at the End of the given Line ft. Operation Firft, On the given Points, with any Opening of your Compares, defcribe an Arch, as s xvu, and thereon let that Opening twice, as from s w x, and from x to v. Secondly, On the Points x and v, find a Point of Interferon, as z : draw the Line r 2, and it will be the Perpendicular required. A Perpendi- cular may alio be erected on the End of a given Line, by either of the following Methods. As for Example; Firft, Let 1 2, Fig. V. be a given Line, and 1 the given Point. Operation. Firft, On I, with any Opening of your Compafles, defcri'oe an Arch, as 3, 9, and thereon let its Radius, from 3 to 4, whereon with the fame Open- ing, defcribe the Arch 3568, and thereon fet up its Radius three Tunes, at the Points ,% 6, 8. Secondly,Draw the Line 8 I, and it is the Perpendicular required. Secondly, Let AD, Fig. VI. be a given Line, and A the given Point. Operation. Open your Compafles to any Diftance, and fetting one Foot in the fiven Point A, fet down the other at Pleafure, as on the Point B, fo that the 'ootin the Point A may be capable to interfecT; the given Line, as in the Point C. A Ifo on the Point B defcribe an Arch, as AG F, over the given Point A. Lay a Ruler from C to B, and it will cut the Arch A G F in G ; draw the Line G A, and it is the Perpendicular required. Thirdly, Let N O, Fig. VII. be the given Line, and N the given Point. Operation. Firft, From a Scale of equal Parts, asacd, Fig. I. take 6 Parts in your Compaffes, and on the given Point N, defcribe an Arch, as M M. Se- cOndly, Take S Parts in your Compafles, and fet them from N to I. Thirdly, Take ro Parts, and on the Point I interfeclthe Arch M M, in the upper N, and draw the Line N N, the Perpendicular required. Now as 64, the Square of 8, and 36, the Square of 6, are together equal to 200, which is the Square of 10 by 10; therefore N M is a Perpendicular to the given Line NO. Fourthly, Let ha, Fig. VIII. be a given Line, and a the given Point. Operation. With 60 Degrees of a Scale of Chords, on a, the given Point, de» fcribe an infinite Arch, as h d ; and then fetting go', from b to c, drawee, 'h� Perpendicular required. Prob. IV. Fig. IX. To ereB a Perpendicular on an angular Point. Let ha c be the angular Point given. Operation. ( r ) Afjjgn two Points, as l c, at any equal Diftance from the giver. Point». (2) On the Points � and c, by Prob, II. fiftd a Peint pf Isterfeftion, as d i and draw da, the Perpend -.cular required, Prob.

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	Of GEOMETRY.
	Of GEOMETRY.	75
		75

Prob. V. Fig. X. and XL To ereft a Perpendicular on the Convexity, and in the Cmcavity of an Arch of a Circle. Firft, Let ef, Fig. X. be the given Arch, and a the given Point. Operation. Set of two Points, as b d, at any equal Diftance from a, and thereon by Peob. II. find a Point of Interfecuon, as c ; draw c a, .the Perpen- dicular required. Secondly, Let bad, Fig. XL be the given Arch, and a the given Point. Operation, (i) Set any equal Diftances on each Side of a, the given Point, as in the laft Problem, and thereon, by Prob. II. find the Point of lnterieiSion, as h. (2} Draw h a, the Perpendicular required, Prob. VI. Fig. XII. To lijeB a Right Line by a Perpendicular. Let a b be the given Line. Operation. On the Points a and b, by Pros. II. find a Point of Interfe&ion on each Side of the given Line, as a'and e, and then drawing the Line d e, it will be a Perpendicular to the given Line a b, and bifacl or divide it into two equal Parts at the Point c. Prob. VII. Fig. XIII. ■ To ere�l a Perpendicular on the Extremity of a Concave Arch, lubofe Center is unknown. Let adb be the given Arch, and a the given Point. Operation. Aflign three Points in any Parts of an Arch, as g d b, and between them draw Right Lines, as gd and db, which by the Iaft Prob. bifeft or divide by Perpendiculars, which will interfe� each other in c, the Center of the Arch} from whence draw ca, the Perpendicular required. Prob. VIII. Fig. XIV. and XV. To let fall a Perpendicular from a given Paint, on a given Right Line. Let a p, Fig. XV. be the given Line, and h the given Point. Operation. Open your Compaffes to any Extent greater than the Diftance from the given Point to the Line, and onh, the given Point, defcribe an Arch iuter- ieding the given Line, in the Points m and b, whereon find the Point of Inter- fe�ion g, and laying a Ruler from h tog, draw the Perpendicular hi, as required. Note, This Operation is to be ufed when the given Point is over, or nearly- over the Middle of a Line ; and the following when the given Point is over, or nearly over the End of a Liae, as the Points, Fig. XIV. Operation. From the given Point e, draw an oblique Line, as e c, which by Prob. VI. bifect in the Pointy, wheieon, with the Radius f c, defcribe a Semi, circle, cutting the given Line in the Point;/, and draw e », the Perpendicular required. Prob. IX. Fig. XVI. , To let fall a Perpendicular, from a given Point, on a Concave Circular Arch, tyjhoje Center is unknown. Let b be the given Point, and aV/the given Arch. Operation. Allume three Points in the given Arch atPleafure, a.sde g, and draw the Linesge and e d, which bxitSt in the Points 0 and c, and thereon erect the Perpendiculars 0 b and c b, which will interfecl each other in the Point b, the Center of the Arch. Lay a Ruler from b, the given Point, and draw a n, the Perpendicular required. Prob. X. Fig. XVII. To divide an Angle into t~vo equal Pa/ ts by a Perpendicular. L�t' b a e be the given Angle. Operation.

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		₇6 Of G E O M E T R Y. Operation. Set any equal Diftance on each Side the Angle, as from a to J, and e, whereon find a Point of Interferon, as n, through which, from the angular Point a, draw the Perpendiculars », as required. Prob. XI. Fig. XVIII. and XIX. To make an Angle equal to a given Angle. "Tis required to make the Angle if h equal to the Angle e a b. Operation. Draw a Right Line, as kf, and open your Compafi'es to any Di- ftance, and on the angular Pointe, deferibe an Arch, as dc; with the fame Open- ing on the Point/, defcribe an Arch atPleafure, as ng : makethe Arch ng, eqaal to the Arch d c, through the Pointg, from the Point/, draw the; Right Lme.fgh, and then the Angle kf h, will be equal to the Angle b ae. In the fame Manner the Angle edf, Fig. XX. is made equal to the Angle bac, Fig. XXI. Prob. XII. Fig. XXII. To continue a Right Line to a greater Length than can be drawn by a Huler at one Operation. Let a be the given Right Line, which cannot be made longer at one Opera- tion, by reafon of the Ruler being of the fame Length. Operation. With the Length of the Line a, on the Point a, defcribe an Arch, as c d ; on which, from the End of the given Line, fet off two Points, as e f, whereon find a Point of In�erfeftion, as h-, unto which, from the End of the given Line, lay a Ruler, and continue the given Line at Pleafure. Prob. XIII. Fig. XXIII. To draw a Right Line parallel to a Right Line at an ajfigned Diftance. Let : k, Fig XXIII. be the given Right Line, and A B the given Diftance- Take the given Diftance A B in your CompaiTes, and on any two Points near the Ends of the given Line, as r and p, defcribe two Arches, as n n and o o, unto which lay a Ruler, fo as but juft to fee their Convexities, and draw the Line m, which will be parallel to ik, at the Diftance of A B, as required. Prob. XIV. Fig. XXIV. To draw a Right Line parallel to a Right Line which ftaall pafs through a given Point. Let eh be the given Line, and h the given Point. Operation. From the given Point b, draw an oblique Line, as b g, at Pleafofs, to cut the given Line in any Point, as�. By Prob. XI. make the Angle c b g, equal to the Angle b gf, and from the Point c, to the Point b, draw the Line a b, which will be parallel to the given Line, as required. Prob. XV. Fig. XXV. To defcribe a Circle concentrick to a given Circle at a gitten Diftance. Let the given Circle be b, and e d the given Diftance. Operation. Draw a Right Line through a the Center of the given Circle, as e f, and make e d equal to the given Diftance, on a, with the Radius a e, de- fcribe the Circle e g c, as required. Prob. XVI. Tig. XXVI. Between two given Points to find two others dire 3/y intcrpofed. Let ad be the two Points given, to find two others directly interpofed, as b ande, by the Help of which a Right Line may be drawn from the Point a to the Point d, with a Rule, whofe Length is lefs than the Diitanca. of a to d. Operation. With any Diftance greater than.half the Length of a. d, en the Points a, d, find two Points of Interfejftion, as e and/, on which, with any Di- ftance greater than half the Diftance between the two Points of Interferon, find two other Points of Interfeclion, as b and c, which will be directly intcrpofed between the given Points a and d, as required. Pr.OB,

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		0/GEOMETRY, 77 Prob. XVII. Fig. XXVII. To divide a right Line into any 'Number of equal Fart si Let E F be the given Line, to be divided into 4 equal Parts. Operation. Draw a right Line at Pleafure as a b, and thereon fet four equal Parts of any Bignefs, as 1 2 3 4, on the Points of a and 4, with the Diftance a 4, make the Seftion », and from n, through the Points a 1234, draw right Lines out at pleafure. This done take the given Line in your Compaffes and fet it from n to b, and to/, and draw the Line b f, which will be equal to the given Line, which will be divided into 4 equal Parts by the Lines n c, n d, n e_tas required. A Right Line may alfo be divided into any Number of equal Parts as fol- lowing, viz. let a b, Fig. XXIX. be the given Line to be divideded into five equal Parts. Operation. From the End b, draw a right Line as b d, making any Angle at pleafure. By Problem XIV. draw a c parallel to b d, or by Prob. XT. make the Angled a c, equal to the Angle db a. On the Lines b d and a c fet off four equal Diftances of any Magnitude as at the Points I z 3 4 on the Line b d, and at 5 678, on the Line a c. This being done, draw the Lines 4 �, 3 4, 2 3, and 1 2, which will divide the given Line «J, into 5 equal Parts at the Points ^^ f e, as required. Prob. XVIII. To divide a given right Line into unequal Parts in the fame Proportion as another Line is divided. Let the right Line A under Fig. XXV. be given to be divided in the fame Proportion as the Line b c, next below it. Operation. On the Points b c with the Dillance b c, make the Seftion a, from whence draw right Lines through every of the Divifions f g i nm. Makead,�a 6 each equal to the given Line A, and draw the Line d 6, which will be equal to the given Line A, becaufe the Triangle da 6 is equilateral, and which will be divided by the Lines a/, ag, cifc. in the fame Number of Parts, and in the fame Proportion as the Line b c. Prob- XIX. Fig. XXVIII. A Circle being given, to find its Center. ■ Let � a b be a given Circle, to find its Center. Operation. Afiign three Points in any Part of its Circumference as/a h, arid draw the Chord Lines/a, and a b, which bifeftin the Points zx, whereon ereft the Perpendiculars zc, and x c, which will interfeft each other inr, the Center; of the Circle. Prob. XX. Fig. XXX. To find the Center and Diameter of a Toiver, &C vohofe Bafie is a Circle being 'without th� Jante. Let the Circle �//reprefent the Out-line of a Cylinder Or round Building, whofe Center and Diameter is known. Operation. Apply the ftreight Side of a ten-foot Deal againft the Ontfide of the Building, as h n, or, for want thereof, ftrain a packthread Line, fo as juft to touch the Building, as the Line h n, touching in the Point k. Set any certain Diftance (fuppofe 10 Feet) from itoh, and from k to n, at which Points, ereft Perpendiculars continued until they meet the Building, as hi, and n I, and mea- fure their Lengths exaftly, which fuppofe to be each 6 Feet. This being done, make a Scale of equal Parts, as Fig. 1. and let every Part reprefent i Foot. Draw a right Line to reprefent hn, which make equal toten Parts of your Scale, and on the Ends h and n ereft two Perpendiculars, making the Length of each equal to 6 Parts, and draw the Lines i k and k 1. Laftly, bii'cft the Lines i k and k I in the Points x z, and thereon ereft the Perpendiculars x a, z a ; which by the laft Pro- blem will interfeft each other in a, the Center of the Building, on which with jhe Radius, a k, defcribe a Circle, which will reprefent the Oat-line of the gwtn h Building,

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		/S Of G E O M E T R Y. Building, and whofe Diameter being rneafured on your Scale of equal Parts, will thew the' Number of Paus, which are the Feet contained therein. Prob. XXI. Fig. XXXI. To find the Center and two Diameters of an Oval or Ellipfis. Let h a it, be a given Oval, whole Center p, and two Diameters are to be found. Operation, Draw at pleafnre two parallel Lines as c e and m g, which bifeft in the Points « and m, through which draw a right Line as In mi, which bifeft in p, whereon defcribe any Circle that will interfect the Sides of the Oval, as c bfd, in the Points c bfd; through the Interferons b d, draw the right Line b d, which bifecl in x ; then through the Points x p draw the longer! Diameter, and through she Pointy, draw the lhortelt Diameter parallel to b d, and p is the Center, as reauired. Prob. XXII, Fig. XXX. Plate III. To drew a right Line through a given Point, that (halt be a Tangent 'Line to a given Circle. Let d be the given Point, through.which the Tangent d b is to be drawn. Operation. Draw a right Line from d the given Point, to a the Center of the Circle, which bifeel in m, whereon with the Radius m d, defcribe the Semi-circle a c d, interiefting the given Circle in c, through which, from d, draw db_r the Tangent Line required. The fame is alio to be underftood of a Tangent Line to an Ellipfis, as Fig. XXXII. Prob. XXIII. Fig. XXXIII. Piute IV. A right Line being given as C d, ta find'another right Line equal thereto. Let d c be the given Line. Operation From the End c draw a right Line at plearureasar, and on the Points a and c, with the Op en ing a r, find the Point of Interfecticn b, and draw a e anda�" out at pleafure ; on c with the Radius c d defcribe the Arch of a Circle de, cutting the Line a c continued in e. On a with the Radius a e delcribe the Arch e f cutting the Line a b continued in the Point g ; then is b g equal to c a', as required» Proe. XXIV. Fig. XXXIV. and XXXV. Plate III. To divide the Circumference of a Circle into Degrees, Minutes, Hours, and Rhumbs,. Let the Circle b a c d, Fig. XXXIV. be given, to be divided into 360 De- grees, the Circle da c e, Fig. XXXV. into 60 Minutes, the Circle db ce, Fig. XXXVI. into 12 Hours, and the Circle erin Fig. XXXVI. into 32 Rhumbs or Points of the Compafs. First, in Kg. XXXIV. and XXXV. draw the two Diameters at right Angles as a d, and b c in Fig XXXIV. and d c, and a e in Fig. XXXV. Set the Radius of each Circle from clog, and from a ton, and then will thofe Quadrants be each divided into three equal Parts. In the fame Manner divide the remaining three Quadrants in each Figures This being done, divider:», ng and ag�nFig. XXXIV. each into three equal Parts, and every Part into ten equal Parts, and then the Quadrent a e c, wili be divided into go equal Parts, In the fame Manner <di>- �vide the Quadrants ab e, h e d, and then the Circle will be divided into 360 De- grees, as required. Alfo divide c », n g and a g, Fig. XXXV. each into five equal Parts, and then the Quadrant a g^n c, will be divided into 15 equal Pans. In the fame manner divide the Quadrants a d_td e, and e c ; and then that Circle will be divided into 60 Minutes, as required. Secondly, To divide the Circle of 12 Hours, Fig. XXXVI. Draw two Diameters at rjght Angles, as a' c and b e, which will divide the C irele into 4 Quadrants, fet the Radius a d, from d to n and to /, alfo from e to � ac� W. E, alfo from c to m and to h, and laliiy from b to 0 and to x, and then will ths Circle do �,t le divided into 13.equal Paru, as required.

			Thirdly,

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		Of G E O M ET R Y. 79 Thirdlv, To divide the 32 Points of the Compafs, Fig. XXXVIL Draw two Diameters at right Angles as e 8, and r n, divide each Quadrant into two equal Parts, and then the whole will be divided into 8 Parts; divide each 8th Part into 2 equal Parts, and then the whole will be divided into 16 Parts. Laftly, divide each 16th Part into 2 equal Parts, and the whole will be ■divided into 32 equal Parts, as required. To proportion the Height of the Figures to the Hours. Divcde the Semi-diameter of the outer Circle of your Dial-Plate into iz equal Parts, give one to the outer Margin for the Minutes, five to the Margia for the Hour¹ s Figures, and the next one to the Margin for the Divslions of the Quarters. The Figures by which the twelve Hours are numbered, are the Capital Let- ters I, V, and X, which are proportioned and made as following. To proportion the Breadth of the Figures, divide their Height into 8 equal Parts, and give one Part to the Breadth of the full Stroke in every F�gwfe, and one Quarter of a Part to the Breadth of the fine Stroke in the V and the X. The Diftance of the I's from each other is equal to their Breadth. The Breadth or Opening of an V at its Top, is 4 Parts, and of an X is t- Parts, as may be feen in Figure XXXVIII. by the dotted parallel Lines. If the Figures itand very high above the Eye, their Graces, which is the arched Finifhings at their Tops and Bottoms, niuil have a Breadth equal to the fine Stroke of an X, that is, of one Quarter of a Part. But when the Dial is near to the Eye, there need not be any Breadth given to them, as in the Figure is represented. The Curvature of every Grace begins, at half a Fa t, above the Bottom, and below the Top of every Figure, as exprefled by the Lines, c d, and a b, and their Projections is half a Partalfo. The Graces to the Fs are all Quadrants of a Circle as h dp, and whole Centers are always on the Lines c d and a b, but the Graces of the V's and X's, are Arches lefs and more than a Quadrant, and whofe Centers are found by this General Rule. From the Point e Fig. X. where the Outline of the Figure cuts c d, the Line <jf the Height of the Graces, erect, the Perpendicular as e m, Make 4 g equal to half a Part, for the Projection of the Grace, and draw the Line eg, winch bife� in �v A, on which creel the Perpendicular 'x m, interfering the Line e re in m, the Center, on which, with the Radius m e, defcribe the Atch eg, which us the Grase required. L E C T U R E IV. On the Confruclion of Plane Figures. Prob. II. Fig. E. PlatelV. TO defiribe an equilateral Triangle, as abc, Fig. E. whofe Sides Jfcali he escl» equal to d a, a given Line ; alfo an Iffceies Triangle as a b c, Fig. F. <i-.:,i :;,',; Safe and Sides /ball be equal to the given Lines d and e ; and like-iuife � Scalenum -Triangle, as Fig. G. v.'hofe three Sides Jhall be equal to the three given Lines, de f. First, make b c Fig. E.equal to the given Line</, on the Points � and c, with the Opening be, make the Point of Interfeflion a, draw the Lines a b, and a e, and they will complete the equilateral Triangle, as required. Secondly, make be, Fig. F. equal to the given Linei-, on the Points �and f with an Opening equal to the given Line d, make the Point of IntetfecLon a, draw the Lines a b, and « e, and they will coinpletethe Ifofceles Triangle, as required. Thirdly, make h e, Fig. G. equal to the Line/", on b, with an Opening equal to the Line e d, and on c with an Opening equal to the Line d make the Section a. Draw the Lints * b and a c, and they will complete the Scaicnum Triangle, as required. Prob. II. Fig. H and I. Plate IV. To ?jiaie a geometrical Square, as Fig. H, wbofe Sides Jhall hi each equal to a L 2 given

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		8o O/ GEOMETRY, givm Line as e, and a Parallelogram as Fig. I. <whofe Length and Breadth Jhall ht equal to tvjo given Lines as e and f. First, Make c d, Fig. H. equal to the given Line e, on d, by Problem III. Left. IILereft the Perpendicular^ equal to c d, on the Points b and c, with the Opening c d, make the Point of Interfeftion a. Draw the Lines a b, and a c, and they will complete the geometrical Square, as required. Secondly, Make a b, Fig. I. equal to the given Line e, on the Point a, ereft the Perpendicular a c, equal to the given Line f, on c, with an Opening equal to a b ; and on the Point b, with an Opening equal to ca, make the Point of Inter- feron d. Draw the Lines d b, and d �, and they will complete the Parallelo- gram, as required. Prob. III. Fig. K and L. Plate IV. To maie a Phombus as a b c d, Fig. K. ivkofe bides Jhall be each equal to the given line e, alfo a Rhombo�des luacb d Fig. L. whofe Sides and Ends Jhall be iqual to the given Lines e f, and nvhofe acute Anglts Jhall he each equal to the given Angle M. First, Make a d, Fig. K. equal to the given Line e, on d, with the Radius d a, defcribe the Arch abc; make a b, and b c, each equal to a a'. Draw the Lines a b, b c, and c d, and they will complete the Rhombus, as required. Secondly, MakeW, Fig. L. equal to the given Line<?, by Problem XI. Left. III. make the Angle d a c equal to the Angle bac, and make c a equal to the given Line/", on the Point c, with an Opening equal to a d. and on the Point d, ivith an Opening equal to c a, find the Point of Intsrfeftion b ; draw the Lines f b and d b, and they will complete the Rhombo�des, as required. Prob. IV. Fig. N and O. Plate IV- To male a Trapezoid, as a b d h, Fig. N. vihofe Height, Top, and Bafi /hail be equal to the three given Lines e Z f, g, and h ; alfo a Trapezia as a e f g, Fig. O, whofe Sides Jhall be equal to 4 given Lines, and one of its Angles as e a g, equal to Q, an Angle given. First, Make a h equal to the given Line g, and bifeft it in n, whereon ereft the Pei pedicular n c equal to h the given Height ; by Problem XIII. Left. III. draw d parallel to a h, bifeft e f in z, and make c b and c d each equal to x e ; draw the Linea d h and b a, and they will complete the Trapezoid a b d h, as required. Secondly, Make a g. Fig. O, equal to the given Line d, by Prob. XI. Left. III. make the Ansle e a g, equal to the given Angle Q, and make e a equal to the given Line d. Cn the Point e with an Opening equal to the given Line i, and on the Point g with an Opening equal to the fourth given Side, find the Point of Interfeftion/. Draw the Lines e f, and f g, and they will com- plete the Trapezia, as required. Note, If the Angle had been required to have been made an internal Angle, then the two Sides,/" e and � o-, muft have been drawn to the Point of Inter- feftion h, as in fig, P, which is a quite different Figure from Fig. O, although the given Angle and Sides are the fame. Ix.is alfo'to be noted, that when four right Lines are propofed, to be the Bounds of a Trapezium, that thofe two Lines which make the Interfe�ion, muft be longer than the Diftance contained betweenthe Extremes of thofe Sides, which make the given Angle, otherwife there cannot be a Trapezium made j for if the aforefaid two Lines, � f, and f g, Fig. O, were but equal to the Di- fiance contained between � and e, the Extremes of the Angle g a e, they would make but one Line, and confequently the Figure would be a Triangle, inftead of a Trapezium ; and if thofe two Lines were lefs than the Diltance from e to g, then there could not be any Figure produced. Therefore 'tis plain, that to make a Trapezium, the two Sides which make the Interfeclional Points muft be greater than the Diftance contained between the Extremes of thofe Sides which contain the given Angle.

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		Of GEOMETRY. 81 Prob. V. Fig. A B C D and S. Plate IV. To defcribe a Circle of any given Diameter, fuppofe ten Feet, and to defcrilt Ovals of the firji, f�cond, third, and fourth Kinds, to any Length required. Operation. Firft, make a Scale of equal Parts, as Z, and let each Part repre- sent one Foot. Take 5 Parts in your Compaffes, and on a defcribe the Circle, Whole Diameter c d, will be equal to ten Feet, as required Secondly, Divide a, f, Fig. B, the given Length of an Oval, into 3 equal Parts at e and b, whereon with the Radius h f, defcribe two Circles interfer- ing each other, in c and g, from which two Points, thro' the Centers e and b_tdraw the Lines ged, gb k, cb m, and c en; on the Points g and c, with the Radius �■ d, defcribe the Arches d k, and n m, which will complete an Oval of the firft Kind. Thirdly, Let d f, Fig. C, be a given Length, as before. Divide d /into four equal Parts, at c e h , on the Points ch, with the Radius c d, defcribe two Circles, touching each other in the Point e; on c h make the two equilateral Triangles ach, and n c h, continuing their Sides out both ways at pleafure as to 5 8 6 and 7, on the Points a and «, which with the Radius n i, defcribe the Arches 5 6, and 8 7, which will complete an Oval of the fe- eond Kind. , Fourthly, Let a k be a given Length, as before. Divide ak into 24 equal Parts, and draw b </and f i, parallel thereto, each at the Diftance of 10 Parts ; draw e h through the Middle of a A,- at right Angles to a A, and make cb, cd, alfo gf, and g i, each equal to 10 Parts, and then will you have completed two geometrical Squares, viz. bc f g and c d gi. Draw their Diagonals, and on their Centers y and a, with the Radius of z d, or z i, defcribe the Arches � a b, and d A i. On the Points c and g, with the Ra- dius � d, defcribe the Arches bed, and f h i, which will complete an Oval of the third Kind. It is here to be noted, That as the Proportion, that the Side of a geometrical Square, bears to its diagonal Line, is yet unknown to all Mathematicians, the Difference between them cannot be afcertained. But however, the neareft Pro- portion that the Side has to the Diagonal, is, as Five is to Seven ; that is, if the Side be five, the Diagonal is feven, and a little more. And therefore when the Length of the Oval is divided into 24 equal Parts, or twice 12, then c d, csV, being c, z A will be 7, and a little more ; and therefore when the Arches d A i_rand b a f are defcribed on the Centers y z, they will exceed the Points a and k, fome fmall Matter. Fifthly, Let e 4, Fig. S, be a given Length as before. Divide the Lengths 4, into four equal Parts, at the Points 1. 2. 3 and through them draw the Lines r t, b n, and s v, at right Angles, to the Line e 4 : make i r, it, alfo 2 b, 2 », and 3 �, 3 <v, each equal to one fourth of e 4, vise, iom, and complete the 3 geometrical Squares, e r 2 t, b In 3, and s 2 i> 4, continuing the Sides n 1, and b \, as alfo the Sides b 3, and n 3, out at pleafure. On the Centers I and 3, with the Radius e I, defcribe the Arches m e h, and <? 4 o. On the Centers b and n, with the Radius nh or n d, defcribe the Arches hd, and ?n 0, which will complete an Oval of the fourth Kind, as required. Prob. VI. Fig. V, W, X, R. T and Y, Plate IV. To make an O-val of any Length and Breadth required, by divsrs Methods. Let the Lines z z, x x, Fig. V. be the given Length and Hreadth. Operation. Firft, make dl equal to z z, and by Prob. VI. L E C T. IIL divided/in two equal Parts, by the Line a r. Mzkexc and a-», each equal to half x x. Make de, equal to x c ; divide e x into three equal Parts, and make eh equal to I Part. Make x t equal to x h, and by Prob. I. hereof, on the Line h t, complete the two equilateral Triangles, hat, and r h t, con- tinuing their Sides thro' the Poinp h and t, at pleafure, Oa the Points b and t, " " " ~ . with

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		82 0/ GEOMETRY. with the Radius t /, defcribe the Arches k 1 m, and� d q ; alfo on the Points a and r, with the Radius r b, defcribe the Arches, b c k, and q n m, which will complete the Oval, as required. Secondly, by a Divifion of tntjo Circles, Fig. W. Let the given Length and Breadth be the Lines x x, z z, as before. Operation. Make the Line i 2, equal to the given Line z z, and divide it into two equal Parts by the Perpendicular 3 6. On a, the Point of Interferon, with the Radius a 1, defcribe the Circle, 1326; alfo on a with a Radius equal to half the Live xx, defcribe the concentf ick Circle 7, 4, 8, 5. Divide the Circum- ference of each Circle, into any and the fame Number of equal Parts (the more the better) as in the Figure where each Circle is divided into 24 Parts. Draw right Lines from the Divifions in the fmall Circle, parallel to the Line I 2, to the Right and to the Left at Pleafure. Alfo draw right Lines from the Divifions as 1 r t x z, in the outer Circle parallel to the Line 3 6, and through the Points of Interfeclion, that they make with the other Lines before drawn, as c d e b i, c�c. trace the Circumference of the Oval, whofe Length I 2, is equal to z z, and Breadth equal to x x, as required. Thirdly by th< Ordinate! of a Circle, Fig. X. Let the given Length and Breadth be as before. Operation. Make b e and a d, at right Angles to each other, and equal to the given Length and Breadth. On c, the Point of Interferon with the Radius c d, defcribe the Circle a c, d, Uc. Divide the Semi-diameter c'f, into any Number of equal Parts, fuppole 4, as at the Points 123; thro' which draw right Lines, parallel-to ni as I g, 2 i, 3 k, which are called Semi-ordinates of the Circle. Divide b c and c e, each into the fame Number of equal Parts, as f c, at the Points 4, 5, 6, thro' which draw Lines parallel to a d. Make 4 7, 4 m (which are Semi-ordinates of the Ellipfis) each equal to 1 g, the Semi-ordinate of the Circle. Make 5 8, and � », each equal to the Semi-ordinate 2 i ; alfo 6 9, and 6 o, each equal to the Semi-ordinate 3 k; then from the Point a, through the Points 7, 8, 9, e 0 n m d, trace one half Part of the Ellipfis. In the fame manner fet off Ordinates on the other Side, and complete the Ellipfis, as required. Fourthly, by the Help of a Line, or String, Fig. T. Let the given Line h be the Length, and the Line <w the Breadth. Operation. Make b f the long Diameter, equal to the Line h, unddn, equal to the Line iv, and at right Angles to b f. Set e f, half the tranfverfe Diame- ter, from d to a, and to g on the tranfverfe Diameter, which are called the Focus Points- of the Ellipfis, wherein fix two Nails, &c. and about either of them, fuppofe the Nail at a, put a double Line of Packthread, &c which (hall reach unto the Point/; then with a Pencil, �V. applied within the faid Line, and held upright, trace about the Circumference of the Ellipfis, which will pafs through the Points b d n, as required. Fifthly, by Help of a Trame!, Fig. R. Let b h and c », be the given Diameters, drawn at right Angles. Operation. Firft, make a Tramel, which is nothing more than two Pieces of Wood, as k i, and x g, fixed together at right Angles, with a Groove in the midftof each, wherein the Pins g e of the Defcribentg a move, as the tracing Pointa defciibes the Ellipfis. The tracing Point?, is generally a fixed Point, but the Points e and g, are moveable Points, and are made to Aide on the De- fcribent at pleafure. The Diftance of the Point e, from the Point a, is always equal tofc,* half the conjugate.Diameter, and the Diftance of the Pointy, from the Point a, i; alu ays equal to half the tranfverfe Diameter. Fix down the Tramel over the two given Diameters, fo that the middle Line of each Groove may lie dbe;;!y over them ; and the Points g e and a, being fixed as afore,- iVJd : Then putting the two Peints e g, into the Grooves, with one Hand move the tracing Point a (wherein generally is fixed a black lead Pencil) and with the other gui�e the Pms or Points e g, in their refpective Grooves, whilft

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		Of G E O M E T R Y. 83 the tracing Point a, makes one Revolution, which will defcribe the Ellipfis required. Prob. VII. Fig. Y. Plate IV. To defcribe an Elliptical Polygon, about a Plantation of Trees, or Piece of Water. Let d I be the given Length, and e d the given Breadth. Operation. Make a Parallelogram, as h b c 9, whofe Length is equal to gf, and Breadth to e d. Bifect, the Sides b b and eg, in the Points e d; alfo the Ends h c, and b 9, in the Points g and f. Divide every half of the Sides and Ends, into any (and the fame) Number of equal Parts, the more the better. In this Example, d 9, and � 9, are divided each into 9 equal Parts, as at the Points 1234, &c. in each Line. Draw right Lines from d to 8, in �9, as alfo from 1 to 7, from 2 to 6, from 3 to 5, from 4 to 4, from 5 to 3, from 6 to 2, from 7 to 1, and from 8 to/; and they will form one fourth Part of the Ellip- tical Polygon. Proceed in the fame manner, to defcribe the remaining three Parts, and they will complete the whole, as required. Note, In Practice this Figure may do near enough to reprefent an Oval ; but ftriftly confidered, it is a Polygon of 4 times the Number of Sides, as are Parts in each half Side. Prob. VIII. Fig. Z. Plate IV. To defcribe an Egg o-vallar Polygon, about an irregular Piece of Water, by the Interfeilion of right Lines. Let the given Length be/ h. Operation. Erect Perpendiculars on the Points/and h, as ce, zndbd, which continue both ways at pleafure. Make � c, and fa, each equal to one third of/ h ; alfo make b h and h d, each equal to three fourths of a c, and draw the Lines a b and c d. Bifeft c d in g, d b in h, a b in e, and a c in f. Then, by the lad Problem, divide each half Side, and half End, into equal Parts, and draw right Lines thereto, which will form the Curvature of the Eg» ovallar Polygon as required. P. Pray Sir, ivhy do you call thefe tnjjo laft Figures Polygons ? for, if 1 miftake not, there are fame Authors who call them Ovals or Ellipfes- M. 'Tis very true, and fo an equilateral Triangle is, by the Ignorant, called a three-fquare Figure, and an Odtagon, an eight-fquare Figure, which is ridi- culous and abfurd, becaufe neither of thofe Figures have any fquare Angles. And as all Ovals are compofed of Arches of Circles, how is it pofiible that right Lines, which form the Bounds of the aforefaid Figures, can produce Arches of Circles ? Therefore if this be confidered, 'tis plain that the Bounds of the afore- faid, and all fuch other Figures, are compofed of a Number of right Lines, which make very large obtufe Angles; and therefore they are either regular Polygons, or Parts thereof; and tho' they come very near to the Bounds of Circles, or Ellipfes of the fame Diameters, yet in fad they are neither. But however, as 'tis cuftomary to call them Arches, I will therefore do fo too, in the following Problems. Prob. IX. Fig. A C. Plate IV. To defcribe a Semi-circle by the Interfeclion of right Lines. Let a c be the given Diameter. Operation. Bifedl a c in b, whereon ereft the Perpendicular b h, 6�\|ual to a b, by Prqb. X. LECT. III. Divide the Angle h b c, into two equal Parts, by the Line be. Divide b c'vaXo 7 equal Parts, and make b e equal to 9 of thoie Parts. Draw the Lines h e and e c, which divide into any Number of equal Parts, as in Prob. VII. hereof, and then drawing the Lines c \, 12,23, �fr. they will form the Quadrant h nc. Proceed in the fame manner, to form ths Quadrant a h, and it- will complets the whole-, as required*' Prqb.

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		84 Of GEO METRY, Prob. X. Fig. AB. Plan IV. To dffcrihe a Scheme .Arch, ntiithout any Refpe�i being bad to its Center. Let a c, be the given Length of its Chord Line, and one half of the Perpen- dicular b, its given Height. Operation. Bifeftar, and ere� the Perpendicular b, equal to twice the given Height. Draw the Lines a b, and b c, which, as in Pros. VII. divide into equal Parts, and draw right Lines of Interferon, which will complete the whole, as required. Prob. XI. Fig, A D, and A E. To defcrihe a Gothick Arch for the Head of a Boor or Window, by the InttrfeSitn of Lines. Let a g, Fig. A D, be the given Ercadth, and e c, the given Height. Operation. Make a g, equal to the given Breadth, which bifeft in e, where- on erect the Perpendiculars c, equal to the given Height. Draw a b, and g d, parallel to c e, and each equal to half? c. Draw the Lines c b, and f d. Di- vide the Lines a b, b c, c d, d g, each into equal Parts, as in Prob. -VII. and draw the interfering Lines, which will complete the whole, as required. Fig. A E, is another Example, whofe Height is lefs than Fig. A D, but its �onftruction is all the fame. Note, If'tis required to have the Curvature of the Hanfes of thefe Kinds of Arches, to be more or lefs flat, the Height of the Lines a b and d g, muft be increafed or decieafed at pleafure, which 3 very little Practice will make yon perfect in. Prob. XII. Fig. A G. Plate IV. To defcribe a Gothick Arch, compofed of real Arches of Circle». Let n g be the given Breadth. Operation. Divide n g into 3 equal Parts, at m 0, whereon with the Radius 0 g, delcribe the Semi-circles g m and 0 ». On the Points » m 0 g, with the Ra- dius m g, defcribe the Arches g r, m t r, oq,s.n�_n q. From q, thro' e, draw the Line q 0 d, at pleafure. Alfo from r thro' m, draw thro' the Line rmb&t plea- fure; alfo, on the Points q and r, with the Radius q 0, more 0 �r, defcribe the Scheme Arches on each Side of e, which will meet the aforefaid Semi-circles, at the Lines b r and d q ; and then will n e g be the Gothick Arch required. Note, The Arches « b c, and c d f, are concentrick to the former, as being defcribed at any given Diftance on the fame Centers. , A Gothick Arch may alfo be defcribed as in Fig. A F, as follows. Let c 0 be the given Breadth. Operation. Divide c 0 into five equal Parts. On the firft Part, at each End, as on b and n, with the Radius n 0, defcribe the Semi-circles c d e, and m I 0. On the Points 0 n c b, with the Radius o b, defcribe the Arches b q, c p, and n p, oq, interfering each other in the Points/ and q; from whence, thro' the Points b and n, draw the Lines q b g, and p n i, at pleafure. On the Points p and q, with the Radius/ /, delcribe the Arches/^, and d i, interfering each other in k, which will complete the Arch, as required. Note, The concentrick Arch, a g h if, is defcribed on the fame Centers as b n, and/ q. Prob. XIII. Fig. C. Plate V. To defcribe an Arch, nxihofe Height is greater than half its Chord line. ' Let c d be the given Breadth, and e b the given Height. Operation. Bifefl: r dm e, and thereon erect the Perpendicular e a, of Length at pleafure. Make e b equal to the given Height ; alfo b a equal to e b, and draw the Lines c a and a d, which divide into equal Parts, and draw the inter- feeling Lines, which will form the Areh as required ; and which is of very great Strength, and much ftronger than a Semi Ellipfis of the fame Breadth and Height, as I (hail demonftrate to you hereafter, when I come to explain the Stresgth and Abutments of all Kinds of Arches. Pro»».

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Of G E O M E T R Y. % Prob. XIV. Fig. AI. P/a/elV. . Ta 'defcribe a Rampart Semi-circular Arch, by the InterfeSlon of Right Lines. Let ap be the given Diameter, and a b the Height of the Ramp. Operation. Bifeci ap in n, whereon erect the Perpendicular n e, of Length at Pleafure. From the Point a, draw a b, parallel to ne, and equal to the pivert Height of theRamp; and draw the oblique Line bp .BvProb. X. LECT. III. divide the Angle e nku into two equal Parts, by the Line» � Divide n p into feven equal Parts, as in Prob. IX. hereof, and make nf equal to nine of thofe Parts. Set up q e, equal to a n, and draw the Lines e f,fp, on the Points b and e, find the Point of Interfeclion c, by making e c equal to e f, and c b to f p, and draw the Lines c b and ce. Divide the Lines "b c, c e, ef, and f p, into equal Parts, and draw the interfering Lines, they will Complete the Semi -circle; as required. Prob. XV. Pig. A L. Plate IV. To dejcrile a Rampant Semi-Ellipfis by the Interfsclion of Lines. Let ch be the tranfverfe Diameter,/V equal t� half the conjugate Diameter; and a b the Height of the Ramp. Operation. Make c h equal to the given tranfverfe Diameter, which bifecTt in g, whereon eredl Perpendiculars, as g d, at Pleafure. Draw ca and e h, parallel t� g �, of Length at Pleafure ; make c b equal to the given Height of the Ramp j al.o makte b a and he, each equal to half the given conjugate Diameter ; arid draw the Line a e. Divide b a, ad, de, andeb, into equal Parts, and draw the interfefting Lines, which will complete the whole, as required. Prob. XVI. Fig. A H. arid A K. Plate IV. To defcribe a Rampant Circle, and a Rampant Ellipfis, by the Jnterfe�ion of Right Lines. Firft, To defcribe the Rampant Circle, Fig. H. Let dfbe the Diameter given. Operation. Make g'i equal to df, arid by Prob. ill. hereof, complete the Rhombus acgi. BifecVac' in b,ci in f, agind, and giiah; then divide 'ab,hc, cf,fi, ih, b g, g d, and da, into equal Parts, and draw th� interfer- ing Lines, which will complete the whole, as required. IL To defcribe the Rampart Ellipfis, Fig. A K. Let ed be the tranfverfe, arid bh the conjugate Diameters ; alf� Iet t�i� Angle di h be a given Angle. Operation. Make g i equal to e d, arid the Angle dih be equal to the given) Angle. By Prob. Hi. hereof, complete th� Rhomboid a cgi, whofe Sides and Ends bifeci in the Points e b d h. Divide a b,b c, c d, di, i h, hg, g e, and e a, into equal Parts, and then drawing the interfefting Lines, they Will complete the ■whole, as required. Prob. XVII. Fig. A. Plate V. To defcribe a Rampant Scheme Arch by the InterfeSion of Light Lines. Let e d be the Chord Line, or given Breadth, c f the given Height �f the Arch, and e a the Height of theRamp. Operation. Make ed equal to the given Breadth, Which bifeci in g, wher��d eredt the Perpendicular g b; of Length at Pleafure. Draw e � parallel to g bj and equal to the given Height of the Ramp. Draw the Line ad, and make f c and c b, each equal to the given Height of the Arch. Draw the Lines a b and b d, which divide into equal Parts, and drawing the incerf�cling Liriez they will complete the whole, as required.

		�-ROBS
	M

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		U �/ GEOMETRY. PftoB. XVIII. Fig. B. Plate V. To defcribe a Rampant Gothick Arch by the Interfeilion of Right Lines. Let ie be the given Breadth, and g b the given Height. Operation. Make/ e equal to the given Breadth, which bifeft in /, whereon erett the Perpendicular � �, of Length at Pleafure ; from the Point B draw the Lines i a and e d, parallel to f b, of Length at Pleafure ; make i h equal to the given Height of the Ramp, and draw the Line he, make ha and ed, each equal to half the given Height, alfo make c b equal toc g, chaw the Lines a b, and b d. Divide the Lines ah, a b, alfo b d and d e, each into equal Parts j and draw the interfering Lines, which will complete the whole, as required. Pros. XIX. Fig. D. Plate V. To defcribe a Rampant Semi-circle by Ordinates. Let cb be the given Diameter, and a a the Height of the Ramp. Operation. Make qdb equal and parallel to the given Diameter c e, on thePoints c e, cre&.the Perpendiculars c a uttdeb, each of Length at Pleafure. Divide'the Diameter ce into any Number of Parts, either equal or unequal, as at'the Pointa I d 6 8,&c. On /, with the Radius/ c, defcribe the Semi-circle cde, and from the Points 1^68, &c. draw Right Lines parallel to the Line c a, of Length at Pleafure. Make q a equal to the Height of the Ramp, and draw the Line a b. Take the Ordinates I 2, 4 3, '6 5, 87, &c. in the Semi-cirele D, and fet them on the Line a b, from 1 to 2, from 4 to 5, from 6 to 5, from » to 7, ate. and from the Point a, through the Points a 2 3 5 7/, '.(Sc. trace the Curve afb, the Rampart Semi-circle required. Fig. E. is a given regular Scheme Arch, from whofe Ordinates the Rampart Scheme Arches dgf..k /,and« mp, are produced at differentKeights of ramping, as ef, hi, and In, where every refpettive Ordinate are equal, in each,, unto 'thofe in the regular Scheme Arch a b c, Fig E. Fi". F. is a given regulaiSemi-Ellipfis, from whofe Ordinates the Rampant Se- tni-Ltlipfis f g e, and Imi, are produced, ac different H eight s in the fame Manner. ■ Prsb.XX. Fig. G, H. Plate'V. To defcribe a Parabola. Not- When a Cone has a Seclion cut parallel to its Sides, the curved Boun- dary of' the Superficies, made by. the Seftion. is called a Parabola. Let xffba a given Cone, and b e the Perpendicular of the given Seftion. Operation. Bifeft the Diameter of the Bale// in/, and from x, the Vertex of the Lone, draw xj>, its Axis, which continue downwards at Pleafure towards d, in Fig. I.'in any Part of the faid Line xp, continued, as at 5, draw I q, parallel toff, and make 5 fc equal to b e. Divide k e into any Number of equal Parts, fuppofe four (but the more the better) as at the Points c p m 1 ; and from thofe Points'draw Right Lines parallel to theKafe �/; mee: the Side of the Cone in the Points <r r h i k. Alio divide 5 a, in Fig. H. into the fame Number of equal Parts 'at the Points 1, 2,3, 4, and through thofe Points draw Right Lines to the right and left at Pleafure, and parallel to lq. In Fig. I. make c n equal to fp, the Semi-diameter of the Cone, and with the Radius n 5,, on the Point », defcribe the Circles lam b, on n in Fig. I. with the Radius's l; s, i w, b u, e g q, in Fig. G, defcribe the Ghdesdfg i, and from the Points 0 p m s, in Fig. G, draw Right Lines parallel to xzd, "interfering the outward Circle in Fig. I. in the Points ub the next in the Points c d, the next in the Points ef, and the next in the Points h g,intcrfecVmg the Diameter / tn, in the Points 0 p j. Then will the Lines ub,c d, cf, h g, k i, be the feveral Ordinates of the Parabola that pailes through its PerpendicB�ar, at its divided Points, 1 2 3 4 ; and therefore making 5 /, 57, cacn Cm ual to 0 a, cr 0 b, in Fig. I. alfo 4 z, 4 », each equal to c «, or n d, alio li, 3 t, each equal to p e arpf, alfo 2 , 2 s, each equal to h a* or q g, and from the'Point /, in Fig. H. through the Points r.j ,x % 1 ( u, to q, trace the Curve of the Parabola required. Kelt,

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		O/ GEOMETRY, 8₇ Note, It is to be obferved, that to defcribe the upper Part of the Curve with �xa&nefs, it is neceflary to find the Points r and iv, as following; divide bp, on & e, in Fig. G. in two equal Parts in o, and draw o r parallel to x d, alfo divide ■x. 2, on the Line x d, in Fig. H. into two equal Parts at i, and draw wr, pa- rallel loxs; on », with the Radius /�> m, in f/j-. G, defcribe the Circle k i, and from the Point o dravv the Line o � /, parallel to x d, cutting / »; in the Point r, make i r, I <w, in -F/f. H, each equal to k r, and through the Points r iv, trace the Curve. By the fame Method you may find more Points if required. Prob. XXI. Fig. K,L,U. Plate V. To defcribe an "Hyperbola. Note, When a Cone has a Section cut parallel to its Axis, the curved Boundary of the Figure, made by the Section, is called an Hyperbola. Let acbbe the given Cone, and dn the Perpendicular of the given Section. Operation. Biieft the Bale cb in t. Continue the Axis a t, downwards at Plea- sure, as to m, in Fig. M. and in any Part thereof, as at�, drawj* parallel tor b, and make y m equal to dn. Divide dn and c m, each into the fame Number of equal Parts, as atx �'g e, and 1234. From the Points e xfg, draw Right Lines parallel to c b, cutting the Side of the Cone in the Poins / i/»2. Make 5 » equal to ct, and through the Point », dravv the Line 0 t, parallel to vy, and equal to c b ; on the Point n, with the Radius c t, defcribe the Circle 0 5 t m_ralfo with the Radius's ms, I r, k q, and i p, defcribe the Circles p q r s. Con- tinue d », the Perpendicular of the Section parallel to the Axis a m, interfeftin"- the feveral Circles in the Points abc de g b ik I. Through the divided Points � 2 3 4, in the Line»2 5, Fig. L. draw Right Lines parallel loy.x, to the right and left at Plcafure. Make c.y and 5 x, in Fig. L. each equal to fa, ax f I, ia Fig. M. alfo make 4, 8 ; 4, 1 2 ; each equal tofb, orfl, alfo make 3, 7 ; 3, II ; each equal to f c, orfi, alfo make 2, 6; 2, 10 ; each equal tof a or f hj, and laftly, make 1,5; 1, 9 ; each equal to half eg, from thePointy through the Points 8, 7, 6, 5, mg, 10, 11, 12, to* trace the Hyperbola required. Prob. XXII. Fig. N. Plate V. ■Upon a given Right Line to defcribe any Polygon; from a Hexagon to a Duodecagon, Lkt an be the given Line. Operation. BifecT; the Line a n in the Point 0, whereon erec� the Perpendicular * //.'. upon the Points a and n v. ith the P�adius a n ; defcribe the Arch x », which divide into fix e-qual Parts at the Points 1,2,3,4,5 ; make x 6 equal to x », alfa x m to x J, x i to x 4, x c to x 3, x d to x 2, and xc to x I. Then will the Points xc de i me, be the Centers of the Circles, 6,7,8,9, 10, 11, 12, which, are capable of containing the given Line, fi,x, feven, eight, nine, ten, eleven, and twelve times, and therefore will be a Hexagon Septagon, Oftagon,�V. But to make this more intelligible, I will illultrate each Polygon fingly in the following Problems. Prob. XXIII. Fig. A. Plate VI. To defcribe a Pentagon, ijcbofe Sides foall be each equal to f g, a given Line. Operation. On the Points^ and/; with the Radius/^, defcribe the Arches ng, and nf; make n c; equal too », the Chord Line of one Sith part of the Arch 11 f,* and on z, with the Radius %f defcribe the Circle^ b eg f ; then making fa, a b, be, eg, g a, each equal '-a gf, draw the Lines a f, f b, f'c, and eg, which will complete the Pentagon, as required. Prob. XXIV. Fig. B. PlateVL To dtferibe a Hexagon, ivhnfe Sides flail be each equal to h <r. Operation. On the Points h and g, with the Radius hg, find the Point of Inter- �e�Uon n, whereon with the Radius n g, defcribe the Circle a b c d g �> make h a, M 2 «b.

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		�8 0/ GEOMETRY. <? b, be, cd, de, and e f, each equal to hg, and draw the Lines h a, a b, b e,. fa', «V, and <■ �, which will complete the Hexagon, as required. Prob. XXV. Fig. C. Plate VI. To defcribe a Heptagon or Septagon, ivbofe Sides Jhall be each equal to a given Line, as y f. Operation. Bifectj � in z, whereon erect the Perpendicular z 7, on the Pointy \vith the Radius^/, defcribe the Arch_y j, make sx equal to one Sixth part of the Chord Line of the Archy s, on x: with the Radius * � defcribe the Circle^ a 3 7 def, wherein from the Point_y fet the given Line y f from y to a, from a to 3, from 3 to 7, C5f. and drawing the Lines y a, a 3, 3 7, �V. they will complete the Septagon, as required. Prob. XXVI. Fig. D. Plate VI. To defcribe an Oilagon, ivbofe Sides Jhall be each equal to a given Line, as p q. Operation. Bifedi p q in 0, whereon erect the Perpendicular or, on the Point 'q, with the Radius qp, defcribe the Arch p x ; make x r, equal to xm, the Chord Line, of one Third-part of the Arch/ x ; and on r, with the Radius r p, defcribe the Circle abed efqp, wherein fet the given Line p q, from p to a, from «to b, from b to c, &c. and drawing the Lines pa, a b, be, &c. they will complete the Octagon, as required. Prob. XXVII. Fig.E. Plate VI. To defcribe a Nonagon, tub of e Sides/hall be equal to a given Line, as e f. Operation. "Rifc&efmb, whereon erect; the Perpendicular h d, on/", with the Radiusf e, defcribe the Arch ea Make a d equal to the Chord Line of half the Arch ea, as a z ; on d, with the Radius df, deicribe the Circle e t sr g men, wherein fet the given Line ef, from e to /, from / to s, from s to r, &c. and drawing the L:.ⁿ.es et, t s, sr, t$"c. they will complete the Nonagon, as re- quired. Prob. XXVIII. fig. p. Plate VI. To defcribe a Decagon, ivboje Sides Jhall be equal to a given Line, as p C. Operation. Oneand^, with the Radius e p, defcribe the Arches a p and a e, and on a erect the Perpendicular ae; make a e equal to the Chord Line of two Third-parts of the Arch a p, and on the Point e, with the Radius e p, defcribe the Circle e n g h ik lm o e, wherein fet the given Line p e, homp to », from nio g, fromg to h, &c. and drawing the Lines p n, ng, g h, &c. they com- plete the Decagon, as required. Prob. XXIX. %. G. Plate VI. To Afcribe an TJndecagon, vjhoje Sides Jhall be equal to a given Line, as ed. Operation. On the; Points e and d, with the Radius de, defcribe the Arches e a, and it a, make a g equal to the Chord Line of five Sixths of the Arch e a, on the Point 'g'; with the Radius � e', defcribe the Circle i kim, t3Y. wherein fet the given "Line e d, from e to /, ft om /' to k, is'c. and drawing the Lines e i, i k, &C they will complete the Undecagop, as required. Prop. XXX. Fig. H. Plate VI. Tt> defcribe a Duodecagon, ivhafe Sides Jhall be equal to a given Line, as g f. Operation. Makers and ad each equal to gf, and, with the Radius df, defcribe the Circle g hi k, &c. wherein fet the given Line^/i from g to h, from h toi, from ;' "0 k, t$c. and drawing the Lines g h, h i, i k, �fe. they will complete the Duodecagon, as required. Having thus �hewn the Conftruftion of each Polygon feparately, you will eafily ur.derfland how to make any Polygon from twelve to twenty-four Sides, by the following Prob. XXXI. Fig.O. Plate V. To majit c Polygon of any Nuvbsr of Sides from twelve to ty:enty-four, upon a given Line, as b c. Operatic».

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		O/ GEOMETRY. 89 Operation. Bifeft b c, in d, whereon ereft the Perpendicular d, a, 24, of Length atPleafure, on the Paint c defcribe the Arch � a, which divide into 12 equal Parts. Take as many of the 12 Parts of b a, as are Sides in the Polygon required more than 12. Suppofe, for Example, a Polygon of fix Sides; upon the Pointa, with. a Radius equal to four Parts, defcribe the Arch 12, becaufe the 12 Parts in the Arch b a, and the 4 J'et from a to 2, are equal to 16 Parts. Upon the Point z, with the Radius of 4 Parts, defcribe the Arch c 8, on the Point 8,With the Ra- dius 8 c defcribe the Circle 16, the Circumference of which will contain the given Line b c fixteen times, and thereby complete the Polygon, as required. The like is alfo to be performed for any other Polygon. Prob. XXXII. Fig. I. Plate VI. Jo maks an Equilateral Triangle, Geometrical Square, Pentagon, Hexagon, Septagon, Oclagon, Nonagon, or Decagon, ivithin a given Circle. Let i da x be the given Circle. Operation. Draw the Diameters i a and d z, at Right Angles to each other, alfo draw the Line da, which bifect in the Point 2, and from h, thro' the Point 2, dravv the Line h 2 b ; through the Point 2 drawr m, parallel to dx, or mak� a c, and am, each equal to a h, alfo draw b a ; make a f equal to ad, and draw de, divide the Arch mac into three equal Parts, and make x m equal to one of thole Parts. Then c m is the Side of an Equilateral Triangle ; da, of a Geometrical Square; de, of a Pentagon ; dh, of a Hexagon;/"^, of a Hep- tagon ; b a, of an Oclagon ; m x, of a Nonagon ; and e h, of a Decagon ; which may be made within the Circle i da z, or Circles equal thereto; as in the Circles K L M N O P, which arc equal to the Circle, Fig. I. and which contain the following Polygons, <viz. In the Circle K is a Pentagon, in L a Hexagon, in. M a Septagon, in N an Octagon, in O a Nonagon, and in P a Decagon. Prod. XXXIII. Fig. A. D. Plate VI. To defcribe any regular Polygon on a given Side, by Help of the Line of Chords, pud knowing the Quantity of Degrees contained in an Arch, >wbofe Chord Line is the Side of the given Polygon. The Number of Degrees contained in an Arch, whofe Chord Line is theSide of an Equilateral Triangle, are 120, of a Geometrical Square 90, ofa Penta. gon 72, of a Hexagon 60, of a Septagon 51 f-, of an Octagon 45-, of a Non- agon 40, of a Decagon 36, of an Undccagon 32 _T⁹_f, and of a Duodecagon 30. To prove that the aforefaid Degrees are the Quantity contained in an Arch, whofe Chord Line is the Side of a Triangle, Geometrical Square, &e. Divide 360, the Number of Degrees in a Circle, by theNumber of Sides contained in the Figure propofed, and the Quotient is the Number of Degrees contained in. the Arch of every fuch Chord Line, which is the Siderepuired. Let it be required to defcribe a Pentagon, as Fig. A. D. Operation. With 60 Degrees of your L;ne of Choids, on z defcribe the Circle a b dih, make a b, b d. d i, i h, and h a, each equal to 72 Degrees, and draw kheLines ab,b d, di,i h, and h a, they will complete the Pentagon as required. Note, If your Line of Chords fnould be of too large or too fmall a Radius, then proceed as follows, <vix. Suppofe it is required to defcribe the fmall Penta- gon p klnm. First, complete the Pentagon a b d ih, as before taught, and draw the Lines s; h, x a, x b, z d, and z i. Bfieft any Side of the Pentagon, as b d, in u : make u t and u each equal to half one Side of the given fmall Pentagon, and draw t i, and ivp, at Rigfif Angle , to a b, meeting the Lines a z and b z, in the Points f and k. Make z l, z n, z m, each equal to z k or 2; p, and drawing the Lines "Ik, kp, pm, mn, and n I, they will complete the Pentagon, as required. Examhe.II. Again, Suppofe the f an all Pentagon p k In m is given, and it is required to defcribe the large Pentagon ab'di h, with a fmall Lijie of Chords. ' FlSRT,

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		₉o Of G E O M E T R Y, First, Complete the fmall Pentagon, and from its Center draw Right Lines, through the angular Points at Pleasure. Continue any Side of the fmall Pen- tagon at both Ends at Pieafure, as the Side kp, to '.aids q and r ; bifeft kp in * : make s q, and s r, each equal to half of one Side of a large Pentagon. Draw th I inesqi, and r a, at Right Angles to qr, and continue them to meet theLines » ,,. and s: ^ in the Points a and d ; make z d, % i, and z h, each equal to fe; or z a, and draw the Lines a b, b d, di, i h, and ha, which will complete th large Pentagon, as required. Prob XXXIV. Fig.R. Plats VI. To dej'cribs any Polygon, on a gi<ven Side, having the Number of Degrees given that are contained in each Angle of the Polygon. The Number of Degree* in the Angle of a regular Pentagon are 108, in a Hexagon 120, in a Septagon I-28 f, in an Oftagon f 35, in a Nonagon 14O, in a Decagon 144, in an Dodecagon 147 f-, and in a Duodecagon 150. Let a b be the giye'n Side. Operation. On the Points a and b, with 60 Degrees of Chords, defcribe the Ardus g f and h i ; make/.; z, and» x, ea.'h equal to 90 Degrees, arid z i, and ■x f, each equal to 18 Degrees ; rhen will the Arches g/, and hi, be each equal to 108 Degrees ; through the Poir^rs f and », draw the Lines a e and b a, each tqual to � b, by Prob. XI. LECT.III. make the Angles a e m, and ba m. each equal to the Angle aba, and draw theLines e m and a m, which will meet in m, and complete the Pentagon as required. And fo the like fjr any other Polygon. The Number r ' es that are contained in the Angle of any Polygon, is found bv f irfg the Number of Degrees contained in the Arch, whofe Ci' rd . Side of the Polygon, from 108, and the Remains is the Quantity of t: - Afigle required. Prob. XXXV. Fig. Q^ PLfeV�. To find the Radius of a Circle capable to contain any Polygon, <whcfc Sides f2a.ll be each equal to a given Line, as a c. Operation. Bifect ac in b, whereon erect the Perpendicular b m ; make ah equal to a c, and on h, with the Radius h a, defcribe the Arch ad c, which di- vice into 6 equal Parts at the Points 12^34, make h n, no,hp,pg,gr,rs, s t, and t m, each equal to the Chord Line of the Arch a n, ap, a g, a r, as, at, am, a\, and draw the Lines a 0, which are the Semi-diameters of Circles that will contain all the Polygons from a Geometrical Square into a Duodecagon, viz.. the Line a 0, 'is the Radius of a Circle that will contain a Geometrical Square, 'the Line a n, the Radius for a Pentagon ; a h, for a Hexagon ; ap, for a Hep- tagon ; ag, for an Oflagon ; a r, for a Nonagon; as, for a Decagon; at, -for an Undecagon ; a m, for a Duodecagon. In the like manner any greater Number of equal Parts being fet above»!, all other Polygons of more Sides than lz may be defcribed. LECTURE V. On the irfcriling andcircumfcribing of Geometrical Figure.:. Prob. I. Fig. T. Plate VI. To injeribe a Circle, as C a b, in any Right-lined Triangle, as i 1 k. Operation, By Prob. XI. LECT. III. divide any two Angles of the Tri- angle by Perpendiculars, as id and A e, interfefting each other in/; from whence, by Prob. Vill. LECT. III. let fall a Perpendicular, as / ,t_son/ ; with the Radius � a, defcribe the Circle abc, which will touch the Sides 11 and It I, in the Points of Contact b and c, and therefore is inferibed, as re- quired. Prob. II. Fig. S. Blote VI. To hfcripe cJCtrac. as v. ! rr, '., tnHbin a GeometricalSfnar/f a..<* bead. Operaties-

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		Of G E O M E T R Y. ₉t Operation. Draw the diagonal Lines bd, and a e, from the Center h ; let fall the Perpendicular he; on the Center h, with the Radius h e, defcribe the Circle n I me, which will touch the Sides in the Points n l m e, and therefore ia inicribed, as required. Prob. III. Fig. V. and W. Plate VI. To inferibe a Circle, as h k 1 i g, tuithin any regular Polygon, as the Pentagon a b c d f. Operation. Let fall a Perpendicular from the Center d, to any Side, as dg, on f e ; with the Radius dg defcribe a Circle, which will touch the Sides of the Pentagon, in the Points of Contaft, hi lig, and therefore is inferibed, as re- quired. Fig. W. is a fecend Example of a Hexagon, which hath a Circle inferibed within it, in the fame manner. P^ob. IV. Fig. X. PlateVh To inferibe a Geometrical Square, as e f � z, within any Right-lined Triangles, as a b c. Operation. On the Point c ereft the Perpendicular c x, equal to c b. From the angular Point a, draw a g, parallel to x c, meeting the Baie bc mg. Drawx^, cutting a c in/'j draw � z, parallel to ag ; alfo fe, parallel to b c ; and e d, parallel to f a ; then will efdz be a Geometrical Square, inferibed withiu the Triangle abc, as required. Prob. V. Fig.Y. Plate VI. To inferibe an Equilateral Triangle, as a b e, in aGeomciricalSquare, as C a d g. Operation. Draw the Diagonal a g, which bifecf in n. On n, with the Radius na, defcribe the Circle ca dg; ong, with the Radi�s�js, defcribe the Arch, b nf. Draw Right Lines from a to h, and to/, which wilt interfectthe Sides of the Square eg and dg. in the Points b and e. Draw the Line be, and the Tri- angle ab e will be equilateral and inicribed, as required. Prob. V. Fig. A. D. Plate V�. To inferibe as Equilateral 'Triangle, <ubeg, &i�tbih a Regular Pentagon, as a b d i b.* Operation. Bifeft any Side, as b i, in two, and er'ecl the Perpendicular z b %_ alfo divide die Angle a hi ir.to two equal Parts, by the Line h %, cutting h z in «, the Center of the Pentagon. On b. with the Radius b z, defcribe the Arch x zc; divide the Arches x k and a; c, each into two equal Parts, in the Points o and m, through which draw the Lines boe and b m g ; alfo draw the Line eg, then will b eg be the Equilateral Triangle inferibed, as required. Peob. VII. Fig. A. PlateVIL 'To inferibe a Regular Pentagon, as n d e h k, <wi/h an Equilateral Triangle, as ai v. , , Operation. Let fall the Perpendicular a k, on v ; with the Radius it i, defcribe the Arch/;jo, at Pleafure. Draw f p, perpendicular to it i,. cutting, the Arch i t s o in p. Divide the Arch i pinto 5 equal Pares, and make po equal to one Part, and draw the Lines a 0 and it 0. bifeft it 0 in /, and draw the Line Ik, continued to/; make it a equal to if, and draw the Line a k, cutting the Line ao'mh. Make k n equal to k h. Make nd and ht, each equal toi, arid then drawing the Lines d », he,* and etc, the Pentagon 3 deb k will be inferibed Within the Triangle i a it, as required. Pros: VIII. Fig. C. Plate VII. To inferibe a Geometrical Square, ajcb h f, ittithin a Pentagon, as d a e n g. Operation. Draw the Line de and (i at right An«les thereto. Make c k equal to f d, and draw the Line«�, which will interfecl eg, the Side of tha Pentagon

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		₉2 0/ GEOMETR Y. Pentagon in/. Draw fib parallel to n g. On the Points / and h, ereft t'-e Perpendiculars � � and be, meeting the Sides of the Pentagon a e and ad, in the Points c and b. Draw f b, and f � � f will be the Geometrical Square inferibed,, as required. Prob.IX. Fig. B. PlateVU. To find the Sides of a Penta-Decagon, or Regular Polygon, of 15 Sides, ixihich may be inferibed in a given Circle. Let cab fin be the given Circle. Operation. Ey Prob. XXXII. L E C T. IV. inferibe the Equilateral Tri- angle a d g, and Pentagon c abfn, fo that one Angle of each Figure meet fa the Point a : then will fig, or n d, be one Third-part offib, or n c ; and as fib and n c, are each one Fifth-part, therefore n d and � g are each one Fifteenth- part, as required. Prob.X. Fig. G. PlateVtt. To circumfcribe a Circle, as a b c e, about a Geometrical Square, ivabce, Operation. Draw the Diagonals, and on the Center d, with the Radius a d, defcribe the Circle a b c e, as required. Prob. XI. Tig. E. Plate VII. To circumfcribe a Geometrical Square, as abed, about a gi-vin Circle, as g f i e. Operation. Draw two Diameters at Right Angles to each other, as fie and g i. Through the Points f e, draw the Lines ab inded, parallel to g i; alio thro' the Points � and /'. draw the Lines a c and b d, parallel tafe, which will meet each other in the Points abed, and form the Georaetiical Square, circumfcrib- jng the Circle, as required. Prob. XII. tig. F. Plate V�t. To circumfcribe a Pentagon, as c b a e d, about a Circle, as x W h f g, and a Circle about a Pentagon. Operation. Firft, by Prob. XXXII. LECT. IV. defcribe the Pentagon th a e d, within the given Circle, and bifeft its Sides in the Points x mi b fig, to* which, from the Center z, draw Right Lines to meet the given Circle in the Points d cb ae. Draw the Lines d c, c b, b c-, a e, and e d, and they will form the circumfcribing Pentagon, as required. Secondly, Bife�t any two Sides, 3s a b and b c, in the Points h and <w, frorn which draw two Right Lines at Right Angles to thole Sides, which will inter- fect each other in <s, the Center of the Pentagon, whereon, with the Radius z a, defcribe the circumfcribing Circle c b a e d, as required. Prob. XIII. fig. C. Plate VI. To inferibe any Polygon ivithin any Circles Let it be required to inferibe the Septagon a 3 7 defy. General Rule. Draw the two Diameters a b and 7 c, at Right Angles, dividing the Circle into four Quadrants. Divide any of thefe Quadrants into the fame Number �f equal Parts as there are Sides in the given Polygon ; then four of thofe Parts wilt be the Side of the Polygon that may be inferibed, as required : fo here the Arch sc 7, being divided into 7 equal Parts, the Side 3 7 contains 4 Parts. Prob. XIV. Fig. D. Plate VU. To circumfcribe any regular Polygon about another Polygon of the fame Kind. Let it be required to circumfcribe th� ijexagori gcal.i x f,-about the H«ga~' gon db m k of. Operation

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		O/ GEOMETRY. 93 Operation, Draw the diagonal Lines d k, b h, m/, to which draw right Lines at right Angles, e c, c a, a I, and x e, which by "their meeting in the Points ecu I i x, will conftitute the circumfcribing Polygon, as required. Prob. XV. Fig. H, Plate VU. To circumfcribe a Pentagon, as o a c y 7., about a geometrical Square, as 1 5 V w. Operation. Continue the Side <w 5 towards d ; bife�t 5 / in i, erect the Per- pendicular i b on the Points 1» and w, with the Radius 5 ;, defcribe the Arches q r and s t, at pleafure. On the Point 5, with the Radius 5 i, defcribe the Arch�V; which divide into 5 equal Parts, at the Points b g f e. Make the Angles i 5 a, and 1 / a, each equal to two Parts of i d. Make the Arches q r, and s t, each equal to one Part, and continue the Line nv r, towards a and y; alfo v t towards m and z ; alfo a 5 towards b, and a 1 towards p, which will interfecl: each other in the Points 0 and c. Make c y, and 0 z, each equal to a c, and draw zy, which will complete the circumfcribing Pentagon oacy z, as required. Prob. XVI. Fig I. Plate VII. To circumfcribe a Pentagon, «jfaorv, about an equilateral Triangle, a! a k p. Operation. On the angular Points a if, with any Radius defcribe Arches, as qxo, lb/, and e db. Divide the Arch dc into 5 equal Parts. Make the Arch c b equal to four Parts of d c. Through the Point b draw the Line a b 0 ax. pleafure. Make the Arch g e equal to, the Arch c b, and through e draw the Line a/, at pleafure. Make the Arch s x 0, and h f, each equal to the Arch b d, and from the Points k and p, through the Points/ and 0, draw Right Lines both Ways at pleafur� ; which will meet the Lines a 0, and a f, in the Points 0 and/". Make 0 r, and � a.', each equal 10a f, or a o-, and join v r, then will fa 0 r <v, be the circumfcribing Pentagon, as required. Prob. XVIL Fig. Z, and A B. Plate VI. To circumfcribe a geometrical Square about any Scalenum, or Ififceks Triangle.* This may be done two Ways. Let e n b, Fig. Z, be a Scalenum Triangle given. Operation I. Continue the Side e n towards d, and through the angular Point b draw the right Line a c, parallel to e d. On e ereft the Perpendicular e' a, to meet the Linea c, in the Point a. Make a c, and e d, each equal to a e, and draw c d, which will complete the circumfcribing geometrical Square, as required. Operation II. Fig. A B. Draw c a through the angular Point/;, and parallel to the Side x n. From the Points » and x let fall Perpendiculars to the Line a c. Make c m, and a b, each equal to. c a, as required, which will complete the. circumfcribing geometrical Square, as required. L E C T. VI. Of proportional Lines. Prob. I. Fig. N. Plate VIL To find a mean proportional Line, between two given Lines. A Mean proportional Line, is that which being multiplied into itfelf, its Product is equal to the Product of the two given Lines multiplied into each other, or it is the Side of a geometrical Square, whofe Area is equal to the Area of a Parallelogram,, whofe Length and Breadth is ecjU,al to the two,given Lines. Let d and g be the two. given Lines. Operation. Draw a right Line, as a c, at Pleafure, make a b equal to the Line r d, and b c equal to the Line e. Bifcft a c ia x, and defcribe the Semi- N, circle

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		$4. Of GEOMETRY. circle ah c: on b erect the Perpendicular h b, which is the mean proportional Line required. Prob. II.. Pig. O. Plate VII. To cut from a given Line, a Part that Jhall be a mean Proportional between �what remains, and a Line propofed, as the Line n. Let n be the given Line, and m the Line propofed. Operation. Draw a right Line, as a g, at pleafure ; make a e equal to the Line?;, and eg equal to m, Bifect a g in r, and on r defcribe the Semicircle ax g; and on e erect the Perpendicular e x. Bifect e g in h, make h c equal to h x, then c e, the Part cut off from a e, equal to the given Line n, is a mean Pro- portional between c a, the Part remaining, and m, the Line propofed. For making I i, in Fig. Q^equal to a c, and i k equal to m ; and the the Semi circle khl being defcribed, the Perpendicular; h (which by the laft Prob. is a mean Proportional to the Lines k i, and i I) will be equal to c e, the Part cut off. Prob. III. Pig. P. Plate Vll. Tvoa Lines being counseled into one Line, and their mean Proportion f pirate, being givoi, to find the Lengths of the given Lines, vohich are called Extremes. Let a f be the given Extremes, connected together without Diitinction, and the Line d, the mean Proportional. Operation. Bifeft « f in b ; on b defcribe the Semi-circle age; on c erect the Perpendiculars/, equal to the Line d; draw ig parallel to a c, cutting the Semi-circle in g. Draw g h parallel to i c, which will divide a c in h ; then is a b, and h c, the two extreme Lines required ; for by Prob. I. h g is a mean Proportional to ah and h c, and is equal to the Line d alfo. Prob. IV. Fig. R. Plate VII. Tvjo right Lines being given, to find a third Proportional. Let k and m be two given Lines. Operation. Make an Angle at pleafure, as d n e. Make n /equal to k, and nh and / a each equal to m, and draw the Line/ h; alfo draw the Linea/, parallel to hf; then will a i be third Proportional required. Prob. V. Kg. 8. Plate VII. Three right Lines being being given, to find a fourth Proportional. Let the Lines i, 2, 3 be three given Lines, and 'tis required to find a fourth, which will be to 3, the third, exactly the fame, as 2, the f�cond, is to the firft. Operation. Make an Angle at pleafure, asngh, make g /equal to the Line I, and g i equal to the Line 2, and � n equal to the Line 3. Draw if, and parallel thereto, the Line n m ; then will i m be the fourth Proportional required ; for i m is to i g, the fame as «/is to � g, and therefore m i is to nfi exattly the fame as i g is to f g. Note, This Problem is nothing more than the Golden Rule, or Rule of Three, geo- metrically pe-fornied. Prob. VI. Kg.'t. Plaie Y H. The mean of three Proportionals, and the Difference of the Extremes being given, ta find the Extremes. Let b c be the mean Proportional, and g e the Difference of the Extremes. Operation. On e erect the Perpendicular e d, of Length equal to b c Bifeft geixih; oni, with the Radius h d, defcribe the Semi-circle k da; and then i 1, and «a, are the Extremes required.

			Paoa.

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		Of GEOMETRY. 95 Pros. VII. Fig. V. Plate VII. To find the Extremes b and f, having tvoo mean Proportionals, as the Lines g and h given. Let the given Line g be equal to 8, and the Line h equal to 4. Operation. Draw a c at pleafure, and on a ere�t the Perpendiculars 6, which, make equal to 8 the given Line^. Make a c equal to twice a 6, and draw the Line bee out at pleafure. Draw c (/perpendicular to a c, and of Length at pleafure ; to which draw a parallel Line, at the Diftance of the given Line h, which will cut the Line 6 c, in the Point e ; from which Point draw the Line e d n parallel to a c, cutting the Line c d in d; then the Lines a c, and c d, equal to the Lines h and/", are the two Extremes required ; for a c equal to 1 6, and c (/equal to 2, multiplied into each other, produce 32, the fame as a 6, equal to 8, multiplied into d e 4, equal to 32 alfo. Prob. VIII. Fig.V. Plate Mil. To find the tvoo Means g and h, having the tvjo Extremes b and i given. Operation. Draw a c equal to the given Length of the Line h, fuppofe I 6, and ereft the Perpendiculars a 6, and c d. Make c d equal to the given Length. oftheLine/, fuppofe 2. Makea�, equal to half a c, and draw the Line 6 c e, of Length, at pleafure. Through the Point d draw the Line n e, parallel to a c, cutting the Line 6 c in e ; then a h equal t� the Line g, and d e equal to the Line h, are the two Means required. Prob. IX. Fig. W. Plate VII. To cut t-zvo Lines, each into tvjo Parts, Jo as that the fiour Segments may he proportional. Let h and q be the two given Line:. Operation. Make a right Angle at pleafure, as azx. Makex « equal to h, and a z equal to q ; and draw the Line a x. Bifeft z x in g, and on g defcribe the Semi circle x c z. From the Point c d draw the Line c h, parallel to 2; x, and c y parallel to a z. Then willij be to y c, as y c is to c b, and^fwillbe to cb, the fame as c b is tob a. Prob. X. Fig. X. Plate VII. To divide a right Line into extreme and mean Proportion. Let abbe the given Line. A Line is ("aid to be divided into extreme and mean Proportion, when the Area produced by the whole Line multiplied into one of its Parts, is equal to the Area produced by the other Part multiplied into itfelf. Operation. Erect the Perpendiculars d, and produce it towards c. Make a c equal to half a b. Make c d equal to c h, and a e equal to a d ; then will the Lines b be divided at e, in extreme and mean Proportion, as required. Demonfiration. Complete the Parallelogram c da b, and draw the Diagonal c a. Make b h equal to b e, and draw b g parallel to b a : alfo from e draw ^/"pa- rallel to c b. Now the Parallelogram h gb a, whofe Length is equal to a b the given Line, and Breadth h b to be, one of the Parts of the given Line, is equal to the geometrical Square dfia e, whofe Sides are each eqnal to* e, the other Part or the given Line. For as the Diagonal c a, divides the Parallelogram cdab, into two equal Parts, arid as the oppofite Triangles, on each Side the Diagonal, are each equal to its oppofite, therefore the Parallelogram g fi mull be equal to the geometrical Square* h; and therefore, if to the Parallelogram f g, we add the Parallelogram g fi, which together make the geometrical Square dfia e, it will be equal to the Parallelogram g h a b, which is the geometrical Square e h, added to the Parallelogram g e ; bec�uf� in both thefe Equalities, the Parallelogram g e is common, as well to the Parallelogram g fi. as to the geo- »etrical Square eh. -, N a Prob.

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		96 �f GEOMETR Y. Prdb. XL Fig. Y. Plate VU. To divide a given Line in any katia or Proportion required. Let i a be a given Line to be divided according to the Proportion of the given Lines k l m n. Operation. From one End of the given Line, as a, draw aright Line, zsae, making any Angle at pleafure. And thereon make a b equal to k, b c equal to /, c c equal to m, de equal to n, and draw the Line e i. From the Points deb, draw the Lines d h, g, and b J, parallel to e i, which will divide the given Line i a, as required. ' Prob. XTI. Fig. Z. PlateVlt. To make upon a give» right Line, two Parallelograms that /hall be in any given Ratio, or Proportion to another. Let b a be the given Line, upon which 'tis required to make two Parallelo- grams, which (hall be to one another as the Line x to the Line z. Operation. From the'Point b, draw the Liri'e� d, making any Angle at plea* fure, and thereon make c b equal to the Line x, and c «/equal to Hie Line z, and draw the Line a d, a!fo draw c e parallel to a d ; then will the Parts b e, and<?n, the Parts of the given Line, be to each other, as the Line x is to the Line z ; and Parallelograms made thereon of any equal Heights, as bf, e a, and g h, be, will be to one another, as the given Line x is to the Line x. Prob. XIIL Fig. A B. Plate VII. The Difference between the Side and Diagonal of a geometrical Square being given, to find the Side of the Square. Let b a be the given Difference. Erect the Perpendicular b c equal to the Difference b a, and draw the Line ac, continued towards d; make c «/equal to c b; then will a d be the Side of the Square required. Prob. XIV. Fig. �. Plate VIII. To cut from a Line any Part required. 'Tis required to cut off two ninth-Parts of the given Line be. Operation. Make an Angle as e a h, at pleafure, and on any Side thereof as on a e, iet off nine any equal Parts, as from a to d, make a h equal to b c, and draw the Line d h ; alfo at two Parts from the Point d draw the Line g, parallel to d h, then will g h be equal 'to two ninth Parts of a h (which is equal to be) as required. Prob. XV. Fig. II. Plate VIII. From a given Point without a Circle aie, to draw a chord Line as l n, in a given Circle, that pall be equal to a given Line, as a b. Operation. Affume any Point in the Circumference as g, and thereon with the Length of the given Linea/;, make the SedYion /, and from g through /draw the Line � Io, of Length at pleafure. On the Center c with the Radius c e de- fcribe the Arch ep, on the Points with the Radius p g, defcribe the Arch mi, cutting the Circle in n and d. Draw the Lines d e and e n, cutting the Circle in h and i; then will either of the Lines d h, or n i, be a Chord Line equal to the given Line a b, as required. Prob. XVI. Fig. IV. Plate VIII. To defcribe a Part or Portion of a Circle, capable of containing an Angle equal to an Angle given, upon a given Line. Let g h k be the given Angle, and f e the given Line. Operation. Make the Angle �<; i equal to the given Angle g h k ; at e on the Line i b er�tt the Perpendicular e b, bifeft the Line e f \v\ g, and erect the Per- pendicular g- «/, cutting the lane b find; whereon with the Radius d e, defcribe the Portion of a Cjrjple fh. " <% then all the Angles that can be made in this Seg- ment as, e cf, fa e, &c, wiil be equal to the given Angle g h k.

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		Of ARCHITECTURE.
		Of ARCHITECTURE.	97
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	Prob. XVII. Fig. IIP. Plate VIII. 'To cut off a Segment of a Circle, capable of containing an Angle equal to an Angle _t given. Let d cb abe z given Circle, from which a Part is to be taken, that fhall contain the given Angle q p f. Operation. Draw the Semi-diameter g e, and Tangent Line be, make the Angle deb equal to the given Anglej/i/, cutting the Circle in d. Then is d b c a e, the Segment required, and all Angles made therein, asdce, dbe, &c. will be equal to the given Angle q f f, as required.. Prob. XVIII, Fig. VI. Plate VIII.' To defcribe afpiral Line, at any given Diftance. Let a b be the given Diftance. Operation. Firft draw a right Line, as h h, at pleafure, and affume a Point therein, as d, at pleafure. Make d c and de each equal to halfa b, and on d defcribe the Semi circle c e, on the Point f defcribe the Semi-circle ef, and on d the Semi-circle / i ; again, on the Point c defcribe the Semi-circle i g, and on d the Semi circle g k. In like manner on the Points c and (/defcribe as many other Revolutions as may be required. Secondly, fpiral Lines may be defcribed con- centrick to each other, as in Fig. p h, next below Fig. VI. as follows. Let q r be the given Diftance. Operation. Draw a right Line, as p h, and therein affume two Points, as a and I, whofe Diftance muft be equal to the given Diftance qr; on. the Point a defcribe the Semi circle b i, and on b the Senii-circles a c, and /' d; then on the Point « defcribe the Semi-circles c i, and d 1, and on the Point b the Semi-circles � e, and If. Proceed in like manner, as in the laft Problem, to make as many other- Revolutions as may be required. Prob. XIX. Fig. V. Plate VIII. To dffcribe an Artinatural Line. Operation. Firft trace by Hand the feveral Curvatures or Turnings at pleafure, which divide into as many Parts as feem each to be the Segment of a Circle, as e c a, it h g, &c. This done, in each Arch affume 3 Points, as e c a, and n b g, and then by Prob. XIX. LECT. III. find the Centers � and m, and defcribe the Curves e c a, and n kbgo. In the like manner proceed throughout the whole, to defcribe all the various Meanders remaining, which will appear with the utmolt Beauty. Serpentine Rivers, and Walks through Wilderneffes, &c. being laid out in this Manner, are the neareft to Nature, and the molt agreeable of all others.

PART III. Of Architecture. LECTURE I. Of the Defcription and Conftruilion of Moldings. THE feveral Members or Moldings of which the five Orders are compofed, are of three Kinds, <oi». fquare, circular, and compound. Firfl, Square Members are Plinths, Fillets, Dado's, Cinctures, Annulets, Aba- cus's, Fafcia's, and Tenia's of Architraves, Freezes, Denticules, Dentuls, and Regula's. Secondly, Circular Members are Beads, Torus's, Aftragals, Ovolo's, Cavetto's. and Apophyges. Thirdly, Compound Members are thofe which are compofed of two or more Arches, as Scotia's, Cyma Recla's, Cyma Reverfa's, Plancers of Modilions, Wf. As fquare Members are nothing more than Parallelograms, I need not fay

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		₉� O/ ARCHITECTURE. fay any Thing of their Conitruclions, and therefore I (hall proceed to fingle and compound Moldings, and give the Etymology of fquare Members as they come in their Order. Prob I. Fig. B. Plate VIII. To dejcribe a Torus. Let <w x be the given Height. Operation. Draw x r, at pleafure, and the Line w parallel thereto, at the Di- itance of the given Height; in any Part, as at», erect the Perpendicular n a, make n c equal to half the given Height, and on c, with the Radius n c, defcribe the Torus required. This Member is called a Torus from the Greek Toros, a Cable, which its Swelling refembles, or rather from the Latin Torus, a Bed or Culhion, becaufe it feems to fwell by the impofed Weight. It is generally placed on a Zocolo or Plinth, D, which is to called, from Plinthos, a fquare Brick or Table, placed the Very lowermoft of all, to preferve the Foot of the Column from rotting ; for ori- ginally Columns were made of the tapering Bodies of Trees. Prob.1I. Fig.Q. Plate VIII. To defcribe an Aflragal ixiith its FilLt. LET<//be the given Height. Operation. Draw fz, at pleafure, and in any Part, as at/, ereft the Perpen- dicular///, equal to the given Height/V, which divide in 3 equal Parts at c and a, through the Points d a e, draw the Lines du, a c,* and e x, parallel to �* ; make/ h, and f g, each equal to e f. On a defcribe the Semi circle de, and on� the Quadrant/ k, which will complete the Aitragal as required. This Member is called an Aftragal from the Greek Aftragalos, the Bone (or more properly the Curvature) of the Heel, and for which Reaibn the French call it Talon, either^of which I think is very proper, when employed in a Pedeflal or Bafe of a Column, but not when placed on the Shaft of a Column, when it does the Office of a Collar, and is therefore by many called Collarino. Prob. III. Fig. O. Plate VIII. To defcribe the Apophyges of a Pilafier or Column. The Apophyges of a Column or Pilafter is that curved Part of the Shaft, which rifes or flies from the Cinfture, and ends in the Upright of the Shaft, as the Arch 6 d; it is alfo by fome Mailers ufed at the lower Part of the Corin- thian Freeze, and of the Dado of a Pedeftal. This Member takes its Name from the Greek Word 'Aro^vyii, becaufe in that Part the Column feems to emerge and fly from its Bafe. In the Tufcan Order, this Member is nothing more than a Quadrant, as ha. Fig. B, whofe Height is equal to its Projection, but in all other Orders it is not lb, and is thus deicribed. Operation. Divide the Projection of the Cinfture e d, Fig. O. before the upright of the Column into c equal Parts, make its Height e b equal to fix of thofe Parts ; draw a b parallel to e d, alio draw b d, which biicct in g, whereon eredt the Perpendicular g a, cutting ba in a ; on « defcribe the Arch b d, the Apophyges required. Note, thefameKuleis tobeobferved in describing the Hollow under the Fillet of the Collarino, at the Top of a Shaft of a Column in everv of the Orders. Prob. IV. Fig. F and G. Plate VIII." To defcribe an Gvslo of any given Height. Let a c, Fig. F. be the given Height. Operation. Firft, draw c d at pleafure, on any Point, as c, e.-cft the Perpen- dicular c a equal to the given Height, through the Pointa draw b e, parallel to d e, on a,, with the Radius a c defcribe the Arch c b ; which is the Ovolo re- quired. Secondly, Let I c, Fig. G, be the given Height. Operation. Divide the given Heigh; into 4 equal Part?, and give 3 of thofe Parts to the Projection. Draw the Lines 3 r, which bifect in d, on which erect the Perpendicular d a, on a defcribe the Arch c 'y. which is the Ovolo required. Ti-ilF

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		0/ ARCHITECTURE. 99 This Member is calred an 0<volo, from the Latin Ovum, an Egg, which 'tis generally carved into, intermixed with Darts and other Devices, fymbolizing Love, &c. It is alfo called Ecbinos or Echinus, from the Greek, as being fome- thing like the thorny Hufk of a Chefnut, which being opened, difeovers a Kind of Oval Kernel, fometing dented a little at the Top, which the Latins call De- cacuminata Ova, and Workmen Quarter Round. P. I remembei that in the laj� Problem you vjas fpeaking of the Apophyges taking its Rife from the Cin�iure, pray tvhat is a Cinilure ? M. A Cinfture is the firft Part of a Shaft of a Column, as a to, in Fig. B. f/at e VIII. which always is placed on the Bafe of every Column, and anciently was nothing more than a broad Iron Ferril or Hoop, to confine and ftrengthem the lowermoft Part of the Shaft, which the Italians call Liftello, or Girdle. The Shaft of a Column is that round plain Part, which is contained between the Bafe and the Capital, of which I Ihall give you a more full Acconnt, when I come to treat of the Parts of an Order. It is alfo called Fuji from the Latin Fuftis, a Club ; Vitruvius calls it Scopus, and by fome Matters 'tis called, Vivo, Fige, and Trunk. Prob.V. Fig. DandE. Plate VIII. To dejcribe a Cavetto of any given Height. Let a c, Fig. D, be the given Height. Operation. Firfi, Draw f � at Pleafure, and in any Part thereof, as at c, ereft the Perpendicular c a, equal to the given Height, and through the Point a draw the Line b g, parallel to e f; make c e equal to c a, and on e with the Diftance e c, defcribe the Cavetto b c, as required. Note, If'tis required to make a Fillet on the Cavetto, as b n, than the given Height mud be divided into 4 equal Parts, and the Fillet made equal to one Part. The Projection of iis under Part c dis equal to one 8th of the whole Height, which is half of h d, or of one Part. This Member is called Cavetto, from the LatinCavus, a Hollow, and Work- men call this Member a Hollow alfo, though I believe not with Refpecl to the Latin, but becaule it is a real Hollow, and as an Ovolo is generally made a Qua- drant, they therefore call that Member a quarter Round. To defcribe a Cavetto a f�cond Way. Secondly, Let hy, Fig. E, be the given Height. Operation, Divide h y into 5 equal Parts, and give the upper 1 to the Fillet make the Projection 1, 3, equal to 4 Parts, and^> n equal to 1 Part, and draw the Line a n parallel to h y ; continue y n out at Pleafure, and draw the Line 3 x n, which bifecl in x, and thereon erecl the Perpendicular xp. On p defcribe the Cavetto « 3, as required. Prob. VI. Fig. H. Plate VIII. To defcribe a Bed Moulding of any Height required. Let a x be the given Height. Operation. Divide the given Height into 8 equal Parts, give 3 to the Cavetto I to the Fillet, and 4 to the Ovolo, and then by Problems IV. and V. defcribe their Carves as required. Prob. VII. Fig. I. PlateMlW. To defcribe a Cymatium, of any given Height. Let a g be the given Height. Operation. Divide the given Height into 4 equal Parts, as at 4 h, and give the upper 1 to the Height of the R�gula. Draw right Lines from the Points 4, 3, and h, at right Angles to the Line 4 b, of Length at pleafure, and draw a g at any Dillance from 4 h, and parallel thereto make n c equal to n g, and draw the Line � g, which bifeft in e, on e c, and e g, make the equilateral Sections d and/, whereon defcribe the Arches � e and e g, which completes the Cymatium, as required. This Member with its R�gula is called Cymatium, from the Greek Kvy.cl.Ttov, Vndula, a rolling Wave, which it refembles, or Kymathn, a Wave. Vitruvius calls it

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		ioo Of ARCHITECTURE. it Epiclheates, and the Italians and Trench, Gola, Geule, or Douane. But when we ipeak of this Molding fingly, without its R�gula or Fillet, we call it a Cyma reMa, and Workmen oftentimes call it a Fore Ogee, to diftinguiih it from Cyma inverfa, which they call a Back Oyee, Prob. VIII. Fig.K. PlateVlU. To defcribe a Cyma inverfa, as b r, of any given Height. Op�ration. Draw the Line n r, at pleafure, in Part as at r, erec\ the Perpen- dicular r b equal to the given Height, which divide into 4 equal Parts, and give the upper i to the Fillet. Through the Points a and b draw right Lines, z.sd b, and c o, parallel to n r, and of Length at pleafure. Make a c equal to a r, divide c a in 6 equal Parrs, and make n r, and e c, each equal to one of thofe Parts ; draw the Line <? g n, which bifedt in g, on the Points n g, and g e ; make equi- lateral Sections, and defcribe the Arches e g, and g n, which completes the Cyma inverfa, as required. Prob. IX. Fig.h. Plate VIII. To dejcrike a Jingle Cornice of any given Height. Let a b be the given Height. Operation. Firft, divide the given Height into 5 equal Parts, give the lower 1 to the Cyma Inverfa/ ; one third of the f�cond to the Fillet e, and the upper 1 to the Regular; and the remaining two Parts and \ to the Cyma Refta d._tSecondly, by Prob. VII and VIII, defcribe the Curves of the two Cyma's, and the Cornice will be completed, as required. ^ Note, That the Projection' of the Cyma Recla, and of the Cyma Inverfa, which is alfo called Cyma Reverfa, is always equal to their own Height. Prop. X. Fig. B A. Flute VIII. To divide and proportion Dentals to any given Height. Let n x be the given Height. Operation. Divide ;he giv.en Height into 8 equal Parts, give the upper one to ns, the H.ighi of the Filler, the next fix to _s <v, the Height of the Dentuls, and the lower one to y x, the Margin of the Denticule. To proportion the Breadths of the Dentuls and Intervals between them, make v q equal to s v, and dividing v q into 3 equal Parts, give two to the Breadth of a Dental, and one to its Interval, whichis called Metoche, which with two Pair of Compaflcs, the one opened to the Breadth of aDentul, and theother to the Breadth of an Interval, fet 01F thofe Distances reciprocally throughout the whole Length of your Molding. If it is required to make Eye-Dentuls in the Intervals, as A A, divide the Height of the Deritui into 5 equal Parts, and give the upper one to the Height; of the Eye-Dentul. Note, This Ornament is generally begun at the projecYing Angle, over an angular Column, with the Form of a Pine-Apple ; or rather, the Cone of a Pine-Tree, as at A g, which is thus defcribed. Make its Breadth zn equal to 'he Breadth of a Dentul, which divide in 4 equal Parts; make k g equal to n z, and draw z g ; make n d, z b, each equal to half» z ; and draw d b, which bifed i.;i e. On e, with the Radius e d, defcribe the Semi-circle dm b. On the Points df, and b �, with the Ra- dius f d, defcribe the dotted Sections next above the Line d b, on which, with the fame Opening, defcribe the Arches bf and fd, which will complete the whole, as required. These Ornaments are called Dentuls, from Dentelli, Teeth, which they re. preient. The Dcnticulus is that flat or fquare Member, on which the Dentul» are placed. Prob. XT. Fig. I i, next under Fig. A B. aforefaid. Plate VIII. To proportion and defcribe an lanick Modilion, of any Height, required. Let a b be the given Height. . Operation. Divide the Height into 8 equal Parts, as at r q, give the upper 2 to the Height of the Cyma Inverfa, with its Fillet,, and the next 5 to the 2 De^th

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		0/ARCHITECTU RE. toi Depth of the ModiJion. Draw d c, for the Side of a Front Modilion, make ce equal to c d, and df equal to de, then is df the Breadth of the Modilion in Front. Divide df into 4 equal Parts ; make f I the Projection of the Modilion in Profile, equal to 6 of thole Parts. Divide the Projection of the Modilion in Profile into 6 equal Parts, at the Points 1, 2, 3, 4, c. Through the Points 2 and 5, draw the Lines 0 m, and c , parallel to f p.* Make 5 ? equal to two Parts and half, and 2 0 equal to one Part : Alfo make 0 m equal to c t, and draw the Line m s t. On the Points m and t, with the Radius t �, defcribe the Arches 0 s and s c ; alfo on 2, with the Radius 2 1, defcribe the Arch 1 o, which will complete the Modilion, as required. This Member is called Modilion from the Italian Modigliani, a plain Support to the Corona of the Corinthian and Compofue Cornice, to which they only belong, altho'now falfly introduced into the lonick. Prob. Xir. Fig. N. and M. Plate VIII. To defcribe Scotia¹1 of any given Heights. Firft, Let ag, Fig. M. be the given Height. Operation. Draw the Linef g, and on any Part thereof, as at g, erecT: the Per- pendicular g a equal to the given Height, and thro' the Point a draw the Line « x, parallel to g f. Divide a g in 3 equal Parts, at the Points d z, and thro' the Point d draw the Line c de, parallel to ax. Make de equal to da. On the Points d defcribe the Quadrant a c ; and on the Point e the Quadrant cf which together form the Curve of the Scotia, as required. This Member is called Scotia, from theGreei ^xoTta, Siotos, Darknefs, which the upper Part caufes by its Proje�ture. 'Tis alfo by fome called Trochilusjtom. the Greek Trochilos, TpoiKa, or Tjojcs., a Rnndle or Pully, whofe hollow Part within the Rope-works hath fome Refemblance of this Member ; and with re- fpeft to its Darknefs, it is by many, th,o' improperly, called a Cavetto. The Italians call it Bajlone. This kind of Sco:ia is adapted to the Attick Bafe. Secondly, Let a d, Fig. N. be the given Height. Operation. Draw the Lines k a and n d, parallel to each other, at the Diflance of a d, and draw a d at Right Angles thereto. Divide a d in 7 equal Parts, and through c, the third Part down, draw h c, parallel to a k. Make c h, and d n each equal to a c ; and drawz' h n, parallel to a d. Make h i equal to h n, and from /' through c, draw the Line i c m. On the Point c defcribe the Arch a m and on /' the Arch m n, which completes the Scotia, as required. Prob.XIII. The Diameter, or Breadth of a Door or Window, being given, to f nd the Breadth of an Architrave that vjill be proportionable thereto. A General Rule. Divide the Diameter, or given Breadth, into 6 equal Parts, and take one for the Breadth of the Architrave required ; and that you may alfo know how to di- vide the Architrave into its proper Members, 1 have given you in Plate VUI and IX. thirty and one Kinds of Architraves, of which thofe marked ABCDEF are Tufcan, GHIKLMNOare Dorick, P QJl S T V are lonick, W X Y Z, A B, A C, are Corinthin, and A D, A E, A F, A G, and A H, are Com- pofite, which in general have the Heights of their feveral Members propor- tioned by equal Parts. As for Example. In Fi%. A. the Height or Breadth of that Architrave is divided into 10 equal Parts, of which the upper 2 and \ is the Height of the Tenia «, and the Remainder is the great Fafcia, with its Hollow. In Fig. D. the Height is divided into fix equal Parts, of which the upper 1 is the Height of the Tenia, the lower 2 the Height of the fmall Fafcia c, and the other 3 is the Height of the greatFafdai>. In the fame manner you are to understand all theothers; and as the principal Partsinto which the Height of every Example is divided are fignified by the equal Divifions and Figures again!! them, and as the Manner of defcribing all the Moldings of which they arecompofed has been already taught, to fay any thing further on the Manner of defcribing them is. O aeedlefs ;,

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		_I02 Of ARCHITECTURE. needlefs s as indeed is what I have already faid, the whole being fo very plain, as to be underftood by the meaneft Capacity at the firft View. L E C T. II. Of the making of Scales of equal Parts, for the delineating of Plans and Eleva- tions of Buildings. The neceffary Scales for our Purpofes are thofe reprefenting, firft, Feet; fe- Condly, Feet and Inches ; thirdly, Modules and Minutes ; and fourthly. Chains and Links. Thofe of Feet, and Feet and Inches, are uted in the making of Plans and Uprights, or geometrical Elevations of Buildings. Thofe of Modules and Minutes are for proportioning of the feveral Members of the five Orders of Columns in Architecture, and thofe of Chains and Links are for making Sur- veys of Lands, as Farms, Parks, is'c. whofe feveral Ufes will be fully illuf- trated in their proper Places. Pros. I. Fig. I. Plate IX. To male a Scale of Feet. Operation. Make a Parallelogram at Pleafure, as a d m e ; open your Com- pares to any fmall Diftance, and fet off lo equal Parts, from m to x h ; alfo make x b and b e, tjfc each equal to m x b e ; then will the Line m e be a Scale of equal Parts,'which may reprefent Inches, Feet, Yards, csV. and which mult be thus numbered, <w'a. as* * is equal to the 10 Parts between m x, therefore at�, place the Number 10, at «the Number 20, &c. being lo many Parts from x. To take off any Number of Feet, lefs than 10, fet one Foot of your Compaffes on x, and extend the other to the Number of Feet lequired. To take off any Number of Feet more than 10, fet one Foot of your Com- paffes in b, and extend the other to the Number of odd Feet that is contained in the given Length more than 10. Suppofe 17 was the given Length : extend your Compaffes from b to 7 Parts beyond x towards m, which is 17 Feet, as re- quired ; and fo the like of any ether Number of Feet, more than >o, 20, &c. To make a Variety of Scales of equal Parts, which is neceffary to have, as that fome Works require a leffer or a greater Scale than others ; therefore, if from the 10 equal Parts, in m x, you draw Right Lines unto the Point a, and afterwards draw Right Lines parallel to m e, at any Diftance, as � r, g q. h p, io, in and I m, you will have made other Scales of equal Parts, of various Sizes, which may fit all Purpofes required, II. To make a Scale of Feet and Inches. Fig. VI. Plate IX. Operation. Make a Parallelogram, as abed, fet off 12 fmall equal Part», from c to e, reprefenting the Inches in a Foot; make e 10, 10 20, 20 30, &c. each equal to the 12 Parts, then is your Scale of Feet and Inches completed ; for « 10, 10 20, are Feet, and the Parts in ce are Inches. To take of a Length of Feet and' Inches, is the fame here as before in the Feet : fo the Diftance of ,3 10, is 15 Inches, of 6 10, is 18 Inches, of g 10, 21 Inches. Scales of Feet'and Inches are alfo made on Two-foot Rules, as Fig. II. in manner fol- lowing, «it's. Make a Parallelogram, as c a -x I,* at Pleafure, and let the Diftance of zfhe made to represent one F.oot. Make/3, 3 1, and 1 b, on the Line %b, each equal to % f ; that is, each equal to one Foot. Draw f g, parallel to c z.. Bife<ft eg in e, and draw toe Lines e z, and ef. Divide g f in 6 Parts, at the Points Ik i h g, and draw Right Lines through them, parallel to e h. and then is the Scale completed ; and the Diftance of %f, which is the given Foot, is divided into 12 Inches, via. The Diftance of g1, is one Inch : h 2, two Inches; «3, three Inches ; It 4, four Inches ; / J, five Inches ; g 6, fix Inches ; / 7, feven Inches ; k 8, cicht Inches ; i 9, nine Inches ; h 10, ten Inches ; g 11, eleven Inches; an/1 � k, one Foot, as before. These kind of Scales may be made either bigger or lefs, at Pleafure, in the very

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		Of ARCHITECTURE. 103 very fame manner, as may be feen at the End a b, where the Foot is made but half the aforefaid. Prob. III. Fig. �V. Plate IX. To make a Scale of Chains and Link f or the flatting of Lands, &C Operation. Make a Parallelogram, is avb<w, and let the Diltance b e repre- fent one Chain, which is equal to four Statute Poles, each 16 Feet and half, or to 66 Feet. Make e d equal to e b, then de is one Chain alfo. Divide a b in- to 10 equal Parts, and through them draw Right Lines parallel to b vu. Divide af, and be, each into 10 Ptrts, and draw the diagonal Lines/"lo, hzo, &c. then your Scale is completed ; and the Diftance of I k, is one Link ; 2 /, two Links ; 3 m, three Links ; 4 n, four Links ; 14 », fourteen Links ; 19 s. nine- teen Links ; zo e, twenty Links, �V. to which one or more Chains-length may- be added, as Occafion requires. At the right Hand End, the Parallelogram / vg iv is another diagonal Scale of Chains and Links, made to half the Mag- nitude of the aforefaid. Prob. IV. Fig.lll. Plate IX. To make a Scale of Minutes, or to divide the Diameter or Module of a Column into 60 Minutes. Operation. Divide the Length of the Diameter into 10 equal Parts, as at the Points 6, iz, 18, &c. on its Ends erecl Perpendiculars, whereon fet up any 6 equal Parts, and draw Right Lines parallel to the given Diameter, which will complete a Parallelogram, as Fig III, whofe upper Side muft be divided into 10 equal Parts, as the given Diameter, as at the Points 6, 12, i8, &c. This done, draw the diagonal Lines, 6, 1 ; 12, 6,18, 12 ; which will complete the whole, and the Diftances taken from the left Hand, perpendicular to the Points i, z, 3, 4, csV. are the Minutes required. Prob. V. To make divers Scales of Chords of any Length or Radius required. Let c c, at the left Angle of Plate IX. be a given Scale of Chords, divided as before taught. Operation. Ereft the Perpendicular ca, of Length at Pleafure, and draw the HypothenufalLine ac At any Diftances from c, draw divers Right Lines pa- rallel to c c, as d d, e e, &c. Draw Right Lines from the feveral Degrees in c c, unto the Pointa, and they will divide all the intermediate parallel Lines d d, e e, &c. in the fame Proportion as the given Line of Chords c c, and confequently each of them will be a Line of Chords, as required. L E C T. III. Of the principal Parts of an Order, and of the Orders in general. An entire Order confills of three principal Parts, viz. A Pedeftal, a Column, and an Entablature. A Pedestal is the firft or lowermoft Part of an entire Order, as e h, Fig. �. PlateXIX. which conflits of three principal Parts, viz. gb its Bafe, gf its Dado, or.Die, and f e its Cornice. Its Name comes from the Greek Stylobates, the Bafe of a Column ; it is alfo called Stereobate, or Stylobate ; but, as Mr. Evelyn in his Parallel obferves, our Pedeltal h Fox Hybrida (a very Mungrel] not z Stylo, as fome imagine, but � Star/do. A Column is the f�cond principal Partof an entireOrder,as be, Fig. I. Plate XIX. which confifts of three principal Parts, alfo, viz. its Bafe de, its Shaft c d< and its Capital b c. The Bafe receives its Name from the Greek Verb (i&lvin', importing the Suftent or Feet of a Thing ; and the Capital from the Latin Capi- tellum, the Head or Top. The Architrave is called by the Greeks, EpiJHkum ; that is to fay, �/>;.upon, and Stylos a Column, which from a mungrel Compound of two Languages C^t>yjj) Trabs, as much as to fay, the principal Beam, or ra- ther from Arcus, Chief, and Trabs, a Beam, we call Architrave. The Freeze takes its Name either from the Greek Zxoip'o^, Zophorus, importing the imagi- O 2 nary

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		104 Of ARCHITECTURE. nary Circle of the Zodiack, depicted with its 12 Signs, or is derived either from the Latin Phrygio, a Border, or from the Italian Phrygio, an embroidered or fringed Belt. The Cornice receives its Name from the Latin, Coranis, a crown- ing, from whence its Fafcia is called Corona, alfo called Sufcrcilium, or rather Stillicidium, the Drip [Corona elmolata Vite) and with more Reafon it is called by the Fremh Larmier. The Italians Call it Gocdolatoio, and Ventale, from its pro- tecting the Building both from Water and Wind, and for which Reafon the Latins call it Mmtum, a Chin, becaufe its Projection carries oft the Rains from the lower Part of the Entablature, 3s the Prominency of that Part in Mens Faces prevents the Sweat of the Face from trinkling into the Neck. An Entaeiature, from the Lalin, Tabul�tum, a Cieling, and by fome called Ornament, is the third, and uppermoft Part of an entire Order, as a b, which likewil'e confifts of three principal Parts, nameiy, its Architrave, Freeze, and Cornke. The principal Parts of Pedeftals, Columns, and Entablatures, are fubdivided and proportioned in fuch Manners, that the Refults of their Comportions fhall give fuch Ufefulnefs, Grace, and Beauty, that are agreeable to the Order they are made to reprefent. The Orders in Architecture were originally but three ; <viz. Dorick, Ionick, and Corinthian, invented by the ancient Greeks ; to which two more have been fince added, called Tufcan and Compofae. The Tuscan Order, for its being the moft robuil and mafculine.'is there- fore placed before the Dorick, and the Rear of the whole is brought up with- the Compofr.e. The Tuscan Order is 10 called from the �fiatick Lydians, who are faid to have firft peopled Italy, and raifed Buildings thereof, in that Part called lufcany. This Order, for its Simplicity or 'native Plainnefs, when well per- formed, and employed at the Entrances of Cities, Magazines, and other Build- ings of Strength, is not in the leaft inferior to any of the other Orders. The general Proportions of this Order are as follow, �viz. the Height of thePedeftai is One-fifth of the whole, its Column 7 Diameters, and the Entablature One- fourth of the Column, or one Diameter 45 Minutes, as exhibited in Fig. I. Plate XfX. The Dorick Order is fo named from Dorus, King of Achajis, who, it is reported, built a magnificent Temple of this Order in the City of Argos, which he dedicated to the Goddefs Jmio, and which, Vitrwjius faith, was the very firft Model of the Kind. This Order, for its Mafculine, or rather, as Scamozzi calls it, Hercukan Afpect, with regard to its excellent Proportion, is to be employed where Strength and Grandeur is required, as at the Gates of Noblemen's Palaces, &e. "The general Proportions of this Order are as follow, <viz. The Height of the Pedeltal is One-fifth of the whole, its Column 8 Diameters, and its En- tablature One-fourth of the Column, or 2 Diameters, as exhibited in Plate XXIII. The Ionick Order is faid to have been invented by Ion, King of Ionia, a Province in Afia, who erected a Temple of this Order, and dedicated it to the Goddefs Diana ; and as this Order is a Mean between the Herculean Dorick and Feminine Corinthian Extremes, it ought therefore to be employed in Por- tico's, Frontifpieces, &�. at the Entrances into Noblemen's and Gentlemen's Houfes. The general Proportions of this Order are as follow, 'viz. The Height of its Pedeftal is One-fifth of the whole, its Column 9 Diameters, and its Enta- blature One-fifth of the Column, or 1 Diameter, 48 Minutes, as exhibited in JVartXXVHI. The Corinthian Order received its Name from the luxurious City of Corinth, where it was invented and made by Callimachus, an ingenious Sta- tuary of Athens, who took the firft Hint thereof from a Bafcet, placed on the

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		0/ ARCHITECTURE. 105 the Grave of a young Lady of Corinth, wherein the Nurfe having put her Play- Toys, according to the Cuftom of thofe Times, and covered the Bafket with a fquare Tyle, a Root of Acanthus, or Branca Urfina, Bears Foot, happened to grow under it ; which putting forth its Leaves around from under the Bafket, as in Fig. V. Plate XXXIV. they turned up the Sides, and inclofed the whole at Bottom ; whilft the Flower-ftalks, in advancing higher, were repulfed by the projecting Ty!e, and obliged to turn under it in a curved Manner. To form this Capital, he made a Vcfe or Bell, to reprefent the Balket, and about it placed fixteen Leaves, in two Heights; from which, in Imitation of the curved Flower-Stems, he fprung Stalks enriched, whofe Curvatures he finifhed with Volutes, and covered the whole with a horned Abacus of Mouldings, in Imita- tion of the Tyle. This Order being the mod rich ar,d delicate of ail the Orders, it fhould therefore be employed within Buildings, as in Rooms of State, l$c. where Magnificence and Beauty are required. The general Proportions of this Order are as follow. Its Pedeltal is One fifth of the whole Height, its Column Io Diameters, and its Entablature is equal to One-fifth of the Column, as ex- hibited in PlateXXXII. The Comfosite Order, called by fome the Roman or Italian Order, is generally made, of all others, the very worft ; for its Capital is nothing more than the lower Part of the Corinthian Capital, covered with the lonick Capi- tal for an Abacus, is much lefs elegant than the Corinthian, as its Entabla- ture is alfo ; and if to thefe be added the Lownefs of its Shaft, which has very little Diminution, and of equal Height with the Corinthian ; upon a jufl View of the whole, it will appear to be rather a Difgrace than a Credit to the Inventor, or at leaft a full Proof of a great Barrennefs of Invention : and that I may not be thought to find Fault with the Endeavours of others, and at the fame time give no better Example, I therefore, in Plate XLI. have given the Compofite Entablature, by Andrea Palladio, with a Compofite Entablature of my own Invention, for Infide Works, which I fubmit to the Judgment of the Judicious. The general Proportions of this Order are exhibited in Fig, I. Plate XXXIX. To thefe five Orders we may add many more, vtx. Firft, The Orders of the Perfians and Cariatides, as Fig. II. Ill, and IV. Plate XLI I. where the Statues of Men and Women are ufed inltead of Columns, of which the firft is crowned with a Dorick Entablature, and the lad with an lonick. Secondly, The French and Spanijh Orders, which are only different from the Corinthian in their Capitals and Enrichments of their Freezes. Thirdly, The Grotefque and Englijh Orders of my Invention, <vide Plates 302, to 310, of my ancient Mafonry. And laftly, the Gothick Order, which makes twelve Orders in the whole. L E C T. IV. Of the Manner of proportioning the particular Parts of the Tufcan Order, by ■ Modules and Minutes, according to Andrea Palladio, and by equal Parts, compofsd from the Maflers of all Nations. Prob. I. To find the Diameter, or Module of an Order, proportionable to any given Height. BEFORE an Order can be delineated, the Diameter mult be found ; and as Columns are employed in four different Manners ; <vix. Firft, alone, without either Pedeftal or Entablature. Secondly, with the Pedeftal only. Thirdly, with the Entablature only. And laftly, with both Pedeftal and En- tablature. Therefore to find the Diameter in every of thefe four Cafes, this is the Rule, -viz. Divide the given Height into the fame Number of equal Parts, as there are Minutes contained in the Height of the principal Parts that are to be employed ; and take fixty of thofe Parts for the Diameter of the Column. The Height of the Column alone, 0 q, Fig. I. Plate XIX. is 7 Diameters : therefore One-feventh of "the given Height, where the Column only is to be 2 employed,

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		io6 Of ARCHITECTURE. employed, is the Diameter required. The Height of the Pedeftal and Column» as bh, equal to ax, Fig. I. Plate XIX. is 9 Diameters, eighteen Minutes and ^, which are equal to 558 � Minutes. Now admit the given Height to be 12 Feet, reduced into Inches equal to 144, and the Inches reduced again into loths, equal to 1440. Then fay, by the Rule of Three Direft, as 558 Minutes, the Number of Minutes contained in the Height of the Pedeftal and Column (rejecting the � of a Minute) is to 60, the Minutes contained in the Diameter of the Column : So is 144a, the Tenths of an Inch, contained in the given Height of 1 2 Feet, to �ji-J-f?» which is very little more than one Quarter part of One-tenth. Now 151 Tenths of an Inch reduced, is equal to 15 Inches, One-tenth, One-fourth of a Tenth, and a very fmall Matter more, and is the Diameter required. And if 1 j Inches, One-tenth, and J of a Tenth, be divided into 60 equal Parts, omitting the fmall Matter more than the * of a Tenth (which will be near enough for Practice) they will be the Minutes of the Diameter, by which the Heights and Projections of the Order may be proportioned. In the fame Manner the Diameter may be found, when the Column and Enta- blature only are employed, whofe Height i p, Fig. I. Plate XIX. is 8 Diameters, 45 Minutes ; as alfo may the Diameter of the entire Order, whole Height ab is 11 Diameters, 3 Minutes, and ±, as expreffed on the Line ltu. This being underftood, and a Diameter being thus found and divided, the delineating of this Order is eafily performed, as follows. Prob. II. To delineate the Tufcan Pedeftal, by Modules and Minutes. Let A, Plate XIX. be a Diameter found, or given (which is alfo called x Module) and divided into 60 Minutes. Before we proceed to this Operation, it is to be obferved, that the Heights of the Members are expreffed on the central Line, to be read upwards, and their Projedlure are placed againft them, to be read level with the Eye, either on the right or left Hand Side. Operation. Firft, Draw a bafe Line, as kr, Fig. III. PlateXIX. and in any Part, as at k, ereit the Perpendicular k k. Make kf equal to 37 Minutes and , as expreffed between k and f ;* alfo make/* equal to 2 § Minutes ; e d to � Minutes ; de to one Diameter, 9 Minutes, � ; ca to 4Minutes � ; ab to 2 Minutes \|; bk to 17 Minutes \|j and thro'the Points kba c d e f, draw Right Lines to the right and left, parallel to the Bafe Line k r. Secondly, Make k r, and/s, each equal to 47 Minutes � ; and draw the Line 1 r. Make//, and ev, each equal to 45 Minutes, and draw the Line <v t. Make d =w equal to 41 Minutes. Make d x, and cy, each equal to 40 Minutes, and draw the Line y x. Make c 41 equal to 41 Minutes. Make a z, and b 45, each equal to 45 Minutes, and draw the Line 45 z. Make b r and k b, each equal to 47 Minutes and J, and draw the Line h r. Then by Prob. V. of L E C T. I. hereof, de- fcribe the Cavetto'sy z, and z <u ; and the very fame being repeated on the left Hand Side of the central Line, will complete the Pedeftal, as required. And as the Members in the Bafe and Capital of the Column, as alfo the Mem- bers in the Entablature, are all delineated in the very fame Manner, there needs no more to be faid thereof, and therefore the next Work is, How to diminifh the Shaft of this or any other Column. But before we can proceed to this Work, it muft beiobfisrved; Firft, That the Heights of the Bafes of Columns in general are all equal to half a Diameter, or 30 Minutes ; as is alfo the Height of the Tufcan and Dorick Capitals. Secondly, That the Cin&ure b, Fig. I. Plate X. and the Aflragal, or Colltrine h k, are both Parts of the Shaft. Thirdly, That fince the whole Column in the Tufcan Order, including its Bafe and Capita], is 7 Diameters high ; therefore taking the Bafe and Capital from it, which together are equal to one Diameter, the Remains, 6 Diameters, is the Height of the Shaft. Fourthly, That Columns in general are diminifhed but in the two upper Third-parts of their Height, the lower Third-part being a Cylinder, Fifthly, That the Tufcan Co- lumn

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		Of ARCHITECTURE. 107 lumn is diminifhed One-fourth of the Diameter of its cylindrical Part; the Dorick One-fifth, the lonick One-fixth, the Corinthian an�Compofttt One feventh, and therefore the Diameter of the Tit/can Column, at its Top, is but 45 Minutes, the Dorick 48 Minutes, the lonick 50 Minutes, the Corinthian and Compoftte, each 51 Minutes f. Prob. Ill, Fig. I. Plate X. To diminijh the Shaft of the Tufcan, or any other Column. Operation. Draw / h for its Height, \ of which is its Diameter. Di- vide lb into three equal Parts, at q and C 5 through the Points / C and b draw Right Lines, at Right Angles, to the Central Line lb. Make C y and C7, each equal to 30 Minutes, and/�, li, and CD, CE, each equal to 22 Minutes and a half; and draw the Lines h D, and A E, on the Point C, with the Ra- dius Cy, defcribe the Semi-circle yiv"]. Divide IC into any Number of equal Parts, fuppofe four, at the Points nqv, and through them draw the Right Lines mo, p r, and s t, of Length at Pleafure. Divide the Arches yz, and 3 7, each into as many equal Parts, as you divide the Line /C, which here is 4, as at the Points 1 z x, and 456, and draw the Ordinates 1 4, z 5, x 6. Make v s, and <v t, each equal to the half Ordinate B 6 ; alfo q p, and q r, each equal to the half Ordinate A 5 ; and n m, and « 0, each equal to the half Ordinate g 4. From the Points h k, through the Paints mp s, and 0 r t, unto the Points y 7, draw the Lines hy, and k 7, fo as not to make an Angle at any Point, and they will diminifh the upper Part of the Shaft, as required. As this Method is general for diminifhing the Shafts of all the other Orders, no more need be faid on this Snbjeft.. In PlateXX. Fig.X and II. is exhibited the particular Members of every prin- cipal Part of this Order, with their refpeclive Meafures of Heights and Projec- tions. Prob. IV. Fig. II. Plate XIX. To proportion the Heights of the principal Parts of the Tufcan Order, by equal Parts. Operation. Divide a I, the given Height, into 5 equal Parts ; the lower one g I, is the Height of the Pedeflal, and the remaining 4 Parts, ag, equal to n r, divided into 5 equal Parts, the upper one is the Height of the Entablature, and the lower 4, the Height of the Column, which being divided into 7 equal Parts, 1 is equal to its Diameter; and thus are the Heights of all the principal Parts determined. PROB.V To divide the Height of the Tufcan Pedeflal into its Bafe, Die, and Cornice, and them into their refpeiJive Members. Operation. Divide^/, Fig. II. Plate XIX. the given Height, into 4 equal Parts, as s <v, give the lower I, to the Height of the Plinth, one Third-part of the next 1, to i A, the Height of the Moldings to the Bafe, and haf the upper itQgh, the Height of the Cornice. To divide the Moldings of the Bafe and Cornice of the Tufcan Pediftal. Fig- IV. Plate XX. Operation. Firft, Divide-* 3, the Height of the Moldings on the Bafe, into 3 equal Parts ; give the upper 2 to the Cavetto, and the lower 1 to the Filler. Secondly, Divide ad, the Height of the Cornice, into three equal Parts ; alfo the upper i, be, into two Parts, and the lower 1, eg, into three Parts. Then giving the upper 1 oi b c, t� the R�gula, and the upper iof eg, to the Fillet, the two Remains will be the Plat-baud and Cavetto. To determine the Projections of theft Members. First, Make the Projection of the Dado k k, equal to half the Height of the Dado and Mouldings on the Plinth taken together, thereby forming a geo- metrical Square, as in Fig. II. Plate XIX. wherein is a Circle inferibed. Secondly, Make lie Projection of the Plinth and R�gula, before the Upright of the Dado, equal to the Height of the Cavetto and Fillet on the Plinth. Thirdly,

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		io8 Of ARCHITECTURE. Thirdly, Divide fh, tbeaforefaid Projeftion, into6 Parts, the firft l flops the two Caveito's at » and o; the third, the upper Fillet m, and the 5th the Platband and lower Filkt/. Prob. VI. To divide the Height of the Tufcan Column into its Bafe, Shaft, and Capital, and them into their refpeclive Members. Operation. Firft, Divide h g into 7 equal Parts, and take 1 for the Diameter. Make eg, and hf, each equal to half a Diameter, for the Heights of the Bafe and Capital. This done, iuppofe G q, and a c, in Fig. XX. to be the Heights cf the Bafe and Capital, as before found. To proportion the Bafe of the Tufcan Column. Divide df, equal to its Height ac, into 7 equal Parts; give 4 to the Height of the Plinth, and 3 to the Height of the Torus; alfo make e d, the Height of the Cincture, equal to 1 Part. To determine the ProjeSion of the Members of the Tufcan Bafe. Divide c 3, equal to the Semi-diameter, into 3 equal Parts, and make c 4, equal to 4 of thole Parts. Divide the Part 3 4, into 5 equal Parts, and a Line as 5 hi, being drawn from the f�cond Part, parallel to the Central Line of the Order, will cut the Central Line of the Torus in /', its Center, and ftop the Cincture at ». This being done, and the Shaft of the Column erected on th Bafe, as before taught, proceed we now To proportion the Tufcan Capital. DiviBE its Height G q, equal to A B, into 3 equal Parts. Divide the upper 1, as E F, into 4 Parts, give the upper 1 to the R�gula, and the lower 3 to the Abacus. Divide the middle 1 into 6 Parts ; give the upper 5 to the Ovolo, and lower 1 to the Fillet. The lower I is the Height of the Hypotra- ehelium, or Neck of the Capital. Novo to fnd the Projcftures of thefe Members, make g i equal to half G g, and divide k I, equal to g i, into 6 Parts ; the firft I flops the Fillet, the 4 Parts J the Ovolo, the fifth Part the Abacus. The Aftragal, to the Top of the Shaft, is thus proportioned. Make q rits Depth, equal to half k », the Height of the Necking, which divide into 3 Parts ; give 2 to the Aftragal, and 1 to the Fillet. The Projective of the Aftragal 0, is equal to m n, <t>iz. to half the Height of the Neck, which is equal to � of the whole Capital's Height, and its Fillet to J thereof. pROB. VII. To divide the Height of the Tufcan Entablature into its Architrave, Freeze, and Cornice, and them into their refpeclive Members. Operation. Divide a A, equal to its Height k G, Fig. 111. Plate XX. into 7 Partb : give 2 to the Height of the Architrave, 2 to the Height of the Freeze, and 3 to the Height of the Cornice. To divide the. Architrave, divide CD, its Height, into 6 Parts, and give the upper 1 to the Tenia, which is alfo caMe�-Diadema, a Bandk't or Fillet to bind the Head, whole Projection dc, is equal to its own Height. Continue its Face to � and b, making each equal to its Projection, and defcribe the Quadrant a c, above the Tenia, for the im- mediate carrying the Rains from it, and the other below it, to ftren<nhen its Projection. To divide the Tufcan Cornice into its Members. Its Height being before divided into 3 Parts, divide the lower 1, de, into 2 Parts, give the upper 1 to the Height of the Ovolo, and the lower 1, h f di- vide into 4ParS3 ; give the upper 1 to the Fillet, and the lower 3 to the Cavetto. 'fhefethree Members taken together, form that which Workmen call the Bed- moulding of a Cornice. Divide the upper two Parts of the Cornice into 24 equal Parts, as b c, give nine Parts and a half to the Height of a Corona, and to the Height of the Ovolo, and the Remains between them i g, being di- vided into three Parts, give z CD the Aftragal and 1 to the Fillet. The Projection of

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		0/ ARCHITECTURE. '109 of this Cornice m I is equal to its Height ; therefore make n o, againft the Freeze, equal to is whole Projection, and divide it into 3 Parts. Divide the firft Part into 8 Parts, as at p ; the firfl: I Part ftops the Projection of the Foot of the Cavetto, the 4th Part its Fillet, the 7th the Ovolo, and the 8th its Fillet next under the Corona. The middle Part being divided into 4 Parts, the third Part from the Left ftops the Drip of the Corona, and the fourth Pare the Face of the Corona. The third or outer Paft being divided into 2 Parts, and the firft 1 Part into 4 Parts, the firft 1 ftops the Fillet x, and the next i the Aftragal y ; and thus is the whole Order completed, by equal Parts, as required. Now to proportion any Part of this Order, to any given Height, thefe are the ^: Rules, viz. I. To proportion the Column and Entablature only, to any given Height, and ta find the Diameter. Rule. Divide the given Height into � equal Parts, the upper one is the Height of the Entablature, and the lower 4 of the Column, which divide into 7 Parts, and take 1 for the Diameter of the Column. II. To proportion the Pedeftal and Column only, to any given Height, and to find the Diameter. Rule. Divide the given Height into 21 equal Parts, give J to the Height of the Pedeftal, and 16 to the Column, which divide in 7 Parts, and take 1 for the Diameter. III. To proportion the Height of the Tufcan Cornice, to any given Height. This admits of two Varieties, 10«. Firft, being confidered as the Cornice of an entire Order ; and laftly, as the Cornice of an Entablature, to a Column only. In the firft of thefe Cafes, divide the given Height into 35 equal Parts, and take 2f, for the Height of the Cornice ; and in the laft Cafe, take 3 Parts, which divide into 3 Parts, &c. as before directed in the Cornice of the Tufcan Entablature. The Intercolumnation of this Order, that is, the Diftance at which the central Lines of the Columns are to be placed from one another, is of divers Kinds, and thofe according to the Ufes they are applied to. As for Example, in a Co- lonade, as Figl. Plate XXII. the Diftance between their central Lines is 5 Dia- meters. In the Frontifpieces, Fig. I. and II. Plate XXI. and in the Arcades A B C, Plate XXII. whole Columns are on Subplinths, they are at 6 Diameters Diftance. And in Arcades of Columns on Pedeftals, as Fig. IV. Plate XXL they are at 7 Diameters Diftance. When Tufcan Columns are placed in Pairs, as a l e f, Fig. II. and de f g h }, Fig. D and E, Plate XXII. the Diftance of their central Lines is 1 Diameter, 45 Minutes. The Intercolumnation of Columns, in Tufcan Portico's, are of two Kinds, tnx. the Middle 5 Diameters, as c d, Fig. II. Plate XXII. and the Sides 4 Diameters each, as h c and d e. L E C T. V. Of the Manner of ' compofing Frontifpices, Arcades, Colonades, and P Mice's of the Tufcan Order. FRontispieces to Doors are either ftreight or circular headed, which la�l L either Semi-circular or Semi elliptical. Semi-circular headed Doors are more graceful than thofe that are Semi- elliptical, which laft is feidom ufed but at fuch times when the Height will not admit of a Semi-circle, as being either too high or too low. When the given Height that an Arch muft rile above the Impofts from which it fprings is more than half the Breadth of the Opening, the Arch muft be aSemiellipfis, made on the conjugate Diameter, as Fig. X. Plate LXIII. But when the give» P Height

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		� lo �/ ARCfilT�C f � R E. Height is lefs than half the Breadth of the Opening, the Arch muft be a Semi � ell�pfis, made on the tranfverfe Diameter, as Fig. IX. Plate XXIII, It is always to be obferved in making of Doors with arched Heads, that their Imports be placed fufficiently above a Man's Height, that they may not obilrucl any Part of the Entrance. Pros. I. Fig A. Plate XXL To make a Tufcan fquare headed Door, <with a circular pitchi Pediment. Draw the Bafe Line, and at any Point as b erect the Perpendicular h e, and draw g h, and k i, parallel to the central Line b c, each at 3 Diameters Diftance» Set up the Subplinths^and k, each I Diameter in Height, and on them ereft two Columns with their Entablature, by Prob. II, or IV. LECT. IV. and give the Subplinths 42 Minutes Projections on each Side of their central Lines. Make the Margins m m 30 Minutes in Breadth, from the cylindrical Parts of the Co- lumns, and from the under Part of the Architrave. Divide the whole Extent of the level Cornice into 9 equal Parts, as is done in Fig. D. Plate XV. and fet up two of thofe Parts from a to e, and draw the Line e i, fqr the upper Part of the raking Cornice. To proportion the raking Members to the raking Cornice, Fig. VII. Plats XV. From the Pointy draw dy, parallel to / iv, alfo x z, parallel to dy. On any Part of x z, as at a, ereft the Perpendicular at, which continue through the level Mouldings. Make a b equal to 0 p ; be equal to p q ; c d equal to q r ; i e equalto r s ; and «/equal to s t ; and through the Points a b c def, draw right Lines parallel to x z, which will be the Members required. And which will have the fame Proportion to the raking Cornice, as the level Memiers have to the level Cornice. To make a circular Pediment. Let:/, Fig.TL. Plate* XV. reprefent the Extent of the whole Entablature. Make x c equal to z Ninths of g i, draw eg, or e i, which bifeft in /or 'h, whereon ereft the Perpendicular � k, or h k, which will cut e x, continued in i the Center, which in Fig. I Plate XXI. is the Point/, on which defcribc the Members found as aforefaid. Prob. II. Fig. IV. Plate XV. To f.nd the Curvature or Meid of the raking Ovolo, that /ball mitre v:ith the level Ovolo. Let n p be a Part of the level Cornice, and a-n the Points from which the raking Cornice takes its rife ; alfo let/ a, and g n, represent a Part of the raking Cornice. On «ereft the Perpendicular nb,\n� continue lato b ; divide iwinto any Number of equal Parts, at the Points 1 2 ;, ts'c. and from them draw the Ordinaces t 2, 3 4, � 6, �?c. Ir. any Part of the raking Ovolo as at c, draw the Perpendicular c m, and make c (/equal to h 0, the Projection of the level Ovolo. Divide c m into the fame Number of equal Parts as are in b n, as at the Points 1 3 5 7, &c. frpm which draw Ordinates equal to the Ordinates in in, and through ths Points 246, (Sc. trace the Curve required. In the fame manner the Cur- \ vature or Mold may be found when the upper Member is a Cavetto, Cyma recta,, or Cyma reverfa, as is exhibited in Fi<>. V, VI. and Vil. Prob. 111. Fig. IV. Plate XV. Tofnd the Curvatiire or Meld of the returned Molding, in an ep;n cr broker. Pediment. Let the Point � be the given Point, at which the raking Molding is to return. Continue up to wards- h z.% pieafure, and from the Point/, let fall the Perpen- dicular � h ; draw/e parallel to h p, and make/ e equal to b a, the Projection cf the level Cornice. Draw, e: parallel to f/j, and divide t.g into the fame i\ umber of equal Parts, as are contained in l n, as at the Points 1 3 5 .7, lsc. from which draw the Ordinates 21, 43, 6r, esc. equal to the Ordinates in b n, through the Points 2 4 6 8, fek. trace the Cnivs required. In the fame manner the Cur- vature or Mold may be found when the upper rviember is a Cavetto, Cyma re�a_; or Cyrna revcifa, as is exhibited in/v, V, Yi. and VII. P:'ate XV-

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		0/ ARCHITECTURE, in Pr o B, IV. Fig. II. Plate XXI. To make a Tufcan circular beaded D<.or ivitb a pitcht Pediment, or Ballnflrade. Setup two Columns with their Entablature as before taught, making the Dis- tance of the central Lines equal to 6 Diameters. Divide n 6, the Height of the Columns, into 3 equal Parts, and fet down 1 Part from n to g, for the Center of the Arch, and draw the Line g t. Make the Breadth of-the Pilafters/ q, each 30 Minutes, from the cylindrical Part of the Columns, and delineate the Imparls and Architrave of the Arch as follows, -viz. In Fig. Ill, Plate XXL a 3 reprefents the Breadth of a Pilafter, make a h equal to a 3, and divide a b in 3 equal Parte at i and g, then the upper i>is the broad R�gula or Fillet, and the lower 1 the Neck of' the Impoft. Divide the Middle Part in 4, give the upper 3 to the Ovolo, and the lower 1 to its Fillet. Make b c equal to half g b, and divide h c in 3 Parts, give 2 to the Aftragal and 1 to the Fillet : And thus are the Heighss of all the Members determined. The Projection of the R�gula on the Ovolo is equai to its Height, as is the Fillec under the Ovolo. The Projection of the Aftragal is equal to the Height tv, and its Fillet to j thereof. To divide the Achitrave of the Arch, divide a 3 into 3 Parts, the inward 1 is the Breadth of 2, 3, the firft Fafcia, half the outer one is the Breadth of a u, the Fillet, and the Remains is the Breadth of n %, the great Fafcia. The Breadth of the Key-ilone n m, on the lower Part of the Architrave, is one eleventh Part of the Semi circle. Now if 'tis required to finilh this Door with a Pediment cither ftreight or circular, proceed therewith as before taught in Pros. I. hereof, and if with a Balluftrade as on the left Side, then by Prob. V, LECT. IV. divided s, the Height, which is equal to the Height of the Pedi- ment, into the fame Parts as the Tufcan Pedeft�l, making the Breadth of the Dado of the Pedeftal equal to the Diameter of the Column at its Aftragal, then the Cornice and Bafe being continued, and the Dado Part filled with Banifters, the whole will be completed as required. To divide the Diftances of the Banifters. Divide the Diftance between the Dado of the Pedeftal and the central Line a b, into 33 equal Parts, give 2 to the half Banifter againft the Pedeftal, 2 to the Intervals or Diftances between the Banif- ters, 4 to the Breadth of each Banifter, and 1 to the half Interval at the central Line a h. The Banifter proper to this Order is exhibited in Fig. ABC Plate LXVIH- with the Proportions of their Members adjtifted by equal Parts. - Not!, If'tis required to complete this Frontifpice (Iriclly, according to Andrea PALLADro'sMeafures, then infteadof the preceding Impoft, we muftinfert either of the Impolis A or B in Plate XLII. where is exhibited all the Impolis to the five Orders by this great Mailer. - , Note a/Jo, If to fuch a Semi-circular-headed Door, 'tis abfolutely neceflary to fet the Columns o,n Pedeftals, then the Diftance of the central Lines of the Column* mufl be increafed unto 7 Diameters, as in Fig. IV- Plate XXL Prob. V. Plate XXII. To make a Tufcan Arcade. Arcades are made in three difi�rent'Manners, «fis. Firft, of fingle Columns aft ABC, fecondly, with Columns in Piers as D E, and laftly with Ruftick Piers in? flead of Columns as F G, and HIK. To form the two firft Kinds of Arcades is no more than to place Columns at ftich Diftances as is expreffed between their central Lines, and to complete them, with their Pilafters, Impolis, and Arches, as taught in the lafl. Problem. Arcades with Piers have their Piers of the fame Breadths as are eqnal to the Breadths of the Pilafters and Columns in the two-former Kinds, as is evident by the dotted Lines continued down to them. ; and the Height of the level Ruf- ticks from which the Arches (pring, is the fame as the Height of die Impofts in the former. The Ruftick^s in the Arches are divided in different Manners, as Firft, Fig. D. where the Arch is divided into 11 Parts; and their Length made equal to half the Breadth, of the Pier. Secondly, Fig. E, where the Key-ftone b. ? ?.. h

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		112 Of ARCHITECTURE. is I eleventh Part of the whole ; the Sides a b, and c d, each equal to half b e, and then the Side o a, divided into 4 Parts give I to each Ruilick. Thirdly, Pig.C is divided in the fame manner as E, but its Pier G being but half the Breadth of the Pier H, the lower Ruftick on each Side is therefore omitted. Fig. B is divided the fame as Fig. D, with its lower Rufticks omitted for the aforefaid Rea- len. Fig. A is divided thefame as Fig. E, and hath its lower Rufticks omitted as in Fig. C, but its Side Rufticks are fquared on their Sides by the central Line of each Pier, and at their Tops, by a Line drawn level from the upper Part of the cir- cular Architrave. The circular Architraves in Fig. A B and C have their Heights equal to half the Thicknefs of their Piers, and their Fillet is equal to I fourth of their Height, as expreffed by the Divifions on the right Side of the Key-ftone in Fig. B. Prob.VI. Fig. I. Plate XXII. To make a Tnfcan Cohnade. To form a Colonade is no more than to range Columns with theirEntablature, at 5; Diameters Dillance as expreffed between the central Lines of the Columns. The Interco'umnation of this Colonade is called Ar�oftyle from the Greek Aracos Rare, and Stylos a Column, by which Vitrwviui fignified the greateft Diftar.ce that fhould be made between Columns that have not Arches between them to affift the bearing of the Architrave. Prob. VII. Fig. II. Plate XXII. To make a Tufcan Pcrtico. Portico's were anciently Porches formed by Columns, fupporting Parts of Roofs, continued out beyond the Uprights of the Ends of Temples, as the Por- tico of St. Paul's Cogent-Garden. But now they are oftentimes placed againft the Fronts of Buildings fupporting a Pediment, to difcharge the Rains, and alfo in Gardens, to terminate the View of a grand Walk, iSc. Divide the given Breadth into 35 Parts, and take z of thofe Parts for the Dia- meter of the Column. This done, fet out the central Lines of the Columns, as ex- preffed between them, and complete the feveral Columns with their Entablature. But as the four middle Columns are finifhed with a Pediment to make the Por- tico, they mult advance 3 Diameters forward before the Range of the Columns a and f, and Pilafters muft be placed behind the Columns b and c, in range with a and/, which indeed (hould be Pilafters alfo. A Pilaster is called by the Greeks, Paraftate, and by the Italians Mem- hrctti, and is nothing more than a fquare Column, and is diminilhed the fame as a round Column, when Handing with Columns ; but when alone, it muft not be diminilhed, nor indeed even when with Columns, as in this Example when ftand- ing at an Angle, as thofe of « and/; becaufe the Quoins of all Buildings fhould i>e erett. Examples for PraBice in tbs Tufcan Order. I. The Height of'the Tufcan Architrave being given, to ji'nd the Height of its Freeze, end of its Cornice. Rule, Make the Height of the Freeze equal to the Height « of the Architrave, and the Height of the Cornice to 3 fourths of the Height of the Architrave and Freeze taken together. II. The Height of tbi Tuf an Cornice being given, to find the Height of the Archi- trave and of the Freeze. Rule, Divide the Height of the Cornice in 3 Parts, and make the Height of the Architrave, and of the Freeze, each equal to two Parts thereof. III. The Height ofa Tufcan Cornice being given, to find the Diameter of the Co- lumn. Rule, By Example II. find the Height of the Architrave and Freeze, and add them to the Cornice ; multiply the Height of the Architrave, Freeze, and Cornice by 4, and divide their Product by 7, the Quotient is the Diameter required. IV. The Diameter ofa Tufcan Column being given, to find the Height of the Cor~ nice. Rule, As 12 is to 9, fo is the given Diameter to the Height of the Cor- nice reauirerj. Tht

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		Of ARCHITECTURE. 113 V. The Height of a Tufcan- Architrave being given, to find, the Diameter of the Column. Rule, Double the Height of the Architrave and it will be equal» the Diameter required, and fo on the contrary, if the Diameter was given and the Height of the Architrave required, then half the given Diameter is the Height of the Architrave. VI. The Height of the Tufcan Entablature being given, to Ji'nd the Height of the "� Capital. Rule, Divide the Height of the Entablature into 7 Parts, and make the Height of the Capital equal to 2 of thofe Parts ; and fo on the contrary, if the Height of the Capital was given to find the Height of the Entablature, divide the Height of the Capital into 2 Parts, and make the Height of the Entablature equal to 7 of thofe Parts. VII. The Height of the Capital and Entablature being given, to find the Diameter. Rule, Divide the given Height of both Capital and Entablature into 9 equal Part», the Diameter will be equal to 4 of thofe Parts. LECTURE VI. Of the Manner of proportioning the particular Parts of the Dorick Order by Modules and Minutes, according to Andrea Palladio; and by equal Parts, compofed from the Mafters of all Nations. THEprinicipal Parts of this Order by Andrea Palladio are exhibited in Fig. I. and its Pedeftal in Fig- III. Plate XXIII. The Bafe, Capital, En- tablature, and Plancere of the Cornice are exhibited by Fig. I. and 111. Plate XXIV. and as they are all proportioned by Modules and Minutes in the fame manner as the Tufcan Order, it is needlefs to fay any more thereof. Problem I. To proportion the Heights of the principal Parts of the Dorick Order by equal Parts. Let ab, Fig. II. Plate XXIII. be the given Height, divide ef, equal to a b, into 5 equal farts, give the lower i to fjie Height of the Pedeftal. Divide the 4 remaining Parts into 5 equal Parts, the upper 1 is the Height of the Entabla- ture, and the lower 4 the Height of the Column, which divide into 8 Parts, and take 1 for the Diameter of the Column. Problem II. To divide the Height of the Dorick Ptdfjial into its Bafe, Die, and Cornice, and them into their refpeSive Members. Let a b, Fig. IV. be the given Height and central Line of the Pedeftal, di- vide c d, equal to a b, into 4 equal Parts, give d 1, the lowed Part to h L, the Height of the Plinth. Divide the next Part into 3, as r s, and give 1 to t s, the Height of the Mouldings on the Plinth. Divide/ s into 8 Parts, give 3 to theCa- vetto G, 1 to the Fillet I, 4 to the inverfed Cyma Recta K ; and the lower 1 to its Fillet L. Make e f equal c to half the upper 4th Part of the Pedeftal's Height, which divide into 2 Parts; divide h g equal to 1 Quarter of ■/"into 3 Parts, give t to the Fillet E, and to the Aftragal D. Divide k i, equal to half ef, into 4 Parts, give the upper I to the R�gula A, and the other 3 to the Fafcia B, The Remains is the Qvolo C. To determine the Projetions of the Members. Ik Fig. II. a Circle being inscribed within the Dado of the Pedeftal, (hews that its Height and Projection are equal, therefore draw the Line q x, parallel to ab, at the Uiftance of half the Height of the Dado F. Make iv iv equal to v x, and through the Point au draw the Line vj p, which is the Projection of the Plinth M, and R�gula A. Divide^ q, the whole Projection before the upright of the Dado, into 8 Parts, and one half thereof as n 0 into 3 Parts ; the firft Part of ««is the Projection of the Fafcia B, and its laftPart, or 4th Part of pq, ofrthe Gvolo C, and the dth and 7th Parts afpq terminante the Aftragal D, and its Fillet E. The firft.Part of p q terminates the Fillet L, the 5th Part the Fillet I, and the 7th Pa» thc.Caveuo G. Problem

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		M4 Of ARCHITECTURE. Problem III., To divide the Height of the Dorick Column into its Bafe, Shaft, and Capitol, and then into their refpe�live Members. Divide the given Height into 8 Parts, x is the Diameter, and as the Height of the Bafe and Capital are each half a Diameter, therefore (as in Fig. If. Plate XXIII.) make q p the Height of the Bafe, and m n the Height of the Capital, each equal to half a Diameter. To divide the Members of the Bafe. Let a/, Fig. IV. Plate XXlV', be equal to a given Height of the Bafe. Di- vide a/"into two Parts, the lower I is the Height of the Plinth : Divide c e equal to half af, into 4 Parts ; give the lower 3 to the Torus, and the upper 1 to the Aftragal, which divide into 4 Parts j and make be the Height of the Cinctiire equal to two Parts. To determine the Projet!ion of the Bafe. Draw the Line h 3 parallel to i k the central Line, and, at the D'ftanee of half a Diameter, divide k 3 into 3 Parts, and make k 4 the Projection of the Plinth equal to 4 of thofe Parts. The Projection of the Torus is always equal to the Plinth in every Order : The Projection of the Cinctiire is equal to a Per- pendicular drawn through the Center of the Torus, as is the Center of the Aftra- gal alio. To divide the Members of the Capital. Let R W.Fig. II. be equal to a given Height of the Capital, divided) into 3 equal Parts, as q 123, and the lower 1 Part is the Height of the Neck : The middle Part equal to#_y, divided into 3 Parts, the upper 21s the Height of the Ovolo, and the lower 1 divided into 3, as a x, the upper » is the Height of the Allragal, and the lower 1 the Fillet: The upper third Part, equal tofw, divided into 3 ; the lower two is the Height of the Fafcia, and the upper 1 di- vided into 3, the upper 1 is the Height of the Fillet, and the lower 2 of its Cyma Reverfa. To determine the PrtiyeSions of thefe Members. Let R W reprefent the central Line of the Column, to which draw the up- right Line of the Column S A parallel to R W, at 24 Minutes Diftance : Make S T equal to half R S, and from any Part of the Neck of the Capital, as at Ai draw the Line A B equal to S T, which divide into4 equal Parts ; the ill Part ter- minates the Projection of the Aftragal under the Ovolo, and % thereof its Fillet, the 3d Part terminates the Fafcia of the Abacus, and \|- thereof the Ovolo. The Aftragal at C is proportioned in the fame manner as the Aftragal to the Tufcan Column. The Shaft of the Dorick Capital is fometimes fluted, either according to the Manner of the Ancients, without Fillets, as on the right Hand of Fij. III. Pints X. or, according to the modern Manner, with Fillets, as on the left Side, in wanner of IcrJck Flutes. 'Fis faid, that the firtt fluted Columns were thofe of the renowned Temple of Diana, huilt at Ephefits, as fome think by the dmaz'-ns, ., which were of Marble, 70 Feet in Height, and whofe Flutings were made in Imi- tation of the Plaitings in Womens R.obes : This Building employed 200 Years to finilh it at the Expence ofall Afia. .The Number of Flutes to the Doric Shaft wjis originally but twenty, as they ftill (hould be made, that their Breadths may be greater than thofe of the ionic and other Orders which are always -24 in Num- ber : And the Reaion is, that as the Doric Order hath a mafculine Afpedl, its Parts ought to be larger and bolder than the Ionic, which veprefents a feminine Slendcmefs. But how juft the Precepts of the Antients may be, fome modern Architects take Liberty to decorate the Doric Shaft with 24 Flutes with Filters, thinking thofe of 20 too large. And indeed, when the Order is made within, a. Building, and near to the'Eye, I think 24 to be better thaxi 20, which are much better iu Columns that Hand abroad* andfgen at a great Diftance.

			P�.OB�

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		Of ARCHITECTURE. 115 PrOB. IV. To divide the flutes, or Flutes and Fillets, in the Shaft of the Doric Column. First, According to therhanner of the Ancients, \&.ihl, Fig. IV. Plate XXIV. reprefent one half of a Part of the Doric Shaft; on i defcribe the Qua- drant 1254, hfc. h, which divide into 10 equal Parts; divide any z of the Parts, as 3 4 5, each into 2 Parts, and on the Points 3 and 5, with three of thofe Parts, make a Scftion, on which defcribe the Curve 3 5. In the fame manner defcribe all the others. Now if from the Points 13579, you draw r^ght Lines parallel to the central Line/ k, and terminate them with Arches, which (hall end level wi.h the upright Part of the Shaft, they will be the perfpeclive Ap- pearances of the feveral Flutings. Secondly, According to the Manner of the Moderns, let c « x x, Fig. III. Plate X. reprefent a Part of the Doric Shaft. Firft, Draw ab the central Line, on a defcribe the Semi-circlech z, which divide into 12 equal Parts, to which draw right Lines from the Center a, and continue them out fomething beyond the Semi-circle. In the Quadrant bz, on the Points � c, 4, 3, 'csV. with a Radius equal to half one Part, defcribe the Qua- drant r 3, and Semi-circles 3 6 t, t 7 v, csV. on the Points r 6 7, &c. with the Radius r 3, defcribe the Arches q 3, 3 t, tv, tsV. which are the Flutes without Fillets. Secondly, In the Quadrant c b divide any one of thofe 6 Parts into 8 equal Parts, and with a Radiusequal to 3 of thofe Parts on the Points b, 9, 13, 17, csV. defcribe the Arches pa, nl\o, /15 m, CSV. which will be the Flutes, and the Intervals 0 p,mn, i I, �sV. left between them will be the Fillets ; and if from the Points pon m Ik, CSV. right Lines be drawn parallel to the central Line, and terminated at the lower Part of the Shaft with Arches as before, they will be the perfpedtive Appearance of the Flutes and Fillets as required. In thefe feveral Manners, the Breadth of Flutes, or of Flutes and Fillets, may be found at the upper Part, and in any Part between the upper and lower Parts of a Co- lumn. It is alfo to be noted, that the Flutings of Columns are fometimes filled for one third Part of the Column's Height, with Staves or Cablings, which are thusdefcribed, viz. on the Points 15, 11, csV. with the Radius 11, 9, defcribe the Arches io, 9, 12 ; 14, 13, 16, CSV. which are the Plans of the Cablings, and which are fometimes enriched with Ribbons, Pearls, and Olives, tsV. ase# h^;bited in the upper Part of this Plate. Prob. V. To divide the Height of the Doric Entablature into its Architraves, Freeze, and CoT' nice, and them into their refpeSive Members. Let d R, Fig. II. PlateXXlV. be the central Line and given Height, which divide into 8 equal Parts, give 2 to the Height of the Architrave, 3 to the Height Of the Freeze, and 3 to the Height of the Cornice. To divide the Architrave. Divide p, the Height of the Architrave, into 6 Parts, give the upper 1 to the Height of the Tenia, the next I divide into 4, give the upper 1 to the Height of the Fillet, over the Drops, and the lower 3 into the Height of the Drops. To divide tbeTriglypbs and Metops in the Freeze, Fig.V. and VT. Plate XLIV. Triclyphs are Ornaments placed in the Doric Freeze, and were firft ufed ia the Delphic Temple, reprefenting an antique Lyre, a m'ufkal Infiniment invented by Apollo. The Word Triglyph comes from the Grt.k Tfiyhvp, fignifying-a three-fculptured Piece, quaff tres habeas Gy/phos, which the Italians call Planctti. A Triglyph conflits of ieven Parts, w. two entire Glyphesox Channels, two Ssmi-Glyphes, andj Spaces or Interftices between them. The Breadth of a Tri" glyph is equal to 30 Minutes, and of a Metop 4; Minutes, which being equal to the Height of the Freeze, is therefore a geometrical Square. Metops are the Intervals orfqoare Parts of the Freeze that are contained be- tween the Triglyphs, and receive their Names from the Greek Meta and Ope, be- tween three, which anciently was enriched with O.xes-Skulls, Inftruments of Sa- crifice, Trophies of War, &c. 2 LiT

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		ii6 Of ARCHITECTURE. Let an q r be the Breadth of a Triglyph, which divide into iz-equal Parts, as at an, from which draw the Lines i t, z 2, 3 t, &c. which continue upwards through the Cornice unto i h, iSc and downwards through the Tenia and Fillet of the Architrave ; make a b and n z each equal to 2 of the 12 Parts in a n, and draw the Line x z ; make b f, i e, k m, and zp, each equal to 1 of the 12 Parts, and draw the Miter Lines e f de, e g, h m, m k, and 0 p, which will complete the Triglyph as required. To form the Drops under the Tenia of the Architrave. From the Points x 2, 4, 6, 8, 10, 1 2, draw Lines towards the Points tt, &c. Hopping them at the Fillet ww, and they will form the Drops as required. To form a Metop, as n b r a. Make n b and r a each equal to nr, and draw the Line b a, then» bra is the Metop required. If it is required to make a hollow Pannel therein, as de h i, divide r a in 6 Parts, and make the Margin about the Pannel equal to 1 of thofe Parts; alfodivide the Margin into 5 Parts, as at A c, and make the Breadth of the Moulding within the Pannel equal to 1 of thofe Parts ; then drawing the Diagonals^/and ce, their Interfeftion is the Center, about which place a Rofe, or any other Ornament at pleafure. Te divide the Cornice into its refpeBive Members, Fig. II. PlateXKIV. Th e Height of the Cornice being 3 Eighths of the whole Entablature, as afore- faid, divide the lower 1 into 3 Parts, give the lower 1 to the Height of the Cap- ping to the Triglyph ; divide the remaining Height equal to b 0 in 4 Parts, and the lower 1 thereof into 6, then the lower 1 is the Height of the Aftragal under the Ovolo, and the next 4 is the Height of the Ovolo ; the f�cond Part of b 0 being divided into 3, theloweft 1 is the Height of the Bells or Drops, the next t of their Fafcia I, and the upper 1 divided into 3, the upper 1 is the Height of the Fillet, and the lower 2 of the Cyma Reverfa ; the third 1 of b 0 divided into 6, the upper 1 is the Height of the Fillet to the Corona, and the lower 5 is the Height of the Corona ; laftly, the upper 1 of b 0 divided into 4, the upper 1 is the Height of the R�gula, and the lower 3 of the Cyma Reverfa. To determine the Projetions e/ the Members in this Cornice. The Upright of the Column and Freeze ,uOS being before drawn, make* M equal to half the Height of the whole Entablature, and from any Part of the Upright of the Freeze draw a Line, as O P, equal to the Projection x M, which, divide into 4 equal Parts at 1 2 3 ; divide the ift Part into 3, the firft I is the Projection of the Tenia in Profile againit the Return, and of the Aftragal, under the Ovolo, which divide into 4, the firft 2 is the Projection of the Triglyph in return, the next 1 of the Capping to the Triglyph over the Freeze, and of the Fillet and Drops under the Tenia of the Architrave. The remaining 2 Parts of the firft 1 of OP, divided into 6, the firft 3 terminates the Ovolo, and the next 1, the Platform K, againft which the Mutnles are placed. The 3d Divifion of O P terminates the Fillet of the Cyma Reverfa, that crowns theMutuIes, and this third Part divided into 3, and thelaft 1 into 3, the firft 1 ter- minates the projecting Mutule L. Laftly, the 'aft Part of O P equal to QJl, di- vided into 9, the firft 4 terminate the Projection of the Corona, and the next 1 its Fillet. . Mutules are a. Kind of Mpdilrons, that are always placed perpendicularly ever the Tr�glyphs, to fupport the Corona, as well of Pediments as of Itraight or level Cornices, and whofe Breadths are always equal to the Triglyphs, as exhi- bited in Plate XXVI. The Word Mutule comes from Mutuli the Latin for Mo- dilion. Th e Figure D E F G is the Plancere or Cieling, which the Italians call Soff.tc, of a Mutule, whofe Sides are each divided into 6 equal Parts, and parallel Lines drawn from them, divides the whole into 36 Geometrical Squares, in whofe Centers the Drops or Bells are placed ; and if from their Centers right Lines be drawn up to the projefling Mutule K L, they will be the central Lines, over which the 6 Drops between K and L are to be placed. Thr

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		O/ARCHITECTURE. ii₇ , The Central Lines of the Drops to H I, the Mutole in Front, are determined .by the Continuation of the twelve Lines from the Triglyph, which alfo makes the Breadth of the Mutule equal to the Breadth of the Triglyph, 'vide Fig. IV. ; Plate XLIV. where ed a h is a complete Mutule in Front, and Fig. III. a Mu- tule in Profile, divided as aforefaid, whofe Drops are drawn to the Points nn, isc. at the Interfe�ions of their Central Lines, with the Line c d drawn through the Midit of the Fafcia a o. In Plate XXV. is exhibited various Manners of making the Returns of the Planceres of the Dorick Cornice, wherein it is to be noted, that Fig. I. and V. which are Returns at external Angles, have but 18 Bells or Drops, each accord- ing to Palladio, and Fig. II. which is a Return at an external Angle, has 36, as at F. zdly, That fometimes Mutules are made fquare, and mew but 28 Bells as at B and D, Fig. IV. which is a Return at an internal Angle, as alfo isF/>.III. whofe (haded Parts ABGCEFG reprefent Parts of Columns, whereby it is feen, that the Mutules D E in Fig. III. D F in Fig. II. and B D in Fig IV. ftand directly over their refpective Columns. The Coffers or holiow Pannels EABC in Fig. II. and A C in Fig. IV. are to be enriched with Rofes, as A, Fir, I. Examples of which are given in Figures A, B, C, D, E, Plate XXXVIII. Prob. VL To determine the Intercolumnations of the Dorick Order. Operation. As the Breadth of a Triglyph is always equal to 30 Minutes, and the Breadth of a Metop to 45 Minutes, therefore the Sum of the Minutes con- tained in the Triglyphs and Metops, that are required between the Central Lines of two Columns, is always the Intercolumnation, or Diftance, at which the Columns are to be placed. Therefore to have 1 Triglyph between, as a b, oie/, Fig. II. Plate XXVII. the Diftance muft: be two Diameters, 30 Minutes. If 2 Triglyphs between, as b c, and de, 3 Diameters, 45 Minutes; if three Triglyphs, as c d, 5 Diameters ; if 4 Triglyphs, as over each of the Arcades, Fig. A B C, &s. 6 Diameters, ij Minutes, csV. Hence it is plain, that in the making of Frontifpieces, &c. to any given Height, the Breadth cannot be confined ; and therefore when fuch a Cafe happens, the Triglyphs and Mutules muft be omitted ; and the Diftance between the Columns flioitld not exceed 4 Diameters. In Plate XXVI. Fig. I. and II. are Defigns of Doors, the firft with a fquare Head, with both circular and pitch'd Pediments over it, the other with a Semi- circular Head, with a Ballullrade and pitch'd Pediment, which are given for Examples, as alfo is Fig. HI. which is half of an Arcade on a Pedeftal. Fig, IV. is the Dorkk Impoft at large, whofe Height a b, divided into 3, the lower 1 is the Height of the Neck, the upper 1 divided into 4, the upper 1 is the Height of the Fillet or R�gula, and the lower 3 of the Fafcia. The middte 1, divided into 3, the upper 2 is the Height of theOvolo ; and the lower 1 divided into 3, the upper 2 is theAftragal, and the lower 1 its Fillet. The Diftance a 2, repreients the Breadth of the Pilatter, and ; s its Upright. Make i h, the Pro- jection, equal to One-third of a i, Make 3 g equal to { h, which divide into 4, then the firft 1 determines the Projection of the two Fillets, to the two Aftragals ; the Third part the Ovolo ; and half the laft Part the Fafcia of the 1 Abacus. The Depth of the Aftragal b d is equal to half the Height of the Neck, divided into 3, give 2 to the Aftragal, and 1 to the Fillet. In Plate XXVII. Fig, I. is a Cclonade ; Fig- II. a Portico ; fig, ABCDE Arcades; with fingl'e Columns, and Columns in Pairs ; and F G HI K, are rufticated Arcades, which are given ai Examplts for Practice. Examples for Praclice in the Dorick Order. T, The Height of the Dorick Architrave being given, to find the Height of the Freeze, and of the Cornice. Rule, Divide the Height of the Architrave into 2 equal Parts; make the Height of the Freeze, and of the Cornice, each equal to 3 of thofs Parts. Q. II.

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		ii8 Of ARCHITE CTURE. II. ThlHeight of the Dorick Cornice being given, to find the Height of the Archi- trave, and of theFreeze. Rule, Divide the Height of the Corniceinto 3 equal Parts ; make the Height of the Freeze equal to the Height of the Cornice, and the Height of the Architrave to Two-thirds of the Cornice. III. The Height of the Dorick Cornice being given, to find the Diameter of the Column. Rule, Divide the Height of the Cornice into 3 equal Parts, and make the Diameter equal to 4 of thofe Parts. IV- 'Ihe Diameter of a Dorick Column being given, to find the Height of the Dorick Cornice. Rule, Divide the Diameter into 4 equal Parts, and make the Height of the Cornice equal to 3 of thofe Parts. V. The Height of the Dorick Architrave being given, ti find the Diameter of the Column. Rule, Double the Height of the A: ehitrave, and it will be equal to the Diameter required. VI. The Height of the Dorick Entablature being given, to find the Height of the Capital. Rule, Divide the Height of the Entablature into. 4 Parts, and make the Height of the Capital equal to 1 of thofe Parts ; and fo on the contrary, if the Height of the Capital was given, and the Height of the Entablature re- quired, it is no more than to make the Entablature equal to 4 times the Height of the Capital. VII. The Height of the Entablature and Capital being given, to find the Diameter. Rule, Divide the Height of the Capital and Entablature into 10 Parts, and take 4 of thofe Parts for'the Diameter required. LE C T. VII. Of the particular Parts oftheloNicK Order, proportioned by Modules and Minutes, according to Andrea Palladio, and by equal Parts, compo/cdfrom the Mafters of all Nations. THE principal Parts of this Order are exhibited by Fig. I. Plate XXVIII. and the particular Parts by Fig. I. and II. Plate XXIX. which in general are determined by Minutes, as the preceding Orders. Prob. I. tig. II. Plate XXXII. To proportion the Heights of ' the principal Parts of the lonick Order by equal Parts. First, Divided/, equal to the given Height, into 5 equal Parts j givethe lower t t� Is, the Height of the Pedeftal. Secondly, Divideaw, equal to the Remains, into 6 equal Parts ; give the upper 1 to the Height of the Entablature, and the lower � to the Height of the Column, which being divided into 9 equal Parts, take 1 for the Diameter of the Column. Prof. II. Fig. IV. Plate XXVIII. To divide the lonick Pedeftal into its principal Parts, and them into their refpeBive Members. First, Draw qvj for the Bafe Line, and svj for the Central Line. Secondly, Divide /' q, equal to s tv, the given; Height, into 4 equal Parts ; give half the upper \ to the Height of the Cornice, and thelower I to the Height of the Plinth. Divide op, equal to the f�cond Pah, into 3 Parts, and the lower 1 equal to x y, into 8 Parts ; give the uppe^ 2 to the Cavett >, half the next 1 to its Fillet, the lower 1 to the Fillet on the Plinth, and the Remains to the inverted Cyma. Thirdly, Divide km, equal tothe Heightof the Cornice, into-4 equal Parts, the 'lower i, divided into 3, the upper I is the Height of the Aftragal, half the next the Height of the fillet, and the Remains is the Height of the Cavetto. The f�cond Part of k n, is the Height of the Ovolo, the next I of the Platform or \Fafci.i, and the upper I divided into 3, the upper 1 is the Height of the Fillet, and the lower 2 of the Cyma reverfa. . : ■ To determine the ProjeBioxs of' the Moid dings. TiiF Diameter being before found, by Prob. I. hereof, divide it into 6 _v equal Fares, and draw'mr_% parallel to s tu, at the Diftance of 4 Parts. Make j' 5, the Projection of the Plinth, and m I the Cornice, equal to_y x, and draw II "' /»

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		Of ARCHITECTURE. _lig 1«, parallel to m r. In any Place againft the Upright of the Dado, as at h, !^ZVL^a.' ^eq^uaV° ^' vvh.ch divide into 4 equal Parts. The firft i terminates the Projeftion of the Platform or Fafcia of the Cornice, the next i the Ovolo the third i the Cavetto's to both Bafe and Cornice ; and which being divided' into 3, as ra', or g h, the laft 1 terminates their Bottoms, then half the firft _tterminates the Fillet * on the Plinth, which completes the whole, as required. Prob. III. Fig. II. Plate XXVIII. To divide the Height of the lomck Column into its Bafe, Shaft, and Capital. The Height hn, equal to//, being divided into 9 equal Parts, give half the lower 1 to the Height of the Bafe. Divide i g, equal to the upper 1, into 6 Parts, give the upper 4 Parts to the Height of the Capital, the Remains be- tween is the Height of the Shaft. Prob. IV. Fig. IV. PlateXXIX. To divide the Bafe of the Ionick Column into its refpeBive Members. Draw 0 z for the Bale Line, and a 0 for the Central Line. Divide hm, equal 'to the given Height, into 3 Parts. Divide nl, equal to the middle Part, into 6; then the lo.ver 1, with the lower 1 Part of h m, is the Height of the Plinth, the next 1 the Height of the Fillet, and the upper 4 of the Scotia. Divide the upper 1 of hm, into two Parts ; divide i k, equal to the lower 1, into 3 Parts, and give 1 to the Fillet under the Torus ; the upper 2, with the upper I offg, i's the Height of the Torus. Make c d, the Height of the Cinfture, equal to One, fourth of/ g. To determine the Proje�iures. Draw//, parallel to a 0, at the Di- llance of half the Diameter before found. Divide 0 p into three Parts', and make p z, the Projection of the Plinth, equal to 1 Part. Divide/ z into 3 Parts, then the firft 1 terminates the Projection of the Fillet '<v; the Center of the Torus <w and the Cincture x. Bifect the laft Part in r, which terminates the Proieftion of the Fillet s, and completes the whole, as required. Prob. V. To divide the Height of the Ionick Capital into its refpeBive Members, Draw the Line 17, tg, for to reprefer.t the Top of the Aftragal, to the Shaft of the Column, and 17 11 for the Central Line. Divide r q, equal to the gi ven Height, into 4 equal Parts ; then the upper 3 of thofe Parts is t-he Height of the Volute and Abacus. Divide the upper 1 Part into 8 Parts; give'the upper z to the Ovolo, the next 1 to the Fillet, and the lower 4 to the Fafcia. Divide L M, the Height of the Volute, into 8 Parts ; make the Height of the Ovolo equal to the fifth and fixth Parts, the Aftragal under it, to the fourth Part and the Fillet under that, to the upper half of the third Part. Make s t, the Height of the Aftragal on the Shaft, equal to one eighth Part of r q, which divide inta 3 Parts, and give 2 to the Aftragal, and one to the Fillet. To determine their Projections. ■ Continue the Central Line towards I at Pleafure, and in any Part of it, as at I, draw a Line at Right Angles, as I K, equal to Three fourths of the Dia- meter, which divide into g equal Parts, each equal to 5 Minutes. Draw the Up- right of the Column, at 25 Minutes Diftance, parallel to the Central Line alfo the Line 13, 30, at 30 Minutes Diftance, which terminates the Pro^;eftioi» of the Aftragal on the Shaft, and the Aftragal to the Capital, whofe End at 13 is the Eye of the Volute. Bifect the Height of the Aftragal to the Ca- pital, and draw its Central Line 12, 29. Divide the Diftance between 2c and .30, in IK,, into 3 equal Parts, and from the f�cond Part draw the Line 2, 22, 16, parallel to the Central Line, which will terminate the Projections oh the two Fillets 'at 22 and 16, and being continued, will interfe� the Cen- tral Line of the . Aftragal 12, 13, in the. Center of the Eye of the Volute. Make u, 10 in the Capital, equal to 35 Minutes of I K, for the Prej.ee- 0^* tioa

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		I20 0/ A R C H 1 T E C T U R E. tion of tie Ovolo. From the Points 40 and 45, in z' i, draw the Lines N 40, and O K, parallel to the Central Line, which will terminate the Projections of the Angles of the Abacus. In any Places, as at a b, and c d, draw 2 Lines, as a b, and c d, between the afore-drawn outward parallel Lines. Divide a b into c Parts, and r^into 2 Parrs; then the 3d and 4th Parts of a b terminate the Projection of the Fafcia and Fillet, the Abacus in Front, and half c d the Fillet of the returned Abacus ; and as the Abacus of this Capital is made cir- cular on each Side, as in the quarter Plan underneath, it is neceflary to fhew how to defcribe the fame. The aforefaid Lines for finding the Projection of theCapital being defcribed, thro' any Part of the Central Line, as at the Point 18, draw the Line Z 18 X, at Right Angles, and make 18 Z equal to 18 X. On the Points Z and X, with the Radius ZX, make the equilateral Section F, on which, with the Radius F 32, defcribe the Arch 31, 32. Make 31 P equal to 11 10, the Projeftion of the Ovolo under the Abacus, then the Point P is the Center of the Plan ; whereon, with the Radios 31 P, defcribe the Quadrant 31,4. In a whole Plan of a Capital, continue the Lines 31 P, and P 4, the z Semi-diameters, out both Ways at Pleafure, and thereon iet the Diftance F I, which will give you the other 3 Centers, on which the Arches of the other 3 Sides may be defcribed ; on the Center P, with the Radius .t, equal to the Up- right of the Shaft, the Projeftion of the Aftragal, and of its Fillet, defcribe the Arches 1 2 3 ; laftly, make X33 equal to X 32, draw the Line 32, 33, whereon defcribe the equilateral Triangle 3Z, 33, 34, whofe Sides will be interfered by the Arches defcribed on the Center F, �sV. and then Right Lines being drawn from one refpeftive Interfeftion to the other, and the like being performed at every of the four Angles of the Capital, the Plan will be completed. The next Work in order to complete the Capital is to defcribe its Volutes, which may be done by either of the following Problems. Prob. VI. Fig. P. Plate XII. To defcribe the ionick Volute. Let ai be the given Height. Divide the given Height into 8 equal Parts at the Points b c defg h, which are alfo numbered, 1234567; bifeft the 5th Divifion e f in x, and on ■,' with the Radius xe,* defcribe a Circle, as ive<vf, which is the Eye of theVolute. Through x draw the Line w <v, at Right Angles to ai, and then complete the geometrical Square ixicvf, and bifeftits Sides in the Foints 1234. Draw the Diameters 2 4, and 1 3 ; and divide each Semi-diameter into 3 equal Parts at the Points 1 234567891021 12, which are the Centers on which the Con- tour or Out-line of the Volute is to be defcribed, as following, niisa. the Point 1 is the Center of the Arch a k, the Point 2 of the Arch k i, the Point 3 of the Ar�h il, the Point 4 of the Arch le, the Point 5 of the Arch c n, the Point 6 of the Arch?;e, the Point 7 of the Arch of, the Point 8 of the Arch p q, the Point 9 of the Arch qr, the Point 10 of the Arch r 1, the Point 11 of the Arch s t, and the Point 1 2 of the Arch / e. To defcribe the inward Line ivhicb ditnimfies the Lift. Divide each third Part of every Semi-diameter of the geometrical Square "jj e m � into 5. equal Parts, as is done in Fig. L. which is the Eye of the Volute �tlaige. The full one, within each of the aforefaid 12 Centers, are theCentert for defcribing of the inward. Line, which Centers are numbered,,13, 14, 15, 16, 17, 1.8, 19, 20,2.1, 22, 23,24. Prob. VTI. Fig. I. Plate XIII. To de/cribetl'e �onick Pointe, a f�cond Way. Let. K. 8 be the given Height. ift. Divide the given Height into 8 equal Parts, and in the fifth Divilion de- fcribe the Eye of the Volute as in the preceding. Through E the Center, draw the Line z�E/b, alfo draw the cblique Lines 1, 5, and 7, 3, each at 45 Degrees Diftance from the Line aFEd 4, which is called the Caibetm. ' ; .' 2 2dly,

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		Of ARCHITECTURE. i2� 2dly, Draw B A, Fig. II. equal to 3 Parts and ahalf ; on A ereft the Per- pendicular A C, which make eqaal to 4 Parts and half, and draw the Line C B, on A, with the Radius equal to half a Part, <viz. equal to E 24, in Fig. I. de- fcribe the Quadrant E^, and draw the Line g B on B, with the Radius B g de- fcribe the Arch g d, which divide into 24 equal Parts, thro' which from B draw Right Lines to meet the Tangent Line C A in the Points 1, 2, 3, 4, $, &c. Make E 1, E2,E 3, E 4, E J, E 6, E 7, E 8, �V. in Fig. II. equal to A 1, A 2, A 3, A4, A 5, A 6, A 7, A 8, (Sc. in Fig. I. On the Points a and I, in Fig. I. with die Diftance 1 E, make a Seclion within the Eye of the Volute, on which defcribe the Arch a\. On the Points 1 and 2, with the Diftance 2 E, make a Seftion in the Eye as before, and thereon defcribe the Arch 1,2. On the Points 2 and 3, with the Diftance 3 E, make a Se�ion as before, whereon de- fcribe the Arch 2, 3, proceed in like manner until the Out-line be completed. To diminijh the Lift of the Volute. Let A F be its given Breadth. Divide a F into 24 equal Parts, and make I a equal to 23 Parts of a F ; 2 b to 22 Parts ; 3 c to 2t Parts ; 4 d to 20 Parts ; 5 <? to 19 Parts ; 6 � to 18 Parts, &c. Proceed then to find Sections for the feveral Arches, which pafs thro' the Points abed, &c as was done for the outward Arch I, 2, fa 4, 5, &c. and they will complete the diminiftied Lift, as required. The hnick Volute was anciently defcribed by 6 Centers, as follows, Fig. III. PlateXlll. Suppose a/to be the given Height. Divide the given Height into 8 equal Parts, and make the Eye equal to the 5th Divifion, as in the preceding Examples. Divide the Height of the Eye into 6 equal Parts, as at the Points 1, 3, 5, 6, 4, 2, which are the Centers on which you may defcribe the Out-line, as fol- lowing.

	On the J 3 with the J /b I defcribe the Yb i e V^ft. ^f" ^fo«» ^thePoint { 4 Radius < 4 e> Semi-circle < ek c > ^re^edf ^ ^V°^lute'

6, To defcribe the inivard Line of this Volute. Divi de each 6th Part of the Eye into 4 equal Parts (as in Fig. A, which is the Eye of the Volute enlarged, for tha better underllanding of the Situation of the Centers) and take the next inward ones for the fix other Centers, on which yon may defcribe the inward Line, as required. Note, It is beft to begin the describing of this inward Line at the Eye, and work outwards ; for if any Miftake ftiould happen in Practice, it is much eafier rectified in the outward Parts than in the inward, where the Parts are nearer to. gether. Prob.VHI. Fig. N. Plate XII. To defcribe an Elliptical Volute of any Height and Breadth required. Let km be the given Height, and f e the given Breadth. First, By either of the preceding Methods, defcribe a Volute, as Fig. H, whofe Height is equal to the given Height, and its Breadth is always equal to ■J of its Height, therefore make ef and* b equal to \ of e a. Divide e a and fb each into 8 equal Parts, and the Lines ab and e f each into 7 equal Parts, and draw the horizontal and perpendicular Lines, which will form c� geome- trical Squares. Secondly, Complete the Parallelogram �?&«, making its Height and Breadth equal to the Height and Breadth given. Divide f b and e-a each into 8 equal Parts; alfo fe and b a into 7 equal Parts, and then drawing the feveral horizontal and perpendicular Lines, as in Fig. H, you will form 56 Pa- rallelograms. Now as the Parts of the elliptical Volute muft have the fame Heights as the like Parts in the circular Volute, therefore make the Ordinates dc,

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		122 Of ARCHITECTUR E. dc, hg, i k, e I, p, r, ty, b z, CSV. in Fig. N. equal to the Ordinates d c, h g, ik. e /, p, r, t, y, &c. in Fig. H. and then every Part of the elliptical Voluie N will arrVdt the j6 Parallelograms in the very fame Manner as the circular Vo- lute H doth the 56 geometrical Squares, and as what is here faid of the outward Line is to be alfo underftood of the inward; therefore, when you have found all the preceding Points through which the Curves are to pafs, apply unto them a thin, pliable Ruler, or with a free Hand trace their Curves, as required. This Ornament is called a Volute, from the Latin, Voluta� volvendo, as that it feems to be rolled upon an Axis or Staff; and the Eye is by fome, from the Latin, called Oculus. Prob. IX. Fig. III. PlateXXIX. To divide the Height of the Ionick Entablature into its Architrave, Freeze, and Cornice, and them into their refp�Sive Members. Divide ax equal to the given Height, into 10 equal Parts, give 3 to the Height of the Architrave, 3 to the Height of the Freeze, and 4 to the Height of the Cornice. To divide the Architrave. � Divide the lower 1 of the Architrave into 4 Parts, give the upper 1 to the Bead, and the lower 3 to the fmall Fafcia, Divide the upper 1 into 4 Parts, give the upper 1 to trie Tenia, the next 2 to the Cyma Reverfa, and the Re- mains to the great Fafcia ; make D H, the Projection of the Tenia, equal to the Height or the Tenia and Cyma Reverfa, which divide into 3 Parts, and give the firft 1 to the Projection of the great Faicia. Divide C D, the Height of the Freeze, into 4 equal Parts, and on the Points C and D, with the Radius of 3 Parts, make the Seclion E, on. which, with the Radius E D, deferibe the fwelling Freeze. To divide the Cornice. The Height of the Cornice, confiding of four Parts, divide hi equal to the t.vo lower Parts into 3 Parts, a.;d the lower and upper Parts thereof each into 6 Parts, as\htn and i k ; give the lower 5 of i k to the Height of the Cavetto, and the upper 1 to the Margin of the Denticule below the Dentules : Give the upper J Parts of h m to the Height of the Ovolo, and the lower 1 to its Fillet. Divide gf, equal to 1 quarter Part of the Height of the Cornice, into 4 Parts, give the lower 3 Parts to the Height of the Corona, and the upper 1 to the Height of its Cyma Reverfa. Divide b n, equal to the upper 4th Part of the Cornice, into 4 Parts, give the upper 1 to the Height of the R�gula, and then dg, equal to the lower 1, being divided into 3 Parts, give the lower 1 to the Fillet between the 2 Cyma"s. And thus are the Heights of ail the Members determined. To determine their Projeilures. The Upright of the Column BCD 10 being before drawn, make B A the Projection of the.Regula equal to B C the Height of the Cornice, and from any Part of C D, as from <v draw a Right Line, as <v<w equal to B A, which divide into 4 equal Parts ; divider^, equal to the 2d Part, into 6 Parts, and ab, equal to the ill Part of vvv, and the ill Part of c d into 5 Parts ; then half the 1 ft Part of a b terminates the Projection of the Foot of the Cavetto, the 3d Part of the Denticule, and \| of the next of its Fillet. Half the 2d Part of<r a'terminates the Projection of the Ovolo, and the 3d Part of v vu the Projection of the Co- rona: Divide ef, equal to the 4th Part of <vw, into 4 Parts, the firft 1 tennis nates the Projection of the Fillet between the 2 Cyma's, To divide the Dentules. Divide xy into 10 Parts, and y z into 3 Parts, give z Parts to the Breadth of each Dentule, and 1 Part to each Interval between them.. . And thus are ail the Parts of the Order proportioned, as required. Prob. X. Plate XXX and XXXI. To determine tbs: hitercolumnations of the Ionick Order,

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		O/ AP. CHIT EC TUR E. 123 It is to be obferved, That alrho' Dentules properly belong to the Ionick Order, yet Palladio, and fome other Mailers, exclude them, and introduce Modilions in their Stead ; and therefore, as the Intercolumnations of the Dorick Order ars determined by the Number of Triglyphs, fo here in this Order the intercolum- nations are determined by the Number of Modilions, or Dentules, that are re-' quired to be placed between them. Firft, To determine Intercolumnations nvhen Modilions are employed. The Diftance between the Central Lines of Modilions is either 30 or 32 Min. Palladio makes them 32 Minutes, and the Breadth of each Modilion 10 Minutes. When the Diftance and Number of Modilions is refolved on, the Intercolumna- tions are eafily found by this Rule, vixt As many Modilions as are required between the Central Lines of any Columns, add fo many times 30 or 32 Minutes together, and their total Sum is the Intercolumnation, or Diftance at which the Central Lines of the Columns are to be placed : Therefore taking 30 or 32 Mi- nutes in your Compafles, fet that Diftance from oae Central Line towards the other, as many times as there are Modilions required ; and if every 30 or 32 Minutes be confidered as 1 Part, and as the Breadth of a Modilion is 1 o Minutes, therefore fetting � Minutes on both Sides of every Part fo fet oft, they will de- termine the Breadth of every Modilion in their refpe&ive Places. When the Di- ftance of Modilions is fixed at 32 Minutes, to have 3 Modilions between thofe over the Central Lines of each Column, the Diftance between the Central Lines rnuft b� 128 Minutes, equal to 4 times 32, or 2 Diameters 8 Minutes : If 5 Mo- dilions, then 192 Minutes, equal to 6 times 32, or 3 Diameters 12 Minutes : If 7 Modilions, then 2$6Minutes, equal to 8 times 32, or 4 Diameters and 16 Mi- nutes : If 9 Modilions, then 320 Minutes, equal to 10 times 32, or 5 Diame- ters and 20 Minutes, �V. In Fig. I. II. andV. Plate XXX. are three Examples, wherein Fig.l. contains X 3 Parts or Modilions, and Fig. II. and V. 14 each, whofe Modilions are at 30 Minutes Diftance, as is feen by the Number of Diameters contained in their re- fpeclive Intercolumnations. In PlateXXXl. Fig. I. is exhibited the Intercolumnation for the Colonade, whofe Columns are at 3 Diameters 44 Minutes Diftance, not 45 Minutes, as in- ferted in the Plate by Miftake of the Engraver, and have 7 Modilions between the Central Lines of every 2 Columns each, at 32 Minutes Diftance between their Central Lines. The Portico, Fig. II. and the Arcades, Fig. III. and IV. , have their Intercolumnations proportioned, fo as to have the Diftances of the Central Lines of their Modilions each 30 Minutes. Secondly, To proportion Intercolumnations tuhen Dentules are employed, Fig III. Plate XXIX. As xy is equal to 25 Minutes, and being divided into io Parts, as aforefaid, 2 of which is the Breadth of a Dentule, and I of an Interval ; it is therefore evident, that each Part is equal to 2 Minutes and a half: And therefore to make the Divifion of Dentules eafy, the Diftance between the Central Lines of Columns muft always contain fome Number of Parts, each of 5 Minutes, as the Occafion may require ; as 1 Diameter and §, wherein there are 18 fuch Parts; or 4Diameters, wherein there are 48 fuch Parts; and 5 Diameters, 6ofuck Parts, as in the feveral Intercolumnations of the Portico, Fig. II. Plate XXXI. Now, if each of thefe Parts be divided into 2 Parts, then each Part will be equal to 2 Minutes and a half, and then giving 2 of thofe Parts to the Breadth of each Dentule, and 1 to each Interval, the whole will be completed, as required. Note, the RakingDentules, in all Kinds of Pediments, muft ftand exaclly over thofe in the level Cornice, in the very fame Manner as the Muttiles in the Dorick Order. The like is alfo to be obferved of Modilions ; and as Modilions are al- ways capped with a Cyma Reverfa, or fome other Moulding, whofe Curvatures or Moulds, on the upper and iower Sides, are both different from thofe of the Front Raking Moulding ; I muft, before I proceed any further, fhew how to de- scribe thofe returned Mouldings to the Caps of Raking Modilions. Pros.

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		124 Of ARCHITECTURE. Prsb. I. Fig. I. II. III. Plate XV. To defcribe the Returned Mouldings of the Caps of Raking Modi lions in Pediments. \ft. Suppose the Ovolo C. Fig. III. to be the Raking Moulding in Front, with which a Raking Modilion is to be capped ; draw the Chord Line a c, and divide it into any Number of equal Parts, fuppofe 8, as at the Points 2, 4, 6, 8, 10, &c. and from them draw the Ordinates I, 2 ; 3, 4 ; 5,6; &c. zdly, Sup- pofe the Lines h b and i c to be the Bounds of the Front Raking Ovolo, and let the Line y i reprefent the upper Side of a Raking Modilion, and d f its lower Side. From the Pointy draw the horizontal Line it/, and from the Point 0, the Line op, make op and ny, each equal to a b, the Proje&ion of the Front Ovolo, and through the Points n and/ draw the perpendicular Lines bl and e f I, cut- ting the upper Line h b, in h, and e, draw the two Chord Lines h i and f e, and divide each into the fame Number of equal Parts, as the Chord Line ac, and from thofe Parts draw Ordinates equal to the Ordinates in C. Through the Points 1, 3, 5, 7, &c. in Fig. A and B trace the Curves h 7 i, and/7 e, which are the true Curves of the Returned Mouldings on the upper and lower Side of the Modilion, as required. Note, The fame Method ofworking will find the Curvatures of all'other kinds of Returned Mouldings j as for Example, when the Front Moulding is a Cavetto, as C, Fig. II. then A and B are the upper and lower Mould, or when a Cyma Reverfa, as C, Fig. I. where A is the upper, and B the lower, as in the z other Examples. Prob. XII. To proportion the Ionick Frontifpieces, Colonades, Portico''s, and Arcades. As by the Practice of the two preceding Orders it is very reafonable to believe, that my Reader is now capable of infpetting into this and the two fucceeding Orders, that is, to readily underftand what is meat by the Meafures affixed to each Part with refpeft to the Intercolumnations, Number of Modili�ns, Breadth of Pilafters, Height of Impofts, �V. I fhall therefore only explain the Impofts, Fig. VI. Plate XXX. and then recommend him to the feveral Figures in Plat-e XXX. and XXXI. for his further Praftice. To proportion the Ionick Impoji by equal Parts. Divide a i, its given Height, into 3 equal Parts, the lower I is the Height of the Neck. The lower half of the middle Part divided into 4, the upper 1 is the Height of the Fillet, and the lower 3 of the Cavetto ; the upper half is the Height of the Ovolo, as is the lower half of the upper 1 the Height of the Faf- cia. Divkle the upper half into 3 Parts; give the upper I to the R�gula or upper Fillet ; and the lower 2 to the Cyma Reverfa. To determine their Projections. Let a b reprefent the Breadth of the Pilafter, and b p the Upright thereof; divide 0 p, equal to the Breadth of the Pilafter, into 3 Parts at t and -v, make/ r equal top rj, divide p r into 3 Parts at x and s, and make r q equal to s r. Then p x determines the Projection of the Cavetto, half/ r the Ovolo, p r the Fafcia, and p q the R�gula. The Aftragai is determined in its Height and Projection, as that of the JJorick. The Heightofthe Impoftin Fig.ll. PlateXXX. isTwo thirds of thelieight of the Column and Sub-bafe, but in Fig. V. it is at 3 times the Height of the whole Pedeitr.l, and the Key-ftones, in both Examples, are One-fifteenth Part of the Semi-circle. The Lengthof Key-ftones are generally made equal to one Diameter, and their Depth below the Architrave is always at Pleafure ; but moft generally about \|- or f of their Breadth, at the lower Part of the Archi- trave. In Plate XXXI. Figures A B C D, are two Varieties of Confoles 0; Key-ftones, in Front and Profile, which may be ufed in the Ionick, Corinthian, or Compolite Arches at Difcretion. Note, The Ionick Impoft by Andrea Palladio is exhibited by Fig. D. Plate XLII.

			PROB.

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		Of ARCHITECTURE. ��f Pros. XIII. To proportion the Dorick and lonick Cornices, to the Height of any Room, &c. First, The Dorick Cornice. Divide the given Height into 50 equal Parts, and give 3 of thole Parts to the Height of the Cornice, which is confidered as the Cornice to an entire Order. But being confidered as a Cornice to an Entabla- ture on a Column, without a'Pedeftal, then divide the Height into 40 equal Parts, and give 3 to the Height of the Cornice. Secondly, The lonick Cornice To find the Height of'a Cornice to an entire Order. Divide the Height of the Rooms into 75 Parts, and give the upper 4 to the Height of the Cornice re- quired. To find the H:ight of the Cornice of an Entablature on a Column only. Divide the Height of the Room into 60 Parts, and give the upper 4 to the Height of the Cornice. Examples for PaBice in the lonick Order. I. The Height of the lonick Architrave being given, to find the Height of the Freeze and of the Cornice. Rule, Make the Height of the Freeze equal to the Height of the Architrave, divide the Height of the Architrave into 3 equal Parts, and make the Height of the Cornice equal to 4 of thofe Parts. II. The Height of the lonick Cornice being gif en, to find the Height of the Archi- trave and of the Freeze. Rule, Divide the Height of the Cornice into 4 equal Parts, and make the Heights of the Architrave and of the Freeze, each equal to 3 of thofe Parts. III. The Height of the lonick Cornice being given, to find the Diameter of the Column. Rule, As 36 is to 50, fo is the Height of the given Cornice, to the Diameter required. IV. The Diameter of the lonick Column being given, to find the Height of the lonick Cornice. Rule, As 50 isto 36, fo is the given Diameter, to the Height of the Cornice required. V. The Height of the lonick Architrave being given, to find the Diameter of the Column. Rule, As Z7 is to 50, fo is the Height of the given Architrave, unto the Diameter required. � VI. The Height of f/v lonick Entablature being given, to findthe Diameter ofthe Column. Rule, As 9 isto 5, fo is the Height of the given Entablature, to the Diameter required. VII. The Height of thelomok Entablature being given, to find the Height of the Capital of ZO Minutes in Height, according to Andrea Palladio. Rule, as 27 is to �, fo is the given Height of an Entablature, to the Height of the Capi- ta! required, and which being doubled is the Height of the Capital of 20 Mi- nutes, as given in Fig. II. Plate XXVIII. VIII. The Height ofthelomck Entablature and Capita' according /0 Palladio being given, to find the Diameter. Rule, As 37 is to 15, fo is the given Height of the Capital and Entablature, to the Diameter required. L E C T. VIII. Of proportioning the particular Parts of the Corinthian Order, by Modules and Mi- nutes, according to Andrea Palladio, and by iquai Parts, cotr.pofedfrom the Maft er s ofall Nations. FIGURE I. PlateXXXU. exhibits the Proportions and Meafures of all the principal Parts of this Order, by Andrea Palladio, and Fig. III. the particular Parts of the Pedefial. Fig. I. and II. Plate XXXIII. exhibits the par- ticular Parts of the Bafe to the Column, with its Capital and Entablature, which, being in general determined by Modules and Minutes, nothing more with re- J'P'�ft to the Formation of their Parts, need be faid, and therefore I fhall proceed to the Divifion of this.Order, by equal Parts. Prob. I, Fig. II. Plate IV. -i � proportion the principal Parts of the Corinthian Order, ant0 any given Height. R Divide

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		ii6 Of ARCHITECTURE. Divide Jw, equal to b z the given Height, into 5 equal Parts; the lower 1 is the Height of the Pedeftal. Divide c s, equal to b r the remaining Part, into6 equal Parts; the upper 1 is the Height of the Entablature, the lower 5 Parts is the Height of the Column, and which being divided into to equal Parts, take 1 for the Diameter of the Column, which divide into 60 Minutes, viz.. Firft, into 6 equal Parts, which will each contain 10 Minutes, and then the�rlt one of them into 10 Parts. Prob. II. Fig. IV. Piute XXXII. To divide the Height of the Corinthian Pedeftal into its Safe, Die, and Cornice, and them into their reffeSlive Meafures. To proportion and divide the Bafe, draw m k, the bafe Line, and i c, the central Line. Divide f k, equal to c k the given Height, into 4 equal Parts. Divided e, equal to the f�cond Part, into 3 Parts; and c z, equal to the lower 1 Part, into 4 Parts, and make b a, xj\ and x v.; each equal to 1 of thofe Parts. Divide b a into 3 Parts, then the upper 2 is the Height of the Cavetto F, and the lower 1 of its Fillet. The two middle Parts of c z is the Height of the inverted Cyma Recta G. Divide 'u> y into 5 equal Parts ; give the upper 1 to the Fillet of the Cyma, and the lower 4 to the Torus H. The Remains i k is the Height of the Plinth. To proportion and divide the Cornice. Make h g, equal to one 8th Part of/ K, the whole Height of the Pedeftal for the Height of the Cornice, which divide into 6 equal Parts. Divide r a, equal to the lower 1 of h g, into 3 Parts : give the lower 2 to the Cavetto, and the upper 1 toits Fillet» Divide op, equal to the third divided Part of h g, into 3 Farts ; give the upper 1 to the Fillet, and the other 2, with the f�cond Part of b g, is the Height of the Cyma Redla. Divide k m, equal to the 2 upper Parts of h g, into 6 equal Parts, and give the f�cond Part below to the Height of the Fillet on the Fafcia B. Divide the 2 upper Parts of A m into 3 equal Parts, as at i ; give the upper 2 to the R�gula, and the Remains is the Height of A, the Cyma Reverfa. To determine the Projetlures of the Mouldings. Draw the Line b I, parallel to c h, at the Diftance of 42 Minutes of the Diameter before found. Make g h equal to g f, and through the Point h, draw the Lin* a m, parallel to b 1, which will determine the Projections of the Plinth I, ani. Cornice at a. From any Point in d f, the Upright of the Dado or Die, draw a horizontal Line, as r s, which divide into 4 equal Parts ; then the firft 1 ter- minates the Fillet on the Torus and Fafcia in the Cornice ; the third Part th« two Cavetto's in the Bafe and Cornice, and one third of the laft Part, the Feet 01 the Cavetto's. Prob. III. tig. IT. Plate XXXII. To divide the Height of the Corinthian Column into its Bafe, Shaft, and Capital. The Diameter being found as before taught, let g r be the given Height. Makeqr, the Height of the Bafe, equal to half the Diameter ; alfo g I, equal to 70 Minutes, for the Height of the Capital ; then / j the Remains is the Height of the Shaft, which is diminifhed one 6th Part at /. ■ Prob. IV. Fig. IV. Plate XXXIII. To divide the Bafe of the Corinthian Column into its rejpc�livt Mmhrt. Draw km for the bafe Line, and/ k for the central Litre. Divide ah. equal to the given Height, into 3 equal Parts, the lower 1 is the Height of the Plinth. Divide b g, equal to the 2 upper Parts of a h, into 4 Parts, the upper 1 is the Height of the upper Torus. Divide c f, equal to the 3 iower Parts of b g, into 2 Parts, the lower 1 is the Height of the lower Torus. Di- vide d e, equal to the upper 1 off/", into 6 equal Parts, the upper and lower Parts is the Height of the two Fillets, and the middle 4 Parts of the Scotia. Draw the Line r I. parallel to i i, at 30 Minutes Diflance, for the Upright of the Column, make/ m equal to 12 Minutes, and / » equal to two third Parts of I my then the Line p n terminates the Projection of the Fillet e, and the upper Torus p. Lallly, the Projection of the Cinilure s, and fillet q, are each equal '2 tO

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		Of ARCHITECTURE. 127 4a trie Projection of the Center of the upper Torus, which is found by fetting half the Height of the upper Torus from p, towards the central Line. This Safe is that which is called the Attic Bafe. Peob.V. Plate XXXV. To divide the Height of the Corinthian Capital, into its refpeBive Members. Let A B I be the centrai Line, and A B the given Height. Thro' the Points A and B, draw the Line a Az, and b B y, at right Angles to A B. At any Diftance below the Point B, draw the Line O P QJ?, parallel to b B y. On any Side of the central Line A B, draw the Line z y, parallel to A B, at fuch a Pittance, as to be clear of the Projection of the Abacus. Divide z y into 7 equal Parts as at the Points 123456, then each Part will be equal to 10 Minutes, becaule the whole Height of the Capital is 70 Minutes. Divide the fecond, fourth, fifth, and fixth Parts each into 2 equal Parts, at the Points Z de and b, and from the Points Z Y X.J WcViT, draw right Lines paral- lel to b B y, as q Z, p Y, 0 X, n d, m W, / c, k V, i b, h a, and g T, which determines the Heights of the Leaves, Stalks, Helices, and Volutes. Divide the upper Part into 2, as on the left hand Side, the lower 1 the Height of the curved Fafcia of the Abacus ; and the upper j, divided into 6 equal Parts, the lower 1 is the Height of the Fillet, and the upper 5 of the Ovolo. Make p 0, the Height of the Aftragal, equal to 5 Minutes, which divide into 3 Parts; gi»« the upper 2 to the Height of the Aftragal, and the lower 1 to the Height of the Fillet; and thus are the Heights of all the Members determined. To determine the Trojr.Bures. Make 2; P, and 25 Q^ on the Line O R, each equal to 25 Minutes, which is equal to 2 Parts and half of zy ; alfo make O P and Q__R, each equal to two Parts of z y or 20 Minutes. Through the Points O P, Q_R, draw the Lines O a, P g, Q^n, and R z ; then Q a, and R z, will de- termine the Projections of the two Sides of the Abacus, and the Lines P^, and Q^n, will be the two upright Lines of the Shaft of the Column. Divide O 10, on the left hand Side, into 8 equal Parts ; then O <iy, the firft three Parts, de- termines the Projection of the Fillet in the Abacus at r; O x, the firft 5 Parts, the Projection of'the Faf�ia at/. and Ovolo at d. O y, the firft 6 Parts, the Projection of the Fillet at j, and O z, 'the firft 7 Parts, the Projection of the Fafcia at v. Make the Projections on the right Hand, equal to thofe on the Left, and then the Abacus will be completed. Make q, the Projection of the Allragal, equal to p 0 ; and s r, the Fillet, unto 2 thirds thereof. Divide p t into 3 Parts, and make p v equal to 4 of thofe Parts. Draw -v x parallel to 11. Draw / n>\ which bifect in wo, whereon raife the Perpendicular tu x, cutting %> x in x, whereon, with the Radius x f» defcribe the Arch <v t. Make h k m, on the left Side, equal ta g t v, on the Right, and then the Aftragal will be completed. On the Point B, with the Radius B 0, defcribe the Semi-circle « N G H IK L M 0, which divide into 8 equal Parts, at the Points N G H I K L M, and from them draw the Lines N A, G C, H D, I B, K E, L F, parallel to the central Line A B, which continue upwards at pleafure, which are the central Lines of the feveral Leaves. Draw the Lines a h, and z q, which determines the Projefture of the two Out-ieaves in the f�cond Range. Divide theDiftance 12, 13, into 4 Parts, and from the third Part, at the Point 14, draw the Line 14 q, which determines the Projecture of the Out-leaf in the lower Range, at the Point 15. This being done, proceed to delineate by Hand the feveral Leaves, Stalk, and H�lice, on the right-hand Side, and when the fame is done» transfer every particular Part thereof unto the left Side, by taking their feve- ral horizontal Diftances from the central Line, and fet them from, the central Line on the left-hand Side; or otherwife, draw parallel Ordinates through on both Sides, and make thofe on the left Hand, equal to thofe on the Right. By either of thefe Methods, you may make the two aides of the Capital exactly the fame. R s Nof&l

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		128 Of ARCHITECTURE. Note, It will be beft, full for to defcribe the Leaves in Grofs, as is done on the right-hand Side, wherein you mutt be very perfect in their Out-lines, before you proceed to divide them into their Palms and Raffles ; and for the eafy di- viding of Leaves into their Palms and Raffles, I have given 7 Examples of Leaves for Piaflice, in Plate -XXXIV. of which the large Leaf D is in a man- ner geometrically defcribed, and whofe Height is toits Breadth, as 7 is to 6, as may be feen by the equal Parts on its left Side, and at its Bottom, which Parts being fubdivided as'in the Figure is exprefied, the Points of the Parts in every Palm are exactly determined. Note, a Palm confifts of 5 Points, as r q q y D, or tiin t A F, or 10 x C G IF. Note a/ft, that when the Learner has formed two or three Leaves in large thus divided, he may then proceeed to make Others of leis Magnitude, by Hand, and omit a!! the albrefaid Divifions by Lines, as R W X Y, which are all Leaves in Front, ferving as well for Pilaftcrs as the Front Leaves of Columns. Fig. S is a Leaf in Profile, and T in an oblique View, fuch as thofe that are between-the middle or Front Leaf, and outer or profile Leaf of a Column. The Figures M and Qjire two Examples of Stalks or Stems for Practice; of which Qjs a Stalk only with its Leaves, and M is complete with its Volute and H�lice. Fig. P is the ancient Ornament with which the Abacus is ufually charged, in (lead of which i have placed a Lion's Mafc, as an Emblem of Majefty, Power, �sV. Prob. VI. Fig, G C D E. Plate XXXIII. To divide the Height of the Corinthian Entablature into its architrave, Freeze, and Cornice, and them into their reffeilive Members. Divide h I, equal to the given Height, into 10 equal Parts; give the lower 3 to the Height of the Architrave, then next 3 to the Height of the Freeze, and the upper 4 to the Height of the Cornice. Divide z h, equal to the Height of the Freeze, into 5 Parts, the lower I is the Height of the iirft Fafcia, with its Bead, which is I fourth Part thereof, the f�cond Part is the Height of the f�cond Fafcia. The third Part, equal to tf, divided into three Parts, the lower i is the Cyma Reverfa between the f�cond and third Fafcia's. The fourth Part, equal to c d, divided into 4 Parts, the upper i is the Bead over the third Fafcia, and the 3 lower Parts, with the two remaining Parts of ef, is the Height of the third Fafcia. The upper or 5th Part equal to a 6, divided into 3 Parts, the upper 1 is the R�gula of the Tenia, and the lower 2 of its Cyma Reverfa. To determine the Proje�iures of thefe Members in the Architrave. Make iv x equal to ivy, which divide into 5 Parts, give 1 Part to the Projection of the f�- cond Fafcia, and 2 to the third Fafcia. To divide the Cornice. Divide k g, equal to its Height, into 5 equal Parts, and i m, equal to the third Part, into 8 Parts. Make y q equal to the two lower Parts of kg, and the lower 1 Part of /' m, which divide into 15 equal Parts ; give the lower 4 Parts to the Height of the Cyma Reverfa, the next 5 Parts and half to the Height of the Denti- cule, againft which the Dentules are placed, whofe Depth are 5 Parts only; the next half Part to the Fillet on the Dentules ; the next I Part to the Aftra- gal, and the upper 4 Parts to the Ovolo. Divide If, equal to the 3 remain- ing Parts of kg, into 3 Parts; the lower 1 divided into 4, the lower 3 Parts thereof is the Height of the Fafcia, againft which the Modilions are placed, and the upper I of the Cyma Reverfa, with its Fillet, with which the Modi- lions are capped. Divide* 0 into 2, the lower 1 is the Margin below the Mo- dilions. Divide s r into 3 Parts, the upper 1 Part is the Height of the Fillet, and the lower 2 Parts of the Cyma Reverfa. Divide* /, equal to the middle Part of If, into 4 Parts, give the upper 1 to the Height of the Cyma Reverfa d d d, and the lower 3 to the Height of the Corona. Divide the upper Part of /pinto 4 equal Parts, and the lower 1 Part thereof into 3 equal Parts; give the lower 1 Part to the Fillet, and then the 4th Part of the upper 3d Part of If, being given to the R�gula, the Remains will be the Height of the Cyma Recta To determine the FmjeSiion of theft Members. Make b, the Projection of the Cornice, before the Upright of the Freeze and Column, equal to k g, its

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		0/ ARCHITECTURE. 12^ its entire Height. From any Part of the Freeze, as A, draw- an horizontal Line, as A B, which make equal to kg, the Projection of the Cornice, and draw the Line b B. Divide A B into 4 equal Parts. Divide c d, equal to the firft Part, into 6 Parts ; then the firit 2 Parts and half determine the Projection of-the Denticule ; the firft four Parts and half, the outer Denticule ; the 5th Part the Fillet over the Dentuies and the ; th Part and half the Aftragal. The f�- cond Part of AB, divided into 5 Pats (which in the Plate is omitted by Miftake) the firit 1 Part determines the Projection of the Ovolo, and one third of the next Part, the Proj.ea.ion of the Oiufide of the outer Modilion. The third Part of A B determines the Projection of the Modilion in Profile at z. Divide e f equal to the lalt Part of A B, into 5 equal Parts, and g b, equal to the 2d and 3d Parts of e f, into 3 equal Parts ; then half the ill one determines the Projec- tion of the Corona at c, and the 2d Part the Fillet of the Cyma Reverfa : And thus are the Heights and Proje&ions ofthe feveral Members of this Order de- termined. The next Work, in order to complete this Cornice, is to divide out the Dentuies and Modilions, and to defcribe the Modilion in Front and Profile. Prob, VII. &.,eC, Plate XXXIII. To divide tie Dentuies in the Corinthian Cornice. Divide the Diftance between the centra.! Line and the Upright of the Freeze into 12 equal Parts, give 2 Parts to the Breadth of a Dentule, and 1 Part to an Interval. Prob. VIII. Fig. G C. Plate XXXIII.' To di-vide the Difiances of Corinthian Modilions. It is generally agreed on by the beft Mailers to place the central Lines of Modilions �t 35 Minutes Diilance, and to. make the Front of each equal to 10 Minutes, whereby their Intervals or Diftahces between are each 25 Minutes, and the Length or Projection of a Modilion is 20 Minutes, equal to double its Front or Breadth. Now as over the central Line of every Column there muft be a Mo- dilion, therefore the Intercolumnation of this Order muft be conformable to the Number of Modilions that are to be between every two Columns ; and to divide the Diitanccs of Modilions, is no more than to take 3c Minutes in your Com- paifes, and to fet off that Ditlance from the central Line of your Column, as often as the Number of Modilions are required. Prob. IX. Fie. IU. IV, _a�d V. Plate XW. To defcribe the Front, Profile and Plan, or plancsre of the Corinthian Modilion, I. To difcribi a Corinthian Modilion in Frjnt. Let the geometrical Square ab hi, Fig. III. be the Out-lines of a Corinthian Modilion, with its Cyma Reverfa and Fillet, whofe Breadth h i, and Depth y i, are given. Bifeft h i in d, and draw the Perpendicular e d. Divide h i into 8 equal Parts, and make the Fillets hi and 7 «"each 1 Part. Bifeft v i in /, and draw. A I parallel to b i. Draw the Lines q m and <v 0 parallel to e d, each at the Dis- tance of half the Breadth of the Fillet h i, and divide the Diftance between them into 8 equal Parts, as at d, and make the fmall Fillets next within the Lines q m. and <v 0, each 1 of the 8 Parts. Draw the Lines m 0 and q v parallel to k I, and each at the Diftance of m %. Take the Diftance to either of the Fillets, and on the Points* and x defcribe the two Semicircles of the Bead. Draw the Lines? q, i> iv, alfo n m and 0 p. Bifeft t fin/, divide f q into 8 Parts ; on � and q, with a Radius equal to 5 Parts, make the Section r, on which defcribe the Arch � ^ In the fame manner defcribe the Arch t f, alfo the Compound Archest) tu, n m and of; which completes the Modilion in Front, as required. II. To defcribe a Corinthian Modilion in Profile, Fig. V. Divide the Length»; � into 3 equal Parts, and the lit one Part into 7 Parts; make 7« / the Height, equal to 8 of thofe Parts, and complete the Parallelogram I b mf. From f, at 4 Parts and J Diftance from m, draw the Line p q parallel to mf. At 4 Parts from m draw the Line k i parallel to I m, whofe Interfeftion, is the Centreof the Eye of the greater'Scroll, and whofe Diameter is equal to the 5 th

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		i₃o Of A R C H I T E C T U R E. 5th Divifion of lm. Fig. D is the Eye of this Volute or Scroll at large, wherein the geometrical Square being infcribed, and each Semi-diameter divided into 2 equal Parts, as at the Points 7, 6, 8, 5, then the Points 1, 4, 3, 2, 5, 8, 7, 6, as they «land in the Figure, are the Centers on which defcribe the Scroll, begin- ning at the Point i. Divide b c, equal to 4 Parts of / m, into 8 equal Parts, and draw the Line d c for the Depth of the fmall Scroll. Make I a equal to 7 Parts of be, and at 4 Parts from 6 draw the Line 4 d B parallel to bf. At 4 Parts asd I from b draw the Line � r parallel to a b, which will interfeft the Line 0 d in the Center of the Eye of the fmall Scroll, whofe Diameter is equal to the 5 th Divifion in b c. Inscribe a Square within the Eye, and divide its Semi-diameters as before, as. in Fig. D, and then the Points 3, 2, 1, 4, 7,6, 14, as they ftand in Fig. D, are the Centers whereon defcribe the fmall Scroll, beginning at the Point 0. Draw the Line op, which bifect in L ; alfo bife�t « L in g and L p in e. Erett the Perpendiculars � A and eB, cutting the Lines ki in A and od in B. On the Points A and B, with the Radius A i, defcribe the Arches i L and L p, alfo the inward Archer, which limit the Breadth of the Lift. III. To defcribe the Plan or Plancere of the Corinthian Modilion, Fig. IV. Make B C and c f each equal to h «in Fig. IIL. alfo make B c and C f each equal to / b in Fig. V. and complete the Parallelogram B c C f. Draw 0 d and / if parallel toC/j each at the Diilanccof hi in Fig. III. Draw the Lines an b and g h max the fame parallel Diftances from B t and C f, as are refpeftively equal to the Projection of the Cyma Reverfa in Fig. III. before b i the Upright of the Modilion, which continue about at the End, and return from B and C. The Beads, with its Fillets r t, and the Cyma's d q r and / k, &c. are defcribed exailly the fame as n m « 0 p in Fig. III. Note, The Manner of dividing the Plancere of the lonick, Corinthian and Com- porte Cornices, and to make their Returns at external and internal Angles, is exhibited by Fig. VII. Plate XLIV. wherein B B reprefent the Plan of the two Modilions next to an internal Angle, and E E of two Modilions next an external Angle, as alfo are H H. The geometrical Squares A C A A F G are hollow PanneU, called Coffers, which are to be enriched with Rofes, as thofe of Fig. A B C D E. Plate XXXVIII. Prob. X. To. proportion the Corinthian Cornice to the Height of any Room required. This admits of two Varieties, <via, Firft, To cenfider the Cornice as the Cor- race to an entire Order ; and, laftly, as the Cornice of an Entablature on a Co- lumn only. To find the Height of a Cornice to an entire Order. Divide the Height of the Room into 75 Parts, and give the upper 4 to the Height of the Cornice. To find the Height of the Cornice of an Entablature on a Column only. Divide the Height of the Room into 60 Parts, an,d give the upper 4 to, the. Height of the Cornice. Pros. XL To proportion Ftontifpiucs, Coionadei, Poiticd's, Arcades, &C. of the Corinthian Order. As the Ir.tercoiomnations of this Order are regulated by the Number of Mo- dilions, whofe Diftances between their central Lines are 35 Minutes, as before obferved, therefore to make Frontifpieces, Colonades, &c. the Diftances of the central Lines muil confift of as many Times 35 Pdinutes as the Nature of the Cafes requires, /Tg-, I. II. and III. Plate XXXVI. are Examples hereof, where the Columns in Fig. I. have 13 Modilions between, Fig. II. 1 2 Modilions, Fig. Ill, «4 Modilions. In Plate XXXVII. Fig. I. coniifts of 13 Modilions, and Fig. A of 12, between the two middle Columns, as before in Fig, I. and II. Plate XXXVI. But as herein Fig, A, there are Columns in Pairs on each Side, their Diiiances have but 3 Modilions between their central Lines, accounting the two.

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		Of ARCHITECTURE. 131 two half Modilions on the Sides of the two central Lines as one Modilion. la Plate XXXVIII. the Colonade, Fig. I. contains 6 Modilions between every two Columns, the fingle Arcades n Modilions, the Arcades of Columns in Pairs 3, and 11 Modilions, and the Portico, Fig. II. contains three Modilions between the central Lines « and b, 6 Modilions between b and c, and 8 Modilions between c and d. Now from the preceding 'tis evident, that the Intercolumnations of this Order mull be as follow, <viz. If it have two Modilions between thofe over the two Columns, theIntercolumnation mull be 1 Diameter 45 Minutes; if 3 Modilions, then the Intercolumnation mull be z Diameters 30 Minutes ; if 4 Modilions, then 2 Diameters 55 Minutes ; if 5 Modilions, then 3 Diameters 40 Minutes ; if 6 Modilions, then 4 Diameters 5 Minutes ; if 7 Modilions, then 4 Diameters 40 Minutes ; if 8 Modilions, then 5 Diameters 15 Minutes ; if 9 ModiLons, then J Diameters and 50 Minutes; if 10 Modilions, then 6 Diameters 25 Minutes; if il Modilions, then 7 Diameters ; and if 12 Modilions, then 7 Diameters 3 5 Minutes. And fo, by the continual adding of 35 Minutes, the Intercolumnatioji for any- greater Number of Modilions may be found, Note, the Intercolumnations for Columns, which have 3, 5, 7, 9, 11 and 13 Modilions between them, as piiblilhed in Palladio Londinenfis, by Mr. Salmon of Colchefler, and revifed by Mr. Edward Hofpus, Surveyor of the London Infurance-Ofrke, are in general falfe, and feem, as that neither of them knew what they were doing ; for by the preceding 'tis plain, that the Intercolumnation for Columns that have 3 Moulions between them, is 2 Diameters 40 Minutes, not 2 Diameters 30 Minutes ; and for Columns that have 5 Modilions between them, is 3 Diameters 30 Minutes, not 3 Diame- ters 45 Minutes, as they have falfly publilhed in p. 87, �V. The Height of Impolis in this Order are two Thirds of the Height from the Bafe Line unto the under Part of the Architrave, as in the preceding Orders, and the Breadth of the Key-ftone is one 1 �th Part of the Semi-circular Architrave ; and as Key-ftones to this Order admit of Embelliihments, I have therefore in Figures a bed e f g h i k, given proper Examples thereof. The Impoli to this Order by Andrea Palladio is exhibited by Fig. F. Plate XLII. and that by equal Parts, by Fig. V. Plate XLIII. which is thus proportioned. fa proportion the Corinthian Impofi by equal Parts. Divide a h the given Height into 3 Parts ; the lower one is the Height of the Neck or Freeze of the Impoli. Divide the middle Part into 3 Parts, and the lower i into 3, give the lower 2 to the Cavetto, and the upper 1 to the Fillet. Divide the upper 1 into 3 Parts, and'give the upper 1 to the Fillet on the Cyma Retta, and the Remains to the Height of the Cyma Re�la. Divide a k, the upper third Part of a b, into 2 Parts, and the upper t into 3 Parts, give the lower 2 to the Height of the Cyma Reverfa, and the upper 1 to the Height of the R�gula or upper Fillet. To determine the Projection of tbefe Members. Draw b e parallel to ah, at a Diftance equal to the Breadth of the Pilafter. Divide d e, equal to the Breadth of the Pilafter, into 3 equal Parts -_y make e 'g equal to one of thofe Parts, and �■ � equal to 1 Third of e g : Divide e g into 3 Parts, and the firft and third Parts thereof each into 3 Parts, then the firft Part from e, determines the Projedlion of z the Bottom of the Cavetto, the next 1 the Fiilet of the Aftragal c, and the next 1 the Aftragal at b, and Fillet on the Ca- vetto at y. The 2d Part of the third Part of eg determines the Proje&ion of the Cyma Recla at x; and e g the Projection of the Fafcia at au : Laftly, b c being made equal to e f, completes the whole, as required. The Height of the Aftragal 0 h, divided into 3 Parts, is equal to halfai 'h the Height of the Neck. The Architrave a b of the Arch is thus divided, <viz. de being already divided into 3 Parts, divide the outer 1 Part into 3 Parts ; give the III Part to v t, the Breadth of the R�gula ; the next 1 to the Ovolo with its Fillet, which is equal

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		i₃2 O/ ARC�ITECTUR E. to J. thereof, and the laft Part to the Cavetto and Bead, which is f thereof. The middle Part of d e is the Breadth of r p the great Fafcia ; and the outer Part di- vided into 6 Parts the firft I Part is the Breadth of the Cyma Reverfa, and th.: other 5 of the fmall Fafcia. Examples for PraBice in the Corinthian Order. I. The Height of the Corinthian Architrave being given, to find the Height of the Freeze and of the Cornice. Rui,e, Make the Height of the Freeze equal to the Height of the Architrave. Divide the Height of the Architrave into 3 equal Parts, and make the Height of the Cornice equal to 4of thofe Parts. II. The Height of the Corinthian Cornice being given, to f nd the Height of the Freeze and of the architrave. Rule, Divide the Height of the Cornice into 4 equal Parts, and make the Height of the Freeze and of the Architrave, each equal to 3 of thofe Parts. III. The Height of the Corinthian Cornice being given, to f nd the Diameter of the Column. Rule, Divide the Height of the given Cornice into 4 equal Parts, and make the Diameter equal to 5 of thofe Parts. IV. The Diameter ofthe Corinthian Column being given, to find the Height of the Corinthian Cornice. Rule, Divide the Diameter into 5; equal Parts, and make the Height of the Cornice equal to 4 of thofe Parts. i V. The Height of the Corinthian Architrave being given, ta find the Diameter of the Column. Rule, Divide the Height of the Architrave into 3 equal Parts, and make the Diameter of the Column equal to 5 of thoie Parts. VI. The Height of the Corinthian Entablature being given, to find the Diameter bfthe Column. Rule, One half Part of the Height of the given Entablature is equal to the Diameter required. VII. The Height of the Corinthian Entablature being given, to find the Height of the Capital. Rule, Divide the Height of the Entablature into 12 equal Parts, and make the Height of the Capital (exclufive of the Ailragal, which is a Part of the Shaft) equal to 7 of thofe Parts. VIII. The Height of the Corinthian Capital and Entablature being given, to find the Diameter of the Column. Rule, Divide the whole Height of the Capital and Entablature into 19 equal Parts, and make the Diameter of the Column equal to 6 of thofe Parts. LECTURE IX. Of the Manner of proportioning the Compofite Order by Modules and Minutes accord- ing to Andrea Palladio, and by equal Parts compofed from the Mafers of ail Nations. THE principal Parts of this Order, according to Andrea Palladio, are exhi- bited by Fig. I. and the particular Parts of the Pedeftal by Fig. III. Plate XXXIX. the particular Parts of the Bafe to the Column and of the Entablature are exhibited by Fig. I and II. PlateXLl. which being in general proportioned by Modules and Minutes as the preceding Orders, nothing more need be faid thereof; and therefore I (hall proceed to the Manner of proportioning the Pans. of this Order by equal Parts. Prob. I. Pig. II, Plate XXXIX. To proportion the principal Parts of the Compofite Order by equal Parts. Divide t r, equal to the given Height, into J equal Parts, the lower 1 Part is the Heightof the Pedeftal. Divide s p, equal to the remaining Part, into 15 equal Parts, and the 11 th Part into 6 equalParts, the 2 upper Parts and � of the next lower Part is the Height of the Entablature, and the Remainder v p is the Height of the Column, and which being divided into 11 equal Parts, I of thofe Parts will be equal to the Diameter of the Column, and its Height to 11 Diameters. Prob. II. Fig. IV., Plate XXXIX, To divide the Height of the Compofite Pedeftal into its principal Farts, and them into their reffefiive Members. Draw vj d, for the Bafe Line, and � ,d, for the central Line, divide k e± equal

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		Of ARCHITECTURE. 133 equal to the given Height into 4 equal Parts, and the 2d Part into 3 equal Parts ; divide z a, equal to { of the 2d Part, into 12 equal Parts, and make a b equal to 5 of thofe Parts, and draw i> b, for the Height of the Plinth ; at 3 Parts above a draw the upper Line of the Torus, and make the Height of its Fillet equal to 1 Part ; give the upper 2 Parts to the Height of the Cavetto, and the next 1 to theHeightof theFillet, then the Remains will be the Heightof the inverfed Cyma Refta. Half the upper Part of ke is theHeightof the Cornice; divide ^ / into 3 equal Parts, and the lower i Part into 6 Parts, give the lower 2 to the Height of the Cavetto, and the next 1 to the Height of its Fillet. Divide the middle 1 of h /into 6 Parts, give the 3d Part of the Height of the Fillet on the Cyma Re- verfa, and the Remains of that, and the lower Part, will be the Height of the Cyma Reverfa. Divide the upper 1 Part of hi into 4 equal Parts, give the up- per 1 Part of the R�gula, and the next 2 to the Cyma Reverfa. To determine the Projections of tbefe Members, The Diameter found as before, being divided into �o Minutes, draw b x, parallel to f'd, at 42 Minutes Diilance. Make x -tv, and b a, each equal to a z, and draw a tv, which will determine the Projection of the Plinth at <v iv, and the Cornice at a. From any Part of b x, the Upright of the Dada, draw a Right Line, as 1.2, which divide into 4 equal Parts; the firft 1 determines the Projection, of the Fafciar d \ of the next 1 the Projeftion of the Cyma Refta at d; the third 1, theFillet on the Cavetto, and on the Cyma in the Bafe at p, and J. of the ]a!l I, the Foot of the Cavetto in the Cornice, and in the Bafe ; laiily, the Projection of the Fillet q, in the Bafe, is equal to the Projection of the Center of the Torus. Prob.iii. Fig. rr. p/^xxxix. To divide the Compofite Column into its Bafe, Shaft, and Capital. The Height dh, being divided into 11 Parts, one of which being the Diame- ter as aforefaid, make h g, the Height of the Bafe, equal to half the Diameter ; an� de, the Height of the Capital, equal to the Diameter, and One-fixtii Pate thereof. Prob. IV. Pig. IV. Plaie XLI. To divide the Bafe of the Compofite Column into its refp.Si-ve Members. Draw kf for the Bafe Line, and c f for the Central Line. Divide a f into 3 equal Parts, the lower 1 Part is the Height of the Plinth. Divide the middle I into J equal Parts, the lower 3 Parts is the Height of the lower Torus, the next I of the Aftragal, and half the next 1 of its Fillet. Divide the upper 1 of a f into c equal Parts, the upper 2 is the Height" of the upper-Torus ; half the next 1 is the Height of the Fillet under the Torus, and the Remains is the Height of the Scoti". To determine the Projcfiures of ihrfe Mini�inos, Draw ih, pavai lel to af, at the Dillance of 30 Minutes, ant] make k h equal to 12 Minutes. Divide kb into 5 equal Parts, thefiril 1 Part and half deter nunc? the Projection of the Aflragal, on the lower.Torus, the f�cond Part ils Filler, the. third Part the Fiiiet under the upper Torus, and its Center s Ko ; and the third Part and half, the Center of the Ailragal on the upper Torus, and its Fillet alfo. The Height of the Aftrajral on the upper Torus is equal to half the Height of the upper Torus, and the Fillet on the Aftragal to half the Height of the Aftragal. Prob. V. Plate XL. To proportion the Part s of the Compofite Capital by equal Parts. First, Set up the Height of the Capital, proportion its Ailragal, Leaves, and Abacus, exaftly the fame as in the Corinthian Capital; and the 20 Minutes contained be ween d, the lower Part of the Abacus, and /, the Top of the upper Range of Leaves, divide as follows, <mz. Divide g s into 8 equal Pain, give the 6th and -til Parts to the Height of the Fillet E. Divide the 5 Minutes between 50 and 55 into 2 equal Parts aty"; then gf is the Height of the Af- tragal D, which is alfo the Heightof the Eye of the Volutes Nand N. Di- vide the upper 5 Minutes contained between 55 and 60 into 4 equal Parts ; S give

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		13$. O/�RGHITECTURE. give the upper i to the Height of the Fillet under the Abacus, and the remain- ing Part if to the Height of the Oyolo C. Now as the Volutes N N are ellip- tical, and have the Centers of their Eyes in that Point of the Line t X, the up- right Line of the Shaft that is cut by the central Line of the AUragal D, and as they aie comprized within a Parallelogram, formed by the upright Lines pro- ceeding from T, the Projection of the lower Part of the Abacus and <w P, as alfo by </r a>, the under Line of the Abacus, and ir the Top of the f�cond Range of Leaves ; therefore by Prob. VI. or VII. LECT. VII. hereof, de- scribe a circular Volute, whofe Height is equal to the Breadth of your Paral- lelogram ; and then from that Volute fo made, by Prob. VIII. LECT. VII. aforefaid, defcribe an elliptical Volute in the afcrefaid Parallelogram, which will be the Volute to this Capital, and which being in like manner performed or» both Sides, the Capital will be completed, as required. Prob. VI. Fig. IJI. Plate XLI. and Fig. I. P&teXUl. To divide the Height of the Compolite Entablature into its Architrave, Freeze, and Cornice. As I have given two Examples of Entablatures in this Order, theorie for the Infides of Buildings, to be feen at a fmall Dillance, and the other for the Out- fides of Buildings, to be fcen at a considerable Dillance, I fhall therefore fpeak particularly thereof. J. Of the Comporte Entablature, to be ufed ivithin Buildings. Fig. III. Plate XLI. Divide /A, equal to the given Height, into S equal Parts ; give 2 to the Height of the Architrave, 3 to the Height of the Freeze, and the fame to the Height of the Cornice. To divide the Height of the Architrave. Divide / c, its Height, into 50 equal Parts ; give 8 to the Height of Z the lower Fafcia, 1 and half to its Bead, 10 to Y the middle Fafcia, 4 to the double Bead X, 1 � to the upper Fafcia, of wb ch 5 muil be given to the Drops V, 3 to the Cavetto T, 1 to its Fillet, z to the Altragal S, 4 to the Tenia R, and 1 tQ its fillet. To divide the Height of the Freeze. Du'iEE n 1'. equal to its Height, into 12 equal Parts, and give the upper 1 to P, its Capital. 'Todi-vi.de the. Height of the Cornier. \ DivtDE k m, equal to its Height, into 70 equal Marts'; give 1 to the lower Fillet, 2 to the AUragal O. 4 and half to the Cavetto N, 1 to its Fillet, 6 to- the.Depticul�, of'winch the upper 5 is the Height of the Dentales ; then give 1 to their Fillet, z to the A�lr�ga'l L, 4 and half'to the Ovolo K, and 6 to the Platform'of the MojdilicEg, of which tire upper � is the Height of the Mo�iiions. Give ? to the Cyma Reveria H, 7. to the Supef-m��ilionJ G. and 1 to the Fillet. Give 1 to the AUragal P, 4 to the Super-aflragal E, and 1 to its Fillet. Give 8 to the Corona D, 3 to the Cyma Reyerfa C, and 1 to it? Fi!{et, Give 2 10 the Allragal B, 8 to the Cyma Rec�a �, and 3 toits R�gula. -'. ;' To 4it'eritiine the PtojfShm of ihrfe Mouldings, Make q E, and C D, each equal to the Semi-diameter of the Column at its AUragal, and draw the Line c. d for the Up ight of the Freeze, which continue tip through the Cornice, Wake the -utuiolt Projection before the Upright of the Freeze, equal to' I; 'vi the Height'of the Cornice. From any Part of the Upright of the Freeze, as at E, draw a horizontal Line, as � i^;. which divide into 4 equal Parts. Divide the firft 1 Part into 3 Paris; then the fir ft t Pa't thereof determines the Projection of the Cavetto and Aftragai at w, and : thirds thereof, the Capital of the Freeze, whofe 'l'ilk-t projects equal to its Height. 'I he f�cond Fart of the firil Part E F. de- termines tile Projection cf the Fillet <v s and 1 fourth of the next t.ird Part, the P'entkulet. Eb, 1 fourth Part of E F, determines the Projection of th*

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		Of ARC HI T�CTUR �. 135 Fillet s, and Center of the Aftragal r ; as alfo the Bottom of the Ovolo K. Divide h d, the f�cond Part of E F, into 8 Parts, or be, its half, into 4 Paru ; then the f�cond Part determines the Projection of the Outride of the Modiiioii at ii. Bifedt df, the third Part of ef, in e. Divide de into 4 Parts, then the firft Part determines the Projection of the Modiliofl in Profile at m ; the f�cond Part, the Super-mcdilion at /, and de the Super-aftragal atz'. Divide /F, the fourth Part of E F, into 7 equal Parts ; then � 2, equal to 3 of thofe Parts, determines the Projection of the Corona, and fh, equal to f of /F, the Fillet of the Cyma Reverfa C. Make y z, the Tenia of the Architrave, equal tof of I of E b. Make^ r, and t x in the Freeze, equal to half'the Diameter at the Safe of the Column. Divide/ x into 6 Parts, and give 2 Parts to the Breadth ofeachDrop, as in the DorickOxdtr. To divide the Uenfules in the Cornice. Divide ab into 24 equal Part; give 2 Parts to the Breadth of each Dentule, and 1 to each Interval. The Breadth df an upper Modilion is equal to id Minutes, and of an under M�dilio� unto 5 Minutes. The Diftance in the Clear between the upper Modilions is 30, and between their Central Lines 40 Minutes; fo that to adjuit the Dilfances of Columns in this Order, we mult place them at 3, 4, 5,65V. times 40 Minutes, and then the Modilions will happen at their true¹Durances. This Entablature, without Ostentation, is the richeltahd ffl�itmacr- nificer.t that has yet appeared in the World. II. Of the* Compofite Entablature, to be tifed rgainjt the Oiitfides of Buildings. Fig. I. Plate XLII. Divide f s, equal to the given Height, into zo equalParts ; give the lower 5 'to the Height of the Architra'/e ; the next 3 to the Height of the Freeze, and the upper 4 to the Height of the Cornice. To divide the Height cf the Architrave. Divide t v, equal to the given Height, into 5' equal Parts ; divide the lower J Part into 4Parts; give the lower 3 to C, the lower Fafcia, arid the upper 1 to B, the Bead. The zi Part of/ v is the Height of A, the middle Fafcia. Di- vide the 3d Part of / v into 3 equal Paits, and give the lower 1 to z, the Cyma Reverfa. Divide y x, the 4th Part of tv, into 4 equal Parts ; give the upper I to the Height of the Bead ,v, and the Remains, with the Remains of the 3d: Part, will be the Height of y the upper Fafcia. Divide the upper Part of / <y into 3 equal Parts ; give the lower 2 to the Height of the Cyma Reverfa, and the upper i to the Height of q the R�gula. To divide the Height hf the Frieze. Divide the upper third Part into ; equal Parts, and the upper � of thofs Parts into 3 Parts ; give the upper 2 Parts CO the Height of' the Altragal », and the lower 1 to the Height of its Fillet». 1 his Freeze may be made either up- right or fwelling, at the Pleafure of' the Architect. To divide the Height of the Cornice. The Height cor.fiSing of 4 principal Faits, divide /' n, the firil Part, into 8 equal Parts ; give the lower 4 Parts to the Cyma Reverfa m, and the upper 4 Parts to the Platform of the under Modilion. of which the upper 3 Parts' mull be given to the Height of the Mc'dilion. Divide//, the f�cond Part of the Height, into 4 Parts ; give 2 thirds of .the lower 1 to the Height of the Cyma Reverfa i, and the upper I being divided into 3 Parts, give the upper z to the Ovol�, and the lower 1 to the Filet. Divide cf, the third Part of the Height, into 4 equal Parts, and the uppcf 1 thereof into 2 Parts ; give' the under I to the Height of the Fillet ^r*. and the three remaining Fort's will be', the Height of the Coronae. Divide the upper fourth Part of the Height of the Cornice into 4 equal Parts, and the lower I thereof into 3 equal Pa.ts; add the lower 1 to the Remains of the third principal Part, which together make the Height of the Mragal C. The i»�P°r 4th Part is the Height off th« ffreguk a.

			S i *f S*

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		i₃6 �f ARCHITECTURE. To determine the Projeilioni ofthefe Mouldings. Draw F O parallel to the Central Line Q_R, make F G equal to F M, from any Part of the Upright of the Freeze, as at K ; draw the horizontal Line K L equal to F G, which divide into 4 equal Parts, and each Part into 6 equal Parts, then the 1 ft Part of K 1 determines the Projection of the Fillet and Center of the Aftragal, the 4th Part the under Modilion, the 5th Part the upper Modilion, and K 1 theOvolo or Capping of the upper Modilion ; the 2d Part of KL being divided into 6 Parts, 4Parcs and \ determines the Projection of the lower Modi- lion in Profile, 5 Parts and J the Super-modilion in Profile, and 5 Parts f its Fil- let ; the firft half Part, between 2 and 3, determines the Projection of the Ovolo under the Corona, whofe Projection is determined by the 3d Part of K L, and its Fillet by the next half Part. The Projection of the Tenia O P is equal to 4Parts of K. 1, and which being divided into 5 equal Parts, give f of the firft 1 to the Projection of the middle Fafcia, and the firft 2 to the upper Fafcia. The Breadth of a Super-modilion is 10 Minutes, and the Interval between every 2 is 2,' Minutes, and which being in every Refpect equal to the Modilions of the Corinthian Order ; therefore when this Entablature is ufed, the Intercolumnations muft be the fame as thofe of the Corinthian Order, of which Fig. I. II. and IV. Plate XLIII. are Examples, and as the firft and laft of thefe Examples are arched Doors, I muft therefore proceed to explain the Impoft and circular Architrave, Fig. V. which is ufed therein. To divide the Compofite Impojl and Architrave. DiviDi a h, the Height, into 3 Parts, the lower 1 is the Height of the Neck or Freeze of the Impoft. Divide the middle 1 into 3 equal Parts, and the lower 1 into 3, give the lower 2 to the Cavetto, and the upper 1 to its Fillet ; divide the upper 1 into 3, and giving the upper 1 to the Fillet, thetwo lower Parts, to- gether with the middle Part, is the Height of the Cyma Recta. Bifect a h in >'.; divide a i into 3 Parts, give the lower two Parts to the Cyma Reverfa, and the upper one to the R�gula. The Aftragal and its Fillet is equal to half m h, the Neck of the Impoft. The Projetions of thefe Members are thus found. Draw h e for the Upright of the Pil after ; divide dc, the Breadth of the Pi- Iafter, into 3 equal Parts, make eg equal to one Part, and g f equal to \ of e % ; make b c equal to e f ; divide eg into 3 Parts, and the firft 1 Part into 3 equal Parts; then the firft 1 Part determines the Bottom of the Cavetto atx, the 2d Part the Fillet of the Aftragal ate, and the 3d the Allragal and Fillet jy ; divide the lalt 3d Part of eg into 3 Parts, the firft 2 Parts determine the Projection of the Fillet at x, and e g of the Fafcia at iv. To divide the Architrave. Divide de, equal toab, the Breadth of the Architrave, into 3 equal Parts ; divide the firft 1 into 3 equal Parts, the outer one is the Breadth of the R�gula, the middle 1 of the Ovolo, with its Fillet, which is a �th Part thereof, and the third 1 is the Breadth of the Cavetto, with its Bead, which is i Part thereof ; the middle 3d Part of de is the Breadth of rp the great Fafcia, and the next Part of thefmali Fafcia, and Cyma Reverfa, which is-i thereof. L E C T. X. Queries on the fixe Orders o/Andre a Palladio, recommended to the Confutation of his Advocates. I. Of the Tukan Order, Plate XX. Qterc I ./^C A N the Cinflurc, which is abfolutely a Part of the Tufcan Shaft, V^/t bejullly confidered as a Part of the Bafe ? <?j z. Are the Parts in the Heights of the Members of Palladio', Tufcan* Bafe, Fig. II. fimilar to the Number of Diameters contained in the Height ot the Colomn I ij 3. Are not the Parts in the Heights of the Tufcan Bafe, Fig. Ill,, fimilar ?» the Number of Diameters in the Height of its Column ?

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		Of ARCHITECTURE. 137 ^. 4. Is not the Neck of his Tufcan Capital too low, and the Projeftion of ks Ovolo and Abacus too little ? ^. �. Is not his -ibacus, and Ovolo under it, too maffive for the Fillet ? i�. 6. If, in the Execution of the Dorick Order, the Triglyphsand Drops are leftout, as often is done, how are the Tufcan and Dorick Arciiitraves to be known from one another, fince that, in both thefe Orders, he has divided each Archi- trave into two Fafcia's ? Q. 7. Is not the Height of his Tufcan Freeze, which he has made equal to A of the Entablature, too little ; for a great Part of its Height being eclipfed, by the Projeftion of the Tenia, the Remains has more the Look of a Fafcia than of a Freeze ? ^ 8. Should any compound Members, as the Cyma Refta of the Cornice, be ufed in this Order, fince that its native Simplicity (which conflits in the Plain- nefs of its fingle Mouldings) is thereby deflroyed ? �>. 9. Which is moft agreeable to the Charafter of the Order, <viz. To finifli the Entablature with the Cyma Refta and R�gula, as Fig. II. or with the plain and bold Ovolo, as in Fig. III. II. On the Dorick Order, Plate XXIV. ^ 10. Is not the Atthk Bafe, which he has given to this Order, much too extravagant, and more efpecially as that anciently this Order was made with- out any Bafe ? Is not the model! Addition of an Ailragal on the Toro&, as in Fig. IV. fufficient to diftinguiih it from theTufcan ? Q, 11. Are the Annulets proportionate or disproportionate to the Ovolo and Abacus ? Have they fo noble an Afpeft as the Allragal under the Ovolo in the Capital, Fig. II ? 4^ iz. Can the Annulets be fcen diitinftly at (a great a Diflance as the afore- faid Allragal ? ^. 13. Is it good Architefture, to make the fame Bed-moulding in the Doric'; Entablature as in the Tufcan ? ^14. Is a driping or oblique Plancer�, as A, the moll agreeable, or the iflolt difagreeable of all others ? III. Of the lonick Order, Plate XXVIII. ^15. Is not the Plinth or his Pedeilal, Fig. III. much too low ? ^16. Should the lonick Architrave be divided into the fame Number of Faicia's as the Corinthian Architrave ? 4J. 17. Is it good Architefture, to make the fame Bed-moulding in the lonick Entablature as in the Tufcan and Dorick? ^ 18. To which of the Orders do Dentules properly belong ? Q. 19. Should the Dorick and lonick Cornices be alike finillied with a Cyma Refta and Reverfa, as in Plates XXIV. and XXIX ? IV. Of the Corinthian Order, Plate XXXII. ^\ 20. Is not the Plinth to his Pedeilal much too low for the Statdincfs of the Order ? ■^ 2 j. Is it good Architefture to make the Shaft of the Corinthian Column. Fig. I. zo Minutes fhorter than the Shaft of the lonick Column, Fig. I. Flnte XXVIII. V. Of the Compofite Order, Plate XXXIX. J^ 22. Is not the Plinth of his Pcdcflal much too low for the Stature of the Order ? Q. 23. As the Corinthian Order, which is more delicate than the Compofke Order, has its Shaft made 20 Minutes fhorter than the Shaft of the Jtuki, why doth he make the Shaft of the �om�ofite Order, whole Capital and Entablature are more maffive than the Corinthian,' 30 Minutes higher than th.- Shaft of the Ionia ? " -

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		i₃8 O/ ARCHITECTURE. 4j. -4- Has the double Aftragal d, in Fig. I. P/a/f XLI. any Similarity or Proportion to the other Members of the Bafe ? Q. 25. Is it good Architecture to proportion the Architrave and Freeze ofthia Order the fame (f a Minute only excepted) as theTufcan ? Qz6. Can any Perlbn believe, that the Fillet on the Freeze and its Aftragal fhould be made equal ? i?. 27. Are not the Greatnefs of the Members in the whole Entablature more proportionate to a Tufcan Column of feven Diameters in Height, than to a flen- der Column of ten Diameters, which lie has affigned ? To thefe I could add much more ; but let thefe fuffice to (hew, that this great Mailer is no more free from Miftakes than another, altho' fo very much ap- plauded by many, who, for want of knowing better, have believed him inimit- able. L E C T. XI. Of the Grotefque Order, Fig. I. Plate XLVII. THIS Order is a Degree below the Tufcan : It conflits chiefly of fquare Members, and is to be ufed in Grotto's, bfc. To proportion the Parts of this Order. Divide a J, equal to the given Height, into 3 equal Parts, and the lower 1 Partint0 7 Parts, give 2 Parts and f to the Subplinth ; divide the upper i Part of a I into 7 equal Parts, and give the upper 1 to the Height of the Ovolo ; divide b k into 5 Parts, the lower 4 Parts is the Height of the Column, and which being divided into 7 Parts, is the Diameter of the Column ; divide b e, the upper 1 Part of b i, into 3 Parts, the upper 1 is the Height of the Corona and Fillet, which is ~ of the whole ; divide f g into 7 Parts, give 3 to the Ar- chitrave and 4 to the Freeze; make g h the Capital equal to J the Diameter, as alfo the Height of the Bafe; make the Height of the Cinfture on the Bafe, and the Fillet under the Capital, each equal to ^ of the Height of the Bafe. To ruft i cat e the Shaft. Divide its Height into 7 equal Parts, and make each Ruftickand eachlnter- val^equal to one Part. The Proje�tion of the Bafeis 40 Minutes, and of the Sub- bafe 45 Minutes, from the Central Line of the Cdumn. The Projection of the Cincture, from the Upright of the Column, is equal to its own Height, and the Projection of the Rulticks is equal to that of the Cinfture. The Shaft is dimi- nifhed -^l_f of its Diameter at the Bafe, and its Capital proje�s before the Upright of the Shaft \ of its Diameter at the Capital. The Projection- of the Ovolo, from the Central Line^ m, is 1 Diameter 37 Minutes \. LEG T. XII. Of the Attick Order, Fig. VIII. Plate XLV. Til IS Order is never ufed but when an Attick Story is placed over the Cct- nice of fome one of the preceding Orders, and is thus proportioned. Divide D G, the Height, into 9 equal Parts, give the upper 1 Part to the Height of the Cornice. To divide the Members of the Cornice. Fig. II. Divide the Height into 10 equal Parts, give the firft 3' Parts to h m, the Height of the Denticule, the next 2 to the Height of the Cavetto , the next 3 to the Height of the Corona iv,* and the upper 2 to <v the Cyma' ReVerfa, with¹its Fillet /. Note, In the Plate the Cyma Ueferfa is, try Mifiake, indde a Cyma lecla, which' the Reader is drfredto correal. The Height of the Denticule, divided info 6 Parts, the Depfh of thcDentules' rr.aft be made �- of thofe Parts, therr Breaths % Parts, and the Intervals each r Paie

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		Of ARCHITECTURE. 139 Part and \. The Projection of the Cornice is equal to its Height. The Height cf the Plinth is \z Parts and §, as alfo is the Breadth of the Pilafter, of thofe 10 Parts into which the Height of the Cornice is divided, and the fraall Torus and Fillet on the Plinth is 2 Parts and §. If it is required to place Balls on the Necks over the Pilafters of this Order, the Height of the Neck mult be equal to the Height of the Cornice ; which be- ing divided into 5 Parts, give 2 to the Plinth, J the next I to its Fillet, and } of the upper 1 to the upper Fillet. The Diameter of the Ball is equal to the Dia- meter of the Pilafter, and theDiftances of the Pilafters are always the fame, as of the Columns over which they (land. UCT. XIII. Fig. II. PlateX. Of wreathed Columns. AS atfome times the Shafts of'the lonick and Corinthian Columns have been wreathed or twiiied, it is therefore neceffary to (hew, flow to defcribe a wreatbid or tivifled Column. Let a b c r be a given Shaft, ift. Bifect a b in G, and draw the Line Gc, make r p equal to r c, and draw iv p parallel to c r. Draw the Diagonal Lines fp, and dr, and make the Triangle dx.c equal to the Triangle p g r, on the Points z and g ; with the Radius g r, defcnbe the Arches pr, and dc; 2dly, Make p 0 equal to p iv, and draw e 0 parallel to c r, alfo draw the Diagonals ep, and 0 iv. Make the Triangle 0 p g equal to the Triangle c b <w, on the Points b and�, with the Radius b d, defcribe the Arches de, and p 0. ylly, Make«7* equal to ov, draw the Diagonals s 0, and en, make the Triangle s f e equal t� the Triangle ni 0, and on the Points/"and i, with the Radius i 0, defcnbe the Arches se. and n 0 ; 4thly, Make n /equal to n t, &c. and fo proceed to re- peat thefe Operations until the whole be completed, as required. L E C T. XIV. Plate XT. Of the Manner of dividing the Flutes and Fillets on the Surfaces of real Pilafters and Columns. "Wy ILASTERSare fluted in two different Manners, <viz. ehher with Fillets JL only, as Fig. N. or with Fillets and Beads at their Angles, as in Fig. M. THt Number of Flutes in the Front of a Pilafter ihould be feven precifely, although fome make lefs, and others more, but thofe are never done by an Artift or Workman. The Breadth of a Flute is to the Breadth of a Fillet, as 3 is to 1. In Fig. N. there are 8 Fillets, and 7 Flutes, which are thus found, -viz. divide the given Breadth of your Pilafter into 29 equal Parts, give 1 to each fillet, and 3 to each Finte. In the other Example, Fig. M. divide the given Breadth into 31 equal Parts, give 1 to each Bead, and the other 29 to the 8 Fillets and 7 Flutes, as in Fig. N. To readily divide the Flutes and Fillets of a Pilofler. Draw a Line at Fleafurc, as a b, Fig. N. and therein let off 29 any equal Parts from a tob. Make the equilateral Triangle azb, and from the 29 Divi- sions draw Lines to the Point z : This being done, fet the given Diameter of your Pilafter from z to d, and to c, and draw the (-ine c d, which wi 1 be divided at the Points e/� h i, �c. into its Flutes and Fillets, as required. For as c d is parallel to a b, therefore the Triangle c d z is �milar to the Triangle z a b, and confequently the Line c d is divided in the fame Proportion as the Line a b. In the fame Manner a Pilafter with Beads and Fillets is readily divided by an Equilateral Triangle of 31 Parts, as dab, Fig. M. To divide at once the juf Breadths of Flutes and Fillets on the Surface of a real Column. Let Fig. F. Plate XI. be the Plan of the Bafe, and Fig. E. of th� Top of a given Column, to be fluted with Fillets. -7 Operation.

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		J40 Of ARCHITECTURE. Operation. Draw a Right Line, as p q, Fig. I. at Pleafure, and having 2 Pair of Compares, open I Pair to any frnall Diftance, fuppofe q r, and the other Pair to one third Part thereof; now as thefe two Openings of the Compaffes are to one another, as the Breadth of a Fillet is to the Breadth of a Flute, therefore from p, towards q, fet off the two Openings, each 24 times reciprocally, that is interchangeably, as firft p r, then rs, then/, equal to p r, &c.* but you muft obferve that the two Openings aforefaid are fuch, that when you have fet each 24 times from p to q, that the Length from p to q be lefs than the Girt or Cir- cumference of you Column that is to be fluted, otherwife your Labour will be in vain From the feveral Divifions fo fet off, on the Line p q, draw Right Lines perpendicular top q, of Length at Pleafure, and then you may proceed to the finding of the true Breadths of your Flutes and Fillets, as following. 1 ft, Strike a perpendicular chalk Line from theAftragal to theCinfture on the Suiface of the Column, and being provided with a narrow ftreight-edged Piece of Parchment, �fe. girt about the Column at its Bafe, and cut the Parchment exactly to its Girt. This being done, apply one End of the Parchment to one Side of Fi�r. I. fuppofe at x, and its other End unto the other outer Line, as at a; then will xa, the ftreight Edge of the Parchment, be divided-by the afore- faid perpendicular Lines at the Points b c d efghiklm. &c. which are the true Breadths of the feveral Flutes and Fillets for your Column, and which being marked on the Edge of the Parchment with a Black-lead Pencil, apply the faid Parchment about the Bafe of your Column, laying one End unto the Chalk Line aforefaid, as at B, and prick eft' the Breadth of every Flute, as at a b, c d, ef, g h, i k, I m, &c. 20ly, Take the Girt of the Column under its Aftragal, and apply it to Fig. L as fiom n to 1, whereon mark the Breadths of every Flute as in the former, and applying one End of it unto the aforefaid perpendxular Line, as at A ; prick ofF theBreadrh of each Flute, as at the Points 1 2, 3 4, 5 6, 7 8, 9 10, 11 12, cifr. and then Chalk Lines being ftruckon the Surface of the Column, from the Di- vifiqns under the Aftragal to thofe at the Bafe, the whole Surface of the Column will be fet out ready for working, as required. Note, To know when a Flute is worked truly Semi-circular in a Pilafter, apply a Square within it, and if the angular Point and Sides of the Square touch theSur- face, and Extremes of the Flute, at the.fame time, as at p q r. Fig. G. Plate XI. the Work is true, otherwife it is falfe. And Flutes that are lefs than Semi-circles are proved by the very fame Method, only inftead of applying a Square, you muft apply a Bevel in the Manner following. As for Example, Let abc, Fig. H. Plate XI. be the Plan of a Flute ixshofe Depth is Irjs than the Radius of the Circle, ofiicbich the Flute is a Segment. Operation. Affume a Point in any Part of the Flute, as at b, s.n� draw the Lines be d, and b a f Nail together two ftreight Pieces of Lath, &c. fo as to make an Angle equal to the Anelefbd, and, to prevent its opening or fhutting 10 a gi eater or Idler Angle, tack on a Brace, as the Piece g e, then will your Level De prepared for Ufe, as the Square aforelaid. �V, By this Method the Height and Extent of any Scheme or rather circu- lai Arch being given, may bedeicribed without any Recourfe being had to the Cerner ; for if the Sides of the Bevel be kept to a and c, the Extent of the Flute, the angular Point/', oy Prob.XVL LECT. VI. Part II. will always fallon fome Pan or other of' the Arch abc;* and consequently if the Point b be applied to the Point», and then moved on towards b, tlience to c (the Sides of the Bevel bei»» a; v.ays kept Aiding clofe to the Points a and c) it will defcribe the Arch u bc,', w&cii is a Segir.eiu of a Circle, and without any Regard being had to its Center. Fig. Jl. and III. Plate XL fhews the Manner of making an Inftrument on Pai'icboard, or Ivory, for the ready letting off the Breadths of Flutes of Columns on s Drawing, without ih.- Trouble of defcribing and dividing of a Semi'circlr.

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		Of A R C H I T E C T U R E. i#t as before taught, which is an Invention of Mr. Edward'Stephens, Cabinet-maker, and thus made. Operation. Firft, defcribe a Semi-circle, as eg, of a larger Size than the Diameter of any Column that you may defignto draw ; divide its Circumference into its proper Flutes and Fillets, as before taught, and then drawing right Lines from them, to the Center a, the Inftrument is completed. Secondly, fuppofe you have the drawing of a Column to be fluted, whofe Semi diameter is equal to P a On the Center a c defcribe the fmall Semi- circle d i, 2, 3, 4, 5, �fc. which will cut the central Lines of the Inftrument, in the Points 1 23456, &c. from which draw right Lines with Black-lead, at right Angles to c g, and they will divide d a into unequal Parts, which are the true Appearances of the Breadths of the feveral Flutes required. And the Edge � a, being applied to the Diameter of the.Lolumn in your drawing, prick off the feveral Diviiions, which wiil be the Breadths of your Flutes and Fillets, as required Fig. III. Is another Inftrument of the fame Kind, made for fetting off tfts Flutings of Dorick Columns, acording to the Manner of the Ancients. LECTURE XV. Of the Manner of placing Columns againfi Walls, and over one another, as the Dorick on the Tufcan, the lonick on the Dorick, cifc. COLUMNS are placed either againft Walls, with a fourth Part of their Diameters infened, as Fig, III. and IV. Plate XXX. when three Quarters of the Body of the Shaft projects before the Upright of the Walls ; or entirely clear from the Wall, as Fig. 111. Plate XLIII. in which laft Cafe, a Pilafter is al- ways inferted in the Wall, as C and E, before the Columns D E; and the In- tercolumnation or Diftance of the Column from the Pilafter, is always the fame as when Columns are placed in Pairs. The Quantity oflnfertion of Pilafters mult be fuch as will be agreeable to the Parts of their Capitals. In the Tufcan and Dorick Orders the Pilafter may project before the Wall, a half, a third, a fourth, a fifth, a fixth, or feventh Part of its Diameter; but in the lonick, Co- rinthian, and Cowpofite Orders, they fhould be half a Diameter precifely, other- wife the Ornaments of their Capitals will be unevenly divided, and have a very bad Appearance. Whem Columns are to be placed over one another, as was the Cuftom of the Ancients, who placed an Order in every Story, we are to obferve, firft, That the Diameter of the Column in the f�cond Story be at its Bafe, equal to' the Diameter of the lower Column at its Aftragal ; and that they Hand exactly per- pendicular over each other, that the upper Solid may Hand on the lower. Se- condly, To place the upper Columns on a continued Pedeftai, whofe Height fhall be fo agreeable to the Windows, as to make the Cornice of the Pedei�a� do the Office of Stools to the Windows ; for when Columns have their Bales placed below the Bottoms of Windows, fo that their Stools being continued ftop againft the Shafts of the Columns, asthofe do at the Royal Banquet'tijig-fioitji at Whitehall', they have a very ill Effect. The Intercolumnation of Orders placed over one another mull be governed by the Triglyphs and Modilions, and therefore to place the Dorick over the Tu/can, regard, muft be had to the Num- ber of Triglyphs in the upper Order, to which theTu/can muft be conformable, as indeed muft the lonick to the Dorick in'fome Cafes, when the Diftances 'of its Modilions muft be made a little more or lefs to bring them into Order; and when the Corinthian is placed over the lonick, the Modilions of the lonick muft be conformable to thofe of the Corinthian. When an open Gallery is made over an Arcade, the Openings between the Columns may be quite down to the Bottom of the Pedeftai in the upper Ordrr, as in Fig I. Plate XLIV. but at fuch Times 'tis beft to place a Balluftrade be- tween the P�deltals, which will be a Security and sn Ornament alfo. T LECTURE

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		�4* O/ ARCHITECTURE. LECTURE XVI. Of the various Kinds of Ornaments for the Enrichment of'the federalMembers of ixhich the five Orders oj Columns are compofd. THE Ornaments that are, and may be invented for the Enrichments of Mouldings, are endlefs ; but thofe that are now in the greateft Efteem, I have introduced in the feveral Members of the laft four Orders ; not that every Order muft be fo fully enriched as I have exprefled, but fuch Parts of them only, as fhall be judged fufficier.t ; and that the Learner {hould not be at a lofs to know what Ornaments are proper for fuch Members, as he may be inclined to enrich." I therefore have been fo profufe, as to give every Member an agree- able Enrichment. And as oftentimes 'tis required to enrich Pannels, Piclure- Framcs, and other Parts of Buildings, I have therefore, in Plates XVI. XVJI. and XVIII. given a great Variety oi Ornaments at large; together with the Sec- tions of divers curious Mouldings for fuch Purpofes, of which take the following Account. I. The Figures E F I are Ornaments called Vitrwoian Scrolls, I fuppofe from Vitrwvius, who might be the Inventor of them. The Diilances of the Spirals is atpleafure; but their Height being divided into two Parts, their Diftance is generally .equal to 3 of thofe Parts, and their Spirals are defcribed by the Me- thods before taught. II. The Figures G H K L M are Interlacings, or Gui'cclns of various Kinds, of which G H K and L are compofed of the Arches of Circles, as is eviden: by Infpection, and that of Fig. M, of parallel right Lines, which form geome- trical Squares of any Magnitude connecled together, by Qnadrants on the Out- fides. The fret Ornament of the Ancients is by fome called Guilochi, of which in Plate XVIII. I have given Examples of 15 Kinds, for the Practice of the young Student, and whofe Number of Parts into which the Breadth of each is to be divided are fignified by Divisions, and numerical Figures againft each. III. The Eggs and Darts, commonly called Eggs and Anchors, as Fig. I. Plate XVI. are thus defcribed. Divide the Height 7 P into 9 equal Parts, at the Points 123456789. Firft, Draw a C and k B, parallel to 7 P, each at the Diftance of 7 Parts; and divide a 7 and 7 k, each into 7 Parts. Through the Point � draw e m, parallel toCB; make � e zn�fm, each equal to 4 Parts, and draw the Lines e 3 and m h. Through the Point 3, ou' the central Line 7 P, draw the Lines e 3 y, and m 3 <v. On the Point/", with the Radius/' 12, defcribe the Semi-circle 0 12p. On the Points e and m, with the Radius m 0, defcribe the Arches 0 <v and p y ; and on the Point 3, with the Radius 3_v, defcribe the Arch f 1 y, which will complete the Out-line of the Egg. Secondly, Draw the Line d I, through the Point 7, on the Line 9 P, and divide the Diftance between the Points 3 and 4, on the Line 9 P, into 2 equal Parts, and draw the Lines d % and I iv. On the Points d and/, with the Radius d h, defcribe the Arches h z and b iv ; arid on the middle Point, between 3 and 4., on the Line 9 P, defcribe the Arch <w z. Thirdly, Draw c g through the Point 8 ; make c 8, and 8 g, each equal to 3 Parts. From tire Points c and g draw the Lines g x and c A, through the middle Point between the Points 4 and Line 9 P. On the Points C and g, with the Radius ci, defcribe the Arches i A and 1 x ; alfo on the middie Point between 4 and 5 aforefaid, defcribe the Arch x P A. Fourthly, Through the Point 2, on the Line 9 P, draw the Line 2 r s ; as alfo draw the Lines i B, flopping at r ; alfo 12 B, and a r ; then one half Part of a Dart will be completed; and in the fame manner complete the other half, and all others. New from hence 'tis plain, that to fet out the Diilances of Eggs and Darts, you muft firft divide the Height of the Ovolo into 9 equal Parts" Secondly, Take 7 of �hofe Parts, and fet that Diftance along your Molding, and then Lines being drawn from Jthofe Points, fquare to the Top ar.-J Bottom of the Qvcto, eyery other Line

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		0/ ARCHITECTURE. i_4? will be the central Line of an Egg, and the others of the Darts, which divide - as albrefaid. Eggs in Ovolo's are oftentimes enriched with Leaves, Hulks, fcf<r. inftead of Darts, as between NOP. Plats XVI. IV. The federal Mouldings for Patmels and Picture Frames, Plate XVII. are thus divided. . I. Of Mouldings for Pannels, Fig. I. divide the Height into 3 Parts ; give two thirds of the upper I to A the R�gula ; the remaining 3d Part, and the middle great Part, to B, the Cyma reverfa ; half the lower Part to C the Aftra- gal : And the remaining half Part, divided into 3 Parts, give a to E the Ca- vetto, and I to D its Fillet. The Diftances of the central Lines a k, c d, ef, &c. of the Leaves, �Y. is equal to the Height of the Cyma B. Secondly, Fig. IL Divide the Height into 4 Parts, give the upper 1 to A the R�gula; the next 2 to B the Cyma Recla; and the lower i divided into % Parts, give the upper 1 to C the Fillet, and the lower 2 to D the Cavetto. Divide b d into 5 Parts, and fet off the central Lines of the Leaves, as a c, Zzfc. each at the Diftance of 7 Parts. Thirdly, Fig. IV. Divide the Height into 5 Parts ; give the upper 1 to A the R�gula, two thirds of the next t to B the Cavetto ; the next 2 Parts, with the Remains of the 4th Part, to C the Cyma Reverfa, and the lower Part divided into 3 Parts, give 1 to the Fillet E, and 2 to the Aftragal D. The Diftance of the central Lines of the Leaves, ^fc, b d, a e, c /, &c. is equal to the Height of the Cyma Reverfa. Fourthly, Fig. V. Divide the Height into 5 Parts, give the upper 1 to the R�gula, the next 1 to the Ovolo, 1 third of the next to its Fillet, the remain- ing 2 thirds, and the next 1 to the Cavetto ; and laftly, the lower I divided into 3, give the upper 2 to the Aftragal, and the lower 1 to the Fillet. Divide a (/into 9 Parts, and make the Diftance of a b, be, &c. equal to 7 of thofe Parts, as aforefaid. Fifthly, Fig. VI. Divide the Height into 3 equal Parts, and the upper 1 into 3 : give the upper 2 Parts to the R�gula A, and the Re. inainder, vvith the middle great Part, to the Ovolo B. The lower great Part divided into 2 Parts, give the upper 1 Part to the Allragal C, and the lower Part being divided into 4 Parts, give the lower 3 Parts to the Cavetto D, and the other 1 Part to its Fillet. The Djllances of the central Lines of the Eggs, &c, are to be found as aforefaid. II. Of Mouldings for Pi�ure Frames. First, Fig. III. Divide the Height into 4 Parts ; the upper 1 divide into 3, give i to the R�gula A, and 2 to the Cyma Reverfa B. Divide the upper half of the next Part into 2 equal Parts : give the lower Part to the Cavetto E, and the upper Part being divided into 3 Parts, give the upper 2 to the Aftragal C, and the lo.wer 1 to the Fillet D. Divide the lower 4th Part into 3 equal Parts, and the lower 1 Part into 3 Parts ; give the lower 2 Parts to the �avetto K, and the upper 1 to the Fillet 1. Divide the upper 3d Part into 2 Parts-; give the upper 1 to the Fillet G, and the Remains to the Aftragal H. Divide b d into 5 Parts, and make the Diftance of the central Lines of the Leaves, as a c, �fe. equal too of thofe Parts, the central Line of the Rofes to the Vitrwuian Scroll in the Freeze F, is direftly in the Midft of the Freeze, and tlie Diftance of the. Centers of each Rofe, as ef, is equal to the Height of the Freeze. Secsndly, Fig. VII. Divide the Height into 3 equal Parts, and each Part into 4 equal Parts; give the upper 1 Part to the R�gula A, the next 2 Parts to the Ovolo B, and the next I to the Fillet C and Cavetto D. Give the middle gr,eat Part, and 3, fourth of the lower great Part, to E the Freeze. Give the next fourth Part of the lower great Part to the Cavetto F, and Fillet G ; and then the Remains », being divided into 4 Parts, on z defcribe the Quadraritj x, and then, making c � equal tojjss, defcribe th� Curves y a, an&ab, which with the Qua- drant^, forms that Moulding which Workmen call the Weljh Ogee.* The Man- lier of defcribing the Guiloeoi in the Freeze is plain to Infpeftion, as alfo are th«. pittances of the Eggs, in B the Ovolo, and Leaves in H the Weljh Ogee. 7 z Tkirdl?

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		144- Of ARCHITECTURE. Thirdly, Fig. VIII. Divide the Height into 3 Parts, and each Part into 4 Parts, as before; give the upper 1 Part to the R�gula A, the next 2d Part and 1 third of the third Part, to the �volo B, the f�cond third Pare to the Fil- let C, and the Remains of the upper 1 great Part, to the Cavetto D. The middle great Part is the Height of the Freeze E. The lower greatPart. being divided into 4 equal Parts, give the upper I to the Cavetto F, and Fillet G ; the next 1 to the Aftragal H. The remaining 2 Parts, divided into 8 Parts, give one to the Fillet I, 5 to the Cyma ReCta K, and the lower 2 to the Fillet. To thefe Examples many more might be added, but as I mull not fweli the Work to a much greater Bulk and Price than is propofed ; and as hp that is Marter of thefe, will be able to invent others without End, I ihall therefore proceed to, LECTURE XVII. Of the Manner ofrufiicating the Shafts of Columns and Pilafters, Plate XLV. The Orders ufually rufticated are the 1'ufcan, Dorick, and Ionick. To rujiick the Tufcan Column, Fig. A and B. DIVIDE the Height of the Column into 7 equal Parts, and give 1 Part to each Ruftick, whole Projetions may be made equal to the Projection of the CinCture as in Fig. A, or equal to the Projection of the Plinth, as in Fig, B, and which in both Cafes may be made diminifhing with the Column, or Up- right, as expreffed by the dotted Lines ; but this lalt has a very heavy Appear- ance, and feems contrary to Reafon, by over-charging the fmalleft Part of the Shaft witlv the greateft Rullicks. To rufticate the Dorick Column, Fig. C and D. Divide the Height of the Column into 8 equal Part.- ; give z to each Ruftick, as I h and d d, and the fame to the Intervals c c. The Projections of thefe Ruf- icks are determined as thofe of the Tufcan. t To rufticate the lonick Column, Fig. E and F. Divide the Height into 9 equal Parts, give 1 to each Ruftick, and to each Interval, and determine their ProjeClures, as in the Tufcan and Dorick. To rufiicate Tulcan Pilaflers, Fig. G and H. Pilasters are rufticated in two different Manners, <viz. either champher'd, zs Fig. G, or rabbeted, as Fig. H. To rufiicate a Tufcan Pilajier ivith champher'd Ruftich, as Fig. G. Divide the Lleight of the Column into 7 equal Parts, and anyone of the Parts, as h y, into 8 equal Parts, give 6 Parts to the Height of the Face of each Ruftick, and 1 to each of its Champhers. The Projection x y of the Ruf- ticks, before the Upright of the Pilafter, is equal to i Part. To rufticate a Tufcan Pilafter, noiih Rabbet Ruftich, as Fig. H. Divide the Height into 7 equal Parts, as before, and one Part into 1 2 Parts, as at a. c. Make the Height of each Rabbet equal to two Parts, and then the Height of the Face of each Rultick will be 10 Parts; or if every two Parts be con�dered. as 1 Part, then each Rabbet will be I, and each Ruftick will be 5, as expreffed by Figures on the right-hand Side. The Projection of the Rufticks, before the Upright of the Pilafter, may be m~de equal to the Projection of the CinCture, or to the Height of a Rabbet ; but this lalt is rather too great, for then the Rullicks will have a very heavy Appearance. LECTURE XVIII. Oft; Block Cornices and ruftick Quoins, Fig. IL III. IV. V. VI. and VI�. Plate XLV». DIVIDE, a z. Fig. II. the given Height into 9 equal Parts, and give the ■ loweft 1 to the Height of the Plinth. Divide the upper 8 Parts into 1.4 Paits; give the upper 2 to the Height of the Cornice, and the lower. 12 to the it 'Rollicks, Divide the Height of each Ruftick into 4 Parts, give .3 to the "Face

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		Of ARCHITECTURE. 145 Face of each Ruftick, aiid §■ of I Part to each Champher. Divide I x, the Height of the Cornice, into 4 equal Parts, and give to each Member, as each Part doth exprefs. The Projection of the Cornice is equal to 2 Parts and -\| of the Cornice's Height. The Length of the ftretching Rufticks are equal to 3 Parts, and of the heading Rufticks to 2 Parts of the Cornice's Height, fet back from the Upright of the Quoin, Fig. III. IV. V. VI. and are V. different Ex- _samples, whofe Parts are proportioned in the fame manner as their feveral Di- vifions and Numbers exprefs. LECTURE XIX. Fig. I. II. III. IV. V. VI. Plate XLVI. Of the Marnier of proportioning the principal Parts of Doors, Windows, and Niches. TO proportion Doors to any given Height, Fig. IV. V. and VI. Firft, Divide the given Height in Fig. IV. and VI. into c equal Parts, - the upper I Part is the Height of the Architrave, Freeze and Cornice, and the lower 4 of the Door. Make^ h, in Fig. IV. and i k, in Fig. VI. each equal to 2 Parts for the Breadth of the Openings, and �� Part thereof is the Breadth of the Architraves x g, and k x- Secondly, Fig. V. Divide the Height into 4 equal Parts, and the upper 1 Part into 4 Parts, then the upper 3 Parts is the Height of the Architrave, Freeze, and Cornice, and the Remainder is the Height of the Door, whofe Breadth is equal to 1 great Part and a half, and its Architrave to � of the Breadth. The Breadth, of the open Piiafters k x, againfl: which TruiTes are fixed as at k, to fupport the Cornice, is equal to \| of the Breadth of the Architrave. Divide the lower 4th. Part, of the upper great Part, into 2 equal Parts, and that gives the Depth from the Cornice, at which the Foot of the Trufs is to be placed. The proper Trufs for the Support of thefe Kinds of Cornices is exhibited in Fig. I. Plate XIV. and is thus described. . To defcribe afpiral Trufs, for the Support of Cornices over Doors, Windows, and Niches. Divide A B the given Height (including the Height of the Architrave, Freeze, and Cornice) into 15 equal Parts, give the upper 4 to the Height of the Cornice, and the lower 11 to the Height of the Trufs. Let the Line M r; reprelent the Upright of the Face of the open Pilafter againft which the Trufs is fixed. Draw "Wen parallel to M z, at the Diftance of'two Parts and f ; aifo draw B » the Safe Line at right Angles to M z. From the Points 8, 4 and 2 in the Line A B, draw the Lines 8 g, 4 G and 2 E, parallel to B z, and of Length towards the Right-hand at pleafure: Thefe Lines laft drawn determine the Heights of the greater and lefler Spirals or Scrolls. Divider e, the under Part of the Cornice, into 9 equal Parts, and a g into 7 equal Parts ; alfo divide G E into 7 equal Parts, and make G y and E 8 equal to 8 of thofe Parts ; and this being done, proceed in every refpeel to defcribe the two Spirals, as you did thofe in the Corinthian Mo- dilion, Fig. V. Prob. IX. Lect. VIII. hereof. Fig. II. Is the Front-view of this Trufs, whofe Breadth H I is equal to B F, mis,, to 1 Part and \|. of the Parts in A B, and which being divided into 8 equal Parts, is defcribed in every Particular the fame as in, mxo p I, in Fig. III. the Face or Front of the Corinthian Modilion. To divide the Heights of the Members in the Cornice. The Height being divided before into 4 equal Parts, divide the lower 2 Parts into 4 equal Parts, give the firft I Part to the Height of the Cavetto V, the next 2 Parts to the Fillet T, the Dentule S, and Fillet R, and the 4th, or upper Part, to the Ovolo Q. The 3d great Part is the Height of the Corona P, and the next and laft Part is the Height of the Fillet O, the Cyma Recta N; and R�gula M. The Projection of the Cornice W X is equal to its Height W e. To divide the Dentules. Divide x x the Height of the Denticule into 6 Parts, and make the Length of a Dentule eqnal.tq 5 Parts. Make the Breadth of a Dentule and an Interval e$ual

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		i4<f O/ ARCHITECTURE. equal to the Height of a Dcntule, which divide into 3 Parts, give 2 to a Dentuic and l to the Interval. IL To proportion IVindoivi and Niches to any given Height, Fiff. I. II. and 111. Plate XLVI. Divide the given Height into 5 equal Parts, the lower 1 Part is the Height of thePedeftal, \yhofe Farts are to be divided according to the Pedeftal of any Order required. The remaining 4 Parts being divided into 5 equal Paits, the upper 1 Part is the Height of the Entablature, and their Breadths, if for Win- dows, into 2 Parts. The Breadths of their Architraves, as m n, Fig, III. is equal to f of the Opening, and of their open Pilafter, to f of the Architrave, as like- wife are the Margins 0 p and y r, Fig. II. when made into Niches. The pro- per Entablatures to be placed over Doors, Windows, and Niches are exhibited by Figures A B C D E F and G, Plate XLVII. But as fometimes the Quoins and Heads of Windows arerufticated, I have therefore in Plate XLV. given five Ex- amples thereof, vvith the Divisions of their Parts, which explains them to the meaijeft Capacity. LECTURE XIX. Of Pediments. PEDIMENTS are employed either for Ornament and UCe, or for Orna- ment only. Pediments for Ornament and Ufe are thole which are made on the Outfides of Buildings, and which mull be entire, that thereby the Build- ings underneath may be wholly protected from the Injuries of Rains. Entire Pediments are made in three different Manners, «y/ss, \ft, Streight, as Fig, IL Plate XLIII. which Workmen call a raking Pediment, idly, Circular, as Fig. I. Plate XLIII. And, ^dlj, Compounded of three Arches, as Fig. II. Plate XLIX. Th e Manner of finding the Height of the Fafiigium, or Pitch of a raking and circular Pediment, being already taught in Proe. I. LECT. V. hereof, I fliall therefore proceed to (hew ; Hon» to defcrfhe a compound Pediment, as Fig. II. Plate XLIX. A Compound Pediment has the fame Pitch as a raking Pediment, therefore to defcribe a Pediment of this Kind, draw the raking Bounds of a pitched Pedi- ment, as B A and A C, bjfecr. B A in b, and A C in d, al lb bifeft A d in c, and thereon erect the Perpendicular c F, cutting the central Line A F in F. Bifect B h in a, and dC in e, on the Points a and e erect the Perpendiculars a E and e D, which will cut the Perpendiculars C D and B E in the Points E and D. On the Points E � and F, with the Radius E B, defcribe the Arches B b, b, A d, and d C, and concentrick thereto, at the refpective Heights of the feveral Members of the Pediment, defcribe the whole as required. Pediments forOrnament are thofewhich are imperfect, and arevulgarly called Broker, or Open Pediments, as Fig. I. II. III. Plate XLVI1I. and Fig, I. and III. Plate XLIX. Thefe Sort of Pediments mould never be ufed without Buildings, becaufe being open in the Middle, they let in the Rains on the Cornice, in the fame manner as if no Pediment was there. It is therefore that thefe Kinds of :nts muff, be ufed within Doors for Ornament only, and whofe Opening is generally made for the Reception ofaBufto, Shield, Shell, &c. .Now feeing that to make an open Pediment without Doors is abfurd, to make an entire Pedi- ment within Doors, where no Rains come, muft be abfurd alfo. In the Trfcan Order, the Length of the raking Cornice, as A G,. Plate XLV1II. being divided into 5 equal Parts, as at 1, 2, 3, 4 ; the Length of the R�gula t G is equal to the 4 lower Parts. The fame is alfo to be cbierved in a circular open Pediment, as Fig. 1. Plate XLIX, But in a Dorici Pediment, the Length of the raking Cornice is to be regulated by theMutules, foras the raking Mutules, as H I, in the Pediment, muft be directly over A 3-, in the level Cor- nice, therefore the Ci&ance fir, the Projection of the Cornice beyond the Upright of the level Mutule K, being fstfiom 4 to b, and the Lin&� 20 being drawn, it cuts

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		0/ ARCHITECTURE. 147 cuts the raking Line s a into 9, making the Length of the raking Cornice re- quired. The Length of the raking Cornice of an lanick Pediment is determined by- placing a Modilion in Profile againft a raking Modilion, as Hagainft G, equal in Projection to � 5, the level Modilion in Profile, and making 5,1; the Projection �f the raking Cornice beyond the Upright of I, 13, the Upright of the raking Modilion in Profile, equal to the Projection of the level Cornice beyond the leveit Modilion in Profile. The laft raking Modilion in the Pediment is always at pleafure, according as the Breadth of the Opening of the Pediment is required ; and therefore it might have been either that over E or F, inftead of G over A. Note, The fame is alfo to be underflood of Pediments of the Corinthian and Compcfite Orders. Pediments are fometimes finifhed with Scrolls, as Fig. III. 'Plate XLIX- which are thus defcribed. Let A B C be the Extent and Pitch of a rakinp Pedi- ment. Bifed B A and AC, in b and g, find the Centers H D G, in the fame manner as you found the Centers E F D, in Fig. II. Draw the Lines b D, and g D, and on the Points H and G defcribe the feveral Members on each Side, as was done in Fig. II. Divide b A into 8 equal Parts. From the third Part draw the Line C D, and �n the Center D defcribe the Arch b c, and Members concentrick thereto ; make c e equal to 3 Parts and A of b A. Divide c e into 8 equal Parts, and on the �th Part from c defcribe a Circle, as the Eye of a Volute or Spiral, and therein find the Centers as before taught, on which turn about the two Cyma's, and finifh the Eye with a Rofe, &c. at pleafure. Note, Sometimes the Cyma Recla is left out of the Scroll, and the Cyma Re- verfa with the Corona only, are turned about to form the Scroll, which has s very good Effect ; and then in fuch a Cafe the Cyma Recta is ftopt, and returned is in an open Pediment. LECTURE XXL Of t ruffed Partition:. WHEN Partitions have folid Bearings throughout their whole Extent, they have no need to be truffed ; but when they can be fupported but in fome particular Places, then they require to be truffed in fuch a manner that the whole Weight {hall reft perpendicularly upon the Places appointed for their Support, and no where elfe. As Partitions are made of different Heights to carry one^ two, or more Floors, as the Kinds of Buildings require, therefore in Plate L. I have given fix Examples, of which Fig. II. V. and VI. are of one Story in Weight, and Fig. III. IV. and VII. of two Stories. The firft Things to be confidered in Works of this Kind, is the Weight that is to be fupported , the Goodnefs and Kind of Timber that is to be employed ; and proper Scantlings neceff�ry for that Purpofe. The Strength of Timber in general, is always in proportion to the Quantity of folid IV�atter it contains. The Quantity of folid Matter in Timber is always more or lefs, as the Timber is more or lefs heavy ; hence it is, that all heavy Woods, as Oak, Box, Mahogany, Lignum Vitte, &c. are ftr�nger than Elder, Deal, Sycamore, &t. which are lighter, or (rather) lefs heavy, and indeed, for the fame Reafon, Iron is not fo ftrong as Steel, which is heavier than Iron ; and Steel is not fo ftrong as Brafs or Copper, which are both heavier than Steel. To prove this, make two equal Cubes of any two Kinds of Timber, fuppofe the one of Fir, the other of Oak, weigh them fingly, and note their refpeclive Weights ; this done, prepare two Pi. ces of the fame Timbers, of equal Lengths, fuppofe each 5 Feet in i .engtrf, and let each be tried up as nearly fquare as can be, but to fuch Scantlings, that the Weight of a Piece of Oak may be to the Weight « f the Piece of Fir, as the Cube of Oui is to the Cube of Fir; then thofe two Pieces

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		�₄S Of A R C H I T E C T U R E. Pieces being laid horizontally hollow with equal Bearings, and being loaded in their Middles with increafed equal Weights, it will be feen, that they will bend or fag eq'ually, which is a Demonftration, that their Strengths are to each other, as the Quantity of folid Matter contained in them. As the whole Weight on Partitions is fupported by the principal Poft, their Scantlings muft be firft confidered ; and which fhould be done in two different Manners, tin. Firft, when the Quarters, commonly called Studs, are to be filled with Brickwork, and rendered thereon; and laftly, when to be lathed and plaiftered on both Sides. When the Quarters are to be filled between with Brick-work, the Thicknefs of the principal Polls fhould be as murh lefs then the Breadth of a Brick, as twice the Thicknefs of a Lath ; fo that when thofe Pods are lathed to hold on the Rendering the Laths on both Sides may be flufh with the Surfaces of the Brick- work. ; and to give thefe Pofls a fufficient Strength, their Breadth muft be in- creafed at Difcretion ; but when the Quarters are to be lafhed on both Sides, or �when Wainfcotting is to be placed againft the Partitioning, th;n the Thicknefs of the Pofts may be made greater at pleafure. The ufual Scantlings for princi- pal Pofts of Fir, of 8 Feet in Height, is 4 or 5 Inches fquare; of to Feet in Height, 5 or 6 Inches fquare ; of 12 Feet in Height, 6 or 7 Inches fquare ; of 14 Feet in Height, 7 or 8 Inches fquare : of 16 Feet in Height, from 9 to 10 Inches fquare. But thefe laft, in my Opinion, are full large, where no very great Weight is to be fupported. As Oak is much llronger than Fir, the Scant- lings of Oak-Pofis .need not be fo large as thofe of Fir ; and therefore the Scant- lings affigned by Mr. Francis Price, in his Treatife of Carpentry, are abfurd ; as being much larger than thofe that he has afijgned for Fir-PoJIs. To find the juft Scantling of oaken Poft, that mall have the fame Strength of any given Fir- Fcfts, this is the Rule. As the Weight of a Cube of'Fir is to the Weight of a Cube of Oak of the fame Magnitude, fo is the Area of the fquare End of any Fir-Poji, to the Area of the End of an oaken Poft ; and whofe fquare Root is equal to the Side of the oaken Poft required. The Diftances of principal Pofts is generally about loFeet, and of the Quar- ters about 14 Inches, but when they are to be lathed on both Sides, the Dif- tances of the Quarters fhould be fuch as will be agreeable to the Lengths of the Laths, otherwife there will be a very great Wafte in the Laths. The Thicknefs of ground Plates and Raifings are generally from 2 Inches and half to 4 Inches, and are fcarfed together, as expreiled in Fig. I. K L M N O P Q R. In the feveral Examples aforefaid the principal Pofts have their Inter-tics and Braces framed into them, as exprefled in Figures F BGHCDAkE, whole refpective Places the feveral Letters in each refer to. L ECTURE XXII. Of r.akid Flooring. THE principal Things to be obferved in naked Flooring is firft the Difpo- fition of Girders, or Manner of placing them in the moft fecure and ad- vantageous Manner. Secondiy, their Scantlings, and laftly, the Manner of trufTir.g them, when their Lengths require it. There are fome Carpenters, who infift that Girders fhould be laid on ftrong Lentils over Windows, and who allcdge that Girders, being laid on Lentils in, Rers, the Piers are endangered at the Decay of thofe Lentils. Others infift, that 'tis beft to lay Girders in Piers, as being the molt folid Bearings, and that if feund oaken Lentils are laid under them, they will endure as long as the Brick- work will remain found. In Buildings, whofe Piers are narrow at the renewing of Lentils, the Piers will be endangered in both thefe Cafes; for Lentils laid over Windows muit be laid into the Piers, on both Sides of a Window, and which, when taken otot, will make large Fra&ureSj chat will be very little lefs dangerous than the other,

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		Of A RC HITECT�RE. 149 Other, and therefore I fliall fubmit this Point to the Difcretion of the Work- men. Lentils laid in Piers between Windows, for the Support of Girders, fhonld have their Length;; equal to the Breadths of the Piers : And thofe laid in Party- walls, or Gable-ends of Building, fhould be equal in Length to the Diftance that is contained between every two Girders. The Thicknefs of Lentils fhould al- ways be equal unto the Height of z or 3 Courfes of Bricks, and their Breadth unto a Brick's Length ; fo that in every of thofe Particulars, they may be con- formable to the Brickwork in which they are. placed, and to that which is raifed on them. And for the better difpofing of the Weight impofed on Girders, Lentils fhould always be-firmly beded on a fufncient Number of fhort Pieces of Oak, laid a-crofs the Wails, vulgarly called Templets, which are of excellent Ufe. Let Girders be laid in Piers, or in Lentils over Windows, it will, in both thefe Cafes, be commendable to turn fmall'Arches over their Ends, that in cafe their Ends are i�i�t decayed, they may be renewed at Pieafure, without difturbing any Part of the Brick-work; and, for their Prefervation, anoint their Ends with melted Pitch and Greafe, �via., of Pitch 4, of Greafe t : and indeed, we e Lentils to be covered with Pitch and Greafe alfo, it would contribute verv greatly to their Duration. It is always to be obferved, that the fhorteft Girders bend dowh, or fag», as Workmen term it,' the lealt, and thereforeit is always belt to lay Girders over the narrow Parts of Rooms, and whofe Ends fhould always have each, at leaft 14 Inches bearing in the Walls, excepting in fmall Buildings, where the Front, �sfe. Walls are but a Brick and half in Thicknefs, when to prevent the Ends of theGirders from being feen without Side, their Bearings cannot much exceed 11 Inches. It is alfo to be obferved, that Girders be fo difpofed of, that the Boards cf every Floor be parallel throughout the whole Floor; for it is as difagreeable to the Eye, to fee the Joints of Boards in the fame Floor, lie different Ways, as it is to fee Steps out of one Room into another, which fhould always be avoided. In the carrying up the feveral Walls of Buildings, it fhould be carefully ob- ferved to lay in Bond Timbers on Templets, as aforefaio, at every 6 or 7 Feet IB I eight, cogged down, and braced together with diagonal Pieces at every Angle, which will bind the whole together, in the molt fubitantial Manner, and prevent Fraciures by unequal Settlement. The Diftances of Girders fhould never exceed 12 Feet, and their Scantlings muft be proportioned according to their Lengths ; as by Experience it is known, that a Scantling of 11 Inches, by 8 Inches, is fufficient for a Fir Girder of 10 Feet in Length, the Area of whofe End is 88 Inches, it is very eafy to find the proper Scantling fora Girder of any greater Length, fuppofe 20 Feet, by tins Rule: As 10 Feet, the Length �f the fir ft Girder, is to 88, the Area of its End, fo is 20 Feet, the Length of the f�cond Girder, to 176, the Area of ks End. Now, to find its Scantlings, that being multiplied into each other fhaH produce 176 Inches, the .Area found, one of them mult he given, <««. either the Depth, or the Thicknefs. In this Example, the given Depth (hall be 12 Inches; there-! fore divide 176 by t2, and the Quotient is 14 inches and 2 thirds, which is the ether Scantling or Breadth required. To prevent the fagging of fnert Girders, it is iifoal to cut them Camber, thaH«f «0 cut them with an Angle in the ivlidit of their Lengths, fo that their Middles fhall rife above the Levels of their Ends, as many half Inches as th� Girder con- tains times 10 Feet. And indeed Girders of the gieateit Lengths, although trailed, fhould.be cut Ctrr.bcr ia-thefams Manner.' ' In Plate LI 1.1 have given three different Examples fo�the tfuf�injjof G h '■ �nd in Plate LUI. Fig. I. a fourth* which being in gei r�-nfpet «fare fubmil the fihoiCs t� the Difcretion of the '■■> jtkmaa,

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		i₅o ^ARCHITECTURE. Th e next in Order are Joifts, of which, there are five Kinds, viz. Common- "Joijis', Binding-Joifts, Trimming-Joijis, Bridging-Joifts, and Cieling-Joifts. Firfl, Common-Jqifts are ufed in ordinary Buildings, whofe Scantlings in Fir are ge- nerally made as follows, mix. Jo:j1s,o� 6 Feet in Length, to be 6 and half by 2 and half; of �Fcet, 6 and half by z and half; of 12 Feet 8, by 2 and half. But in large Buildings, the Scantlings are made larger, where it is common to make Joifts of 6 Feet, 5 by 3 ; of 9 Feet, 7 and half by 3 ; of f 2 Feet, 10 by 3. As Oak is much heavier than Fir, it is cuftomary to make the Scantlings of Cak-Joifts larger than thofe of Fir, but I believe it to be entirely wrong, for the Reafon before given relating to the Strength of Timber. Secondly, Binding-Joifts are generally made half as thick again as Common-Joifts of the fame Lengths, which are represented in Fig. V. and VI Plate LI. by nmqp, feV. and which are framed flufli with the under Surfaces of Girders, to re- ceive the Cieling-Joifts, and about three or four Inches below their upper Sur- faces, for to receive the Bridging-Joifts; fo that the upper Surfaces of the Bridging' Joijis may be exactly flufh or level with the Girder to receive the Boarding. In Fig. IV. Plate LI. A reprefents the Seilion of a Girder; bb, &s. Parts of two Binding-Joifts, tenoned into the Girder, a a, ISc. the Ends of Bridging- 'Joijls ; e e Boarding on the Bridgings ; d d, &c. Mo� tifes in the Binding- Joijts to receive the Tenons of Cieling-Joifts ; as alfo are the Mortifes, b c, be, &c. But thefe laft are thofe which are called Pulley-Morcifes, into which the Ciel- ing-Joifts are Aid. To underftand this more plainly, the Figures////are add- ed, which reprefents the Sections of fo many Binding-Joifts ; g g, �5V. the Seclions of fmall Joilts between them ; x x a Side-view of a Bridging-Joift ; and h h h Cieling-Joifts, tenoned in the Binding Joifts, flufli with their Bottoms, as aforefaid, to receive the Lath and Piaffer. The Dittance that Binding-Joifts fliould be laid at, fnould not exceed 6 Feet, tho' fome lay them at greater Di- llances, which is-not fo well, becaufe the Bridging and Cieling-Joifts muft be made of larger Scantlings, to carry the Weights or the Cieling and Boarding, and confequently a greater Quantity of Timber muft be employed. But how- ever, as this Particular is at the Will of the Carpenter, I fhall only add, that the Scantlings for Bridgings of Fir, having 6 Feet Bearing, fliould be 4 by 3 Inches; thofe of 8 Feet Bearing, � and half by 3; and thofe of 10 Feet, 7 by 3. The Pittance from each other is generally about 12 or 14 Inches. The Fig. ABCDEFGHI, exhibits different Kinds of Tenons for Binding- Joifts, which are to be pracYtfed as Occafions require. The Figures V. and VI. exhibit the View of a Floor over two Rooms, wherein the Girders F F are laid in the Piers C A D B. In Fig. VI. the Binding-Joifts n m q p, &f<r. and brimming Joifts arereprefented fingiy, without the Bridging-Joifls ; andin Fig.V. the Bridging-Joifts are laid on the Binding-Joifts, as when ready for to receive the Boarding. This Example is given only as a Specimen of thefe Kinds of Plans, that from thence the young Student may the better know how to reprefent Plans of floors, when required, The Figures II. and III. are Examples of Floors made of fliort Lengths, which 1 have given for the Diverfion of the Curious. L E C T. XXIir. Oft Roofs and their Coverings. BE F O R E we can proceed herein, a Plan of the Building to be covered mull be made, by which we may acquire a juft Knowledge of the Dimen- i,, ns of every Part that will be contained in'the whole Defign, before any Part of the real Work be begun : and by which we fhall alfo be taught how to per- form every Operation at once in the leaft time, and to account for or eftimate the Quantity of Timber that will be employed. Supfose amrs,Fig. II. PlateLll. be the Plan of a regular Building tobe cover- ed, which is 50 Feet by 25 Feet in the Clear within ; iiviT. make a Parallelogram, by

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		Of ARCHITECTURE. i₅* by a. Scale of equa! Parts, vvhofe Length (hall be 50 of thofe Parts, and Breadth 25 Parts, which will reprefent the Infide of the Building. Secondly, without the Side and Ends of this Paralellogram, draw Right Lines parallel thereto, at the Diftance of the Breadth of the Raifing, fuppofe 1 Foot, equal to 1 Part of the Scale. Thirdly, as the Diflance at which Beams are laid mould not exceed 10 Feet, on account of the Lengths of Cieling-Joifts which are framed in between them ; therefore divide the Length of the Plan, with as many Beams as are ne- ceilary,, as at the Points b i k I, and / <vxy; and draw the central Lines of the Beams b t, i v, k x, and /y ; as likewife the central Lines of the Plan 1 10, and z. iv, and the Bafes of the Hips az, rz, and 87;.', 8 s. Fourthly, confider the Height of the Pitch, which let be equal to 6, 5 ; then the Lines 5 k, and j x, are the Lengths of a Pair of principal Rafters, the Angle 5 k 6 is the Angle or Mould for their Feet, and the Angle 6 5 k for their Tops. On the Points 2 and 8, erect, the Perpendiculars z, 3 ; 2, 4 ; and ,8, 7 ; 8, .9. Draw the Lines a 3, 7-4, m 7, s 9, which are the Lengths of the four Hip Rafters; the Angle 2 a 3 is the Angle or Mould for all their Feet, and the Angle a 3 2 for all their Tops, and which, with the Lengths of the principal Rafters being meafured on your Scale of equal Parts, will give you their true Lengths in Feet and Parts of Feet. This being done, make your Raifing equal to the Magni- tude of the Building, and brace its Angles, as n », t$c.. which will be a very great uren.gthcning to them. Divide out the Diitances of the Beams, and cog them down on the Raifmgs, as at cdef, which is a fecure Method to tie the Building together. Set out the Mortifes for the deling-joi/is in the Beams, fo . that the under Surfaces of the Joifts may be fluih with the under Surfaces of the Beams, and obferve that the Diitances of the Cieling-Joijls be agreeable to the ufual Lengths of Laths, that no Wafte be made thereby in the Lathing. The like Caution fhould alfo be taken in theDiftance of Rafters, for very often the Tyler is injured very greatly in the Wafte of his Laths. When the Lengths and Angles of the Principal and Hip Rafters are thus discovered by the Plan, we muit then confider the proper Scantlings for them, and for the Beams on which they ftand. When Beams exceed 20 Feet Extent, it is always beft to trufs them up in one or more Places, as their Lengths may re- quire. Beams fhould never exceed 1 c Feet in their Bearings, nor Rafters more than 10 Feet, and efpecially in Roofs of very low Pitch, vvh�fe Covering has a .much greater Prefl'ure on their Rafters, than thofe of higher Pitches, and which may therefore in fome Cafes exceed 10 Feet. The Height or Pitch of a Roof fhould be agreeable to the Building it covers, and to the Kind of Materials it is to be covered with. The Kinds of Covering in England ire four ; -viz. Lead, Pantyles, Plain TyUs, and Slates. Firit, Coverings of Lead ate, of all others, the moil beautiful, but theExp.cnce being the gteateft, it is therefore never ufed but for to cover mag. niikent Buildings. The Height of Roofs covered with Lead is at Pleafure, bu-t now it is generally ufed for Roofs that are very low, and which is common!; 2 Kinths oi the Building's Breadth, which is called Pediment Pitch. Secondly, Coverings _c.f Pantyles may be alfo ufed to low R.oofs, but the general Pitch is 3 Eighths of the Building's Breadth. Thirdly, Coverings of plain Tyks and S lat. s have generally the higheft Pitch, on account that, when they are laid on low Roofs, the driving Rains will enter between them. The Pitch allowed for thefe Kinds of Coverings is that, whole Rafter's Length is equal to 3 Fourths of the Building's Breadth, and which is called true P'tih. To form trie Truffes for principal Rafters, we muft divide the Length of the Rafter into fome Number of equal Parts, each to-contain about 10 Feet ; and at thofe Parts place fuch Collar Beams, Prick.Pods, and Struts, as are fufficient to fupporc them. In Plate Llil are 15 Cefigns for the truffing of principal Rafters, whofe Beams extend ic, 30, 45, 60, and 73 Feet, and whofe feveral Pitches are made agreeable to the aforeiaid Coverings. Fig. � and R are Extents, if Feet each, thefrft for Lead, the laft for Pantyles, which U 2 require

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		i₅2 Of ARCHITECTURE. lequi-c no Help from Collar-Beams, c5V. but Fig. T, of the fame Extent, being higher, and confequently has longer Rafters, muft be helped by a Collar-Beam placed between them, and for tne fame Reafon Fig. K, whofe Beam extends 3oFeet, muft have two Collar-Beams, whilft Fig C and D, of the fame Extent, whofe Pitches are lower, and Rafters are fhorter, will each do with one Collar- Beam. When the Extent of Beams are fuch, that, the Lengths of Collar-Beams will be too great, which (hould never exceed i 5 Feet at the moft, the Weight of the Rafters and their Coverings muft be fupported by Prick-Pofts and Struts, framed ■ into King-Polls, by means of which the Beams will be miffed up fecure, and tha whole Weight ftrongly fuflained. For this Purpofe all the remaining Examples in this Plate, and thofe in Plate LI V. are given, which being in general confpi- cuous, requires very little more Explanation. In Plats LIV. the Figure E exhibits the Manner of framing the Foot of a principal Rafter into the End of a Beam, where a is a Part of the Rafter, f �, a Part of the Beam, and c d, the Tenon of the Rafter's Foot in its Mortife. TI18 Fig. C exhibits the upper Part of a King-Poft, with its Joggle d d, into which e e, the upper Parts of two principal Rafters are framed, whole Shoulders b b mult be made truly fquare to the Joggle. The Fig. B exhibits the Manner of fram- ing the lower Parts of Struts, as h e, into the Joggle of a King-Poft, as at a bd, whofe Shoulders (hou Id alfo be fquare to the Joggle, or as nearly fquare as poffi- ble ; n v is an Iron Strap, to bind the Beam g g unto the King.Poft B, which is bolted through the King-Poft at nn. As the common Method of framing the Truffes of principal Rafters of lar^e Roofs, is to lay the whole Weight of the Beam and Covennc upon their Feet, they therefore mould be fecured at the Beam with Iron Straps, to prevent their flying out, in cafe that their Tenons (hould fail. According to this Method all the '1 ruffes in Plate LIII. are made ; but as I apprehended this Method was cap- able of Improvement, I therefore confidercd. that if under the lower Parts of principal Rafters, there be diicharging Struts framed into the Beams and Prick- Foils,'as a b, e /, Fig. A. Plate LIV. they will difcharge the principal Rafters from the greateft Part of the whole Weight. The Trfcis, Fig. F, hath its Struts turned the contrary Way to all thepre- ceding^ and the whole Weight is taken off the Rafters, by the difcharging Struts c c and h g, for th� whole Weight that hangs on the King Poll is At- tained by the Struts «ja'and b f, which are f attained by the Prick-Polls c d and hf, which are fuflained by f he difcharging Struts ce and bg. In the fame Man- ner the Weights of the Truffes Fig. G, M, R,P,S, andT, are difcharged by their difcharging Struts, which are ffiaded to diflinguilh then, from the others. The Truffes H L N arc for Buildings that have arched Cklings, which are tied in, by their Hammer Beams I i, in Pig. H, ek, and'/"/, in Fig, L, and d i, ana d,* in Fig."N, which muft be made-very fecure >j Straps gnd Beits, as at/- and V, ^: i,ti% H. The Truffes G and I admit of Gcrrecs. But the Top of Fig. 1, which is called a Trunk-Ro�f, muft be covered ». ith t cad. The! ruffes O Q_R and S are Trufies for Ivi Roofs ; thofe of O R and S are wholly fupported by their King Polls and Struts, but that of Q^mufthave its Guiter at a fupported either with a Parts-Wall ortruffed Partition, as Fig. K, whofe principal Polls are a a, ts'c. the Gutter-Plate dd. fe\'. and Struts cc. The Trufs, Fig. D, as alfo"; Fig. 3, Plate LV. arc for the Roofs of Churches, which are fuppofed to be fupported within Side by Columns at b and c Tufiriextand lait Kind of Roofing, whofs Timbers are ftreight, is that of Spires on the Towers of Country Churches, as Fig.G. Plate LVJ. The Height or Pitch of Spires is from 4 to C of the Tower's Diameter on which thejr Hand. .As as the l�verai Hips have an equal Inclination, they do therefore trufs up each other. The Bale of a Spire is generally an Octagon, whofe Manner of framing is exhibited by Fig. A, which if made of good Oak, and fecurely (jolted down on the Heads of eight principal Foils, fixed oji the Sides of the Tower.,

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Of ARCHITECTURE. 153

Tower, will ftand unto the End of Time, could the Materials endure fo Ion?.
The f�cond Example, Fig. C, has its Spire placed on anOswRoof at e f framed.
together as Fig. B, which is reprefented at large, and whofe Bafe^ /�> is framed
together, as Fig. D. The third Example, Fig-. H, whofe Spire is placed on a
Lanthorn, is fomething more difficult than the preceding, and therefore Fig. F
is given to (hew the Manner of framing the Lanthorn, and Fig. E the Cirb to
the Lanthorn's Head.

As I have thus given a brief Explanation of thefe feveral Sorts of Truffes for
ftreight Rafters, it will be neceffary to fay fomething of the Scantlings of Beams
and Rafters before I proceed any further.

I. Of the Scantlings for Beams.

If the Length \ 4c / .. _c ...
c o );' its scantlings

or a Beam < 60 >

(hould be

of Fir be

II. Of principal Rafters.
Inches.
5 by 6
If the Rafter \ 36 � its Scantlings \ 7 6 � , _t.

be of Fir, -2 41 > at Top (hould

Bottom

and its Length / 60 1 be

III. Of final/ Rafters.

reei.

, of \ /its Scantlings
be 7 C fhould be

Inches.

3 y- ' ■

If the Lengththe Rafter b

5 3

.6 3

Cip.cular Roofs are the next that come under our Consideration, which are,
Firft, Cylindrical, as Fig. A. Plate LV. Secondly, Spherical, as Fig. G and N.
Thirdly, Spheroidical, as Fig. D, which two laft are vulgarly called Domes.
Fourthly, Trumpet-mouth'd, as Fig. C A. Fifthly, Bell-Roofs, as Fig. IJL
Sixthly, Bottle or Ogee Roofs, as Fig. M. And Laftiy, Compound Roofs, as
Fig. C and L. And as by Infpeftion it is plain, thatthefe Roofs in general have
their Truffes formed by the fame Principles as the preceding, I need only add,
that Fig. V is the Plan of a Spheroidical Dome, whofe feveral Truffes are con-
nected together at their Tops by the horizontal Braces, abed, on which the
Lanthorn D is erected.

Fig. H is a half Plan of the Spherical Roof or Dome, Fig. G, whofe Pur-
loins cf d, and c hgi k. are reprefented by the concentrick Semi-circles 5 348,
and 6 127, and the Bale of each Trufsby the central Lines q%», r z, sx, ta,
andji 'v. The feveral Pubs,' or principal truffed Rafters, mult diminifh as their
Bafes at, x s, &c. and may either be framed into a horizontal Cirb at Top, as
nv z x ay, or connected together as in Fig. F, on which the Lanthcrn F may be
erected.

Now as by the preceding we have taught how to find the Lengths of our fe-
veral Rafters, to give them their proper Scantlings, and to fupport them and
their Beams, in fuch a Manner as the Nature of the Work fhall require, I fhall
now proceed to (hew,

Hotu to lay out Roofs in Ledgement, Fig. IV. Plate LVI�,

To lay out a Roof in Ledgement is no more than to lay out the Skirts and
Ends ; but thereby is taught now to find the Lengths and,Angles of every par-
ticular Part, and con-eouuvly the Quantity of the whole.

Example

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		�54 G/ ARCHITECTURE. Example I. Le t a h c d, Fig. IV. Pate L VII. be the Plan of a Raifing to a fingle regular fcip'd Rcof, wherein zy, I 3, 2 2, n p, are Beams ; on and op, the Out-lines of a Pair of principal Rafters, 0 r the Height of the Pitch, r b and r d, alfo am and c m, the Bafe of the four Hips, r s and r q, each e-qual to r 0, the Height of the Kip Rafters, whofe Lengths are s b and q d. On the Ends a c and b d make the Ifofceies Triangles bed and a f c, whofe Sides be, de, and a f, f c, are each equal to sb, the Length of a Hip-Rafter. Continue thecentralLinesof the Beams xy and np to /and x, and tok andiv, making iz,lm, yiv, and^jv, each equal to the Length and Breadth of a principal Rafter ; and draw the Lines ai, k /and //-, alfo civ, "jjx an�xd: This being done, draw in fuch other principal Raf- ters as are requifite, and between them the Purloyns, as 8, 9, 6, 5, 7, �fc. at Difcretion, observing not to place any two Purloyns directly oppofite, whofe two Moi tifes would weaken the Principal very much ; laftly, between the principal Rafters draw in the fmall Rafters, and then the Lengths and Anglesof every par- ticular Part of the whole Roof will be determined, and from which a juftEfti- rnate of the Quantity of Timber that will be employed therein (Regard being had to the Dimensions or Scantlings of the feveral Parts as aforefaid) may be made. In Fig. VI. the Angle O P R being equal to the Angle op r, in Fig. IV. there- fore the. Angle at P is the Bevel of the Feet of the principal Rafters, as the Angle at O, lor the fame Reafon, is the Bevel �f their Tops ; and the Aiigle S BR, Fig. V . being equal to the Angle s b r, in Fig. IV. therefore the Angle at B is the Bevel of the Feet of the Hip-Rafters, and S is the Bevel of their Tops. The Fig. A B, on the left Hand, exhibits a Joint made by a Purloyn and a Hip, as by ai, and the Purloyn 12, 14, the Meafure of whofe Angle is the Arch if 3, ij. Fig. VII. reprefents a Pair of principal Rafters truffed up, on whofe Prick-Polts is placed a Cupola, as e f i g h. THEnextin Order is, to find the Angles of the Jack-Rafters againft the Hips, and to back the Hip Rafters. As Jack-Rafters are parallel to one another, therefore all their Angles againft the Hips are the fame. To make the End of a Jaci-Rafitr ft to the upright Side of a Hip-Rafter. There are two Angles to be formed, that is, the one upon the uppei Surface of the Jack-Rafter, the other on its Sides from theEnds of the former. The Angle ©nits upper Surface is the Angle made by the upper Edges of the Jack and Hip; and which is that, that every Jack Rafter makes with the Hip-Rafter in the lodge- ment, as every oi the Angles between e and d. Therefore from your drawing inLedgement, fet your Bevel to oneof thofe Angles, and the l�vera! Jack-Rafters �being cut to their refpeftivc Lengths, at their upper End» on their upperSurfaces, apply that Bevel, and defcribe the upper Angles. This done, tak;; the Mould S, made for ihe Tops of the principal Rafters, and apply it agatnft theSides of each Jack Rafter, Et the Ends of the Angle on their upper Sat faces, and by its upper Edge draw Lines; then from the Line of the upper Angle, through the Lines on the Sides, faw through the Ra'fter, and that Cut will be the Angle required. To find the Angle of the Bad of a Hip-Rafter. From the Point c let fall a Perpendicular, as c h, on the Hip fa ; make c � equal to c h ; alfo make a i equal to a c ; draw the Lines c g and g i, and the Angle c g i will be the Angle or Back of the Hip required. Example II. Fig. V. Plate LIX. This f�cond Example is of a regular double Roof, which is hip'd as the pre- ceding, with Valleys within-ude.' The Out-lines of this Plan are a f g k, wherein h B, B E, E / and i h. are the Ridges, a B, E �, ; k and h g are tne Hips, h A C, B A C, D A E, and D A i are the Valleys, A C, D A the Gutter, fq the Height of the Pitch, p q and qr a Pair of principal Rafters, v'f and t k Hip-Rafters. By the lalt Example, lay out the Ends/ 'i i i, and ali'jg, alfo the Skirts a b e f and g A I k; continue � II to c, k 1 to d, a h to .-, and/ e to d, and tecaufe .the Lengths of the Valleys aie eqeal

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		Of ARCHITECTURE. i₅₅ equal to the Lengths of the Hips, therefore make Ht, I d, be, and de, each equal to one of the Hips, as I k, and draw the Lines c d and c d: This beinf done, draw in all the principal and fmall Rafters at Difcretion, and then the whole will be completed, as required. Example III. Plate LX. This third Example is of an irregular double Roof, whofe Ends are hip'd, and whofe Plan is tpzzyz, wherein rs, so, ol, \m,mnj, and <or are its Ridges, t s, p o, 2 i, z m, <vy, and r x are its Hips, r q, s q, and nv <f are the Valleys, iv q a Gutter, and m i o n a Flat 5345 and 578 are two Pair of principal Rafters, t s, r x are the Bafes of the Hip Rafters, / 1 z, and 1 3, _x, p 0, and 2 1 are the Bafes of the Hip-Rafters, 2, 1 4, and p r ; z m is the Safe of the Hip-Rafter k 1 6, and njy of the Hip-Rafter y I 7. On the Points s and r creel: the Perpendiculars .r 1 2, and 1-13, each equal to the Height of the Pitch, and draw the Lines 12/, and 1 3 x, which are the Lengths of thofe two Hip Rafters. In the fame Manner, on thcPointsa, 1, m, �v, erec� Perpendiculars of the fame Height, and draw the other Hip-Rafters : This dont;, by the firft Example lay out the whole in Ledgement and fill tip the feveral Skirts and Ends f g b i, Ik, c a, and d e, with their principal and fmall Rafters, which will complete the whole, as required. Note, If the drawing be made on thick Paper, and the whole be cut out by the Outlines, you may, by bending the Drawing on the Lines of the Eaves and Ridges, fold up the whole, and thereby form a real Model of the Work to be done. Example IV. Plate LVIII. This Example is of an irregular Roof, whofe feveral Angles are bevel, where- in t s aq is the Plan, 1 1 e ; iz, I; 13,/; and 1 4, 0 ; are the Beams over which the principal Rafters are to ftand. Let the Line c n be the Bafe of the Ridge, which is to be placed at Pleafure, and let tc, ac, and n s, nq be the Bafes of the 4 Hips; on the Points cgkn erecl: the Perpendiculars c d, gf, k 2", and n m, which make each equal to the Height of the Pitch, and draw the Lines d 11, de ; � 12, f h ; i 13, il; m\\,mo; which will be the Lengths of the feveral principal Rafters. At the Points c and n, ere� the Lines nr, np, and cv, cb, perpendicular to the Bafes of the Hips, and each equal to the Heights of the Pitch, and draw the Lines tar, a b, and r s, pq, which are the Lengths of the feveral Hip-Rafters ; make s x and x q, the Sidis of the Scalenum Triangle s x q, equal to rs and pq, alfo / nv and iv a, equal to tv and a b, which will complete the Ledgement of the Ends. Make 1 4, z, equal to the principal Rafter 14 m, and s z equal to the Hip r s ; alfo make 0 z equal to the principal Raker 0 m, and qz equal to the Hip p q ; a!fo makeifj/ equal to the principal Rafter de, and ay equal to the Hip iv a ; .alfo make 8 y equal to the principal Rafter^ 8, and t y equal to the Hip t v. Make y W and y Y each equal to c g ; alfo WX and YZ each equal to g k ; alfo Xz and 'Zz each equal to k n. Draw the principal Rafters 12W, 13 X, and b Y, / Z. Laftly, draw in thePurloyns 21, 22,20,23,24» at Difcretion, and they will complete the whole Ledgement, as required. As the Beams lie oblique to die Raifings, therefore all the principal Rafters muft be backed, which is thus performed. Let 'd c, Fig. E, reprefent that Part of the Railing that is at the Foot of the principal Rafter de ; alfo let C E reprefent a Part of the Beam Lie; and b the I ">wer Part of the Rafter d e ; and make the Angle D E C be equal to the Angle de c. . FnoMthe Pointjr, in Fig. E, erecl the Perpendicular y x ; then the Font of the Rafter being made equal to the Angle D E C on the left hand Side, fei o't the Diilance z x, and from the Poins z ftrike a Chalk Line up the Side of the Rafter parallel to its upper Edge, and rhen a Fletch being cut off from v, the right har-d Angle to the Chalk Line aforefaid, the Rafter will be backed, as required. In the uir.jc Manner the other Rafters f h, il,ko, i$-c. reuft-be backed-, as cxpriflud

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i_S6 O/ARCHITECTURE. expreffed by the F/gures H L and O. And the Angles DEC and E D C, in Fig, E, being equal to the Angles dec and e de, &e. are the Moulds for the Top and Foot of the Rafter de, &c. The fame is alfo to be underflood of the Moulds of the l�verai Hip-Rafters, in Figures T V C, R N S, D C E, and P NQ> whofe Angles are equal to the refpeftive Angles of' the Feet and Tops of thoi'e Hip- Rafcers agninlt which they are placed. The next and lail Work is to back the Hip-Rafters, which is done by this general Rule. Through any Part of the Bafe Line of a Hip-Rafter, as the Point \ o in nq, craw a Right Line as g 8, at Right Angles, cutting the Out lines of the Raifing in the Points g and S. From the Point to let fall a Perpendicular on the Hip- Rafter^ q, as io, o ; make 10, z, equal to 10, o, and draw the Lines gz and 2 8, then the Angle g 2 8 is the Angle of the Back of the Hip/> q, as required. L E C T. XXIV. Of the Manner of defcribing An^le Brackets and Hip-Rafters in pohgonal Roofs. AS Brackets are ufed very frequently in Buildings. I fhall therefore fhew how to find the Curvature of any Angle Bracket by one general Rule, as lol lows. Let A, in Fig. VI. Plate LIX. be a Front Bracket given, whofe Height is db, its Projection a b, and its Curve a Cavetto; and let the fhaded Parts bd reprefent an Angle of a Building, againft which the Cove is to be fixed. Draw the Lines a h and hi parallel to the two Sides of the Building, at the Diftanceof the Projection of the Front Bracket; and draw 7 a¹ the Bafe of the Front Bracket, and f h the Bafe of the Angle Bracket ; divide 7 c into any Number oi equal Parts, as at the Points 6, 5, 4, 3, 2, 1, and draw theOrdinates 6, 8 ; 5, 9 ; 4, 10 ; 3 li; &c. divide h f into the fame Number of equal Parts as 7 f is di- vided, which will be done by continuing the Ordinates of 7 c, until they meet hf in the Points 6, 5,4, 3, &c. whereon ere� the Ordinates 1, 13 ; 2, 12 ; 3, 11; &c. equal to theOrdinates 1, 13; 2, iz; 3, it, &c. on the Line 7 c ; and through the Points 13, 12, 11, io, 9, 8 /, trace the Quarter of an Ellipfis, which is the Curve of the Angle Bracket required. B y the fame Rule, all other Kinds of Angle Brackets may be defcribed, and which is very evident. By fig. I. II. III. IV. VII. VIII. IX. which exhibits all the Varieties of Brackets, at acute, right, and obtufe Angles, and wherein the Front Bracket in each Example is expreffed by the Capital A, and the Angle Bracket by the Ca- pital B. The Curvatures of Hip-Rafters ta polygonal Roofs, that is, thofe whofe Plans are Polygons, as the Figures I L 1V1 N, Plate LVI. are alfo found by tranfpofing the Ordinates of a principal Rafter (which muft be given) upon the Bafe of a Hip-Rafter. Suppose, in Fig. I. a d to be the Bafe, over which the Cavetto principal Rafter c dis to (land, and let a e be the Safe of a Hip-Rafter. Divide a d'vaxo equal Parts, and draw the Ordinates z, 1 ; 4, 3 ; &c. on the Line ad; divide « fin the fame Manner as a d, and on the Line a e draw the Ordinates 1,2: 3. 4; 5, 6 j CSV. and from the Point b, through the Points 2, 4, 6, 8, <3c. trace the Curve of the Hip-Rafter as required. In the fame Manner, in Fig. L. the principal Rafter c d being given, the Hip-Rafcer b e is found ; as alfo are the Hip-Rafters b e in Fig. M, and c e in Fig. N. the principal Rafters being firlt given. L E C T. XXV. Of the Formation of the Heads of Niches.

	N	IC H � S, quafi Nidi, or Ncfls, of old Concha,, were a Kind �f PluieUs, or fmall Tribunals, and are fo called by the Italiins to this Day, wherein

	Statues arc placed to protect them from the Injuries of Weather. The Heads of Niefos are made 4 difrexe&� Ways, as, fif, with Brick? ; fecondlj, with Store ; thirdly.

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		Of ARCHITECTURE. 157 thirdly, with Ribs or Quarters, lathed and plaiftered, or covered and lined with (lit Deal, �sV. and, lafily, with divers Thicknef��s of Plank glew'd upon one another. Those made with Ericks or Stone are built upon Centers of Wood, which are the very fame as thofe which are covered with flit Deal, and are of two Kinds, wz. the one femi-circular, the other femi-elliptical. I. To make the Center f or the Headofa feini circular Nicb,- Fig. VII. Plate LX. Make a femi-circular Raifing, equal to the Plan of the Niche, and cut out as many Ribs as are neceflary, each equal to half the Curve of the Raifing, and of the fame Curvature ; cut out the curved Front, whole Breadth is at plcafure, and whofe Curve muit be equal to that of the Raifing ; This done, fix your Front- piece on the Ends of the Raifing, and then the Distances of the feveral Ribs being fet out on the Raifing, as at the Points cdefghikl, fix thereon the feveral Ribs, which connect together at a, and then will they be ready t� receive their Covering and Lining alio, if required. To cover or line thz Head of a Niche, Fig. K. Plate LVI. Let a f c be the Plan of the Head of a femi-circular Niche, and complete the Circle afc d. Draw the Diameters a b c, and dbf, continued out towards e at Pleafure. Make f r, and/" s, each equal to I fourth of a f ; then/" s will be equal to half a f and draw the Lines h b and s k Divide b d into any Number of equal Parts, and draw the Ordinates 1,8; 2,9; 3, 10 5 &c. 'and on the Points where thofe Ordinates cut the Semi-diameter b d, with the Ra- dius of each Semi-ordinate, defcribe Semi-circles, as the dotted Semi-circles in the Figure. Make e f equal to the Curve a f. Make fp equal torn;/» equal to a 2, fn equal to a 3, f m equal to a 4, � /equal to a 5, f k equal to a 6, and fq equal to a 7. On the Point e defcribe the Arches 13, 14; 11, 12 ; 9, 10, ciff. Bifedl the half Part of each of the dotted Semi circles, as/r in /', I 8 in 2, 3 9 in 4, 5 10 in 6, 7 11 in 8, 9 12 in 10, i1 13 in 12, and, 13 14 in 14. Make//», and/g, each equal to half the Arch//; pi, and ^2, each equal to half the Arch I 2; » j, and 0 4, each equal to half the Arch 3 4; and fo in like manner, n J, and 5 6, to half the Arch 5, 6, &e. From the Points, through the Points 13, 11, 9, 7, �sV. and 14, 12, 10, �fc. trace the Curves e b and e g ; then four fuch Pieces, as e hg, will cover the Head of the Niche, as required. Note, If the Niche be to be lined, then the Diameter of the Circle,, bein» made equal to the infide Diameter of the Niche, the lining may be found in the fame Manner. The fame Method is alfo to be ufed, for the covering or lining of a Semi eiliprical headed Niche, as is plainly feen by Fig. O, where every of the (ame Operations is performed on the Plan of an Ellipfis, and where e h s is. the Covering for 1 eighth of the whole Hemifpheroid. A fometimes the Niches ara made femi-polygonal, it is neceffary to fhew their Covering alfo, and which is of great Ufe in the Covering of polygonal Roofs,.as thofe of Banqueting.Houles, Turrets, Mc. Let Fig. L. Plate LVI. be a Plan given, whofe principal Rib or Rafter is ( d, and Hip b e. Make the Length of if equal to the curved Length of c d, and draw the Lines g a and b a. Draw the Ordinates to the principal Rib t don its Bafe a d. Make the feveral Diftances k i, 1,2; 2, 3, on the Line Af equal to the feveral Parts of the princial c d, as they are divided by the Ordinates, making/- i equal to the firft Part from d; 1, 2, equal to the f�cond Part, 2, 3 equal to the third, (3c. Divide k a in the fame Proportion as a d, a,t the Points 1, 3, 3, isc. through which draw right Lines parallel to g h, to terminate at the Lines g a and h a ; alio through the Points 1, 2, 3 ; in the Line � f draw right Lines at Pleafure, and parallel to g h. Then making the Lines 1,, 7 ; 2,8; 3, 9; fjrV. on the Line if, equal to the Lines 1, 7 j 2, 8 ; 3, 9i (yVi on the Line i a ; and from/, through the Points 13, 12, 11, &c. to h, trace the Curve//». In the fame Manner trace the Curve/g.- Then the Piece f g b, being bended up, and laid on the two Hips that (land over the Linc^ a X iv.d

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		i₅8 O/ ARCHITECTURE. and h a, will be the Covering for that Side of the Roof or Niche, as re- quired. Note, The Covering to the two Ogee Roofs M and N, and the Cavetta Roof I. are found in the fame Manner, as is evident to Infpe�ion. II. To make the Center of the Head of a femi elliptical beaded Niche, Fig. IX, X, XL Plate LX. Let b � d, Fig. XI. or e b g, Fig. X. be the Plan of an elliptical-headed Niche. Firft, Make the Raifing and Front, each equal to the Plan, and fix them together. Secondly, Cut out the middle Rib, which is a Quadrant whofe Radius is equal to a f, and fix it on the Raifing at/", and to the Front-piece at a, as in Fig. IX. which will keep the Front-piece in its true Pofition. This done, fet out the feverai Diftances of the other Ribs as at g b i k, CSV. in Fig. XL and draw the Lines g a, h a, i a, and k a. Thirdly, If the Lines g a, h a, i a, and k a, be each considered as the femi-tranfverfe Diameters of fo many Ellipfes, whofe feverai femi-conjugate Diameters are each equal to the femi- conjugate Diameter a/, then one half Part of every of thofe Semi-ellipfes will be the true Curves for the l�verai intermediate Ribs, that are to Hand on the Raifing, at g b i k, &c. and which being connected together, as-at a, in Fig. IX. and either covered or lined, by the Rule before delivered, the whole will be completed, as required. III. ^ll o make a femi-circular headed "Niche, ivith the Thicknejfes of Boards, Plan fa, &c. glewed upon one another, Fig. XIV. Plate LX. First, let c a e be the Face of the Niche, defcribed on a Wall or flat Pan- nel, &c. Divide its Height i a, into fuch equal Parts, as will be agreeable to the Thicknefs of your Plank, as at the Points 4, 7, &c. thro'which draw right Lines parallel ta e e. On the Edge of your Plank fix a Center, and de- fcribe a Semi-circle thereon, equal to the Plan of your Nich ; apply � Square to the Center, and draw a Line on the Edge to the other Side, to find the oppofite Center, whereon, with a Radius equal to 4 6, defcribe another Semicircle; then with a turning Saw, cut through from I Semi-circle to the other, and then your firft Thicknefs is made. Secondly, on the Edge of your next Piece of Plank fix a Center, and thereon defcribe a Semi-circle equal to the laft. Apply a Square to the Center, and find the oppofite Center as before, whereon with the Radius 7 g, the half L�ngth of the Line that paffes through the next equal Part, defcribe another Semi-circle ; and with a turning Saw, cut through from one Semi-circle to the other, and then is your f�cond Thicknefs made. Pro-' ceed in like manner with all the remaining Thickneffes, obferving to make the under Semi-circle of every Piece, equal to the upper Semi-circle of the next laft, and which being glewed together, when the whole is dry, clear off the Infide, with a circular fmoothing Plane, whofe Curve is fomething quicker than the Curve of the Niche. IV. To make a femi-eilipiica! headed Nietf, tuiih the Thicknefs of Boards, Planks, &c. glewed «pen one another. Fig. XV. Plate LX. "Let d b e reprefent the femi-elliptical Niche required. Divide its Height a b into equal Parts as before. Make a b c, Fig. XIII. equal to b a e, Fig. XV. Makea c, and c d,* at right Angles, and each equal to a 6, the Height of the Niche, Fig. XV. and on c defcribe the Arch a 3, which reprefents the middle Depth of the Niche. Divide a c, Fig. XII. and a b. Fig. XIII. (which ars each equal to b a, the Height of the Nich, Fig. XV.) into the fame Number of equal Parts, and from thofe Parts draw Lines parallel to c d, and b c ; then will the Parallels in Fig. XIII. be femi-tranverfe Diameters, and the Parallels in Fig. XII. will be femi-conjugate Diameters of the feverai Ellipfes, which are to be defcribed on the upper and under Surfaces of the feverai Ticknefles of' Planks, {yV. in the very fame Manner as the Semi-circles in the preceding Ex- ample, and which being glewed together in iike manner, will form a i'emi- elliptical headed Niche, as required.

			LECTURE

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		Of ARCHITECTURE. i#> LECTURE XXVI. Of Timber Bridges. BRIDGES of Timber differ very little in their Truffes from thofe of Roofs, as is evident by the feveral Defigns in Plate LXI. and LXII. In Plate LXI. T have three Defigns ; that of Fig. IV. is an Aperture equal to 30 Feet ; that of Fig. I. to 45 Feet ; and that of Fig. V. to 60 Feet. The Fig. fl. is a Seclion of the feveral Profiles, whofe Breadth is equal to 50 Feet. The Piles that fupport the Truffes of thefe feveral Defigns are fuppofed to rife a fufficient Height above the flowing of the Water, fo that the Joints in the feveral Truffes erecled thereon may not be affected thereby ; and when the Depth of. a River is fo great, that the Length of Piles above the Bed of the River mud exceed, when driven, 25 or 30 Feet, then Super-Piles muft be erected upon horizontal Beams, mqrtifed down upon the Heads of the lower Piles, as in every of thefe Examples. The Scantlings proper for Piles to fuch Bridges mould not be lefs than one Foot in Diameter, at the Middle of their Lengths The Fig. III. reprefents Part of the Plan, with the Bafe of two Truffes, a and b, whofe Diftances in the Clear mould not exceed 10 Feet; be- caufe on them the Joifts which carry the Floor of the Bridge are laid. The under Piles muft be Ihod with Iron, that they may the better penetrate through the feveral Stratums of Earth, into which they are to be driven. Before Piles are driven, the whole Weight of the Framing that is to come on them, and the Weight of the Planking on the Joifts, Clay, Gravel, Pavement, �V. mould be eitimated nearly to the Truth ; otherwife the Piles cannot be driven with any Certainty, and which is thus to be performed, <viz. Divide the total Weight to be fuHained, by the neceffary Number of' Piles, and the Quotient will be the Weight that each Pile is to fupport. Then each Pile being driven until it re- fill a Force much greater than the Weight it is to fupport, it may be depended on, that afterwards there cannot be any Settlement by the Weight it is to fuftain. The Scantlings for the Beams of Truffes fhould be about 12 Inches by 9 Inches, as alfo mould be the feveral King polls. But the Struts and Joifts need not exceed 9 by 6 Inches, and the Plank on the Joifts being made 3 Inches in Thicknefs, will be fufficient. Before the Timbers are worked (which is fup- pofed to be of the belt Oak) 'tis bell to cut them out to their Scantlings, and lay them in a running Water for a Month at the leaft, to foak out the Sap, which is very deftru&ive, and then dry them throughly over a Saw dull Heat, &c. before they are worked. If this be carefully done, and the Work kept dry whilft working, and being truly framed, there will be no fagging in the Work, as ufually happens by the lhrinking of the Timbers, when they are not thus fhrinked before working ; nay, I have experienced, that Timbers {o pre- pared have always fvvelled afterwards, and made the Joints much clofer than, when firft put together. It is alfo advifable, for the better preferving of the Tenons, that every Mortife and Tenon be well covered over with a good Body of white Lead, and boiled Linfeed Oil, which will endure along Time, and will not permit any Rains to enter the Mortifes, to the Prejudice of the Tenons. The Ends of the Joifts mould alfo be covered with brown Paper, dipped in Pitch, and Sheet-lead laid over the Paper. And for the more effectual preferving of the Plank and Joifts, the Plank ought to be covered with a (Irong Clay firmly ramm'ddown unto about 9 Inches in Depth, on which the Road of Gravel and Pavement, or Gravel only, of a fufEcient Thicknefs is to be laid, with aRifing in the Middle, to difcharge hafty Rains to the Sides, as exhibited by B, in Fig. 1. Plate LXII. In Plate LXH, are two other Defigns, each of 100 Feet Opening, which I Jnade for the New Bridge at IVeftminJhr ; but believing that Interelt was pre. dominant to real Merit, I therefore declined to trouble the Honourable Com mif- X z fioners

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		�6o Of ARCHITECTURE. fioners therewith, as I have now the Publiek, in hopes that they may be of forhfc help to Invention, if not worthy of being put into Practice, over Rivers, where large Openings are required. The Defign, Fig, It. is of prodigious Strength, as being-a double Trufs, and whofe Timbers are fo fixed together, that not any Pm of the whole can fag the hundredth Pait of an Inch, they being prepared before worked, as afore- faid. . Fig. I. is a Section of the Breadth of the Bridge, wherein A A, �sV. repre- fents the feveral Truffes, for the Support of the Jbifts and Roads. A and C reprefents the Foot-ways, each 10 Feet in Breadth ; and B, the Horfe-way, 30 Feet in Breadth. As the Officers of the Struts a d e t b I c k i, &c. are obvious to every difcerning Kye, I need not fay any Thing thereof. � The Fig. V. contains a double Defign. the Struts on the Side G being dif- ferent from thofe on the Side H. Both thefe Defigr.s are of immenfe Strength ; and as the whole is laid on Stone or Brick Piers, which rife above the flowing of the highslt Tide, a Bridge of this Kind will be of very great Duration. As there is fome Difficulty to lay Foundations for Stone Piers in Rivers that,are affected by Tides, and as in wooden Bridges the molt early Decay is in that Part of trie Piles, that are affected by the rifing and falling Waters of the Tides, there- fore to avoid both thefe Inconveniences, fach Piers may be thus erected, tiia. Con- fider the Weight of a Pier, and the Weight that the Pier is to carry. Af- fign the Place in the River where the Pier is to ftand : Bore the'Ground for 15 or 2c Feet in Depth, that a Judgment may be formed how long the Piles muft be. This done, drive a Range of Piles, dove-taild together, at about 1 ^ Inches, without the Upright that the Stone Pier is to be erected, all round the Limits of the Pier, and the like exactly under the Upright of the Pier. Thefe two Ranges of Piles form within the Ground a firong Enclofure, about the en- compaffed Earth on which the Pier is to ftand. Within the Limits enclofed drive as many Piles as fhall be thought fufficient to carry the Weight, and which fhould be driven nearly all equally ; that is, firlt, to drive them all to fuch a Depth, as to keep them upright in their Places. Secondly, to drive them all about 2 Feet lower, and then all two Feet lower again ; and fo on, un- til each Pile be firmly driven, as aforefaid. By this regular driving down all the Piles together,.they will caufe the inclofed Earth into which they are driven, to be equally comprefied, and of much greater Compactnefs than it was before, as being confined by the double Ranges of Piles firft driven. When all the Piles are thus driven, their Heads mult be fawed level, at about 18 Inches below the Surface of the low Water ; and to render them imperifhable, the ■whole muft be filled up with ftrong Clay, let down in large fquare Pieces, worked very ftifF, and well ramm'd, which is a Workeafy to be performed, al- though the Depth of Water fhould be 20 F'eet. When this is done, prepare 2 double Floor of Oak Timbers, free from Sap, each Floor about 10 Inches in Thicknefs, pin'd down one on the other, fo that the upper Timbers lie at right Angles a-crofs the lower. Fix this Floor on the Piles, and thereon erect the Stone-work, to any Height required, 'i'he next Work is to �1! up the Space between the outer Range of Dove-tail'd Files, and the next inner Piles, to pre- ferve the inner Range from being injured by the Flux and Reflux of the Tide ; and which being firmly performed, the whole Fcuudation will be rendered as imperilhable, as were all the Piles driven into the very Bed of the River, as be- ing fecur�d from the Actions of both Air and Water. The outward Range of Dove-tail'd Piles are all that are liable to decay, and as their Office is no more' than to fupport the outward Cafe of Clay, which is there placed to preferve the next inner Range of Piles, they are eafily and foon repaired, as their Decay» pecur. -■Note, The outer Range of Piles malt be made of fuch a Length, as to rife fomething above the Level of High-water; and horizontal Beams being mortifed dawn on their Htads, with horizontal.Ties la^d through the Thickaefs of the Pier

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		Of ARCHITECTURE. \6i Pier In fmall Arches turned for that Purpofe being cogg'd down on the Beams, they will be a lading Prefervative and Defence to the Piers, againll all the Infults of tempeftuous Weather and Navigation that can happen. Note, If the Depth of Low-Water be any thing conliderable, it will be a very fecure Way to drive a Range of oblique Piles, jnft within the Limits of the up- right Piles, as Braces, to Heady the next within, from inclining either way by the Weight of the Pier. If inflead �f Timber Truffes, 'tis required to make Arches of Stone, a fuf- ficient Number �f Piles muftbe added within every Pier, that, with the others, Wiil be capable to carry the additional Weight of the Arches. Note alfo, That Piers built with well burnt Bricks, kid in Terrace, on aBsfc- ment of large Blocks of Stone, about 3 Feet in Height, will be much cheaper than being made entirely of Stone, and of longer Duration : For Well burnt Bricks do not decay fo fall as Portland Stone, which is very evident by St. Paul's Cathedral, where the Stone in many Parts of the South Side is already decayed more than the 10th Part of an Inch. LECTURE XXVII. Of Brick and Stone Arches to Windows, Doors, &C. I. Of freight, circular, elliptical, Gothick and rampant Arches in fir tight Walls, Plate LXIII. IN this Plate is exhibited 13 Kinds of Arches, of which Fig. I. II. III. IV. V. VII. VIII. and IX. are Arches of Brick-work, and the others of mili- tated Stones. In Fig. I. and III. the Dillanceof the Center, to which all the Joints have their Sommering, is equal to the Breadth of the Window ; but thofe of Fig. II. and IV. is the Center of a geometrical Square, whofe Side is equal to their Breadth. Fig. V. is a femicircular Arch, whofe Joints fommer to its Center. Fig. VII. and IX. are femi-elliptical Arches, the firft on the conjugate Diameter, and the laft on the tranfverfe Diameter. The Courfes in Fig. VII. arc divided on the inner Curve h f m, and outer Curve a en, into the fame Number of equal Parts, as alfo is the right-hand Side ai Fig. IX. whofe left-hand Side has its Courfes Sommering to c and / the Centers of the Ellipfisi Fig. VIII. is a Gothick Arch, whofe Courfes have the fame Sommerings as thofe of Fig. IX. In all theleCafe-s the only Thing to be obferved is, that theNumber of Courfes into which each is divided be an odd Number, that thereby the middle Courfe may be perpendicular, and that the Breadth of each Courfe on the upper Part of the Arch be fomething lefs than the Thicknefs of a Brick, to allow for rubbing. The ruliicated Arches, Fi�. VI. X. XI. and XII. have the fame Sommering as thofe of Fig. V. VII: VIII. and IX. To divide their Key-fiones and Rufiicks. Divide each half Arch into 9 equal Parts, as in Fig. V. givei to half the Key- flone, the next 1 g to its Counter-Key, and 2 to each Rullick and Interval, as the Figures exprefs.,. The like is alfo to be obferved in all the other Arches. The Arch, Fig. XIII. is "a rampant Semi circle, whofe Curvature may be de- fcribed by Prob. XIX. Le�. IV. Part II. or as following. Let � h he the Breadth, and f g the Height of the Ramp; draw g h, and in the Middle of f h erect the Perpendicular ^ a, of Length at pleafure ; alfo draw the Line � r paral- lel to/h. From the Point of Interfeclion made by the Lines g h znd/h, fet ap half the Breadch of the Opening to a, and draw the Lines a g and a h. Bifect g. a in m, and a h in 0, and erecl the Perpendiculars m n and 0 p ; then the Point «is the Center of the Arch g d, and ph the Center of the h.rc\idb, which divide into Rulticks as in Fig. VI. Then the Length of the Rufticks mud be equal to J of the Opening, and of the Intervals to \|- of the Rullick, as exhi- bited by �lJ, Fig. VI. II. Offlreight, circular and elliptical Arches in circular Walls, Plate LXIV. The firft Work to be done is the making of the Centers to turn thefe Kinds of Arches upon, which may be thus perforiu'd. Let G H I K be the Plan of a circular

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		��2 �f ARCHITECTURE. circular Building, and at Fig. VI. 'tis required to make a Center for a Semi-circti- �ar Arch to the Window, whofe Diameter without is a d, and within » m. Bifett a din f and defcribe the Semi-circle a p d. Divide a d into any Number of equal Parts at the Points 642, csV. and draw the Ordinates 6, 6 ; 4, 4 ; 2, 2 ; fcfr. Divide» m into the fame Number of equal Parts, and make the Ordinates 6, 5 ; 4, 3 ; 2, 1 ; &"c. equal to the Ordinates 6, 6 ; 4, 4; 2, 2 ; cirV. arid through the Points 531^, &V. trace the Curve n k m, then a p d and « /t m wil! be the two Ribs for the Center : This being done, place the Ribs perpendicular over the Lines a d and n m, and cover them, as Centers ufually are, and then applying the Edge of a Plumb-rule to the divers Parts of the In-fide and Out-fide of the Window's Bottom, the Top of the Rule will give the feveral Points at which the In-fide and Out-fide of the Covering is to be cut off, fo as to ftand exactly over the In-fide and Out-fide of the Building, and then the Center will be completed, as required. To divide the Courfes in the Arch of this Window. On a flat Pannel, &c. draw a Line, as b e, Fig. VII. make a f 0 equal to the Curve a c d, alfo make a b and o e each equal to the intended Height of the Brick Arch. Make f p in Fig. VII. equal to e p in Fig. VI. alfo make a b and de in Fig. VI. each equal to b a in Fig. VII. then the Points b and e will be the Ex- tremes of the Arch. Make p r in Fig. VII. equal to b a the given Height of the Arch, and through the Points b r e and a p 0 defcribe two Semi-elliples, which divide into Courfes as before taught, and which will be the Face of the Arch required. To find the Angles or Bevels of the Under part of each Courfe. Continue the Splay-Backs of the Window m d and n a until they meet in F. On F, with the Radius F « and F a, defcribe the Arches n y 11 zn�af s, making ny <v equal to the Girt of the Arch n k m. Make » 6, »4, » 2, ny, &c. on the Arch r. y <v, equal to » 6, »4, n z, n'y, dsV. on the Curve n k m, and draw the Lines 6 F, 4 F, 2 F, y F, U5�. make the Ordinates 6, 5 ; 4, 3 ; 2, 1 ; y x, &c. on the Lines 6 F, 4 F, &c. equal to the Ordinates 5,6; 3, 4 ; j, 2 ; h i ; &c. on the Line n m, and through the Points 5, 3, 1, x, �V. trace the Curve �ox n. In the fame Manner transfer the Ordinates 5, 6; 3, 4; 1, 2 ; c,f; &c, on the Line a d to the Arch s f a, as from j to 6, from 4 to 3, &c. and trace th� Curve sea; and then will the Figure ny w s c a be the Soffito of the Window laid out, and which being divided into the fame Number of equal Parts, as the under Part of the Arch a p 0, Fig. VII. and Lines drawn to the Center F, as is done in Fig. II. to the Center A, by the Lines 2, 2, 2, &c. thofe Lines will give th� Bevel of every Courfe in Soffito, as required. Fig. V. is another Example of a femi-elliptical Arch, whofe Front is Fig. IV. Alfo Fig. II. is a third Example of a Scheme Arch, whofe Front is Fig. I. And Fig. VIII. is a fourth Example of a ftreight Arch, which in general are performed by the aforefaid Rule. To find the Curvature of every Courfe in Front. Suppose therufticated femi-circular-headed Window, Fig. IX. be (landing in the Side of a Cylinder, whofe Sides are the Lines Q_T and P V, continue out the Sides of each Ruftick until they cut the Sides of the Cylinder in the Points Q_R S T and N O P, &c. then the Lines O^N, R O, Q^N, (jfc. will be tranf- verfe Diameters of fo many Ellipfes, whofe conjugate Diameters are each equal to the Diameter of the Cylinder, which defcribe as in Fig. X. and draw their con- jugate Diameters k /, / m and n 0 ; make the Dif ances e 5, »; 3, / I, on each Ellip- sis, equal to a g the Semi diameter of the Window, Fig. IX. alfo make the Diftances 56, 3 4, 1 2, on each �llipfes, equal to g 10 the Height of the rullick Arch; then the Segments of the feveral Ellipfis, j, 6 ; 3,41-1,2; at Z X A, will be the Curves of the feveral Courfes, as required. Fig. Ill- reprefents the Manner of covering the Out-fide of a Cone, the Arch c a being made equal to the Circumference of the Circle e, which is equal to the Bafe of the Cone : This Figure is exhibited here to fhew, that the Soffito of a femi-circular-headed Window, whofe Splay is continued allround, is no more than

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		Of ARCHITECTURE. 163 than the lower Superficies of a Semi-cone ; for if the Splay was continued in � very Part, it would meet in a Point, as the Lines id b and /' e h, Fig. VIII. and form a Semi-cone as aforefaid. This is illuftrated by Fig. V. Plate LXVII. where / <i> iv reprefents the Sec- tion of a Wall, in which is placed a circular Window, as Fig. A, whofe Splay is exprefs'd by ac and f b: Now, if c a and�_/"be continued, they will meet in /, on which, with the Radius i c, defcribe the Arch c I, alfo the Arch b m. Make the Lengthof the Curve c /equal to the Circumference of dp, the outer Circle of the Splay, and draw the Line / m ; then the fhaded Figure k being bent about and fixed within the Splay, it will exactly fitevery Part thereof: But as the bend- ing of Stuff of any confiderable Thicknefs is impracticable, therefore divide the whole into Parts, as at I, 2, 3, 4, 5, 6, &c. which glew, or otherwife fix toge- ther, equal to the Curvature of the Window, at Pleafure. Fig XI. reprefents the ancient Manner of making ftreight Arches of Stone, in Places where no Abutments can be had, whofe Vouffoirs are joggled together, and their fpreading prevented by Iron-bars tooth'd into the Head of each, run in with Lead, as at e c e, and c. LECTURE XXVIir. Of Centering to Arches and Groins, Plate LXIV, TO defcribe the Curvatures of Groins is the chief Thing to be done va, Works of this Kind, which is moil eafily performed, as follows. Example I. Fig. A. Let a c e/be a fquare Plan, whofe Vault is to be interfered by two Concave Semi-Cylinders. Defcribe the Semi-circle aba, which divide into Ordinates, as I, 2, 3, &c. Draw the Diagonal a e, which divide into the fame Number of Ordinates, and make them equal to the Ordinates of the Semi circle, and through; their Extremes trace the Semi-ellipfis age, which is the Curve of the G^oin re- quired. In the fame Manner the Groin k g e is found, whofe interfering Arches are kb i and i d e; as alfo are the Groin Curves of Fig. Q_S and B. The fi- gures D and E are both fingle femi-cylindrical Vaults, in whofe Sides are fmall in- iefting Vaultings over the Heads of Windows or Doors, which are thus de- fcrib'd, Fig. D. Draw as many Ordinates in the given Arch at one End as are neceflary, as the Ordinates 1, 2, 3, 4, J, which continue until they meetrf e the Side of the Bafe of the fmall Arch, and from thofe Points draw Lines perpendi - cular thereto, of Length at pleafure. On d i, the given Breadth of the fmall Arch, defcribe the interfering Curve of the fmall Vault of any Kind, as re- quired, as a b i ; divide the Bafe of one Groin, as e i, into the fame Number of equal Parts as/, the \| Breadth, and erec\ Ordinates thereon, equal to the» Ordinates on d x, and through their Extremes trace the Curve if, which is the Curve of that Groin required. By the fame Rule all other Kinds of interacting Arches may be found, although they cut the ftreight Vault on any oblique Angle inftead of aright Angle, the Bafe of the fhorter and of the longer Groin being divided into the fame Number of equal Parts, and the Ordinates in each being made refpeftively equal. The other Examples at q n, in Fig. D, and atk gvndr p, are given for a further Inflection, to illuftrate the Truth of this. Rule. To find the Lengths and Atigles of Boards for the Covering of Centers, Fig. NOP. Suppose b d k I, to be the Plan of a Vault, whofe interfering Arches are the Semi circle bed, and the Semi-ellipfis d b I ; continue b d, both ways, and make it equal to the Girt of the Semi-circle bed, and the Center i draw the Lines i a and i g ; then the Triangle ai gis the Covering for one End, and the Board- ing being cut with Angles, equal to the Angles made by dotted parallel Lines, and the Lines a i, and i g, will be the Bevels ; and their Lengths being taken from the Lines a i, and i g, unto the Line a g, will be their Lengths, as required. Continue dm, both ways, and make e z».equal to, the Girt of the Semi-elli, fis d h m,

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		i�4 0/ ARCHITECTURE. J h m, and draw the Lines / e, and /' m; then the Triangle iem is the Covering for one Side, whofe Bevels and Lengths are to be found as before. Nate, The Figures RTVXY, exhibit a Me:hod for defcribing the Cieling pf a Vault in Piano, as pub'ifhed in Mr. Price's Treatife of Carpentry, which is as follows. Firft, abed, Fig. X. reprefents the Plan, Fig. Y and V. the two .interfering Arches. Draw the Bafes of the Groins a g d, and c gb, make the Length T equal to the Girt of Fig, V. including the two Piers a I, and m b, make the Length of the Parallelogram Fig. R. equal to the Girt of the Semi- ellipfis lm n, alfo make its Breadth equal to the Girt of the Semi-circle i e k ; draw Ordinales at pleafure from the Ellipfis^/V. V. to divide the Semi-tranfvcrfe Diameter of the Plan i g, in the Points 12345678. Draw e k, in Fig. R. through the Center g, divide gf, in Fig. R, in the fame Proportion as half the Semi eilipfjs/w, and through the feveral Divifions draw Ordinates, equal to the circular Curves that ftand over the dotted Lines included between the Lines a g, and g c, in Fig. X, and then Lines being traced through the Extremes of thole Ordinates, the Figure included by them and the Line d i, will be the Covering to the Part age. But if the Lines g f and g k, in Fig. R, be each divided in the fame Proportion as the Semi-trantverfe Diameter 1 g, in Fig. X. and right Lines be drawn through them, as Mr. Price in his Treatife of Carpentry directs, their Interfeclions will not form the Covering for age, in Fig. X. nor will the Paral- lelogram a e lb. Fig. R, be the Covering to the two Interfering Arches of Fig. X. as he miftakenly has aflerted. To dfcribe curved Giohs, Fig. K I F, Plate LXV. Let a b c d be the given Plan. Continue a e and b d, until they meet in the Point 1 in the Line e f. Bifedl a e, and b d, and defcribe the two Semi-circles age, a.nd b k d. Divide the Diameter of either Semi-circle as b d, into any Number of equal Parts, fup- pofe ten, and draw the Ordinates 5, 4, 3, 2, 1, tsV. on the Point 1 in the Line e f; from the feveral Parts in the Diameter b d, defcribe concentrick Arches to the Line a c, divide the Arch a 5 b into the fame Number of equal Parts, as the Di- ameter b d is divided into, and from the Point I in the Line e /draw right Lines, which will interfeft the aforefaaid concentrick Arches, in the Points through which the Curves c i b, and a i d, the Bafes of the Groins muft be traced. To defcribe inner and outer Ribs. Draw a b, Fig. F, equal to the Girt of the outer Curve a 5 h, alfo e f equal to the inner Curve e e d, and divide each into 10 equal Parts, from which creft Ordinates equal to the refpeclive Ordinates in the Semi-circle b k d, and through their Extremes trace Curves, which will complete both Ribs, being fo bent or worked, as to Hand on the Curves a 5 b, and eed. To fnd the Curvatures of the Groins. Make the Bafe Line of Fig. H. equal to the Curve Line a i, alfo make the Eafe Line of Fig. W. equal to the Curve Line/ d. Divide each into 5 equal Parts, and thereon raife Ordinates, equal to thofe in the Quadrant h b i, and through their Extremes trace Curves, and which being bent or worked fo as to ftand on the Curves a i, and i d, they together will form the circular Groin n i d, and the other being found in �he fame Manner, will be the Groins as required. Fig. C. exhibits the Manner of framing truffed Ribs far the Centers of large Arches, Stone or Brick, whofe Parts are to be put together, as th^ Arch is raifed on the Sides. The Struts 3 n 0 are fuppofed to be placed on upright Tim- bers at a End i, which at the taking down of the Center are to be tal>en away. As in the fpringing of the Arch there is very little Weight that b�ars on the Center, therefore the firft horizontal Beam b h muft be placed at fome consi- derable Height above the fpringing of the Arch ; and the Struts, 3 » 0, are ef- ficient to carry its Weight. When the Arch is raifed up to b and h_t then the f�cond horizontal Beam c g muft be raifed with its feveral Bafes, Struts, and Dif-, charges yzex iu <v _n, which together will ft rongly refill the Weight on the Sides, for

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0/ ARCHITECTURE. ,65

for as the Braces, &c. »n the one Side have their Dependance on the other Side
nothing can injure them * when the Arch is brought up to eg,,then the iu>pe?
Part may be completed. The Mortifes in the feverai Parts of this Trufs mull
be all Pully-Mortifes, that when the Arch is key'd in, each Tenon may be driven
Out of its Mortife, and every Part taken down gradually at Pleafure.

LECT. XXIX.
Of St air-Cafes.

WITH regard to the great Varieties of Buildings I have in MateLXVl. gheb
!2diiferent Defigns for Stair-Cafes, from which the ingenious Workman
may form fuch others as his Occalions may require. f{g ^ _n _a Triano-ular �
D C, and D E, are Circular ; D I, and UK, are Elliptical ; D L. Oflanaular �
DM, Semi-circular ;DF, a Trapezia; D, a geometrical Square; DA.andDfi'
are Parallelograms, which in general may be made fit for any Nobleman's Palace'
Before a Stair-Cafe is made, we (hould confnler, fir ft, the Height of th�
Floor to which we are to afc.nd. Secondly, the Rife and Number of Steps that
are neceflary for the Height. Thirdly, to divide the Number of Steps by fuch
half Spaces (or Breathing Places) that are neceflary ior repofm*v on the'vVay
Fourthly, that the Space above the Head, commonly called Headway be fpa-
cious ; and taftly, that the Breadth of the Afcent be proportionable to the whole"
Building, and fufKcient for the Purpofe intended, fo as to avoid Encounters by
Perfons afcending and defending at the fame time. The Height of Steps (houU
not be lefs than 5 Inches, nor more than 7 Inches, except in fuch Cafes where
Neceflity obliges a higher Rife. The Breadth of Steps (lioula not be lefs than io
Inches, nor more than 15 or 16, S�thtf' forrre allow 18 ,nchc-s, winch I thi"k is
too much. 1 he Light to a Stair-Cafe fhould always be libera!, to avoid Slips
Falls, &c. and which may proceed from the Sides, from a Cupola or Sky.light
at the Top, as the Situation will bell admit. Before this Kind of Work is be-
gun, it is belt to make a Plan, and to lay out the whole in Ledgement as follows

Le*- t, 0. 9, 11, Fig. D G. Plate LXVi. be a given Plan.

Make </.r equal to the Breadth of the "Wceflt: which may be made from 7. Feet
and |, to 10 ireer. Draw db, b a, and am, parallel to the Out-lines �f the Plan
Divide a b, b a, and a m, each into fuch a Number of Steps, whofe feverai"
Heights are equal to the whole Height to be afcended; within the Parallelogram
abmd,irvn the Thicknefs of the hand rail. A-dd into one Sum the HeMts of
the l�verai Steps, between b and d, and at thatbiftance, draw q,-, parallel" to 0 1 ■
draw the Hypothenuial Liner r, and continue out the Plan of each Step to rrr-et
the Linerr at ,; fet up the Height of the firft Step, and draw it parallel to *�/
until it meet the Bafe Line of the l� Step; then let up .the Heigh taf the ?d Stop*
and draw it paraliei to 0 s ; proceed in like manner to fet up the Heights of ail
the remaining Steps unto r: make op equal to 0 q, and draw 2 p parallel to to j
at the Point 2, begin to fet up the Step* unto the Point 1, and draw v 1 parallel
to t 0 : Make t to equal to t n>, and draw <w 8 parallel to 19 ; at g b�gttt to fet
the Steps as aforefaid unto i, then will / s be equal to the Height of the Stor"
and the feverai Figures ffft, op-Ti vt-, t tsofiyfy, will'be the Sides of the
Stair-Cafe laid out in Ledgement as required.

Th? Plan, Fig. B, is in like manner reprefented by Fig. C, wkich may bei
corfidered as its Seclion, wherein / m is the Heigh: to be afcended, g b the firflj
Flight, bo the J Space, hi the f�cond Flight, i n the J Space,' ik the lalt
Flight, \vho^re Landing, as Workmen term it, is I h.

i�ete, The parallel dotted Lines between gh and ik reprefent Strions of Wood-
which are cafed underneath to reprefent folic! Steps.

The Fig
ients
She M

Q_ represents the half Space of one Plight of Stairs, Fig-. P. reore-
^ with its Banifters,. and Fig O. repreients Fig. P. completed with:
ngs of its Hand-rail, Bale, tsc.

Th e next Thing- to. be �pnfi.dered is the Manner, of placing tlie Neyvsfeto, Stairs,.

^^H

ar/.

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		j66 Qf ARCHITECTURE. and half Spaces. In fig. E. Plate LXVI. the half Spaces are made.fqnarc to the Angles of the Newels, which caufes the Hand-rail of the fir ft Flight to drop the Height of z Steps below the Rail of the 2d Flight. In Fig. F. the Stairs arc fet to the Middle of the Newel, which caufes its Rail to drop two Steps, and in Tig- G. thev are placed to the Outride of the Newel, and drop but one Step. Laftly 'vaFig- H ' the Stairs are fet half their Ereadth clear without the Newel, which'caufesthe Rails to meet, as in Fig. O. �-.".,. To prcferve a Re: ularity in Fig. I and K. which have large Mouldings, fet the Stairs the Breadth of half a Stair clear or. the O «fide of the Mouldings. Jc is alfo -o be obferved, that as it is ufual to place half �allujiers againft Newel, therefore when it happen? that the Interval or Space is too great, then the Newel fhould be augmented, as in Fig.* K. , Jig. L. exhibits the regular Method, and Fig. M. a fhameful Method of join- ing Rails'and Ballufters, which lalt is to befeen in the Stair-Cafes at the Weft- End of the Pariftl-Church of 5/. Martins in the Ftrlds, London, and which was executed under the Direftion of Mr. James Gibbs, Architect. Fit N exhibits the Manner of dividing the Heights of Raking Ballufters by continuing theMembers of the (freight Isailulters; a'�Fig X.andY,PlateLXV�ll. exhibits the Manner of placing llreight and raking Ballufters over each other. The Fi�ures IKLMNOP Q_are d vers Examples of Baliulie.s as were ufed by the Ancients; as alfo are Fig. i' V W and R divers Gmlo.his and Orna- ments, which were often ufed inftead of Ballufters. and which, when well exe- cuted are very grand. It was the Cuftom of the Ancients to begin the Balluftrade of a grand Stair- Cafe with a Pedeftal, as Fig. S Plate LXVT1. which to a hvge Stair-Cafe is yet the moft grand Manner, but many modern Architects, who think themfelvcs wifer place a twilled Rail at the lowermoft Stair ii.ftead of a Pedeftal. InVmall Buildings a twilled Rail is very proper, but in magnificent Buildings I think them vaftly inferior to a noble Pedeftal. To defcribe a fwificd Rail is the next Worn in Order, which may be performed as following. Let the Lines B D E, Fig. IV. P'hte LXVII. reprefent the Edges of the two lower Stairs of a Stair-Cale. _ Dvide b 9 the Tread of the f�cond Stair, into 9 equal Parts, continue the LineD towards the left at Pleafure. Draw N F, parallel to 9 b, at the Diftance of 7 Parts alfo draw the Line 14 d, at the Diftance of 3 Parts, then db is the Bread-h of the Hand rail. Draw A n parallel to 9 b, at the Diftance of b 9, then the Point n is the Center of the Eye of the Scroll Or1 the P�int a defcribe the Quadrants b c, and de, which is the Length of the tufted Part of the Rail, the remaining Fart to», the Eye, being level. On » defenbe the: Circle z xfi, vvhofe Diameter w p muft be equal to d b, the Breadth of the Hand rail. Divide the Radius n t into 4 equal Parts, and through the firft Part at ., draw the Line r / cutting the Line N F in'x ; on* defcribe the Quadrants cf, and eg, make 0 /equal unto 2 Parts of n p, and draw the Line t s parallel to A n. On the Point t defcribe the Quadrants//,, and ., make � w equal to 3 Parts of »>, and through the Point w, draw the Line/-, parallel to rx ; on « defcribe the Quandrant.b -a, and on w the Quadrants/, and then is the Plan completed... To defcribe the Mould for the fivijl. Continue b 9 towards M, and F N towards b, in Fig. I. alfo draw L I, parallel to b N, at the Diftance of N K, in any Part of N b, as at c, draw the J ine a'�at Right Angles to b N, and on c defcribe the Semi-circle a b f, make a d Mi ft, each equal to the Rife of one Stair and draw the Line/ c t. Make , N equal to c t, divide I c into any Number of equal Parts, and draw the Or- el' ibt'it 1 � 16 2 � k 3 ; &c- divide c N into the fame Number of equal farts'as lib c and make the Ordinates thereon equal to the Ordinales on b r, Ld through their Extremes trace the Curve N/, which ,s the Curve of the Out- fide of the Mould, Make b k equal to the Breadth of the Hand-rail, and on �

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		0/ARCHITECTURE. 167 With the Radius c i, defcribe the inner Semi circle. Make c b equal to rt ⁰ⁿⁱ/' ^the Semi-diameter of the inner Semi-circle, make Ordinates, which" transfer on c b, as before, and through their Extremes trace the Curve of the Mould which will complete the whole, as required. For the Out-lines of the Plan of the twilled Part of the Rail J c and de are Quadrants, therefore the outer and inner Curves of the Mould will be both a quarter Part of two Ellipfes ; becaufe the twilled Rail, ftiidly confidered, is no other than the Seftion of a Cylinder, as L M I K, whofe Diameter af is equal to twice a b in Fzg. IV. and its tranfverfe Diameter equal to dt, and conjugate Diameter to a f. The Twift of a Rail over a circular Bafe at a half Space, as ab fig- II U ^ltj7 r^Tr^inS f ^.P^ingvasbeingth* fourth Part of'an Ellipfis, made by the Seflion of a Cyhnder, whofe Diameter is equal to twice a c The Manner of making the Knees and Ramps of Rails, is the next that is to be confidered, which are thus defcribed. LETff/r, qt, s<v, and w, be 4 given Stairs. From, the Middle of the lower Stair, draw the raking Line/./, fo as to be parallel to a s* «,, the Notes of the Stairs : Alfo draw kb parallel to f/, at the Diftance of the Rail's Thick nefs. Continue ts to g, and make fb equal to �>, and draw a d parallel t� «. From 'he Point p draw the under Part of the Knee, parallel t_8f\|aal.o Ik,* at theD.ftance of the Rail's Th.cknefs, and then the Knee will be com- pleted. Divide the Angle n.f b into 2 equal Parts, by the Line/, cuttin th- Lmeab in a On a, with the Radius a b, defcribe the Arches _? b, and ic which is the Ramp required. Now this Rail being fet upon the BaUuder, ,_n h' afligned Height, fo for the Points I and b, to ftafd over'the Points ltd I ' Will becompleted, as required. ' " Fig. IX. is the Bafe of' a Newel-pod, whofe Sides are fluted in various Man nets, as exprefled at a b c d, �, and Fig.* VI. is a View of the Moulding of a" Hand rail for a common Stair Cafe. ^s To ji'nd the Mould of a fiwifted Rail to a circular or elliptical Stair Cafe Fig. VII. and VIII. Plate LXVII. ■ � o^T f ,^D' if?' Y '^L ^be ^the ^P!an °^f ^a critical Stair-Cafe, whofe Bafe » a Circle, and who e Starrs wind about the Cylinder a bd, &c tZvuI of the Sta.rs being divided, continue out. the Diameter da, towards the left Hand, as to/, of Length at Pleafure. Make « � equal to the Girt of the Sem- circle* ^which divide into the fame Number of equal Parts as there"� Stairs in the Plan of the Semi-circle a b d, as at the Points 1 2 z / F* from which ereft Perpendiculars, as 1 a, 2 a, 3 _a, &V of Length ar Pit f Confider the Rife of a Stair, and make the Perpendicular/., equal to the Rifr of all the 12 Stairs that go round the Semi-ckle ab /^dS��ttPef pend.cular/^ ,nto 12 equal Parts, as at the Points 1234, �V. from which draw Lines parallel tofd, continued out towards the right Hand, at Pleafure Which will interfea the Perpendiculars on the Line/W, in the Points ac acac, isc._ and which are the Breadths and Heights of the Treads and Rife* of the .2 Stairs, at the Side of the Semi-cylinder a b d ; for was t�e whole Figure^/* applied about the Semi-cylinder, then the Parts ac ar hft would be in the refpedive Place of each Stair. Let _a _e reprefent the Breadth of the Handrail, and the Semi-circle e 10 rits Baie, dyer which«w? is to Hand Divide its Diameter ec into any Number ofeqSt S fst I 2 3 4. ^-and draw the Ordinates :, 6 ; ₂, ₇ . 3, 8 ; ₄, ₉ , k S& nine upwards, fo as to meet the horizontal Lines drawn from the Perper-dicu ar^/, ,_n the Points 28, 27, 2�, 25, ©V. through which trace the �,ee Curve 28 ,4, «, which is the Sectional Line of the Cylinder over which it �ands. Make the D.ftances t_S, 21 ; io,,j._; 18,-,3 /,7 12- I^dloJ equal to die Ordinates ,0, 5 ; o, , _K 8, 3 _w, ₂ . jj^, ? '_a_d ^ & I-omsao ,19. 18. 17.16, to «, on the Line/,/, trace the Curve, 20 To J which is the infwe Curve of the Mould, and whofe Out-curve 21. '_a,* be'in
			^Y * JutfHENK m
			^Y * JutfHENK m

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		168 Of A R C H I T � C T U R E. made concentric thereto, will be the Mould required, whofe End 21, 20, when fet up in its Place, will ftand perpendicular over its Bafe b lo. Note, This Mould, tho' made but for one 4th Part of the Cylinder, will ferve for the Whole, by repeating the fame, or adding 3 or more others of the fame Kind to the Ends of each other, as often as there are Revolutions in the Cy- linder. Fig. VIII. is the Plan of an Elliptical Stair-Cafe, whofe Mould ik is defcribed in the fame Manner, and therefore needs no other Defcription. L E C T. XXX. Of Compartment!for Mormmev.tal Infcriptions and Shields, alfo divers Ornaments far Buildings and Gardens. AS in the preceding Leftures I have explained the principal Parts of Build- ings, I (hall now conclude this Part with fome particular Ornaments, which are in common Ufe, and which are as neceffary for the Enrichment of Drawings, as of Buildings themfelves. In Plates LXIX. and LXX. are contained 14 Defigns of Compartments for Monumental Infcriptions, Coats of Arms, to be placed in open Pediments, &c. InPlateLXX�. is contained, firft, 11 Kinds of Vafes, as ABCDEF G H I K L, for the Enrichment of Piers to Gates, Parapet Walls, &c. as alfo are the Balls P Q, and Pine-apple R. The Figures MOS arc Defigns for Flower-pots, which are to be employed as Ornaments, in fuch Places where; Vafes will be too large. As the principal Parts of thefe Ornaments are pro- portioned by equal Parts, as exprei�ed in divers Places between them, the young Student will fee how eafy it is to make them to any given Height. The Fig. W Y, A B, A C, have their principal Parts determined by equal Parts alfo. Figures W and Y afe Defigns for Chriftening Fonts; and A B, A C, for Pedeftals to horizontal Dials ; and indeed, when horizontal Dials are very large, the Figures W and Y may be employed to their Pedeftals. Fig. X. is a Kind of Pedeftal, called a Terme, from Terminus, the God of Bounds or Land-roarks, who being anciently made (landing in a Sheath, thefe Kinds of Pedeftals were taken for the Support of Bufto's, and are thus pro- portioned to any given Height. Divide the given Height into 10 equal Parts ; give the upper i to the Height of the upper Altragal, Fillet, and Cavetto ; and the lower 1 to the Height of the Plinth, Fillet, and inverfed Cima. The Projection of the great Aftragal is 2 Parts on each Side the central Line, and of the (mall Aftragal in the Bafe, one Part on each Side, from which the other Mouldings take their Projections, as common in Columns. To fuite tbefe Pedeftals. Divide the Breadth into 21 equal Parts, give 1 to each Fillet, and 3 to each Flute. The Fig. N. rtprefents a Harpye, a fictitious Monfler, faid to have the Hear! cf a Maiden, a'ad Body of a Bird ; and if fuch are made in Stone or Metal, hav- ing the Bodies of Turtle-doves, Owls, and Magpyes, they wiil be pretty Em- blems of the Indecency, Wifdoro, and bablLog Nonfenfe of Women. The Figures Z. A'D� A E, and A F, rcprefent the Monfler called Sphinx, whofe Head and Bresft is like'that of a Woman's, its Voice like a Man, its ,Body like a Lion, and Wings as a Bird ; but fometimes their Wings are omitted, as Fig. AD and A E. The Figures T �nd'Y are two Kiads ofObelifks, for Lamp poils,-'if e. the or.e fquare, the other octangular; and F:g: A G h the Defign of a. She!! for to enrich the Head of 3 Niche.

			PART

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169

PART IV.

Of the Mensuration of Superficies and Solids.

AS the Foot is the Standard Meafure of mod Nations, I (hall therefore pre-
fix to the following Rules a Table of Foreign Feet, carefully compared
with the Englijb Foot, wherein it is fuppofed, that the Englijb Foot is
divided into rooo equal Parts, as alfo into 12 Inches, and each Inch into 10
equal Parts.

Decim.

1,000
i,o68
1,066

,94²,946

1,033
,920
,948

1,001

1,162
»944
,965

1,026

,967

,970

1,007

1,016

1,824*

Inc.
12
00

ioths.
o

Engl. Feet.
Englifh foot.
Paris the Royal Foot.
Paris Foot, by Dr. Bernard.
Amfteidam Foot.
Antwerp Foot.
Leyden Foot.
Strafburg Foot.
Frankford ad Msenam Foot.
Spanifh Foot.

il
11

00
11
11
00
01
11
11
00
11

Venice Foot.
Dantzick Foot.
Copenhagen Foot.
Prague Foot.
Roman Foot.
Old Roman Foot.

Greek Foot.

China Cubit. ■

Cairo Cubit.

00
00

II
06

� � Babylonian Cubit. , I

Old < Greek Cubit. , 1

�

02 4
02 3

(_ Roman Cubit. , 1

Turkifh Pike. 2,200 2

Perfian Arajh. 13>^I97 3

LECT. I.
Of Rules for meafuring the Superficies of geometrical Figures, Plate LXXII.
Rule I. To meafure any ■plain Triangle, Fig. A B C D.
S I : half the Bafe c d or hi, Fig. A or B : : b d or g i, the Perpendicular,
: the Area j or as I : the whole Bafe m s, ok z y, Fig. C or D : : -* the
Perpendicular: Area.

' To find the Area of any plain Triangle, having the Sides only given.
Add the 'hree Sides together ; from the half Sum fubtract each Side feverally,
arid note their Differences. Multiply any two of the Differences together, and
their Produ� by the other Difference. Multiply the lad Product by the half Sunt
of3 Sides, andtheSquare Root of their Produdtk the Area required.

Rule II. To meafure a geometrical Square, or Parallelogram, as the Figures E F.
As 1 : c d the Length : : a c the Breadth : Area.

Rule III. To meafure a Rhombus, or Rhomboides, as the Figures G and H.
'As 1 : a d, equal to c e the Length : : b c the perpendicular Height : Area.

Rule IV. To meafure a Trapezoid, as Fig. I.
As I : f, the Bafe f e, : : the perpendicular Height b f : Area.

Rule V. To meafure a Trapezia, as Fig. K.
As i : a Diagonal, as b g: ■ half the Sum of the 2 Perpendiculars a a and
f e : Area,

Rule

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		fjo Of the Mensuration of Superficies an J Solids. Rule VI. To< meafure any Polygon, as the Hexagon L. As I : J the Circumference a k, : : \| the Diameter eg, equal to ba\ Area. Rule VII. To meafure any irregular right-lined Figure, as Fig. M. Divide the Figure into Trapeziums, as de, e f, cd, be, and the Triangle I a e, whole Areas find by Rule I. and V. and their Areas added together is the Area required. Rule VIII. To find the Length of an Arch of any Circle, as a c d, Fig. S. Divide the Chord Line into 4 equal Parts, make the Chord Line of a b equal to � Part, then hd is nearly equal to half the Arch Line required : Or thus arithmetically ; Multiply ac, the Chord of half the Arch, by 8 ; from the Pro- duct fubtraft ad Divide the Remains by 3, and the Quotient will be equal to the Length of the Arch Line acd required. . Or thus, From the Chords a c and c d, fubtraft the Chord ad. Divide the Remains by 3, and then the Quo- tient added to the Chord Lines ac and c d, the Sum will be nearly equal to the Arch Line acd, required. Rule IX. To meafure a Quadrant, «bee, Fig. O. As 1 : f the Ar�h c e, : : a Side, as b e : Area. Rule X. To meafure a Semi-cirele, as' a d c, Fig. O. As I :� the Arch adc, : : the Diameter ac : Area. Rule XI. The Diameter of a Circle being given, to find its Circumference. As 7 : 22, : : the given Diameter : Circumference required. Or, as 113 : 35j, : : the given Diameter : Circumference required. Or, as 1 : 3,141593, : : the given Diameter to the Circumference required. Or, as 1,00000,00000,00000,00000,00000,00000,00000 : is to 3,141,9,26535,89793, 23846,26433,83279,50288, fo is the Diameter given, to the Circumference re- quired. Rule XII The Circumference of'a Circle being given, to find its Diameter. As 22 : 7, : : the Circumference given : Diameter required. Or, as 355 : IJ3, : : the Circumference : Diameter. Or, as 3,141593 : 1 : : the Circum- ference to the Diameter. Rule XIIl. The Diameter of a Circle being given, as a C, Fig. N. tb find its Area. I. By Van Cui.ek's Analogy. As 1 : ,7854, : : the Square of the Diameter : Area. II. By MetiusV Analogy. As 452 : 355, : : the Square of the Diameter : Area. III. By Archimedes'.! Analogy. As 14 : 11, : : the Square of the Diameter : Area. Rule XIV. The Circumference of a Circle being given, to find i's Area, As 1 : ,07958 : . the Square of the Circumference ; Area. Rule XV. The Area of a Circle being given, to find its Diameter. As I : 1,2732, : : the Area : Diameter required. Rule XVI. The Area of � Circle being given, tofindits Circumference. As 1 : 12,56637, : : the Area : Circumference required. Rule XVII. The Diameter of a Circle being given, to find the Side of a Square nearly equal to the given Circle. As 1 : ,8862 : : the Diameter : Side required. Rule XVIII. The Circumference of a Circle being given, to find the Side of a Square nearly equal to the given Circle. As I : ,2821, : : the Circumference : Side required. Rule XIX. Tb Dia?neter of a Circle being given, to find the Side of a Square* inferibed. As I : ,7071 : : the Diameter : Side required. Rule XX. The Circumference of a Circle bang given, to find the Side cf a Square inferibed As i : ,2251, : : the Circumference : Side required. Rule

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		Of //^Mensuration of Superficies and Solids. 171 Rule XXI. The Area of a Circle being given, to find the Side of a Square in- fcribed. As I : ,6366 : : Area : Side required. Rule XXII. The Side of a Square beinggiven, to find the Diameter of its circum- fcribing Circle. As 1 : 1,4142 : : the Side of the Square : Diameter required. Rule XXIII. The Side of a Square being given, to find the Circumference of its circunifcribing Circle. As 1 : 4,443, : : the Side of the Square : Circumference required. Rule XXIV. The Side of a Square being given, to find the Diameter of a Circle nearly equal to the Square As I : 1,128 : : the Side of the Square : Diameter required. Rule XXV. The Side of a Square being given, to find the Circumference of a Circle nearly equal to the Square. As 1 : 3,54;, : : the Side of the Square : Circumference required. Rule XXVI. To find the Diameter of a Circle, as c e, Fig T. having the Chord Line a b, and Height c d, of the Segment a c b, given. Square a d, and divide the Product by cd, the Quotient will be equal to de, then c d, more de, is the Diameter required. Rule XXVII. To meafure tbeSeBor of a Circle, «;cba, or d a e f, Fig. R, As 1 : § the Arch Line, : : the Radius da, or c a : Area. Rule XXVIII. To meafure the Segment of a Circle, as a b C, Fig. P. Imagine Lines to be drawn from a and c, to the CenterP ; and abeP will be a Sector ; which being meafured by Rule XXVII, and the fuppofed Triangle a cP being deducted Irom it, the Remains will be the Content of the Segment required. To meafure the great Segment of a Circle, as d e f. Imagine Lines drawn from d and e, to the Center P, as da and a e, in Fro-, R. Then to the Area of the Seftor da if, found by Rule XXVII. add the Area of the Triangle da e, by Rule I. and their Sum is the Area of the greater Segment required. Hence it is plain, that the Center of a given Segment of a Circle muft be known before its Area can be found. Rule XXIX. To meafure the Zone of a Circle, as a d e f b c, Fig. Q. To the Paiallelogram d f a b, add the Segments de f, and abc, and their Sum is the Area of the Zone required. Rule XXX. To meafure the Superficies of any irregular cwvilineal Figure, as the Figure V. Divide the curved Bounds into Segments, as n p a, a b c, ede, efg, g h /', i k I, l m ». To the Area of the right-lined Figure na c e gi In, add the Area of the Segments npa, c de, g h i, i k I, a,nd from the Sum fubtract the Areas of the Segments abc, efg, and n in I, and the Remains will be the Area of the irregular Figure required. Rule XXXI. To me�fure an Ox Eye, as Fig. W. Draw the Line a d, then add the Area of the Segment a c.d to the Segment *ah d.* Rule XXXII. To meafure any fpherical Triangle, as XY Z, and A. Fig. II. First, Fig. X. to the plain Triangle a ce, add the Segments abc, ede, and a e m, their Sum is the Area required. Secondly, Fig. Y. to the Area of the plain Triangle a bfi add the Segments a c b, and b df, and from t:ie Sum fubtraft the Segment e a f, and the Remains is the Area required. Thirdly, Fig. Z from the plain Triangle a e c, fubtra&the Segments e a d, and Sa c, and to the Remains add the Segment ecn, the Sum is the Area required. Fourthly, Fig, A. from the plain Triangle, a d f, fubtraft the Segments c b e, the Remains is the Area required. Rule XXX1IL To meafure any mixtilineal Triangle, «BCD E, Fig. II. First, the Triarjgle C, from the plain Triangle, c a d. fubtraft the Seg- ments act, and ' c c d^ the Remains is the Area required. Secondly, the Triangle ..... ' ' " ' "D,

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		172 Of the Mensuration of Superficies and Solids. D, to the Triangle c a e, add the Segments b c a, and c e d, the Sum is Ehe Area required. Thirdly, to the plain Triangle E, add «he Segment b a c, the Sum is the Area required. Rule XXXIV. To meafure compounded regular Figures, as F G H, Fig. IF. First, the Fig. F. to the geometrical Square abed, add the Semi-circles e and/, the Sum is the Area required. Secondly, the Fig. G. from the geometrical Square 1234, fubtraft the Quadrants lab, zed, b.$g;-tf\, the Remains is the Area required. Thirdly, the Fig- H. from the Parallelogram 1234. Sub- tract the Triangles \bc, d2e,a$h, andfgq., the Remains is the Area required. Rule XXXV. To meafure Egg and Heart Ovals, as Fig. OPQ^ First, the Egg Oval, Fig. O, to the Trapezoid a dfb, add the Semi-circle a c d, and the Segments a f, J'bg, and d b, the Sum is the Area required. Se- condly, Fig. P. to the plain Triangle a e d, add the Semi-circle a be, and the two Segments a d, and e d, the Sum is the Area required. Thirdly, the Heart Oval Q^ To the plain Triangle a bg, add the two Semi circles a de, ceb, and two Segments afg, and b gf, the Sum is the Area required. Rule XXXVI. To meafure an Ellipfis, as the Fig. I K. As ! : ,7854 : : the Square of two Diameters : Area. The Area of every Ell:pfi3 is the mean Proportional between the Area's of its circumicribing and inferibing Circles,! as in Fig. N. For. as the Area of the circurnferibing Circle a b f 'm: the Area of the Eilipfi: a gp x : : the Area of the Ellipfis a g p x : Area of the inferibed Circle b g 0 x. Rule XXXVII. To meafure the Segment of an Ellipfis, as e f i, Fig- M. or d g n, Fig. N. First, The Segment of an Ellipfis whofe Bafe is parallel to the conjugate Diameter, usefi, Fig. M.is in proportion to theSegment dfn, of the fame Height of the circumscribing Circle ; as b m, the Diameter of the circurnferibing Circle : c k the conjugated Diameter of the Ellipfis : : the Area of dfn, the Circle's Seg- ment : eft, the Area of the Segment of the Ellipfis. Secondly, the Segment of an Ellipfis whole Bafe is parallel to the tranfverfe Diameter as d », Fig. N. is in proportion, as the Area of the inferibed Circle bg 0 x : the Area of the Ellipfis agp x : : the Area of the Segment of the inferibed Circle : d g n the Area of the Ellipfis required. Or as� x the Diameter of the inferibed Circle : a p the tranfverfe Diameter : : the Area of the Segment of the inferibed Circle ; Area of the Segment of the Ellipfis. The Fig. K and L, are each a Semi-Ellipii-¹» the fir ft on the tranfverfe, and the l�ft 011 the conjugate Diameter, whofe Arcns are to be found by confidering each of them as a whole Ellipfis, and take j th« Area fo found, for their Areas required. The Fig. IK fheixis how ta defcribe any El ipfis by the Help of three freight Laths, & C. as following. Be'aw the 2 Diameters ay"and b n at right Angles, to their given Lengths. Make » d, and ne, each equal to half the tranfverfc Diameter, then a and e are the two Focus Points, whereon fix two Laths, as on Centers, as d% and e'h, each equal to the tranfverfe Diameter. To their Ends b and g fix a third Lath, covual :.c the Diil.'iace of de, fo that the Ends at/_> and g may be moveable as the J-�int oi a Two-foot Rule. Then the 3 Laths being moved about the 2 Focus Points, their feveral Points of Interfeilion will trace out the Ellipfis required. Rule XXXVII�. To meafure the Areas of a Parabola, as Figures R or S. L'ye'iy Parabola is equal to 2 Thirds of its inferibing Parallelogram. There- fore as i : df, Fig. R, or a f Fig. S : : a d, Fig. R. or b a, Fig. S : a 4th N.urft- rrr> 2 Thirds of which is the Area required. LE C T. I. s Cf Rales far meqfuring the Solidity of all Kinds of Bodies, and their Superficie:, Rule I. To meafure the Solidity of the Cube R, or the Parallelopipedon W. AS 1 : the Area of any End or Side : : the Depth or Length irom that Ei»4 or Side : the Solidity requires!,

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		Of the Mensuration of Superficies and Solids. 173 The Superficies of the Cube R, is the geometrical Squares 123456, Fig- S. and of the Paralleiopipedon W, the Parallelograms I X, 4 5, and geometrical Squares 2 3, Fig. X. RULE II. To ?t:ecfure the Solidity of any Frifm as the Figures, V, A B, and A D. As 1 : the Area of one End : : the Length : Solidity required ; the Super- ficies of the triangular Prifm V, is the Parallelogram 1 2 5, and Triangles 3, 4, Fig. Z. Of the hexangular Prifm A B, the Parallelograms I 23456, and Hexagons 7 8. And of the Trapezia Prifm A B, the Parallelograms 2, 3, .4, 5, and Trapezias I, 6. Rule III. To mcafure the Solidity of a Cylinder, ivhofe Bafe is a Circle, as Fig. A. Plate LXXIV. or an Ellipfis as Fig. I. Plate LXXIII. As I : the Area of one End : : the Length : the Solidity required. The Su- perficies of the elliptical Cylinder I, is the Parallelogram I nmo, (whofe Length is equal to the Circumference of the Cylinder) and the 2 Ellipfes c k d a, and e igf. And the Superficies of the circular Cylinder A is the Parallelogram a, and two Circles D C. Rule IV. To meafure the Solidity of a Tetrahedron, as Fig. T Plate LXXII. the Pyramis A G and A F, and Cone, Fig R. Plat : i. XXI V In every of thefe Bodies, as 1 : the Area 0/ its Bafe : : f of i's Altitude trie Solidity required. The Reafon hereof is, that every Cone i equal to {of its eif- cumfcribing Cylinder ; that is, to a Cylinder of the fame Bafe and Altitude. Sa likfwife every Tetrahedron and Pyramis i< equal to { of it: circu'mfciibingPnfm, whole Baf� and Altitude is the fame as thofe of the Tetrahr.dron and Pyramis, and therefore it follows, that as 1 : the Area of the i'a'e of a Cylinder, or Priim,.: : the Length of its Axis : a 4th Number, one 3d of which is equal to the Sohilit/ of the Cone or Pyramis inkribed therein. The Superficies of the Tetrahedron is the equilateral Triangles 1, z, .3.-4. The Supeficies of the fquare Pyramis A F is the geometrical Square" A E, and the lioiceles Triangle a e b,'h gd, d h c, and a cf. The Superficies of the octangular Pyramis A G is the Octagon A F, and Ifofceles Triangles a b ede f g h i and the Superficies of the Cone is the Setlor b h if. and Circled /. Note, The Length of' the Arch k if is equal to^- the Circumference of the Bafe of th« Cone. And the Radias h h, tob f, th� Side of the Cone. Rule V. To meafure the Sojidity of a Spheric as Fig. T. Plate LXXIV., As 21 : 11 : : the Cube of the Sphere's Axis : Solidity required, or as 1 : ,5236 : : the Cube of the Sphere's Axis: Solidity le^uiredj tor if the Axis of a Soutre be 11, its Solidity is ,3236. Every Sphere is equal to a Cone, whofe Axis is equal to the Radius of the Sphere, and its Baf: to the Area of the Sphere. Or every Sphere is equal to two thirds of its circunifcribing Cylinder. Therefore, as 1 : the Area of a grea Circle of the Sphere : : the Diameter : 4th Number, two thirds of which is the Solidity of the Sphere. As a Cone is equal toJ-of a Cylinder of equal Bafe and Altitude, and as a Sphere is equal to \|- of a Cylinder of equal Diameter and Alti.ii'd�, 'tis there- fore evident that a Cone whole Bafe is equal to a great Circle of a Sphere,- and its Axis equal to the Axis of the Sphere, its Solidity is equal to J'the Solidity of the Sphere And a Cone, whofe Axir is equal to the Semi axis of the Sphere, and the Diameter of its Bafe to twice the Diameter of the Sphere, willbe equal to the Sphere ; as alfo is a Cone whole Axis is equal to twice the Diameter of the Sphere, and the Diameter of its Bafe equal to the Diameter of the Sphere. Rule VI. To meal tire the Superficies of � Sphere. The Area of every Sphere is equal to four great Circles theieof, fo the Area of the Sphere, Fig. T. Plate LXXIV. is equal to the Circles V W X Y. Or as' I : the Diameter ; : the Circumference to' the Area required". 2» Ifjte

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		174 �f the Mensuration of Superficies and Solids. Note, The Area of a circumfcribing Cylinder is to the rea of the infcribed Sphere, as 3 is to 2 ; and which is the fame Propor ion_t that the Solidit) of the Cylinder has to the Solidity of the Sphere. Note, If the Covering or Area of a Semi-fphere be laid put as taught in the CovenngoftheHeadsofSerrii.circularNici.es, in LF.CT XXV. heieof as is exhibited in Fig. M, by fuiux, &t. and the Area of a Part, at of Z, be multi- plied by twice the Number of Parts laid out, the Product will be the Superficies of the Sphere required". Note a!Jo, i be l�verai occult Arches 'n thi Fig. are no more than a Repetition of thofe in Fig Iy. Pinte LVI, which 1 have infected here again, for the eafier UnCerltanding the Manner of deferibing th. fever»! Parts/ v iv x, &C which are the Superficies of the Semi iphere laid open. .Rule VI. To tncafure the Solidity of any Si-gmmt of a Sphere, <w 1 1,7* Fig. M: Plate LXXIII'. I. The Diameter and A\^Jitu(l< of tb Fruftum being given- To 3 times the Square of / 3. 'he Semi diameter of its Ba+e, add the Square of 3, 1, its Altitude. Multiply the Sum by the Height, and thatProduft again by ,5236, the Product, cutting off 4 Decimals to the Right hand, is the Solidity, required. II. The Axis of the Sphere k g, and I 3, the Height of the S gment given _ From 3 times the Axis, fubtraft twice the Height of the Segment, multiply the Remainder by the Square of the Segment's Height and that Product by ,52 \b, the Product, cutting off 4 Decimals to the Right hand, is the Soliaity required. Rule VIII. To meafure the Solidit]/ of any Frufum of a Sphere, as h kb, Fig. A L,'Plate LXX1V. From the Solidity of the whole Sphere, deduft the Segment h m i, and the Remains is the Solidity of the Fruftum required. Rule. IX. To meafure the Zone of a Sphere, as h k d e. Fig. A L. Plate LXXIV. Frqm the Solidity of the whole Sphere, deduct the two Segments h m k and, d e h, the Remains is the Solidity of the Zone required. Rule X. To meafyre the 7 one of a Sphercid, as Fig. L, Plate LXXIII. Multiply the Square of hi, the conjugate Diameier, by a n, the tranfverfe Diameter, and that Produft by ,5236, the Produft, cutting off the 4 Decimals, is the Solidity required. Note, Every Spheroid, as a ce g, Fig. Q) Plate LXXt:I. is equal to 2 thirds pf a Cylinder, as a d n f whofe Diameter is equal to the conjugate Di «meter, and Height to the tranfverfe Diameter. Rule XI. To meafure thi Soli /if y of the Segment, or Fruftum of any Spheroid, Inscribe the Spheroid in a Sphere ; then as the Solidity of the Sphere is to the Solidity of the Spheroid, fo is any Part of the Sphere to the like Part of the Spheroid. Rule XII. To meafure the Solidity of a parabolic Conoid, as Fig. N. Plate LXXIII. This Solid is generated by the Revolution of a Semi parabola, on its Axis, and is thus meafured, viz. Multiply the Square of its Diameter, by .7854, and jts Product by half the perpendicular Altitude, the Produft (cutting off the 4 Decimals] is the Solidity required. P.ULE XIII. To meafure the Solidity of the Frvftum of a parabolick Conoid, as fa eg, Fig. N. Plate LXXIII. Multiply the Sum of the Squares of a c and/ g, the leffer and greater Diameters, by ,3927. and that Produft by the perpendicular Height of the Fruliurn, the lait Produft is the Solidity required. Rule Xi\^?. To meafure the Solidity of a parabolick Spindle, as Fig. W. Plate LXXIII. Multiply the Square ofg- I its greateft Diameter, by ,41888, (being -j?- of ,7854) and that Produft b-y b a its Length, the lalt Product, cutting off the De- cimals, is the Solidity required, 2 Rule

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		Of the Mensuration <� Superficies and Solids. 175 Rule XVI. To meafure the 'Solidity of a Fri/Jlum of a paralolick Spindle, as d f g 1, or of a Zone, as d f m o. Multiply the Square o\ g /, the greateft Diameter, by 1,57.08 ; alfo mul- tiply the Square of df the left r Diameter, by ,7854; alfo multiply the Square of the Difference of the Diameters, by ,31416; then from the Sum of the two former Pioducfs, fubtra�t the laft Product, and multiplying the Remainder by l Third of the perpendicular Length, that Product is equal to the Solidity of th� Zone if m 0, whofe half Part is jequal to the Fruftum dfg I. Rule XV!. To meafure the Solidities of the fi-ve regular Bodies, viz. The Tetra~ hedron, Fig. T. Plate LXXII. The Oitahedron, Fig O, Plate LXXlII. The Hexa- hedron or Cube, Fig. R. Plate LXXII. The Icofahedron, Fig. T. and Dodecahe- dron, Fig. R. Plate Lxxrir. If the Side of each Body be confidered as 1 Or unity, their Solidities are as follows, iiz. Solidities Superficies. Tetrahedron 0,1178511 � 1,732051 Octahedron 0,4714045 � 3,464102 Hexahedron 1.0000000 � 6,000000 Icofahedron 2,181695 � 8,660254 Dodecahedron 7,663119 � 20,645729 To find the Solidities of either of th'efe fiodies, As I : the folid Content in the Table, : : the Cube of the Side of the like Body to be meafured : Solidity required ; or if each Face be confidered as th� Bale of a Pyramis, whofe Vertex is in the Center of the Body, then one fuch Pyramis being meafured fingly, and its Solidity multiplied by the Number of Faces contained in the Body, the Product will be the Solidity of the Body re- quired. To find the Superficies of either of theft Bodies. As 1 : the fuperficial Content in th; Table, : : the Side of the like Body to be 'meafured : fuperficial Content thereof; or if the Area of one Face be firff found, and multiplied by the Number of Faces contained in the Body, the Pro- duct will be the fuperficial Content of the whole, as required. Note, The Superficies of the Tetrahedron is the Fig. V. of the Cube, the Fig* S. Plate LXXII. as has been already obferved. Of the Octahedron, the 8 equi- lateral Triangles 12435867, Fig. P; of the Dodecanedron, the 12 Penta- gons I Z 3 4 5 6 7 8 9 10 11 12, Fig S ; and of the Icofahedron, the 20 equi- lateralTriangles 1234, &c. Fig. V'. Plate LXXIII; and which being de- lineated on Paper or Pall-board, as exhibited in the feveral Figures, and thert cut out and folded up together, will form the feveral Bodies in juft P/oportion. Rule XVII. To meafure the Solidity of any Fruftum of a Pyramis or Cone, nu bof� Bafe is right angled to its Axis, as the F ruft urns of Pyramis''s Figures A C and E f Plate LXXIII. and the Fruftum of a Cone, Fig. S. Plate LXXlV. Multiply the Area of the greater End by the Area of the leffer End, and, extraft the fquare Root of the Product. Add .the fqua>e Root to the Areas of both Ends, and the Sum mukiplied by § of the Ffuitum's Length, the Product is the Solidity required. The Superficies of the Fruftum of the triangular Pyramis A, Plate LXXIII: h the 3 Trapezoids, abed, d h bf, cf 1 3, and two equilateral Triangles '■ i 3, and c df, in Fig; B. The Superficies of the Fruftum of the pyramid G is the four Trapezoids a b 58, 8 2 d 7, 6, 7 e f 25 4 6 ; and the two ge�aie_tri�ai Squares, 1234, and 5867, Fig. D. The Superficies of the Frultum e'f the octangular Pyramij, Fig. E, are the four Trapezoids on each Side, and th.- two' Oclagons<3v, and a 5 g f, &e. The Fig. F, is aifo the Superficies of tbs o-'t- afiguiar Frulblm E, where the Trapezoids � 23456789 �. s i:.L$idSs. The ©flagon F its Bafe, and the Qftagon 4 its Top. % r. T«f

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		jj6 Of the Mensuration of Superficies and Solids. The Superficies of the Fruftum of a Cone, Fig. S. Plate LXXIV. is the im- perfecl Superficies deqfer, and the Circles n n, and h k I. Rule XVIIJ. To meafure the Solidity of a Prifmoid, or Frulium of an irregu- lar Pyramid, ivbofe Ends are dfproportionable, Fig. G. Plate LXXIII Tofb, add half b a, w.iich multiply by h g, the greater Breadth, and referve trie Product. Tobd, add. half/b, which multiply by c d the leffer Breadth, to which add the former Product referved, and the Sum being multiplied by { of the perpendicular Height, the Product is the Solidity required. The Superficies of this Fruftum is the 4 Trapezoids 1245, ^ant* ^l^^e ² ^^a* rallelograms 3 and 6, Fig- H. Rule XIX. To meafure the Solidity of an oblique Fragment of a Cylinder, as a c, Fig. P. Plate LXXIV. As 1 : the Area of its Bafe a, : : half its Length ; the Solidity required. The Superficies of this Fragment is the Ifofceles Triangle/ g e, the Ellipfis b, and the Circle d. Note, Fig. Z. is a double Fragment, whofe Superficies is the two Eilipfes e and /, and geometrical Square 1 gmn; and Fig. O is the Out-fide of de a b, which is a Fragment of a Fragment of the Cylinder b d gb. Rule XX. To meafure the Solidity of a Cylinder, ivbofe Ends are oblique to its �xh,as Fig;L. Plate LXXIV. By Rule XIX. meafure the Fragments a and b feparately, and add their So- lidities to the Solidity of the Cylinder p q, the Sum is the Solidity required. The Superficies of this Cylinder, is the double Trapezoid cfbg, h g-ki, and the two Eilipfes c and d. The Figures EIK are other Examples of this Kind, whofe Superficies produce different Figures, according to the various Sections oi their Ends, which I have added for further Examples of this Kind. Rule XXI. To meafure the Fragment of'a Cone, as b d c, Fig. A B, Plate LXXIV. As 1 : the Area of its Bafe, : : y^y of its Altitude : its Solidity required. The Superficies of this Fragment is the curved Figure c 8 e i, the Circle q 0 r, and the Ellipfis al de, Fig. A X. Rule XXlI. To meafure the Fruftum of a Cone, nvhofe Ends are obliaue to the Axis, as Fig. A C, and A D, Plate LXXIV. First, meafure the Fruftum, as a Fruftum whofe Bafe is right-angled to the Axis, and from that Solidity deduct the Fragments that are deficient at the Ends, and the Remains will be the Solidity required. The Superficies of thefe Fruftums are laid out as following, fig. A B. On a defcribe the Arch cm I, &c. e, equal to the Circumference of the Bafe of the Cone, which divide into 8 equal Parts, at the Points m I k i, &c. and draw the Lines a m, � I, a k, &c. Draw b. 1 parallel to d c, and divide 1 c into 4 equal Parts. Make a 5, all, each equal to a 4. ; make «6, a 10, each equal to a 3 ; make a 7, a 9, each equal to a z; make a 8 equal to a 1. Through the .F.oints 1 ;, 10, 9, 8, and 7, 6, 5, trace the Curves e 8, and 8 c ; then the Fi- ' gu're c 8 e i c is the Superficies of the Side. The Superficies A C, and A D are . deforibed in the fame Manner.

			PART

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			177

		PART V. Of Plain Trigonometry, Geometrically per- formed. LECTURE I. Of the Solution of plain 'Triangles. I. Definitions. FIRST, plain Triangles.are right-angled or oblique-angled. Secondly, a right-angled Triangle is fuch a Triangle as hath one right Angle, and two acute Angles, as the Triangle A, Plate LXXV. whofe Angle beaks. right Angle, and the Angles c b a, zxx�bac, are both acute Angles. Thirdly, an oblique Tnangle is fuch a Triangle as hath one obtufe Angle, and two acute Angles, as the Triangle B, whofe Angle b c a is obtufe, and the Angles c b a, and cab, are both acute Angles. Fourthly, In every right angled plain Triangle, that Side which fubtendeth (or is oppofite to) the right Angle, as b a, in Figure A, is called the Hypothenufe ; and of the other two Sides, the one, as e a, is called the Bafe ; and the other, as c b, is called the Perpendicular,. Fifthly, in every oblique plain Triangle, as Fig. C, the longeft Side is gene- rally called the Bafe, as c a ; but fometimes one of the other two Sides is made the Bafe. Sixthy, in every right-lined Triangle, the Sum of the Degrees con- tained in the three Angles, are equal to 180 Degrees; therefore if you have any two Angles given, you have alfo the third given, it being the Complement to 180 Degrees. Seventhly, And as in a right-angled plain Triangle, the right Angle contains 90 Degrees, therefore if anyone of the two acute Angles bs given, the other acute is alfo given, becaufe it is the Complement of the other acute Angle to 90 Degrees; or of the other acute Angle and right Angle to 180 Degrees. Eighthly, In all plain Triangles whatfoever the Sides are proportional to the Sines of their oppofite Angles. The Solution of plain Triangles has always confifted of 12 Cafes, but herein I have reduced them unto 8 Cafes, of which 4 are of Triangles right angled," and 4 of Triangles oblique; and which anfwer every particular exa&ly the fame, as thofeof other .Authors divided into 12 Cafes. I. Of right-angled plain Triangles. In the Solution of right-angled plain Triangles, there are always two Parts given, as two Sides ; or an Angle and one Side ; to find a Side or an Angle re- quired. Case I. Fig. A. Plate LXXV. The Bafe C a 80 Feet, and Perpendicular c b 60 Feet, being given, to f.nd the acute Angles c b a and bac, and the Hypothenufe. Make f «(by a Scale of f eet) equal to 80 Feet and cb equal to 60 Feet, and draw b a, which is the Hypothenufe required. With 60 Degrees of Chords, on the angular Points b and a, defcribe the Arches e </and gf, which being meafured on the Scale of Chords, e dsv\\\ contain 52 Deg. 30 Min. and ^/37 Deg. 30 Min. which are the Angles required. Case II. Fig. A. Plate LXXV. The Hypothenufe b a 100 Feet, and the Bafe c a 80 Feet, being given, to find the acute Angles, and Perpendicular b c. Mak»

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		�j% Of Plain Trigonometry, Geometrically perform J. Make c a equal to 80 Feet ; ereft the Perpendicular c bof Length at pleafure ; on a, with the Lengthof 100 Feet, incerfeft the Perpendicular at b, and draw the Line b a , then meafure the Degrees in each Angle, as in Cafe I. and b c will be the Perpendicular required. Case 111. Fig,,A, Plate LXXV. 'The Bafie c a 80 Feet, and the Angle cab, oppofite to the Perpendicular 37 Degree 30 Mitt, being given, to find the Perpendicular t b, and H)pothenvfe b a. Make c a equal to 80 Feet; ereft the Perpendicular e b of Length atPJeafurt; make the Angle bac equal to 37 Deg. 30. Min. and draw the Line « b, which will cut the Perpendicular in b, then b c is the Perpendicular, and b a is the Hy- pothenufe required. Case IV. Fig. A. Plate LXXV. The Hypothenuje b a IOO Feet, and the Angle c b a 5 2 Deg. 30 Min. oppofite to the Bafe, being given, to find the Length of the Bafie c a, and of the Perpendicular c b. Draw b a equal to ioo Feet ; make the Angle bac equal to -z Deg. 30 Min, and draw b rof Length at pleafure ; make the Angle bac equal to the Comple- ment of the Angle c b a, and draw the Linea c, wmch will cuti cine; then c a is the Bafe, and be the Perpendicular required. II- Of'oblique-anghaplain Triangles. In the Solution of oblique angled plain Triangles, there are always three Parts given, as two Sides and ari Angle, or two Angles and a Side, to find a Side or an Angle required. Case I. Fig. B. Plate LXXV. Two Sides, and an Angle oppofite to one of the Sides, being given, to find the third 'Side. This admits of three Varieties, as, Firft, The Bafie b a IOO Feet, and Side b c 50 Feet, with the Angle b a c 28 Drg. oppofite to the Side b C, being given, to find the Side c a co Feet'. Make b a equal to 100 Feet; on b, with the Length of �o Feet, delcribe the Arch d c at pleafure ; in any Part of b a, as at h, make an Angle, as h h f, equal to the given Angle 28 Degrees ; from a, draw the Line a c parallel to h e, v, bich will cut the Arch d c in c, then the Line c a is the Length of the Side required. Secondly, The Bafie c a, Fig. C, 100 Feet, and Side b a ;o Feet, v.it.b the Angle C b a 110 Degrees oppofite to the Bafie c �, being 'gitten, to find the Side c b 60. Feet. Make b a equal to co Feet ; make the Angle c b a equal to 1 to Degrees, and draw � c of Length at pieafure ; on c, with the Length of the Bafe 100 Feet, interfeft the Line b c in c, then c b is the Length of the Side required. Thirdly, The tiuo Sides c b 60 Feet, and b a 50 Feet,with the Angle b c a 28 De- grees, oppofite to the Side b a, being given, to find the Length of the Bafie c a 100 Feet. Draw e a at Pleafure ; on c make the Angle a c b, equal to the given Angle 28 Degrees, and make cb equal to 60 Feet; on b, with the Length �f 50 Feet, interfeft the Line f a in a, then c a is the Length of the Bafe required. Case II. Fig. C. Plate LXXV. The Bafe c a �0O Feet, ana the Side c b 60 Feet, vcith the Angle b c a 28 Peg. contained between them, to find the third Side b a, and the Angles c b a and b a C. Mare c a equal to 100 Feet; make the Angle b c a equal to 28 Deg. and the Side c b equal to 60 Feet, draw the Line b a, which is the third Side required , then meafure the Angles c b a and b a c, as in Cafe I. of right-angled plain Triangles. Case Ilf. Fig. C. Plate LXXV. The three Sides C a ICO Feet, c b 60 Feet, and b a 5 c Feet, being given, to find all the Angles. By Prob. I. Lect. IV. Part. II. complete the Triangle b c a, and by Case I- of right angled plain Triangles, find the Quantity of each Angle. Case

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		Of Plain Trigonometry, Geometrically perform d. 179 Case IV. Fig C. Plate LXXV. Two Angles, as b c a 28 Dfg. b a c 42 Z)eg. cW one Side, as c b 60 Feet, being given, to find the other two Sides b a 50 Feet, and C a IOO Feet. Make c b equal to 60 Feet ; make the Angle b c a equal to 28 Deg. and the Angle bac equal to 42 Deg. continue out the Lines b a and c a, and they will interfeft each other in the Point a ; then c a and b a are the two Sides required. Note/The Doftrine of plain Triangles, performM by the Tables of Logarithms, Sines, Tangents and Secants, being more difficult to be underitood by Learners than the preceding, and as to have added thofe Tables would have fwelled the whole beyond its intended Bulk and Price : I therefore omitted the Analogies and Tables, which, if this Work be favourably accepted, I will publilh here- after in a feparate Volume. LECTURE II. Of Menjuration of Heights and Difiances. THE proper Inftruments for thefe Purpofes are a Quadrant, as Fig. D, and a ten Feet Rod, Chain, &c. Prob. I. Fig. F. Plate LXXV. To taie the Altitude of an Qbjeft, as 'the Obelijk b n, by the Help of a Quadrant, Move from theObjft, until, looking through the Sights of the Quadrant to the Top of the Objea, the Plumb-line cut 63 beg. 26 M:n. on the Limb, as at h : then the Height of your Eye being added to your Diftance from the cen- tral Line of the Objeft, is equal to \ the Height of the Objeft : Or move back- ward, until the Plumb-line cut 45 Deg..as at i, and the Height of your Eye added to your Diftance at before, the Sum is the Height required. And fo in like manner moving backwards, until the Plumb-line cut 33 Deg. 20 Min. as at k, then ^? of the Dillance is the Altitude. And at /, where the Plumb-line cuts 26 Des. 34 Min. the Dillance is double the Altitude. H any Obftruciion is between you and the Objeft, fo that you cannot meafure to its Bale, then o-o nearer, or farther, until the Plumb line cut 26 Deg. 34 Min. a;, tt /, and there make a Mark on the Ground ; move backward in a right. Line with yourfiri! Stati <n and the Objeft, until the Plumb-line cut 18 Deg. 26 Mm. as at m, then the Diftance between your two Stations / and m, is equal to the Altitude required. Prob. II. tig. G. Plate LXXV. To find the Altitude of an Object, by knowing the Length of its Shadow. Set up a Stick of any known Length, fuppofe 3 Feet, as de: Let the Length of the Shadow of the Objeft be b e, and of the Stick eg; then as the Length of the Shadow of the Stick is to the Height of the Stick ; fo is the Length of the Shallow of the Objeft to the Height of the Objeft. Prob. III. Fig. H. Plate LXXV. To taie the Altitude of an Objea that is accejfible, by the Help of a ten Feet Rod and a Stick only. Let the Obehjh a. h be an accejfible Objea, tvhofe Altitude is required. Erect a ten Feet Rod in any Place, as at m, and a Stick, as nf, equal to_vthe Height of your Eye, at any Diftance in a right Line with the Building ; look from the Top of the Stick, level to the Building, and againft your Ray of Sight, at the 1 o Feet Rod, make a Mark, as at e ; cauie a f�cond Perfon to flide a Piece of Paper up the 10 Feet Rod ; fo that, looking from the Top of the Stick/; to b the Top of the Objeft, you fee the Top of the Paper, as at d, at which place make a Mark : This done, meafure the Diftance of the two Marks on the 10 Feet Rod e and d, alfo the Diftance e f ; then as e � is to e d, fo is e � the Dif- tance of the Stick from the Objeft, to e a the Height of the Objeft above the Level-Line cf, to which add the Height of the Stick nf, and the Sum is the Altitude required. Prob,

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		i8o Of Plain Trigonometry, Geometrically perform''J. Prob. IV. Fig. G. Plate LXXV. To taie the Altitude of an Objid that is inaccejjib'e, by itt Sbadoiv. Suppose the Shadow of the Object reach fiom b to e, and at the fame time, the Shadow of a Staff reach from e to g ; at about two or three Hours after, when the Sun is rifen considerably higher, place down a Mark at the End ot the GbjecVs Shadow, which fuppofe to be at c ; alio, at the fame time, make a Mark at the End of the Shadow of the Stick, fuppofe at f ; now, as the Triangle df g is fimilar to the Triangle ace, and as the Triangle de f is fimiiar to tne Tri- angle a c b, therefore, as / g is to the Height ot the Staff// e, fo is c e to the Height of the Object required. Prob. V. Fig. I. K. Plate LXXV. To meafure the Altitude of a Hill or Mountain, by the Help of a Spirit-Level and Station-Staffs. (i.) Erect your Level truly horizontal on the Top, as at c, and directly * againft the Iniirument, let a f�cond Per.fon hold up a fiiding Station (biff, with a Vane fixd thereon, which he is to move up, until, looking through the Sights of your Level, you fee its upper Edge, as at » : This done iet the lecond Perlon write down the Number of Inches and Parts of Inches that his Vane ii above the Ground at»/, let a thirdPerlon wiite down the Number of Inches and Parts of Inches that your Iniirument is above the Surface of the Ground at �. (2.) Re- move your Level down the Hill, as to 4, and your 2d Affiliant to k, and let your 3d Affiliant eredt his Station-ftaffat m, the Place where your 2d Affiliant fir ft ftood : This done, fix your Iniirument truly horizontal, and looking to your 3d Affiliant at m, let him Aide up his Vane until you fee its upper Edge, at which Time he is to fet down, under the Height of the Iniirument obferved at �, the Inches and Parts of Inches that his Vane is then above the Ground ; alfo look to the Station-ilaffcf your 2d Affiliant, and caufe him to Aide up his Vane, until you fee its upper Edge, as at I, and let him place down the Inches and Parts that his Vane is above the Ground, under his firft Height obferved at m. Proceed in like manner at every other Obleivation, as may be required to defcend unto the Bottom at b. (3.) Let each Affiliant add into one Sum, the Heights of his feveral Observations, and then that of your 3d Affiftant's being fubti acted from that of your 2d Afliilant's,. the Difference is the Altitude ot the Hill required. Prob. VI. Fig. P, O, M. Plate LXXV. 5 0 meaf�re an inacaf/ible Dijiance. Inaccessible Distances r.-.ay be mealured by many Methods, as, Eirit, To find the Dijiance of the tivo trees 7 and 8, Fig. P, v.hich is renier d in- acceffible by the River b b. Assign any Point on the Ground, from which you can meafure directly un- to the two Objects 7 and 8, as the Point 9 ; continue 7, 9 unto II, and S, 9 unto 10, making the Diftance of 9, 11 equal to j, 9, and the Ditlauce of 9, 10 equal to 8, 9, then tJie Diftance from 10 to n is equal to the Diftance of �7, 8 required. Secondly, To find the Diftance of the Tree at r, in Fig. M, from the Point v, �which is ) endct-'d iiinaccejfble by the River h b. Imagine a Line to be drawn from v to r, and thereon erect the Perpendicu- lar 11vu, of any Length, and let r v be continued at Pleafare towardsy, which may be done by (training a Pack-thread Line from v towards y, in a right Line with v r. In any Part of the Perpendicular v in, aflign a Point as <w, and at any Diftance from you, place a Stake in a right Line between vj and r, as at j ; alfo another on the Perpendicular, at any Diftance from vu, as at t. Make the Tri- angle 10 t x, equal to the Triangle vu s t ; and continue vu x, until it meet the line vy in J ; ^xn the Diftance vyis equal to the Diflance v r, required. Thirdly,

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		Of Plain Trigonometry, Geometrically perform'd. 181 Thirdly, To find the Difiance of the tivo Trees, \Z and 13, Fig. O, 'which is rendered maccef/ible by the River d. Affign a Point at 16, from which you can meafure to both the Obje&s. Place two Stakes at any Diftance in right Lines, from the Point 16, to the two Objects» as at the Points 14. and 1^, and meafure the Sides of the Triangle 14, 15, i6> alfo the DStances from the Point 16, to the Objects 12 and 13. On Paper, with � Scaleof Feet, make a Triangle, whofe respective Sides are equal to the Meafures of the Sides of the Triangle 14, 15, 16, and continue out the Sides, refpecting the Sides :6, 14, and 16, 15. each equal to the Meafures of 16, 12, and 16, 1 3. Then the Ddlance between the Extremes of thofe Lines, being me�fured oil your Scaleof Feet, will be the Diftance required. PtioE. VII. Fig. N K L. Plate LXXV. To meafure an inaccejjible Difiance by Help of a geometrical Square, right-angled or equilateral Triangle. Fir/?, To meafure the Difiance � b, Fig. N. m.hich is rendered inaccefpbk by the River C, by Help of a geometrical Square. Imacne a right Line to be drawn from c, to the Object b, which continue towards 4. On the Point 5 erect the Perpendicular 5 z, of Length at Pleafure, and therein affign a Point as z, where with a Piece of Board make a geometrical Square, apply its Angle over the Points, and dire:t its Sides z I to the Object; alio at the fame time caufe an Affiliant to- move along the Line �, 4, until b/ the Side of the geometrical Square 2; 3, you fee his Station llafF erect, at 4. This done, meafure the Sides of the Triangle ,'24, arid then as the Side 5 4, is to the Perpendicularc 5, fo is the Perpendicular z $, to 5 b, th� Diftance re- quired. Secondly, To meafure the Difiance 1 kj Fig. L. �which is rendered inacceffibh by the River b. Being furnilhed with a Piece of Board that isan equilateralTriangle, as/»j#, ripply one of its Angles over the Point/, and direct a Side, as l m to A, and at the fame time direct an Affiliant to fix upa Station-ftaffin a direct Line with the other Side In, at any Diltance from you, as at p, and then let up a Mark in the Point /. This being done, move along the Line I p, until by the Sides of the equilateral Triangle, you can fee both the Mark fet up at /, and the Object at k, which you will do at the Pointy ; then the Diftance of lp is equal to the Diftance Ik required. Note, In the fame Mannet ah inacceftible Diftance, as f a, Fig. K. may be found by � right angled plain Triangle, as cfg, whofe Sides if, aadfg, are �qual, as is evident to Infpectiorj. Prob. VIII. Fig. Q^ Plate LXXV. To meafure the Difiances of divers Ob/cits, that are inacc/fible at two Stations, by the Help of a common fmall I able, or Joint/tool, and a /height Rule, tvitb per- ptndlcular Sights fxed at each End thereof. Let. the l�verai Objects be abed, and the two Stations / k, atlco'Feetj Yards, &c. Distance. Being furnilhed with a ftreight Rule, about two Fe�t, dr two Feet arid rfaif in Length, with perpendicular Sights fo fixed at each End, that the Slits of the Sights ftand perpendicularly over the thin Edge of the Rule (>vhich is generally called an Index) and a fmall Table or Stool, that hath a fmooth and even Sur- face, proceed as foil ws, wise, with a Scale of Feet, is'c draw a Line in th� Middle of the Table, as i k, equal to too Feet, th� Diltance between tKe cwd Stations ; and then being at one of the Stations, as ?t i; lay the Edge of the Index to the Line ii, and move the Tible, until through the Sights of the In^ dex, you fee the other Station k, and there fix your Table faft. On the Point » on your Table fix a Pin, and applying the Edge ot your Index to the Pin; look through the Sights, to the fiilt Ooject St a ; ar.d craw S Line frotri the Pin, b<- the .Edge of the Index at Pleafure, as i a. Move your Index in like Manner; to..ever_iv of the remaining Objects, drawing Lines frofii the F.n, to

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132 Of Surveying LANDS, &c. wards each Object, as at firft. This done, remove your Table unto /�, your f�cond Station, and placing the Point i on your Table, towards the firft Station lay your Index to the Line k i on your Table, and move the Table, until through the Sights you fee your firft Station, and there fix your Table fait. Fix a Pin iri the Point k on your Table, and then applying the Side of the Index to the Pin, direftthe Sights unto every of the Objects, and draw i ines as before, at the firft Station, which will intellect the former in the Points a be d. and whole Diftanccs (or theDiftances from the two Stations/and k) being nieafured on the fatneScale, by which the Line i A was drawn on the Table, will be the true Distances of each Object required. Note, By the fame Method of working, the Plan of any open Field mny be taken, if the Angles are confidered as fo many different Object:, and can be ail fcen at each Station.

	I

		PART VI. Of Surveying LANDS, &c. TH E ufual Inftruments for this Purpofe are generally the Plain Table, Theodolite, Circumferentor, and Chain ; but as the three firft are Inftru- ments of great Expence, beyond the Reach of common Workmen, for whole .Sake I have publifhed this Work, I fhall therefore give fome few Examples, to fhew how, by the Help of a ten Feet Rod, or Chain, and a Joint-ftool orTable, they may make the Plan of any Piece of Land, that is not ot very great Dimen- fions, with the utmoft Exactness. N.B. The Chain is that which is called Gunter i Chain, whofe Length if equal to 4 Statute Poles, or 66 Feet, divided into too Links, each 7 Inches r�s in Length. Proe. I. FtgiS. 'Plate LXXV. To male the Plan of an Irregular Side of a Field, «jihgfedcab. Make an Eye-draught on Paper, expreiiing the feveral Angles, and therein draw the occult Line ia; as alfo the l�verai perpendicular Off-fets 12 b, 42 g, 56/", (s"c This done in the Field, mealure in a right Line from i, towards af and when you come againft the Angle h, as at the Point 12, write down on your Eye draught, theDiftance nieafured from », as alfo theLengthof the Off fet ! 2 b, which place on the Off let. Proceed in like Manner, to meafure the re- maining Diftances to every Off-fet, and the Length of each Off-fet. This done, t'raw a Line on Paper, and with a Scale of Feet, fet off from i all the feveral Di- stances, .as » 12, i 42, i 56, c?V. and from tliofe Points erect Perpendiculars, makingea�h equal to their r�lpectiveMeafures in the Eye-draught, and then right Lines, as /,?>, bg.g f, cs'e. being drawn from» to�. from b to g, from g to/, &c. they will be the Pian of the irregular Side of the Field, as required. Note, If the Side of the Field be curved, as Fig. R. then take Off-fets at every remarkable Bending, ashxhgetk, �V. which meafure and plan as before, #nd through tiicir Extremes truce the Curve, As required. Proe.II. Ftg.V. Plate LXXV. To 7nake the Plan rfa Field by the Help of a Chain onlyi as Fig. a C d g f e. Make an Eye-draught of.the Field, and divide it into Triangles. Meafure 'lie Sides of the Field, and of every imaginary Triangle,, ' which place on each (cfpUiive Side, with a diagonal Scale �f Chains and Links, as expreflcd tfy Fig,-

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		Of Surveying LANDS^r. 183 IV. Plate IX. By Prob. I. LEC T. IV. Part II. delineate all the fevera Triangles, as reprefented in your Eye-draught, asd they will complete the Plan of the Field, as required. Prob. III. Fig.Y. Plate LXXV. To make the Plan of an irregular curved Field by Help of the Chain only, as bcdefghik. First, Fix up Marks, fuch as Pieces of Paper fixed into the flit Ends of Sticks, at proper Places, as b c d efghi k, and imagine Lines to be drawn from one to the other, as bc, cd, de, e f, Sec. Affign a Station towards the Middle of the Field, as at a, and imagine right Lines to oe drawn from thence, unto the l�verai Marks at b c d ef, isc. which will divide the whole'into ima- ginary Triangles. Make an Eye-draught as before directed, expreiiing every Triangle, C3V. By Prob. II. hereof meafure and delineate the feveral Triangles, and by Prob. I. meafure and delineate the Off-fets on the Out lines of the feveral Tri- angles, necefiary for defcribing the curved Boundaries, which will complete the whole, as required. Note, Chains and Links are thus written, vix. 3 Chains, 75 Links, as from b to a, thus, 3 : 75, and 2 Chains, and 10 Links, as from c to a, thus, 2 : 10, &c. Prob. IV. Fig. A C. Plate LXXV. To make the Plan of a Field, ivhofe Angles cannot be all f een under three Stations, as at a d c, by Help of a Table and Chain. Assign three Stations in the Field, as ad c, at any Diftances, fuppofe ad, at 3 Chains Diftance, and dc at 3 Chains and 35 Links. Draw a Line on your Table by your Scale of Chains and Links, to reprefent 3 Chains, theDiilance between the Stations a and d. Place your Table in the Field, over the fiationary Point a, and laving your Index on the Line a d, move the Table about, until you fee the Station d, and there make your Table faft. Fix a Pin in your Table, at the Point a, and laying your Index thereto, direct the Signes to the feveral Angles ntnowjx 3, and draw right Lines from the Pin towards each Angle. Mea- fure the Dillances from your Station a, unto every of the Angles, and from your Scale of Chains and Links let from the Pin, on each Line, as a m> an,ao, am, tfc. their reipe�live Lengths, as 2 : 75 ; 3:75; 3:6c; bV. and draw the Lines mn, no, ov,vw, hm x, aVid x 3. Move your Table to the f�cond Station d, and laying your Index on the Line ad, move the Table about, until through the Sig-hts you fee your firll Station at a, and there make it faft. Fix a Pin in your Table at the Point d, and laying your Index to the Pin, turn it about until through the Sights you fee your third Station at c ; and by the Side of the Index draw the Line<?\-, which make equal to 3 Chains 25 Links, the Diftance of the third Station c from d. Alio, from the Pin on the Table, direcl: the In- dex to the Anglej, and draw the Line dy, equal to its meaiured Length, and join the Side 3y. Remove your Table to c, the third Station ; lay the Index on the Line dc, and move the Table about, till through the Sights you fee the Station d, and there make it faft. Fix a Pin in your Table, at the Point c, and laying your Index thereto, direcl the Sights to the Angles » hi I, and draw Lines towards each Angle, equal to their refpeclive Meafures, from the Station 'c. Then the right Lines y z, zb, hi, il, and l m, being drawn, they will complete the Pian, as required. Prob. V. Fig. A C. Plaie LXXV, To make the Plan of a Field, by going about it �without Jfde_t by Help of a Table and Chain. First, Goabout the Field, and at proper Diftances makechoiceof Stations, as at a,p, 'j, r, !,g, whereat fix upSticks with Paper as aforefaid. Then beginning at any one Station, as at a, meafure the Diftance from a to g, and from a to p. Draw a Line on one Side of your Table, on which fet from your Scale of Chain? A a ? and

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		? 84 Of Surveying LANDS, &c. and Links, the Length from a to g, place your Table over the Point a. Lsy your Index on the Line reprefenting the Linea^-, and move the Table abont, until through the Sights you fee the Mark at�, and there make it faft. Fix a Pin in your Table at the Point a, and laying your Index to the Pin, dir ft the Sights to the Mark 3t p, and by its Side draw the Line a p, equal to its Length before meafured. By Prob.I. hereof, on the Line ag, meaf.re and delineate the Off fets bo, c », dm, alfo the Off let k i, from the Off-fet am ; then e i,- and/ h, alio the Off-fets t v and i nv, on the Line a ^. Thro the Extremes of the aforefaid Off fets draw the Lines to i;, <v c, on, nm,tnt, I », and ; h. Place your Table oyer p, and 'laying the Index on the Line pa, move your Table about until through the Sights you fee ihe Mark ar a, and there make it fall. Fix a Pin in your TabL at the Point p, and laying your Index to the Pin, direft the Sights to the Mark q, and by its Side driw the Line p q, which make equal to the Diftance that the Mark at q is from the Station at/.. Meafure and delineate the Off-fet q x, and draw the Line «il» #. Repeat thefe Operations at the Stations qr s, and you will complete the whole, as required. Note, By the fame Rule, the Plan of a F;eld may be made, by goine about it ■yvichin-fide, as fignified in F/'�-. Y. by the ftationary Diftances, 3 Imn o j r t >� Prob/VI. Pig. T. Plate LXXV. To makt the Plan of an inclofed Road, Street, &C. Firjf, Make cho.ce of proper Stations as at s t and i\ at which Places fix up Marks as aforefaid ; meafure the Diftances t s, and t<v, draw a Line on your Table, to reprefent the Line t w, on which, from your Scale of Chains and Links, fet its meafured Length. Place yourTable over the ftationary Point t, and laying the Index to the Line / <v, move theTable about until thro' the Sights you fee the Mark at m, and there make it faft. Fix a Pin in your Table at the Point. /, and laying the Index to the Pin, direft the Sights to the Mark at s, and by its Side draw the Line t s, equal to its meafured Length. By Proe. 1. hereof, meafure and delineate an Off-fet againft every Angle contained in the two Sides of the Road or Street, and right Lines, being drawn to their Extremes, v ^:'i he the Plan of the Road or Street, as required. Proe. VIII. Fig. XX- Plate LXXV. To make the Plan l/ an irregular Wall by the Help of a Ten-foot Rod only. Firjl, Make an Eye-draught as W W, and thereon fet down the Length of every r�fpeclive Side contained in X,X; and then proceed to measure the Angles as following ; <siz. (l) To meafure the Angle x a e, imagine the Side X a, to be continued io Feet, as from a to h, alfo fet to Fee: from a to c, and meafure the Diftance be, which iuppofe to be J Feet. Place the Meafures of this Angle on your Eye-draught, as at abc. ( z) To meafure the Angle a e i, fet IO Feet on each Side the angular Point e, 'as to d and f and meafure the Line � d. which fuppofe to be 20 Feet, place thefe Meafures on the Eye draught, as at de f. Proceed in like manner to, take the Meafures of all the remaining Angles, at i mp r <w, �fc. To delineate this Plan from the Eye-draught. Make W a, equal to 21 Feet, the Length of X a, on a in Fig. W. with a -Radius equal to 10 Feet of your Scale, by which you delineate the Plan. De- scribe an Arth as b c, make b c equal to � Feet, and through the Point e dravy be equal to J Feet, and through the Point c �r&w a e equal to 32 Feet, the Length of the Side a e. On the Points, with a Radius of 10 Feet, defcribe the Arch df. and therein fet 20 Feet from dto f. Through the Point / draw the Line ei, equal to 23 Feet, the Length cf the Side e i. Pnceed to defcribe the remaining Angles and Sides, in the fame Manner, which will complete the Plan, as required. ' Prob. IX. Fig. A B. Plate LXXV. To make the Plan of a Serpentine River. T.rf, Aiiign ilationary Durances, as fe d ac b,zx\i fix up Marks as aforefaid. Make an Eye-draught of the whole, meafure the ftationary Diftances, and fet ' their.

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		Of MECH A NICK S. 185 their Meafures on their refpective Places in the Eye-draught. Draw a Line on your Table, to reprefent the Line a b, which by your Scale of equal Parts make equal to 3 Chains 20 Min. its meafured Length ; place your Table over the ftationary Point a. Lay your Index to the Line a b on your Table, and move the Table about until through the Sight you fee the Mark at�, and there fix your Table faft. Fix a Pin in your Table on the Point a, apply your index to rhe Pin, and difeft the Sights, firll to c, and then to d, drawing Lines on the Table towards rhe ftationary Mark; c and d, which by your Scale of Chai' s a d Links make equal to the refpeftive meafured Lengths ; «waft c a, 6 Chains %o Links, and a d, 4 Chains 75 Links. By Prob. I, hereof meafure and delineate proper Off fets, and through their Extremes trace the Curvature of the River. Remove your Table to the Station d, fetting up a Mark again at a. 1 a; the In- dex on the Line, re'prefenting the Lire a d; move the '1 able about until thro' the Sights you fee the Mark at a, and make the '1 able faft : fix a Pin jn the Table at the Pointa, apply the Index to the Pin, and dnettmg the Sights to the Station e, draw the Line de, which make equal to 8 Chains 36 Link;, it- mea- fured Length. Then by Prob. I. hereof, meafure and delineate the Off-fets to the Side of the River, which are here delcribed by dotted Line;, and thro' their Extremes trace on the Curvature of the River. Remove your Table to the Station e, and repeating the fame Kind of Operation asatd, you will complete the whole, as required Note, When the Weather is dry, you may feal down a Sheet of Paper fmooth on the Table, and make your Plans thereon, but if the Weather be mmft or wet the Paper will not do, nor indeed fo well as the Table in dry Weather ; be- caufe Paper is always fhrinking or fwelling very fenfibly, as the Temperature of the 7\ir is more or iels dry, which the Table does not, in fo great a Degree. Prob X. To find the Quantities of Lands in Acres, Roods, and Poles, nubqfe Dimerfions are tak.n if Gunter'j Chain. Rule, Place your Dimenfions, and multiply them to- gether as in Decimal Multiplication, as in the Margin, 27 92 From the Product cue off 5 Figures to the Right-hand, the 39 57 Remains to the Left, when any, are Acres. Multiply the � -----~�■ Figures cut off by 4, the Roods in an Acre, and from 195 44 its Product cut off five Figures to the Right as before, the 1396 o Remains to the Left, when any, are Roods. Multiply 25128 the laft i, Figures cut off, by 40, the Poles in a Rood, and 8376 from the Product cut off 5 Figures' to the Right, the Re- Acres 110)47944 mains, if any, to the Left a'e Poles. So in this Example the . _______ Produd is no Acre?, 1 Rood, and 36 Poles, which is 4 thus, written, A. R. P. ,� no 1 36 Roods 1)91776 40 Poles 36)71040

			PA R T VII. Of M e c h a N 1 c K s. L E C T. I. Definitions of Matter, Gravity, and Motion. Y Machanicks is meant geometrical Rules for demonftrating Motion, and the Effect of Powers or Forces in removing the Matter of Bodies. 2. Matter

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		i86 Of MECH ANICKS. 2 Matter is an impenetrable, divifible, and paffive Subftance, and there- fore has Extenfion and Refiltance, which are the Properties of all Kinds of Bodies, and whofe univerfal Principle is Gravity. 3 Gravity is that Force by which Bodies are carried, or tend towards the Center of the Earth, and which is in Proportion to the Quantity of Matter they contain. Gravity is abfolute, accelerate or relative; Gravity Abfolute, is the whole Force by which Bodies tend towards the Center of the Earth. Gravity Accelerate, is Force of Gravity confidered as growing greater as it approaches the attracting Point, as in Bodies falling. Gravity Relative, is the Excefs of the Gravity in any Body above the fpecifiek Gravity of a Fluid, as of Air or Water in which it move. 4. Specificic Gravity is the appropriate and peculiar Gravity or real Weight which any Species of natural Bodies have, and which arifes from the more or lefj Compactneis of the Matter of which Bodies are compofed. 5. Motion is that Force by which a Body continually changesits Place, and therefore is a continual and a fucoeffiv� Mutation of Peace. Motion is either Abfolute or Relative. Abfolute Motion* is the Change of the Locus Abfelutus ofany r�oving Bo�y ; and its Celerity will be mealured by the Quantity �f the abfolme Space which the moveable Body hath palled through. 6. Relative Motion is a Mutation of the vulgar or common Place of the moving Body ; and fo hath its Celerity accounted or meafured by the Quantity of relative Space which the moveable Body moves over. 7. Celerity is the Swiftnefs ofany Body in Motion, and that Force which isinBodics moving, and whereby they continually move, is called their Momentum, ivhicii arifes from their Weight or Quantity of Matter, and the Velocity' of their, Motion wherewith they move. 8. The Motion of all Bodies is naturally recti linear, and therefore the Ve- locity of a Body will be conltantly the fame, if no external Caufe obitruft the Motion, or make any Alteration in its Line of Direction. 9. The Line of Direction is that Line wherein any Body or Power endeavours to move, that is to fay, it is the Line of Motion that any Bodr goes in according to the Force impreffed upon it. And the Change of Places, or continual PaiFaga of a Body along fuch a Line, is called its Local Motion. 10. Velocity is that Affection of Motion which io meafured by comparing together the Quantity of Space which a Body hath palled through, and the Time in which it was pafling that Space. Thus equal Velocity is that, whereby equal Space is palled over in equal Time. So if two Bodies are put in Motion at the fame Inttam of Time, and both pais the Length of one Mile in an Flour, L~'c, their Velocities are then iaid to be equal. Greater Velocity is that whereby ei- ther a greater Length is pafied over in the i'ameTime (as when either of the afore faid Bodies travel two Miles in an Hour) or an equal Length in lefs Time.- As when the aforefaid Body travelled one Mile in half an Hour, &c. Hence it follows, that if t>\o Bodies are put in Motion at the fame Time, and one travel a hundred Miles, whilft the other travel but fifty Miles, that Body which travels 100 Miles, moves with double the Velocity of the other ; the like is to be underftood oi Velocities trebled, quadrupled, c5V. 11. As the natural Motion of falling Bodies arifes from the Principle of their Gravity or Weight, and is found by Experience tc be a Motion uniformly acce- lerated, and being attended with the fame Gravity or Weight, at every Degree of Velocity, it therefore comes to pafs, that the Spaces through which Bodies fall perpendicularly, are, as the Squaras of the Times wherein they fall, ac- counting from the Beginning of the Fall, As for Example, Fig. I. Plate LXXVI. The perpendicular Deicent of Bodies is at the Rate of IJ Feet in the firft Second of Time, and in every fucceeding Second the Spaces areas the Squares of the Seconds, �viz. If a Body be 5 Seconds of Time a falling from a to �. and

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		,0/ MECHANIC K S. ' 187 and in the lirft Second it falls I 5 Feet, as from a to b, at the End of the f�cond Secend of its Falling, it will have fell 4 times a b equal to 60 Feet, as to 4, which is equal to 2 multiplied in 2, the Square of the Seconds orTimes in falling. Soin like manner, at the End of the 3d Second, it will have fell 9 times 1, Feet, equal to 135 Feet, which is equal to 3 multiplied into 3, the Square of the Se-, conds or Times in falling; and in the fourth Second, 16 times 15 Feet, equal to 240 Feet, as to 1 6. Hence it is plain that the Increafe of Motion in every Mi- nute, Isc. is according to the Series of the uneven Numbers, Wis. 1, 3, 5, 7, 9, 11, &c. which are the Differences of the Squares, 1, 4, 9, 16, 2�, �V. 12. As the Motions of Bodies are accelerated in falling, their Forces are thereby increafed in the fame Proportion. And therefore if the Body a, in fall- ing from a to b, has a Force at b equal to 1 Pound Weight, it will have a Force at 4 equal to 4 Pounds Weight ; for as its Velocity from a to 4 is three times a3 great as from 0 to b, it will therefore have a Force 3 times greater at 4 than when at b, and fo in like manner in its falling to 16 its Force will be equal to 16 Pounds, and at 25 to 2; Pounds, &c. 13. And it is alfo to be obferved, that equal Bodies falling on inclined Planes, whole loweft Parts are in the fame Level, have the fame Force and Velocity at the End of their Falls, as when let fall perpendicalar, but employ a longer Time in their Defcents. So if the Body b, Plate LXXVI. defcend in the perpendi- cular Line b g,ox in either of the oblique Lines b f or bh, it will have the fame Force at � or A, as at g, but it will be longer in falling from b tof than from b to g, and longer from b to b, than from b to/, &c. 14. If a Body defcend on an inclined Plane, as d' b, Fig. C. it will by its ac- quired Velocity afcend another Plane of equal Inclination, as be, unto the fame Height, allowing for the Refinance of" the Air, and Friftion of the Plane. 15. If Bodies fall in the Lines c f, df e f, b f, af, fcV. defcribed in the Ciicle, Fig. B. they will from the Points in the Circumference ab c d e, come to the Baie fat the fame time. For as the Lengths of their Lines of Defcent are to one another, fo is their Velocities to each other. 16. If a Body, asb, Fig.Y.. be thrown perpendicularly upward with any Force, the Velocity wherewith the Body afcends, will continually diminifh, tillat Length it be wholly taken away ; and from that Inftant of Time the Body will defcend in the fameLine with fuch an increafmg Velocity, as to fall from a to c, with the fame Force and in the fame Time, as it was thrown up from c to a. The like is alfo in Bodies thrown up on inclined Planes ; for if in Fig. C. the Body a be thrown from b tod, with a certain Force, and in a certain Time, it will by its own Weight return again to b, with the fame Force and in the fame Time as it afcended. 17. If a Body defcend in the Arch of a Circle, as c, Fig. D. in the Arch d e, the Velocity will always be anfwerable to the perpendicular Height b e, from which the Body fell; but the Time of the Body's Defcentwill be greater from - to e, than from b to e. 18. Now from hence it follows, that the Body a, Fig. F. to dt-fcend the Arch Line a c, or the Chord Line a c, will require more Time than were it to fall in the Perpendicular b c, but will m all the Defcents have an equal Force at c. > L E C T. I. Of the Latus of Nature. IT is to be obferved, that all the Varieties of Motion of Bodies in general are conformable to the following three Laws. Law I. All Bodies continue in their State of Ref, or Motion, uniformk in a right Line, excepting they are obliged to change that State, by Fercet imfreffed; and therefore it follows, 2 FtRj^.

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		j88 Of M E C H A N I C K S. First, If a Body be abfolutely at Reft, and unfurnifhed with any Principle, whereby it could put itfelf into Motion, it will for ever continue in the fame Place, till acted upon by an external Body. Secondly, Wnen a Body is put into Motion, it has no Power within itfelf to make any Change in the Direction of that Motion, and therefore mull move forward in a right Line, as I have before oblerved, without declining any Way whatever. Thirdly, All Bodies, endeavour to remain in their State of Reft or Motion, and therefore fome actual Force is required to put Bodies out of a State of Reft into Motion, or to change the Motion which they before received. This Qua- lity in Bodies, whereby they fo preferve their prefent State of Motion or Reft, till fome active Force dilturb them, is called the Vis Inertia of Matter. It is by this Property that Matter, unaclive of itfelf, retains all the Power imprefl'ed upqn it, and wiil not ceafe to a&, until oppofed by as great a Power as that which firft moved it. Uw IT. Jill Change of Motion is proportional to the Po-iver of the moving Force impreffed, and is alivays made according to the right Line in ivhich that Force is imprejjed. That is to fay, firll, If in one Minute of Time, two Bodies, as a c, Fig. G. move from a and B, towards/ and d, with equal Velocities, fo that when the Body a is arrived at b, the Body c, which moved from B, may act its full Force againft the Body at b ; then will the Line of Direction of the Body a, which was in the Line ad, be changed into the diagonal Line- be, of the geometrical Square fb e d ; and by the Action of the Body c, on the Body b, the Volocity of the Body b will be fo accelerated, as to pafs, in the f�cond Minute, through the Dia- gonal b e, the Side of whofe Square is equal to a b, the Space which the Body b travelled through in the firft Minute. Again, if at the End of the f�cond Mi- nute, when the Body b is arrived at e, another Body ftrike againft it at j-, with the fame Velocity as b then has, then will the Line of Direction of the Body b, in the f�cond Minute, which is b i, the Diagonal continued, be changed into the Diagonal e n, of the Square n i k e ; and by the Force of this f�cond Body, the Velocity of the Body at e will be fo accelerated as to pafs, in the third Minute, through the Diagonal ne, the Sides of whofe Square is equal to the Space which the Body b travelled through in the f�cond Minute. If at the End of the third Minute, when the Body b is arrived at the Point h, it be again acted upon by a third Body at m, with the fame Velocity as the Body at n then has, then will the Line of Direction of the Body at n, in the third Minute, which is the Diagonal e n, continued to p, be changed into the diagonal Line » r, of the Square ro p n ; and by the Force received from this third Body, the Velocity of the Body at n Will be fo accelerated as to pais, in the fourth Minute, through the Diagonal n r, the Sides of whofe Square is equal to the Space which the Body travelled through in the third Minute. And if at the End of the fourthMinute, when the Body is arrived at r, it be again acted upon by a fourth Body, as s, whofe Velocity is equal to that which the Body b then hath, the Line of Direction of the Body at r, which then is the Diagonal n r continued to x, will be changed into the diagonal Line r m, which is directly retrograde, or contrary to its firft Line of Direction from a to b ; and by this laft additional Force, tneVelocity of the Body at r will be fo accelerated, as to pafs through the Diagonal r'<v, of the Square jr<v r t, in the fifth Minute. In this Manner, by the continual Actions of Bodies, whofe Velocities are alike increaf�d, at the End of every Minute, tne Velocity of a Body may be fo increafed, as to travel ten thoufand Millions of Millions of Millions of Miles in a Minute. Secondly, That the Change of Direction is always proportional to the Force JmprelTed, is evident by all the preceding Lines of Direftion of the Body b, for the diagonal Line b e is the fame to the Line b d, as it is to the Line f b- That is, the Angle» fb e, and e b d, are equal, and confequently the Diagonal l> t, . . which!

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		Of M E C H A N I C K S. 189 Ahich' is the f�cond Line of Direction of the Body b, is perpendicular to the Angle b d, and therefore is a proportional to the Force impreffed at b. The like is to be underftood of the Diagonal n e, which is perpendicular to the Angle i e h, alio of the Diagonal r n, which is perpendicular to the Angle 0 u p. and of the Diagonal /■ nj, which is a Perpendicular to the Angle t r �, tsc.* v That the Increafe or Diminution of Motion, or the Velocity with which any Body is moved by the Action of a Power upon it, is proportional to that Power is evident ; for if i apply a certain Power to a Body, that will make it move with fuch Velocity, as to pafs in one Minute coo Yards; to make two iuch Bodies pafs coo Yards in one Minute, will requiie a Power double to the 'former, becauf� there is double the Quantity of Matter, to be removed in the fame Time. And on the other hand, if this double Force be applied to either one of the aforefaid Bodies, which arc fuppofed tp-be equal, iis Velocity will be doubled, and cor.fequently it will travel a thou!aftd Yards in one Minute. Hence 'tis plain, that the Degree of Motion, into which any Body is put out of a State of Reft by any force or Power, will be proportional to that Power ; that is, a double Power will give twice the Velocity, a treble Power three times the Velocity, a quadruple Power four times the Velocity, �V. Law 111. Repulfe, or Reaction is al-jja'p equal, and m contrary Direction to Impidfe or JlSion, i e. I he Ailicn of tiKO Bodies upon cash other is always equal, and in contrary Directions. When any Body arts upon another, the Action of that Body upon the ether is equalled b) the contrary Re-action of that other Body upon the Firir, and are both contrary in their Directions. The Re-action of Bodies is caufed by their Elafiicity, which all Bodies is Nature have in Some Degree or other, though none are perfectly Eiaftick. If the Eody a, Fig.* C, Plate LXX VI. defcend obliquely to b, and firike the horizontal Line at /;, it will by its Elafiicity rebound up towards c, and the Angley'b c, which is called the Angle of Reflection, will be equal to the Angle d k e, the Angle of Incidents. The Eiaftieity of a Body is a Springinefs of its Parts, in the Recovery of its Form immediately after its Form has been altered by another Body acting against it ; as in Woo', when its Figure, after being pre/Ted down is changed, ic vviii, when the Prcffure is taken away, ipring up to its natural State as before ; fo llkewiie a Bladder, blown full of Air, by being prefied on any Part, its Form is changed, du: the very In flam. of Time that the preffure is removed, it will, by the Spring of Air within, -e- cover its former Figure ; and every Force fo applied has at the fame Time an equal fprirtging Force acting again:: it, which is the Re action of the Bod;-. So an Hoop of Iron or Wood, truly circular, as b g, Fig. I. by being (truck: on, or let fall or, the Ground, will at the Inftant of the Stroke a~ Fail, be changed into an E�ipfis as c f e g, but by its Elafiicity, or Springinefs of Parts, it will recover it.'elf into a Circle agsin. The Action and .Re-action of Bodies on Water is very eafily underiiood ; for if & and r, Fig. H. reprefent two Beat� of equal Magnitude and Weight, floating on a flagrant Water, and � Man ftand - ing' in b, by means of a Rope, pull the Boat e unto him, the \ effel e wiii re- act, and at the fame Time pull the Veffel b towards it, with the fame Force, fo that both Vefiels will meet at a, which is the Middle between both. Now 'tis very plain that, that if ;he V'effel c did net react the fame Force on the Perfoir' in the Veflel b as the Force of the Perfon in b aits on c, they would not mecf at a. Now, finee by this 'tis plain, that Action and Re-action are equal, therefore a Body at Reft cannot be removed by any Force, that is lefs than its Weight ; and as I have, in the falling of Bodies, demonuratcd the [r.cieaie of Forte, it is therefore to be underftood, that ail manner of Force given by PreflurS, Blows; Lif.ings, Puilings, Drawings, fc'c. is ecu^l to f�tn� certain Wergh't! For if I pur a Found Weight into & Scale, and with my Hrr.d prefa cow;, cbe other; fo* E b �*

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		ipo 0/ MECHANICK S. as juft to balance the Weight, the Force of my PreiTure is then equal to a Pound; and fo in like manner I may continue to increafe that Force on the one Side, againft. Weight in the other, until 1 prefs the whole Weight of my Body on the Scale: And which being the greateft Force of this Kind, that lean make, there- fore no Man, with a fmgle Pulley, can raife any Weight greater than that of his own Body, unlefs his Body is confined to the Ground. Again, If with a Hammer I ftrike a Blow in an empty Sca'e, fo as juft to raife a Pound Weight in the other Scale, the Force of that Blow may be faiff- to be equal to one Pound, although in reality 'tis fomething more, otherwife it could not juft raife the Weight above the Level of the Scales. In this Manner, the F'orce �f Blows may be made equal to any given Weights, and by this Method of linking into an empty Scale, againft Weight increafed or diminifhed, as Occafion may require, the Force of any Blow may be nearly and eafily difcovered ; and fince that Bodies at reft cannot be removed, or put into Action but by means of Forces or Powers fuperior to their Weights, therefore to remove heavy Bodies there has been an abfolute Neceflity of inventing divers Kindsof Powers, which with the Strength of a few Men will raife and remove Bodies of very great Weights at pleafui e. LECTURE III. Of the mechanical Powers in general. THE Powers ufed for thefe Purpofes are ufually reckoned in Number fix', viz. Firft, Libra, the Balance. Secondly, Feats, the Lever, or Leaver. Thirdly, Trochlea, the Pulley. Fourthly, Axis in Peritrocbio, or the Axis in tho Wheel, and in the Wind-lace. Fifthly, Cuneus, the Wedge; and fixthly, Cochlea, the Screw. But as I proceed, I fhall prove the Balance, the Pulley, and the Axis inPeritrochio., to be no other than Leavers, and the Screw to be no more than a Wedge, fixed about the Body of Cylinder; therefore the fix Powers are reducible unto three. All the Effe&s of thefe Powers may be judged of by this Rule. When two Weights are appliedtc any of thefe Powers, the Wtights ivill equipondi rate y if'when put into Motion, tlxir Velocities be reciprocally proportional to their rrfpec- tiiie Weights. First, Reciprocal Proportion is when in four Numbers, the fourth is leiTer than the f�cond, by fo much as the third is greater than the firft, and ince verfa. The whole Effect of thefe Powers, to raife or fullain great Weights with a fmall Power, is produced by a Diminution of the Velocity of the Weight to be raifed, and incrcafmg that of the Power, in a reciprocal Proportion of the two Weights and their Velocities. That is, by giving as much mce Velocity to the Power, as it weighs lefs than the Weight, that the Quantity of Matter fixed at each End of a Leaver or other Power, being multiplied by its Velocity, may iliew that there is an equal Quantity of Motion at each End ; and therefore it will follow, that when equal Motions act with contrary Directions, they caufe an Equilibrum. . Sec9hdltt, an Equilibrium is when the two Ends of a Balance hang fo ex- actly level, that neither doth afcend ordefcend, but both keep in a Pofnion pa- rallel to the Horizon, which is caufed by their being both charged with equal Weight, as the Bodies d e, hanging at the Ends of the Balance a b, in Fig. M. In every Body there are properly three Kinds of Centres, iriz. Its Center of Magnitude, its Center of Motion, and its Center of Gravity. Fisrt, The Center of Magnitude of any Body is that Point which is equally diftant from its extreme Parts, as the central Point a, of the Sphere, Fig. L, fev. Secondly, the Center of Motion of any Body is a Poir.t about which any Body moves, when fattened any ways to it, or made to revolve or turn about it. So the Body e, in Fig. N-. being fattened with a String to the. Point a, and made Z J tO

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Of M E C H A N I C K S. ipr �o turn about it in the Circle c b if, the Point a is the Center of Motion to the Body e. In the following LeSures on the Balance, Leaver, and Axis in Peritrocbio, the Center of their Motion is called the Fulcrum. Thirdly the Center of Gravity of any Body is that Point on which, if the Body be fupported or fufpended irom it, the Body will reft in any given Situation. In all regular Bodies, whefe Matter is equally the fame throughout, the Cen- ters of Magnitude and Gravity are in the fame Points, butin irregular Bodies not fo ; and therefore in irregular Bodies the Center of Gravity will defcend, till it gets under the Center of Motion, unlefs it be perpendicularly over it ; and from hence we are taught a Method for finding the Center of Gravity of any irregular Body, as follows, <viz. Sufpend or hang up fuch a Body fuccelhvely, by different Sides, and with a Plumb-Line, let fall from the Center of Sufpenfion, fo 3S to touch the Body in each Cafe Obferve where thofe Plumb-lines would in- terfecl each other, being continued through the Body, and their Point of Inter-? feclion is the Center of Gravity required. 'Tofindthe Center of'Gravity common to tivo, or more Bodies, covnefled together by an inflexible Rod, or Rods, Fig. V. and Z. Plate LXXVI. First, Let the Bodies a e. Fig. V. connected together, by the inflexible Rod a c of any known Length, be given. Divide « f in b ; ft) that a b is to b c, as the Body c is to the Body a ; then the Point b is the Center of Gravity re- quired. Secondly, Let b d g, Fig. Z, be three Bodies, whofe refpeclive Centers of Gravity are joined by the Lines b d, b g, and dg. The Line b d, being fo divided in c, that b c Sears the fame Proportion to c d, as the Body d bears to the Body b c is the Center of Gravity common to thofe Bodies, as before in Fig. V. Draw the Liner�, which divide in � ; fo that c f fhall be lofg, as the Weight of the two Bodies b and d are to the Body g ; then the Voint f will be the Center of Gravity common to the three Bodies b, d, g, and they being fufpended at that Point, will hang in a horizontal Pofition. To find the Center of Gravity of a Hemifpbere, Fig. Y. Mare b c equal to \|- of its Radius ; then the Point c is the Center of Gravity required. The Center of Gravity in Geometrical Squares, Parallelograms, Rhombus's, and Rhomboids, is the Point in each Figure, where the two Diagonals interfect each other. All the Parts of Homogeneous Bodies hai'e an equal Preflure about their Cen- ters of Gravity, and therefore when the Center of Gravity of any Body cannot defcend, the Body will remain fixed. This is manifeft by the Geometrical Square a h df, Fig. A B, whofe Center of Gravity is the Point c, and which cannot de- fcend, until the Diagonal af, railed on the Angle/; has pafl'ed the Perpendicu- lar;'/, which will carry c, the Center of Gravity with it beyond^, the Perpen- dicular of its Ba^re, when it will confequently defcend. The fame is alio to be obferved of the Rhombus r t I », whofe Center of Gravity is q, and which mult be removed in the Arch q p, beyond o, the perpendicular Limit of its Bafe In, before it can detcena ; bit the Rhombo�des xy \ iv, whofe Center of Gravity z, being without vnv, the perpendicular Limit of its Bafe I v ; its Centers; will defcend in the 'irch z 2, and confequently the Fig. z y 1 iv. cannot (land ou the Bafe 1 w. From hence 'tis plain, firft, That ail Bodies, whole Centers of Gravity are within the perpendicular Limits of their Bafe, cannot fall. Se- condly, That all Bodies, whofe Centers of Gravity are beyond the perpendicular Limits of their Bafe,cannot itand. Thirdly, That the lefler the Bafe of any Body is, theeafier it will be moved out of.ics Poiition ; becaule the leaft Change is ca- pable of removing the Line of Direction beyond its Bafe. This is the Caufe, why a Ball, whole Bale is a Point and a Cylinder,, whofe Bafe is a Line, are rowl'd eafily by a fmall Force, on a horizontal Plane,.

	Bb 2
	Bb 2	Is

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		*>gz* Of M E C H A N I C K S. ' In the following Lectures it is to be obferved, First, That when a power applied can fuftain a Weight by the Means of a¹BaisteCa, Leaver. Pulley, Ijfc. if an Addition of Power, i ho' it be as little as ca,n be imagined, be made, it will overpoife or raifc the Weight. Secondly, That the Weight of Leavers, Pulleys, &jV. and their Friction is not fuppofed to be any Thing, altho' Rules will be given for to find both. Thirdly, That a Leaver is confidered as a right Line, and the Pin on which a-Pulley moves the i'jme. Fourthly, By Power applied is meant a Force, as that of Weight, Water, Wind, Cifr. Fifthly, That whatever any of thefe Powers gain in Strength they lofe in Time. L E C T U R E IV. Of the Balance. THERE are three Kinds of Balance, >vi«n The common Balance as ufed to common Scales. The Staiera Roniana, Roman Balance, or Steel-yard;, and the Falfe Balance. First, the common Balance is no other than aBeam divided into two equal' Parts, as b �, at c, Fig. O. (and by the enfuing Lecture will appear to be a Leaver of the firft Kind) which inftead of refting on its Fulcrum at c, the Center of its Motion is there fufpended. The two half Parts b c, and c f, are called Bracbia's. To have the Balance horizontal the Center of Motion muft be fomething above the Center of its Gravity ; for was they to be both in one Point, which they would be, was the Beam to be a right Line, asat; then thole Weights which equipondeiated when the Beam hung horizontally, would alfo equipon- derate in any other Pofnion ; whereas, when the Center of Motion is placed a little above that of Gravity as aforefaid, if the Beam be inclined either way, the Weight moll elevated will furmount the other and defcend, caufing the Beam to Aving, until by Degrees it recovers its horizontal Pofition. The Reaion is very plain. Suppofe a i, Fig. P. be the Beam of a Balance put into an obiique Petition, and the Perpendiculars a c. una i g, be drawn from its Extremes a and i, to the horizontal Line c h, 'tis evident that c e, the Diftance of the Perpendiculars c, is greater than eg, the Diftance of the Perpendicular � i ; and as the Weight m is equal to the Weight o, the Weight m will therefore raifc up the Weight o. But was the Ballante a right Line, as b /;, having its Center of Motion and of Gravity both in the Point e, then the Dillances de, and e h, of the Perpendiculars b d, and h k, would be equal, and the equal Weights/ and n, would equiponderate in that obiique Pofition, which the Beam a e i cannot do, becutfe the Center of its Motion is above the Center of its Gravity, which caufes the upper Pointe to be the Diftance of c d, without the Perpendicur b d, and the lower Point i to be the Diftance of g b, within the Perpendicular � k, and therefore c e is longer than e g, by twice c d. The Proportion that the Power has to the Weight in the common Balance is as i, the Length of one Brachia, is to i, the Length of other Brachia ; fois the Power applied, to Weight required, to equipoife it. 1:. The Statera Romana, or Roman Balance, commonly called the Steel-yard, fig. R. and C>. Plate LXXV1. " This Sort of Balance is called the Roman Balance, from its being ufed in sa at Marne, and it being originally made about 3 Ftet in Length, and of :->fod, 'cHras'tberefbre calte� vu'<Steel"yavd,* and i« thos mad� j pr�paie a fmall fquaie Bar of iron or Steel, as 1 z a, Fiv. R. of any abewgth, and of equal Thick- nefs; and let the Point a be 'he Ganter of Motiot». Make the flat End b c of <»ch Solidity, as to balance the Part 2 z g. At ;i:;y lDiftan.cc from a fix a Point as c, on which, thefeveral Things to be weighed are to be fufpended. Note,

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		Of M E C H A N I C "K S. 1513- Note, The Peint c is here fixed below the fireight Line 11 b, for the fame Reafon ai in the common Balance. Draw c h, perpendicular to the Line 12 b ; make the Divifions, a \ ; 1, 2 ; 2,3; 3,4; ferV. each equal to a b. Then I Pound Weight, applied at 1, will equipoife i Pound at c ; alfo 1 Pound Weight at 2, will equipoise 2 Pounds at c ; ail') I Pound Weight at 3, will equipoife 3 Pounds at c, and 1 Pound at iz, will equipoife \z Pounds ate, &c. For as a b equal to Orfe Part, is to a 12, \z Parts, fo is 1 Pound Weight at 12, to 12 Pounds (as the Body f) at r ; aud therefore the Point a is the common Center of Gravity of the two Weights, be- caufe 13 the Sum of the two Weights, is to 1, the lead Weight, as the Length. of the Balance is to one Part, the Diftance of the great Weight, from the Center of Gravity. To find the common Center of Gravity of tivo Bodies applied to a Beam of a knovjn Weight and Length, ixhich is not balanced, as Fig. R. 'was fppoj^d to be, by the more folid Part b c. Let ,i b, Fig. Q_be divided into 13 Parts, let the Body x be 1 Pound, and the Bod ^12 Pounds, and let the Point a be their common Center of Gravity, and the Weight of the Beam equal to 3 pounds. On a the common Center of Gravity, hang the Weight /, equal to the Weights a: and k, and at h, the Center of Gravity ot the Beam, hang the Weight g, equal to 3 Pounds, the Weight of the Beam. Then as the Sum of the Weights g and / 16 Pounds, is to 3 the leffer Weight g ; fo is the Diftance h a, of thoi'e two new Weights, 5 � ; to I j_rthe Diftance of n, from the true Center of Gravity required. . i [I. A 'alie Balance, as Fig.* S. has its Beam unequally divided, as c e, and ed, which are to one another as 9 is to 10, is'c. and its Scales being alfo in the fame Proportion, they will therefore equiponderate as the juft Balance, and whatever is weigh.-d in the Scale, hanging on c, wili be ^ lefts Weight than it really ought to be; but this Cheat is immediately df covered by changing the Scales. LECTURE V. Of the Lever, commonly called the Leaver. ^f \ ^NH E R E are three Sorts of Leavers, which are diftinguifh'd by the different .J. Manners of applying the Power and Weight. A Leaver of' the firft Kind is that, whofe Fulcrum is between the Power ap- plied, and the Weight that is to he raifed, as Fig. A Q^ Plate LXXVI. where the Power is applied at a', the Weight at c, and the Fulcrum at a. Hence 'tis piain, that the common Balance Fig. O. the falfe Balance Fig, S. and the Roman Balance Fig. R. are all Leavers of the in ft Kind, becauie theirCenters of Motion as Fulcrums, are between their Powers and Weights. To know what Weight can be raifed by a Leaver of the firft Kind, this is the Analogy. /Is the leffer Brachia a c is to the greater Brachia � a, fo is the Poii'cr applied at d to the IVctght it wi/l equ'ipof _at c. Therefore a little more being added to the Power at b, will raife tie Weight required. The Length of a Brachia is the Diftance of a Power, or of a Weight, from a Fulcrum, and is always equal to a Perpendicular let fallen from the Fulcrum, upon the Line of Direction of the Power or Weight. So b i, Fi?. A N, is the Diftance of the Power at d; becauie 'tis perpendicular to the Line of Direction. d b, of the Pou er at d, itl like manner the Line i e, which is perpendicular to e b, the Line of Direction of the Powers is the Diftance of the Power at?, as alfo is ai the Diftance of the Ppwer at c. Hence 'tis plain, that the greateft Power is that at d, whpfe Line of Direction is right angled with the Leaver b k, and which is yet more evidently fo by the Power applied at�, whole Diftance from the Fulcrum is no more than h i, equal to the Perpendicular if The like is alio, to be underftood of bended Leavers, as fig, A F, A E, AG, and A L. It

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		194 O/ MECHANIC K S. It matters not, whether the Brachia's of a Leaver be ftreight or curved, as Fig. A M, and A [, for in both thefe Cafes the Diftanees of the Powers, r.nd of the Weights from their Fulcrums are the chord Lines of the Arches, and not the Arches tkemielyes. The nearer the Weight is to, and the farther the Power is from the Fulcrum, the lefs will be the Power, and the lefs will be the Height that the Weight can be raifed ; for if the Body k in Fig. W. he removed nearer to the Fulcrum from op unto n m, it will not require i'o great a Power at -, to raife it, as when at op, nor can it be raifed fo high, as when ate/; for if two equal Bodies be placed at nm, and op, and s the End of the Leaver s p, be forced down to t, ;he Body o p will be raifed to a q, and the Body n m, but to c b. Whmi a Body is on the End of a Leaver, as the Body'« o I c, Fig. A K, (o as co have its Center of Gravity above the Leaver, and is equipois'd by a Power at <o, whofe. Line of Direction is perpendicular to the Leaver />v ; that Power will be increafcd as the Body is railed, as to p a, and decreafed as the faid Body is let lower tofg', for in the frit, the Center of Gravity of the Body at/is brought nearer to the Fulcrum, and in the laft at k it is farther. When a Body, tix'd to the End of a Leaver, has its Center of Gravity below the.Leaver, as the Body 8, ii, io, iz, Fig. A H, to raife the Body as to 7, �, the Pom er mud be increafed ; bar to iet the Body down as to i6, 14. the Power mult be decreaf--d ; for tis evident that 13, 14, the centra! Line of' the Body at 16, 14, is nearer to the.Fulcrum than 3, 1, the central Line of the Body at 7, 5, and ccnfequently will be equipois'd at b, by a leifer Power as <-, than that of g, required at/. Thess being underltood, the Nature of Leavers in g�nerai will bemadeeafy, a,s in the following Problems doth appear. Problem I. The L'ligtb and Weight of a Beam, ivhicb has a Body of knonun Weight f xd to or.e . End, being given, to find the Center of Gravity on the Beam, on 'which one Part of the Beam Jball equipoife the other Part, and the given Body alji. Rule, as the Sum of the Weight of the Balance and of the Body is to the Length of the Balance ; fo is the Weight of the Body to the leffer Brachia; or fo is the Weight of the Balance only, to the greater Brachia. Prob. II. Fig. T. Plate LXXVI. T--WO Bodies as e g, of inoivn Weights, of ivhicb g is hung at b, to the End of a Beam oj Icnoion Weight and Length, wherein the Fulcrum is fixed at a, to find a Point as c, to hang the Weight e, fo that the Weight e, and the Weight of the Balance, fall equipoife the Weight g. Let the Length of the Peam be 14 Inches, its Weight 2 Ounces, and the Fulcrum a one Inch from b, let the Body g be 15 Ounces, and the Body e 1 Ounce: divide the Beam in the Middle at d, and there hang the Body � equal to 2 Ounces the Weight of the Beam. Then as a b, one Inch, the leffer Brachia, is to a d, fix Inches, the greater Brachia, fo is the leffer Body f, 2 Ounces to 12 Ounces, which is a Part of the Body g, whofe Weight is 15 Ounces, which is 3 Ounces more, than the 12 aforefaid. To find the Point c, where the Body e, equal to 1 Ounce, will equipoife the aforefaid 3 Ounces : Say, as the Body e one Ounce is to 3 the remaining Ounces in the Body g, fo is I the leffer Bra- chial a, to 3 the Diftance of the Point c, from the Fulcrum a. Then the Body f, equal to 2 Ounces, is to 12 Ounces in the Body g, as the Body e, equal to 1 Ounce, is to the 3 Ounces in g, and therefore the Bodies � and e, being fix'd at d and c, will equipoife the Body g, on the Fulcrum a. A Leaver of the f�cond Kind is that, whofe Fulcrum is at one End, the Power at the other, and hath the Weight between them, a Fig. X. Plate LXXVI. where a r is the Leaver, a its Fulcrum, r, the Place where the Power is to be applied, and m », and 0 p, Weights placed between them to be raifed. To know what Weight can be raifed by a Leaver of the f�cond Kind this is the Analogy. As the Difiance of the Weight from the Fulcrum is to the Diftance of the Power from the Fulcrum, Jo is the Power to the Weight, that will equipoife it. Hence

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		Of MECHANICK S. 195- Hence 'tis plain, that if a Leaver as d I, Fig. A O, be divided into 4 equal Farts at efi, if the Body c be applied as a Power equal to 1 Pound, ie will require 2 Pounds to equipoife it in the Middle at f, becaufe I Pound wili be fufta ned by the Fulcrum at /. And for the fame Rcafon the Body at s muit be I Pound and -j, and that au' rnuft be 4Pounds. A Leaver of the third Kind hath its Fulcrum atone of its Ends, the Weight at the other, and the P'nver applied in fome Part between them, as in Fig. A P, where n e is a Leaver whofe Fulcrum is at e, its Weight at », and Power applied between them as at /■ h g, the equal divided Parts, as in Fig. A O. To know what, Weight can be raifed by a Leaver of the third Kind, this is the Analogy. As the Length of the Leaver is to the Dijlance of the Power from the Fulcrum, fo is the Power applied to the Weight it will equipoife. Now as the Power is applied between the Fulcrum and the Weight, therefore the Power muft always be fuperior to the Weight ; for if the Body m be equal to 1 Pound, it will require a Power equal to 2 Pounds at h; of 1 Pound and f at i, and of 4 Pounds at g, to equipoife it. To thefj three Kinds of Leavers fome add, what they call a Leaver of the 4th Kind, a.s Fig'. \ L, which in Fact is no more than a Leaver of the firft Kind, -as having its Fulcrum c between the two Brachia's b c and c d. LECTURE VI. Of the Pulley. AN upper Pulley adds nothing to the Power, for in Fig. A, Plate LXXVII. to fuftain the Body /at <-, there rnuft be a Power applied by « at a, which is equal to the Weight of the Body/; becaufe a d, the Diftance of the Power from d the Centre of the Pulley, is equal to d c, the Diftance of the Body from the Centre ; and from hence 'tis plain, than an upper Pulley is a Leaver of the full Kind, becaufe confidering its Diameter, as the Length of the Leaver, its Centre is the Fulcrum, and as both the Brachia's a d and d c are equal, therefore an upper Pulley is of no other Ufe, than to communicate the Motion of the Rope to an under Pulley. An under Pulley, as Fig. I. doubles its Force, for if the Body � weighs z Pounds, 'tis plain, that the Power applied at d can fuftain but half the Weight ; becaufe the Line on the Hook a fulrains the like Quantity. Now if the Dia- meter l d be truly confidered, it will appear to be a Leaver of the f�cond Kind ; for as the Pulley is always rifing on the Line at b, therefore the Point b is the Fulcrum ; and as the Line is always lifting at d, therefore that End of the Dia- meter is to be confidered as the Power, and as the Centre of the Pulley is in the midft between thefe Points on which the Weight hangs, therefore a Power equal. to 1 Pound at d, will equipoife a Weight of 2 Pounds at c. For as b c I, the Diftance of the Weight from b the Fulcrum, is to b d 2, the Diftance of the Power, fo is 1, the Power applied, to 2, the Weight it will equipoife. And in all Tackles of under Pulleys, the Power will be to the Weight it ful�ains, as 1 is to the Number of Ropes applied to the lower Pulleys ; fo in Fig. B, the Power at� is to the Weight, as « is to 2 ; in Fig. C, as 1 is to 3, in Fig. D, as i is to 4 ; in Fig. E as 1 is to 5 ; and in Fig. F as 1 is to 6. Weights may be fuftained by Pulleys, with a fmall Power, the Pulleys being applied as in Fig. G, where the Body /, equal to 1 Pound, will equip life the Bodyj, equal to 8 Pounds. Foras 1 Pound applied at tn, by Means of the upper Pulley i k, will equipoife 2 Pounds at e, fo 2 Pounds applied at p will equipoife 4 Pounds at c, and 4 Pounds applied at r, will equipoife S Pound:, at a, L'!c. For as 1 at m, is to 2 at e, fois 2 at p, to 4 at c, and 4 at r, to S at a. A Weight may be alfo fuftained by Pulleys with a fm-ail Power, the Pul- leys being applied as in Fig. M ; for if the Power at ot be equal to 1. Paand, aj d againft it bs hung the Body / equal so 1 Pound, :hsy will together equip >iie the

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		196 0/ M E C H A N I C K S. the Body�, equal to 2 Pounds, and the Body g, with the Power 1, and Body / equal to I, which together are equal to 4 Pounds, will cquipoiie the Body k equal to 4 Pounds, &c. In Fig. H, the Power at », equal to 1 Pound, equi- poifes 1 Pound of the Body /-, which together, by Means of the Pulley e f, equipoifes z Pounds more of the Body h, and thefe together being equal to 4 Pounds, by means of the upper Pulley b d, equipoifes 4 Pounds more in the Body k: fo that in this Example, the Power at i equipoifes feven times its own Weight. LECTURE VII. Of the Axis in Peritrochio, commonly called the Wheel and Axis. THIS Inltrument is no other than a Wheel fixed on a Cylinder, as d i, on a b, Fig. W, Plate LXXVII. The Central Line a b of the Cylinder is called the Axis, and the Wheel d k i is called the Peritrochio. If b d, and ef, be fixed on an Axis as a h, directly oppofite and parallel, and confidered as the two Brachia's of a Leaver, then the Axis a b, on which they are fixed, will be the Fulcrum ; and if b d be confidered as the Radius of a Wheel, as dc, Fig. W, and ef, Fig. T, the Radius of a Cylinder, on which the Wheel is fixed, as ef, Fig. W, 'tis plain, that this Machine is a Leaver of the firft Kind ; and therefore, as e f, the Radius of the Cylinder, Fig. W, is to d C, the Radius of the Wheel; Jo' is the Power to the Weight ; and,<when Spokes or Teeth are fxed in Wheels, then, as theDij'r.nce of the Extremes of t hof e on the Pinion or fmaller Wheel, from the Axis, is to the Dijiance of the Extremes of thofe on the greater WheeL fo is the Power, to the Weight. By the Multiplication of Wheels, very great Weights may be raifed ; an Example of which I have given in Fig. K, where the Body q, equal to 1 Pound, equipoifes the Body r, equal to 105 Pounds. By means of the four Wheels nfo e, on whofe Cylinders are fixed the fmall Wheels g e b, whofe Teeth work in the Circumference of the large Wheels, the Radius of every fmall Wheel on the Cylinders is 1 Foot. The Radius of the great Wheels are as follows, viz.. The Radius of the Wheel c is 2 Feet and half; of the Wheel o 3 Feet; of the Wheel m 3 Feet and half, and of the Wheel n 4 Feet. Now the Powerq to the Weight >■ is elms cacul�ted ; Firit, As I, the Radius of the fmall Wheel b, is tc 2 and half, the Radius of the great Wheel c ; fois I the Poiver Q, to zand half'the Weight that it idll cquipoife at o. Secondly, As 1, the Ra- dius of the finall PVheel e, is to 3, the Radius of the great Wheel o, fo is 2^ the Poteer applied at O, by the fmall Wheel b, to 7 J the Weight that aviliiquipoife at g. Thirdly, As I, the Radius of the fmall Wheel g, // to 3 and \| the Radius of the great Wheel m, fo is 7 and f, the Pi�ver applied at g by the fmall Wheel e, to 26 and\ the W eight that will equipoifc at n. . Fourthly, As I, the Radius of the Cy- linder p, is to 4 the Radius of the great Wheel n, fo is 26 and-, the Pciver applied at n, by the fmall Wheel <&, to 105 theWeight r, that nmil but equipoife the Body q .equal to I Pound. The Application of a Power to a Wheel is always the greatelr. when applied at right Angles to its Radius, as thePowerf/, Fig. L, Plate LXXVH, which is perpendicular to the Radius cf. and at the Diltance of c fLom the Fulcrum c; therefore when a Power is applied obliquely, as b d to the Radius e d, the Powe* is leflened in Proportion, as f c is to e c. LECTUR E VIII. Of the Wedge or inclined Plane. A WEDGE is the moll plain and finiple Inftrumerst of all the mechanics! Powers, and is put into Action by the acting or ftriking of another Body upon it, which is called Perettfl&t*

			T'.S

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		Of M E C H A N I C.K S. 157 - The Center of Percuflion is a Pointon the Top Surfaceof a Wedge,- which is directly agajnft the Cen'er of the Body (truck thereon ; fo the Point f, Fig. A C, is the Center of Percu-fuon, as being direflly againfi a, the Center of the Body, or Mallet b c, whofe Line cf Direction is b d. It is to beebferved here, a- in the preceding Powers, that the greater! Force is made, when the linking Body Falls perpendicular upon the upper Surface of the Wedge, as the Mailed c, 'on the Wedge/ in the Body d, Fig. A D, whole Line of Direction is c d. To underttand the Power of the Wed :e, which is fuppofed to be right-angled, as a b c, Fig X, the Lengths of its Bale b c. and of its perpendicular Height b c, mult be known; for as the perpendi u'ar Height b «, equal to 2, is to the Bafe b c, equal to 4, fo is a Force equal to to Pounds, to 20 the Weight it will raife ; and ther. fore, the longer the Bale is, with refpecl to the Height, the leiTer is the Power required ; and the Ihorter the Bale is, the greater the Power mult be. For fnppofitig the 1 liangle c e g, to be a Wedge of equal Weight with a b c, whc.'le Bafe <:» is equal to 3 : Then as 2 is to 3, fo is 10 the �forefaid Power applied to r�, which is 5 leis than 20, the Weight railed with the fame Power by the Wedge: a b c, and therefore to raife a Weight of 20 Pounds with the Wedge eg e, the Power mull be increafed to 13 Pounds \. For as 2' is to 3, fof is 13 \ :o 20. But note, That in ail thefe Calculations, it is fuppofed, that tfieie is no Obflruction by Pnftion, but that the Surfaces of Planes, Wedg�s, CSV are perfectly fmooth. Bodies may be raifed by the Means of one Wedee as the body d unto e, by the Wedge a b c, Fig. Z. if there be a refilling Body, asy'�', that will admit the Wtdge a b c, to pais along the Line b g to k : or when two Wedges mutually refill the Weight of the Body to be raifed, as a b c, and f e f, Fig. O, which being equally driven by each other's Sides, will raifa the Body O unto the Line a c. To raife a Body from the Ground, as ah b g, Fig. N, by means of the Wedge efe, is the fame thing as to fplit a Body afunder, as Y, by the Wedge b d; for if the adhering of the Parts of the Body together, which are to be difunited by the Wedge, be confidered ss Weight, the Power, in both Cafes muff, be equal, and the Force with which a Wedge will fo lift a Weight-, or difunite the Parts of a Body, by a Blow upon iis End, will bear the fame Proportion to the Force, wherewith the Blow would ait on the Weight,, if direfily applied to It, as the' Velocity which tiie V- edge receives from the Blow btars to the Velocity where- with the Weight is lifted, or the Parts of the Body difunifed by the Wedge. Bodies may be eqnipoifed on an inclined Plane, as the Body c, Fig. P. Plato' LXXVJ1 by a Weight of !efs Force, as the Body a, provided that the Body � be to the Body e, as the perpendicular Height cf the inclined Plane is to us Kypothenufe. LECTURE IX, Of the Screw. THIS Power i; nothing mere Chan a Wedge, or an inclined Plane, fxrvl about the Body of Cylinder, as Fig. A B. Plate LXXVil, of it may be' confidered as a Cylinder cut into continued inclined concave Surfaces, as s t, tu tv y x, bounded by divers circumvolving Helixes or Threads, as c d, A A-, 0 I, zg,- &c. The Screw is applied in two different Manners ; as firfl, to work in a hollow Screw, which is called the Female Screw orNut, fixed in fome particular ner, as the Nature of the Occaficn requires ; and fometlmes to the Teeth of a Wheel, as to the Spindle of the Flyers of a Kitchen Jack, (5.-. The Force of a Screw is according to the Angle that the H�lix" or Thread makes with the Bafe of the Cylinder, for as' it is really a Wedge, tjsctefe more acute the Afcent of the Thread is, the Iefs Power is r�q&irej to riif� a For, as the Heigh; o^c the Thread vu one half Revolution, h to th� S^-emi--circ'.im- C 1: ff:f-.VS

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		�08 �/ MECHANICKS. ference of the Cylinder's Bafe, fo is the Power to the Weight; becaufe the Height of the Thread is confidered as the Height of a Wedge, and the Semicircum- feience of the Cylinder's Bafe, as the Bafe of a Wedge; and as this Power i> worked by a Leaver of the iecor.d Kind, it may be made of prodigious Force : Suppofe a Screw of 7 Inches Diameter, whofe Circumference is 22 Inches, have its Thread to rife 1 Inch in half a Revolution, then the Power of f:ch a Screw will- be as 1, the Height of the half Revolution of the Thread, is to n, the half Circumference of the Cylinder, fo will the Power be, to the Weight it will cqui- poife. And if a Leaver of 10 Feet in Length, have its End put into the Cylin- der of the Screw, fo as to be juft at the Axis of the Screw, which is done by putting 3 Inches and a half of the Leaver into the Cylinder, then the Axis of the Screw will be the Fulcrum of the Leaver, and the Outlide of the Cylinder will be the Weight to be removed. Now as in the remaining Length ol the Leaver, <viz. 9 Feet 8 Inches and a half, equal to 116 Inches, contains 5 md half, the Dillance of the Weight from the Fulcrum, 33 times and 7 ; e efofe the Power of the Leaver only is, as 1 is to 33 and -'-. Now fuppole a Man's Strength to be equal to 100 Pounds, then as 1 is to 33 <ind ', fo is 100 to 3300! ; and as the Force of the Srew is as 1 is to 11, io is 3300', the Power applied on the Screw by the Leaver, to 36,301 Pounds ^ its Equipoiie ; which by a fmall additional Power continued, may be raifed to the Height of the Screw. LECTURE X. Of the Velocities 'with which Bodies are raif i, and the Spaces through which they and their Foivers move. II THAT any Engine gains in Power, it lofes in Space : In the Leaver, Fig. VV W' P'^ate LXXVI. if .t r be double to r p, the End s being moved down to t, muft move with twice the Velocity, that the End p will do, in mov- ing to a, and the Arch p q will be but half the Arch s t. The fame is alfo to be obferved in the Leaver ar, of the f�cond Kind, Fig. X. fer in raifing its End r to g, the Body at m n, removed to c b, the End r will move with double the Velocity of the Body m n, for the Arch >� g is double the Arch» b. In the raifing of a Weight by one or more under Pulleys, the Space through which the Power muff pals, is to the Space through which the Weight muft rife, as the Power is to the Weight ; fo in Fig. F, Plate LXXVI I. as 1, the Power at x, is to 6 the Weight at W, fo is 1 to 6, the Space through which the Power mull pafs ; and therefore to raife the Body W, 1 Foot in Height, the Power x muft defcend 6 Feet, and confequently mult move with 6 times the Ve- locity of that of the Weight. The like is alfo in the Wheel and irs Axis ; for to caufe 1 Revolution of the greateft Wheel n, on which the Body r is fixed, the little Wheel <r mull make 42 Revolutions ; and if the Diameter of the Cylinder p be 2 Feet, the Weight will be raifed 6 Feet f. But as the Diameter of the fmall Wheel f is 5 Feet, the Power q, equal to I Pound, muft pafs through a Space equal to 4.2 times 15 Feet �§■ its own Circumference, 'equal to 66o Feet, or fo much Rope muft be drawn at q from off the Wheel c. As I have already noted, that the more acute the Artgle of a Wedge is made, thelefsForce is required; therefore whatever is gained in Force by the Acute- r.efs of the Wedge, fo much is loll in Space or Time, becaufe the more acute a Wedge be made, the greater Length the Wedge muft be, to rife equal in Height with another Wedge, whofe Angle is lefs acu e; and in the aforeiaid Example of the Wedge and Leaver the Power mull revolve 30 tin.es in a Circle of 20 Feet Diameter, W'hofe Circumference is 62 Feet �, to raife the Weight 5 Feet «a Height, which Space is equal to 188 j Feet, �.

			PART

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		s 9?

PART VIII. Of Hyp rost a ticks. THE Vford Hydrojiatich, isderivedfromu JVp Water, and�-ar.'M the Science of Weight, from <rTi« to weigh.* As to fully illuftrare this Science in every oi its Particulars, would not only fwell this Volume much beyond its intended Bulk, but would contain many Particulars which are not immediately ufe'ul to Workmen, for whom this Work is defigned, � (hall therefore only fpeak of inch Parts, as are abfolutely receffaiy to be Wncerftoed by Workmen in general. Befor» we proceed to this Subject, I muft firft explain the Nature and Pro- perties of Air. Air is an invifible fluid Subftance, which not only environs the whole Globe of Earth and Water, but is alfo contained in the Interfaces or Pores of all Bodies. Its principal Properties are Fluidity, Traufparency, Rarfficaticn, Condensation, Elajlicity, and Weight or Gravity. That Air is a fluid is evident by its yielding to every Force; that'tis tranf- parent is evident to every Eye ; that it may be rarefied is evident by the Experi- ment of an empty Bladder tied clofe at its Neck, and laid before a Fire, which will fo rarefy the little inclos'd Air, as to make it extend the Bladder to its ut- inoil Stretch, and at laft break through it, with a Report equal to a Gun. And by Computation it is prov'd, that the Air at 7 Miles Altitude from the Eauh is 4 times rarer or thinner than at the Surface ; at 14 Miles Altitude 16 times rarer ; at 21 Miles 64 times i at 28 Miles 256 ; at 35 Miles, 1014 times ; at 70 Miles aooat. 1,000,000 ; and fo on in a geometrical Proportion of Rarity, compared with the Arithmetical Proportion of its Altitude. Vide Sir J/aac Newton's Op- ticks, Page 342. By various Experiments it hath been proved, that Air may be fo condenfed, as to take up but _?'₅ Part of the Space it poffefs'd before, and Mr. Boyle found its Spring or Elarbcity fo great, as to dilate or expand itfelf fo as to take up 13769 times a greater Space than before. This Power of Elaftiei'y is accord-; ing to itsD = niity, and its Deniity is found by Experiments, to be equal to its Compredion. The Weight or Gravity of the Air has been proved by divers Experiments of the Air pump, and Barometer, and 'tis found, that a cubical Foot of Air at the Earth's Surface is 830 times lighter than a cube Foot of River Water, and therefore its Weight is fomethkig more than 1 Ounce and _T\|^§g- ; but the Weight of a Column of the Atmofphere, on a fquare Foot of the Earth's Sur- face, when the Air is the heavieft, is found to be equal to 2259 Pounds A<vair- dupcife, (at which Time the Mercury will rife to 31 Inches) which is 15 Pounds and 11 Ounces, on every fquare Inch. But when the Air is lighted, then the Mercury is r�ifad but to 28 Inches; then the Weight of the Atmofphere on every fquare Foot is but 20.5 Pounds, and on every fquare Inch 14 Pounds and { Ounce. The g�eateft Extent of that Part of the Air which is called At mof there, from the Surface of the Earth and Seas, is about 45 Miles in Height. The. Weight of the Air is greater the nearer it is to the Earth"s Surface, which is caufed by the great Weight of the Air next above it.

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		too Of H YDROSTATICKS. To find the Weight of a Pillar nf the Atmtfpbere. Take a glafs Fubje, cf about.3 Keet in Leng-h, apd.abogt -^ or ₇% of an Inch in Diameter, her�netica�ly fe��ed at one End: rill it full Bf¹Q�i"ckfitvet', immerfe the open End, in a fmall Bafon of Quickfilver, and then holding the Tube perpendicular, the Quickfilver within the Tube will fubfide or run out into the Eafon, until it be fuipended at fome Height above z3 inches perpendi- cular Height. The Reafon why the Quickfilver will be ^ro fufpenoed, is, That the Top of the Tube being fealed, the PrefTure of the Pill ot the atmofphere, perpendi- cularly over the Top. of the Tube, if made or, the 'iff' ci t: e i ube only, aid not on any Part of the Quickfilver within it ; and if i: be confidered, that eve. y Part of the Quickfiiver's Surface ip the Bafon about the Tube, equal to the Eafe cf the Tub:, is prtffed by th� fame Weight of'Air as ihat on theTopo* the Tube, 'tjsevident that ihe Preflure of any oneof ttrofePans iseq-ual to the Weight of the Quickfilver preiling«n its own bale ; therefore the Quickfilver cannot defcend Sower, and therefore the Weight of the Quickfilver in the Tube is equal 10 the Weight of a Pillar of the Atmofphere of its own Dia- meter. On this Principle depends the raifing of Water out of Wells by the Help of a common Pump. In Page 24 mav be feen, that a Cube Foot of Quickfilver weighs 874 Pounds 3%, and a Cube Foot of River Water 62 Pounds A; therefore Quickfilver is fumelhing more than 14 tiroes heavier than River Water, and therefore in a. recurved Tube placed wkh the Ends upwards and open, cne Inch-of Quickfilver will keep in Equilibrio 14 Inches of Water. Nov/ to find how high Well-Water can be raifed by a Pump in any Place, obferve how many Inches the Quickfilver will rife in the Tube as alorefaid ; and Jo many times 14. Inches Water may be raifed by a Pump, becauie every 1.1. Inches Height of Water is but the equipoife of an Inch of-Quickfilver. There» iore when a Pillar of the Atmofphere is eqeipo'is'd by a Pillar of Quickfilver, whofe Height is 30 Inches to equipoife a like Pillar of the Atmofphere « ith a Pillar of Water cf the fame Pafe, its Altitude mult be 3; Feet, which is ;o �i-?es 14 jucher, and which ii generally the created Height that Water can be iiiatle to rife b\ the Help of a Pump. The Antlia or common Pump, Fig-. Q. Plate LXXVII. is a Machine of a verv long Date, which is faid to be the Invention of Ctifcbe 1 a Mathematician of Alexandria, about 120 Yeats before Chrifi. This Machine made of Lead confifts of a fucking Pipe, as c p, folder'd to the Bottom of a larger Pipe or Barrel, as at ₈ m, bi t being made ot Wood ii ro mwe than a com 11 on Pipe, open at both End» j but be it made either of Lead or Wood, at a pioper Diftance be'ow its Top, as at/ m, is placed a Valve as 1'. which opens .:. iv; rds . within the upper Part of the Barrel i fitted a Pijl ,-01 Bucket, as/;, jttft a; big as ihe Bote of the Barrel, in which aifo is a Vah'e, that opens upwards, To this Pi her: or Bucket is f xed an Iron Rod. as c k, wliii ^:; by a Pin is fixed to the End of the Handle e f ; butas thereby the Red ^;:- d au.n out of a Perpendicular, tho'there maybe a Joint in the Rod hear the Piiton, the Power mull be greater than was the Rod to rife tip and down perpcndicnltsrly-, which may be eafily effe&ed by the Arch h d, fixed ^;" the upp i P: 1 of the Handle, and bv two Chains fixed from a to d, and I 001 < to b, whi h ,vi 1 rife up and force down the Piiton truly perpendicular anfl wi-'h the lead Fri�ioh. Nov/ the Manner 1 .' the 'in . )rmance is eafily underftood, for when il e Pij on j: forced down towards y, and : Qu tntity of Watt.' poured in at the 'ito, ihe2 valves being then fhur, and the external "iir being feparated from t^: at within i e fucking fiveop, ivl oie Find p is before imrnerfed in'Water, ♦hereto c is fooa as : Piiton with the Water poured on iris raifed, the Air within the fac! ing Pipe by the Fort : of the ¹ tmofphere on the Surface of the 'hv.i;;.' i* the iVtii is puGstd ■'�■ � ic igh tue Valve at /, ar.d fills that Part of thg

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		Of H YD R O S T A T I C K S. 201 the Barrel, in which the Piflion afcended, at which [nftant the Valve 3t / is font. Now as much Air as is contained between the Valve at n m, and the Bottom of ion, fb much Water at the fame irritant afcemta) at the lower Fart of the fucking Pipe. The Pifton bein; ; gain forced down the Barrel towards n m, the confined Airunder it is compelled to force ope., 'he'Valve at g, ts the Pifton defcende ; and it being lighter than the Water, is by the Water pufhed up into the external Air, and the Valve of the Pifton is inftantly fliut. 'Then the Pifton being raifed, the Air fucceeds, and the Water below afcends after the �' ir by the Piefl'&re of the Atrnofphere aforeluid ; and fj by a few Repetitions the whole Air is pumped out, and the nicking Pipe and barrel filled with Water. Now to raife the Water as the Pilion is forced down the Barrel, the Valve at n m* being then fliut, the Water under the Pifton, as before was laid of the Air, in that Part is compelled to open the Valve of the Pifton, and admit the Pifton to defcend into it, which Valve is Dim the ver/ Inftant that the Pifton is down; and then the Pifton being raifed as its Valve is then 11.ut, that Water cannot A turn back, and is therefore lifted up by the Pifton, in the upper Pare of the Barrel, fo as to be received at the Spout i, and at 'he fame. Time the Valve at n m is forced open by the afcending Water in the Pipe o f ; and the lower Part of the Barrel being again tilled, the Valve at n m flints and retains it fur the next Defcent of the Pifton, and thus, the Aftion of the Pump may be con- tinued in, railing Water at Pleafure. 'I'he Sjpben or Cram, a b, Fig. R. Plate LXXV1I, is nothing more than a recuived or bended Pipe, having one Side 'i ing�i than the other. And as the afcending Liquid is forced upinto the fherter Side, (the Aii be'rog firft exhaufted) by the Pr�ffure of the Atmctfphere as before in the Pump, therefore Mercury will ren from one Veffei to another by the Means of this Infiniment? provided that the Bend of the Syphon is not more than 30 or 31 Inches above the Surface of the Mercury', and Water, or Wine, i' the Height of the Bend doth not exceed 55 Feet ; but in both thefe Cafes the Mouth of the defcei ding Tube muit be Something lower than the Surface °'' ^trle Mercury, or Water, into which the (hoi t Tube is immerfed ; for if th - descending lube be equal to the afcending Tube, the Fluid will lerhain in the Syphon, unlefs fome externa! Caufe more than the Atr force it ou.; becaufe the Weight ol the Fluid on both Sides are pqaal. By this Method Watei may be carried over H-ills, asexpreff�d i 1 Fig. V. Piat� LXXVII. if their perpendic»iai Height above the Surface of rue Water, as e r be kis than 35 Feet. By the PreHui e of the Atrnofphere it is, that Mercury will afcend to the fame Altitude io all Kinds of Vef��ls and in any Situation, as is (hewn in Fir. S. Plate LXXVII provided that th. ir upper Parts be perfectly clofe, fo as not to admit any Air to er.tcr in, and by the PreiTure of the Ataiolphere it is, that Water in Rciervoirs is forced to enter the Conduit-Pipes for conveying of Water to any Fountain, iz\-, that is below the Horizon or Level of the Rcfervob, be the ljjliai ce ever fo great.

			F I N 1 S.

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B O

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		BOOKS printed for C. and R. WARE. Sciences. Neceffary for all Mafons, Joiners, cifr. or any who defire to attain to a Knowledge of that noble Art. In two Volumes in Octavo, the third Edition. Tranflated by Mr. Chambers. Price to s. 6 d. To which are added, 200 curious Copper Plates by the late ingenious Mr. Sturt. IV. A new and compendious Syflem of Practical Arithmetick. Wherein the Doctrine of Whole Numbers and Fractions, both Vulgar and Decimal, is fully explained, and applied to the feveral Rules or Methods of Calculation ufed in Trade and Bufinels : And by mewing and comparing the natural Dependance upon, and Agreement of, one Rule with another, the whole is render'd more eafy than heretofore, and the Learner is initru&ed in the Vulgar and Decimal ODerations together, which at the fame time demonllrates the Reafon as well as C-he Practice of both. . By William Pardon, Gent. Price 4 s. V. The Builder's Pocket Companion : Shewing an eafy and practical Method for laying down Lines, for all Sorts of Arches and Curves ufed in Houfe-Building, Ship Building, Gardening, &c. alio to make the Centers or Ribs for Vaults and Cieiings, and Brackets for Coves, either regular or .re- gular. Together with true and concife Rules, to find the Lengths, Bevels, and Moulds for the Back of an Hip in any Kind of Roofs, whether Square or Bevel, Hexagon or Pentagon, &c. let their Rafters be ftraight, or Curves of different Sorts. To which are added, the five Orders of Columns, with the Entabla- tures and Pedeftals, the Proportions whereof are taken from the immortal Andrew Palladio, and laid down after William Halfpenny's practical Method : With feveral other uieful Problems, never before printed. By Michael Hoare, Carpenter. The third Edition, corrected. Price 2 s. 6d. Vf. The Builder's Jewel ; Or, The Youth'sInstructor, and Work- man's Remembrancer. Exprefling fliort and eafy Rules, made familiar to the meanell Capacity, for Drawing and Working. 1. The five Orders of Co- lumns entire ; or any Part of an Order, without Regard to the Module or Dia- meter. And to enrich them with their Rullicks, Flutings, Cablings, Dentules, Modilions, &c. Alfo to proportion their Doors, Windows, Intercolumnations, Portico's, and Arcades. Together with fourteen Varieties of Raking, Cir- cular, Scroll'd, Compound, and Contracted Pediments ; and the true Forma- tion and Accadering of their raking and returned Cornices ; and Mouldings for Capping their Dentules and Modilions. 2. Block and Cantaliver Cornices, Ruihck Quoins, Cornices proportioned to Rooms, Angle Brackets, Mouldings for'! abernacie Frames, Pannelling, and Centering for Groins, Trufs'd Partitions, Girders, Roofs, and Domes. Witn a Section of the Dome of St Paul's, London. The whole illuftrated by upwards of 2C0 Example?, engrav'd on :oo Copper- Plates. The Tenth Edition. By B. and T. Langler. Price 4s. 6d. VII. The Workman's Golden Rule for Drawing and Working the Five Orders in Architecture. Wherein their Pedt/lals, Columns, Entablatures, Impofis, and Arches, are taken from the bell Examples of the Antients, and*- proportioned by equal Parts, in a more concife, accurate, and eafy Manner, than has been done ia any Language. For the Inuruction of Appkentices and Journeymen Mafons, Bricklayers, Carpenters, Joiners, Carvers, Turners, Painters, Plai�erers, Cabinet- Makers, CSV. (and fuck Masters) who are un � acquainted with fo much Architecture, as is abfoluieiy nectifkry for them to underfland, in their refpective Profelhons. And Others, who defire a Juft Knowledge of the Fundamental Rules of that nobie Art. By B. Langley, Architect. Price is. 6 d. Bound. VIII. Arithmetick and Measurement, improved by Examples and plain Denionllrations : Wherdn are laid down the different cultomary Perches,' and o:her

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