BROWN (John) Tl.e Descn-.-fioiTand Use
Quadrant, being a
Particular and General i strnment useful,
at Land or Sea, both for Observation and '

Use of the Quadrant, shewing the Na-
^quot;'^'■Artificial, and TnstrumLtel way
of Making Sun-DiaJs, ; also the Use
^ the same Instrui^- -.t in Navigation

MoreUniverfally ufeful, Portable andCon^
veiiient, than any other yet dtlcoverecL
With its Ufesin
cy^rithmetlck..

London, priiired by John Darby^ for John Wing-
field, and are to be lold at his houfe in Crutched-
Fryers ; and by John Broten at rhe Sphcar and
Sun Dial in the M/nories and by John Sellers
at t'ie Hermitage-ßairs in Waffing. 1671.,

quot;^l^tendly J{eader, Thou haft once
more prefented to thy view, a
further Improvement vind ufe of the
SeUor^ under the name of the Tnan-
guler !^adrant, fo called from the
lhape thereof.

In the ^far i(Jlt;Jo, it was ray lot,
firft, to apply and improve this fpr-
mer Contrivance of Mr. Samuel Fo-
Jler on a ^imdrant^ to a joynt R ule
or Sedor and did, in 1661, publifli
my prefent Thoughts thereof, in a
fmall Difcourfe, under the name of
the Joynt l^ule.

Since then, through my perfwafi-
ons,and aflriftance,another Piece was
publifhed 1667, by l.T, under thé
name of the Semi-Circle on a SeSlor:
But neither of thefe, that is to fay,
neither my own nor his, fpoke what
A anbsp;I would

1 would have it fpeak neither have
I hopes ever to produce a Difcourfe
either for method or matter, worthy
or becoming fo excellent, univerfal,
and ufeful an Inilrumcnt, for the
moft Mathematical Occafions, being
for acuratenefs, conveniency,cheap-
nefs, and univerfality, before all o-
thers. For,

1.nbsp;If it is made of Wood, if the
Wood keep but ftreight, it is as true
to be made ufe of as'of Metal.

2.nbsp;It may be made of any Radius
or bignefs, and yet in little Room in.
comparifon of other Quadrants.

4.nbsp;As to the Projedion for Hour
and Azimuth, particularly ufing on-
ly two Lines of Natural Sines, the
Thred and CompafTes for thofe two
difficult ( and many more eafie) Pro-
poficions.

6. The convenient Contrivance
that happens to it, of three Inftru-
ments in one, i//^. A Senior, ^lua-
drant, and Guvte' 's J^le • all three
conveniently in one.
. The confideration of thefe things,
and the love and wil iagnefs I al-
wayes had,to the commanicating of
them to others, hath put me on this
hard task of writmg this Colledtion
of the ufe thereof,

Wherein I do moft heartily beg
thy Pardon and Acceptance, to ac-
cept in good part the willing endea-
vours of my poor Ability, which I
doubt not but to have from moft that
know me For,firft, my infuificiency
in the Tongues, Arts, and Sciences :
Secondly, my Meannefs and Pover-
ty in the World, for thefe Imploy-
iiients, which take up fo much of a
mans time, and ability, to perform
them to purpofe, may plead my ex^
cufe ; for firft, here is the Produd

To the 'Readerquot;.
of more than Two years Improve-
ment of more than vacant Hours
\vith the great difadvantage of ta-
king three Weeks at times, to do
that which three Dayes together
might have as well, if not much bet-
ter, performed ^ And at laft, to call
the Affiftance of two others, to un-
dertake the Charge thereof, to mid-
wife it into the World.

Thus, as Widows Mites are ac-
cepted, which are offered in fmceri-
ty fo I hope will mine, though at-
tended w4th much diforder, as to
Method more uncouthnefs, as to
Stile and Matter ; What it is, it is
as at firft Compoling, for I could ne-
ver get Time nor Liberty, from my
daily Trade and Calling , to tran-
fcribe it twice.

Yet was it not done at any time
carelefly, but with good will and a
free intent of plainnefs and ufeful-
nefs for the publick good of others,
as Well as uiy own recrcation and
deliehr,nbsp;' The

To tie J^adeft
The Gunters Rule, the Quadrant,
and Seftor, I need not commend,
they are fo well known already but
this I will add, a better Contrivance
and more general hath not yet to
my knowle^e been produced , nor
a Difcourfe where the ufe of all the
Three together hath fo been hand-
led,nor many more Examples,l^ongh
Mr. Ifindgate and Mr. Patridge have
done fuflficiently for the Gunteu-
Lines, and UuGunter for the Se-
ftor, and Mr. Collins with the Qua-
drant, and all of them diftindly far
beyond this yet this Difcourle ot
all the Three together, may give

The Difcourfe of Dialling, is ga-
thered from Mr. Wells and yet
thofe that fliall read Mr. Wells, and
this may often-times think other-
wife for I affure you, I faw not one
leaf of his Book all the while it was
doing., but, I hope, itmaypleafe

To the Reader:
in moderate fort, an ordinary capa-
city, both for plainnefs, conveni-
ence, and variety.

The cutting of the Regular Bo-
dies, I learned from Mr. John Leake^
and the way is ready, convenient,
and €xalt;a, and worthy of remem-
brance.

The way of Meafuring Superfi-
cies and Solids, from Mr. Gunter :
and my conftant Experience in thofe
Imployments^ and the Learner may
here be fupplied with what is often
complained on, x//'^. the Interpre-
tation of Hard-words, as much as I
could call to mind, or think to be
convenient for that purpofe.

In the 1 ƒ th Chapter, I have ga-
thered many Cannons from Mr.
Collim his Workes, and applied
them to the rnangder §luadrant 5
and been more large than needs in

To the quot;R^adeu
fome places, yet I hope to the con-
tent of fome inquiring Perfons.

The bufinefs of Navigation , I
fear, may prove moft defeâive -, for
my part, I never yet faw Grave fend,
much lefs the Streights of Gibralten
but for Obfervation and Operation,
the Inftmment will do as well as
any, if well made and applied.

So for the prefent, I reft and re-
main, ready to ferve you in, and
fupply defedts by well making of
thefe Inftruments, at the Sphear and
Sun-Dial in the Great Minories.

At length my pains bath brou^t to
the things I long intended^ {pafs
And doubt not but in every place,

wiU plea fed with it be.
For firft the 'Xymes young may find,
fome terms to be exj)lai7ied i Which

■with Gunters Rule and Bow.
The Travifs Quadrant and Crofs-
the Davis Quadrant too; (ftaves.
Their ufes all to more than halfs,

of lying in lefs room,
A fault that SaylorE m'^fi abide,
when they on Ship-bor^dcome.
In the next place, the quot;Rudiments

of Geometry exM
The right Sines Ö thnr complements,
and how they lie comj^aU, Withi

in Round ««a? Plain Triangles ;
Which ferve to deck Urania i^ueen,
as Jewels, Beads, and Spangles,
Inthe next place AxithrnQtick^
by Numbers and by Lines 5
In way es that rvon't be far tofeek^,

]{oofs. Chimneys, Walls and floor.
Computed and reducd at once,
in Thicknefs, Lefs or More.
The cutting Platoe's Bodies five,

and Dials on them fix:
Their Meafure and their Magnitude,
in C ircle circumjcribed 5 Whoß

and Diggs , have been dejcribed,
Tiamp;c« fl^/« Aftronomy,
are many Propofitioni,
Which fitly to th' Rule 1 apply t

avoiding repetitions.
And after, in the plea/ant Art,
o/^ Shadows, I do wander.
To draw Hour-lines in every part,
both upright, over, and under:
Andalltheufual Ornaments,

at one, or at tnoo Stations,
Performed by thofi wayes that make,
the fen^eft Operations,nbsp;And

The Argument of the Book, '
And alfo ready Rules to uje,
the Logarithmal Table 5
Which may prove ready Hints to thefei

that are in thofe mojl able :
And many other ufeful Thing,
is fcatteredhere and there,
Inbsp;Which formerly by Me bath been,

accounted very rare.
And laflly, for the Saylors fake, ,,
I have fpent many an Hour,

wore ufefiil than allothet:
.S(?»-Inftruments that they do ufe,
' at Sea for Obfervaticn ;

^ Age a8. line S.for Komhrds Rmhoidei.
1 P.73.I.!aft.f.337,r.247. p.75J•I.^l7.'•.8•
^F.nbsp;p 87.1-i4.r. multitliedby. p.89.1.14. f-s^yifii^.n

12 ro 7, r./com 7 to II. p,i46.1.:i.f ii Se5/on,r.
jnbsp;13 SeS/on. p.ifS l.laftjdeleW. p.i^o.l.ii.f.72»

X.brick- 1.20. f. ■, r. I p.iS^-I.iZ-t.
inbsp;Ceiling,r.Tileing p.zoi.]Ai,{.^zLjnks i.S^Linkf-

are to come in after 134- 5. Alfo, the two lines
over 3-54S. fliould come in after 3.54$. p.157.
l.ig.f. 2496,r. 249-6. p.370.1 3.f.Jj/ier. Co-fine.
p,383.1.22. addjJj the general Scale, p.384.1.14.
t. = S. 0. r. == Co.ftne. p.414.1.ii.f- f
r. on, p.420.I.22.f.7i r.3i. p,429.1.i5« f. Decli-
nation, r. Suns Right Afcention.

The De/cription, and fome lijes of the
TiiaDguiai'Quadrant, ortheSeüor
made a ^adrant s being an excel-
lent Injtrumentfor Obfervations dnd
Operations at Land or Sea, perfor-
ming all the Ufes of the Fore- jtaff,
T)2\h-^adrant, GunterV-jSowj
G\mttxs-Crofs-flajf,Gnx\.ttts-§^d-
drant and SeUor, with far moré
ton-veniency and as much exallwfs
as any, or all of them will do^

llrftjit is a joynted Rule (or Senior)
made to what ien^th or Radius
you pleafe, ( as to p, 12, iB,
24, 30, or 3 (J inches Length,
■vvhen it is folded or fhut together 5 the
fliorter of v/bich Lengths is big enough for
Land ufcs, of Paper draughts; the four lall
for Sea ufes, orObfervatioiis,) To which is
Bnbsp;added,'

added, a third Piece of the fame length of
the Seélor, with a Teiinon at each end, to
fit into two Mortice-holes at the two ends
of-the infide of the Scftor, to make it an
^Equilateral Triangle 5 from which fltape,
and its ufe, it is properly callcd a TrUngnUr
^Quadrant.

2. Secondly, as to the Lines graduated
thereon, they may be more or lefs, as your
ufe of them, and as the coft you will be-
ftow, fhall pleafe to command: But to make
it compleat for the promifed Premifes, thefe
that follow are necelTary to be infcribed
thereon, as in the Figure thereof.

• And firft you are in order hereunto to
confider, The outer-edges of the (Seélor or)
Inftrument, the inner-edges, the^Quadran-
taUfide, the Seéfcr-fide, and the third or
loole-piece, alfo the fixed or Head-leg, the
moving-leg, the head, and the end of each
leg, alfo the head and lég center 5 of which
inore in its proper place.

1.nbsp;And firft, on the outer-edge is plated
the Lines of Artificial Numbers, Tangents,
Sines, and verfed Sines, to as large a Radius
as the Inftrument will bear.

2.nbsp;Secondly, on the in-fide or edge, on
fhort Rules is placed inches, foot me2fure,
thelineof 112, orfuch-like. Butonlarger
Inftjuments, a Meridian line to one inch,

or half ah inch (more or lefs) for orie de-
gree of the iEquinoftial, for the drawing of
Charts, according to Mercator, or any o-
ther more ufefül Line yoü fliall appoint fof
your particular purpofe.

3. Thirdly,in defcribing the Lines on the
two fides 5 firft I fhall fpeak to the Seétor-
fide, where the middle Lines all meet at thé
Center at the head where the Joynt is: the
order of which ( when the header joynted
cndlyeth toward your left hand, theSeétor
being fhut, and the Seftor-fide upermoft •)
is thus :

1.nbsp;The firft pair of Lines, and lying next,
to you, is the Line of Sines, and Line of
Lines, noted at the end with S, and L : for
Sines and Lines, the middle Line between
them that runs up to the Center, and wkere-
in the Brafs center pricks be, is common
both to the Sines and Lines in all PaEallel
^fes, or entrances.

2.nbsp;The Line next thefe, and counting
from yoij^ is, the Line of Secants beginning
at the middle of the Rule, and proceeding
to 60 at the end, and noted alfo with Se for
decants, one of which marginal Lines con-
ïhiued, would run to the center as the
tber did.

3* The next Lines forward,and next the in-'
'itr-edge on the moving-leg are the Lines of
B z ^ Tangents^'

Tangents; the firft of which, dnd lifxt rd,
you, is the Tangent of 4j, being the largefl
Radius, (as to the length of the Rule: ) the
ether is another Taijgent to one fourth pare
of the length of the other, and proceeds to
76 decrees,a little beyond the other 45': the
middle Line of thefealfois common to both,
ill which the Center pricks muft be. At the
end of thefe Lines is ufually fee T. T. for
Tangents.

4. On the other Leg of the Se£^:or, arc
the fame Lines again, in the fame order
counting from youj wherein you may note,
That as the Lines ot Sines and Lines on one
Teg, are next the outward-edge; on the
other Leg, they are next the inward-edge :
fochat at every, or any Angle whacfoever
the Seitor.ftan^ at, you have Lines, Sines,
and Tangents to the fame Radius: and the
Secants to juft half the Radius, and confe-
qumtly to the fame Radius by turning the
Compafles twice ; Alio any Tangent to the
greater Radius above 45^, and under 76, by
turning the Compaffes four times, as after-
wards will more appear: Which contrivance
is of excellent convenience to avoid trouble,
and fave time ; and happily made ufe of,
in this contrary,manner to the former wayes
of ordering them,

to the greater Line of Tangents on the
head-leg, is placed the firft 4f degrees of
the leflcr Tangents, which begin from the
Center at 4y degrees, becaufe of the llrait-
nefs of the room next the Center, where
they meet in a Point : yet tliis is almoft of
as good ufe, as if it had gone quite to tlie
Center, by taking any parallel Tangent
from the middle or common Line on -the
great Tangents, right againft the requifite
Number counted on the fmall Tangent un-

Sixthly, next to this will not be amifs
to adde a Line of Sines, to the fame Radius
of the fmall Tangent laft mentioned, and
figured botIr wayes for Sine and co-Sine, or
fometiines verfed Sines,

7.nbsp;Seventhly, next to this a Line of E-
qual Parts, and Chords, and the Secants in
3 pricked line beyond the little Tangent of
4j', all to one Radius : To which (if you
fleafe) may be added, Mr- Faflers Line Soil,
^nd his Line of Latitudes ; but thefe at plea-i
fure.

8.nbsp;Eighthly, on the outermoft-part of
both Legs next the out-fide, in Rules of half

inch thick and under, is fet the Line of
■Artificial verfed Sizies, laid next to the Line

Artificial Sines, on theoureivedge ; but
^^ the Rule- bs thick enough to bear four
B 3nbsp;Lines,

Lines, then in this place may be fet the Me-
ridian Line, according to Mr. Guntery coun-
ting the Line of Lines as a Scale of Equal
Parts. Thus much as for the Sedor-fide of
the Liftrument.

4, Fourthly, The laft fide to be defcri-
bed is the Quadrantal-fide of the Inftru-
ment, wherein it chicfly is new. Therefore
I fliali be as plain as I can herein,

To that purpofe I fhall in the defcripti-
on thereof imagine the loofe piece, (or third
piece) to be put into the two Mortife-holes,
which pofition makes it in form of an jfiqui-
lateral Triangle, according to the Figure
annexed, noted with ABCD^ wherein
A B is for brevity and plainnefs fake called
rhe Moveable-leg, D B the Head or Fixed-
leg, D A the loofe-piece, B the Head, A
and D tlie ends, C the Leg-center, at the
beginning of the general Scale; the center
at B the head-center, ufed only in large In-
ftruments, and wfien you pleafe on any 0-
other.

Fir ft. On the outer-edge of the moveable-
Leg, and loofe-piece, is graduated, the 180
degrees of afemi.circlej C being the center
thereof.

And thefe degrees are numbrcd from o\6o
on the loofe-piece toward both ends, with
lo, 20, 30,40, e^-r. and about on the
moveable-!cg,. with 20, 30, 40, 5:0, o'o,
70, 80, andpo at the head : Alfo it is
ttumbred from (Jojo on the raoveable-leg,
with 10, 20, toward the head 5 and the o-
ther way, with lo, 20, 30,40, yo, (To on
the loofe-piece; and fometimes alfo from
the Head along the Moveable-leg, with
10,20, 30, ^c. to po on the loofe-piece ;
and the like alfo from the end of the Head-
leg, and fometimes from 60 on the loofe-
^ece both wayes, as your ufe and occafion
ftall require.

SecorMy, Onthe Quadrantal-fidCof the
ioofe-piece, but next the inward-edge is
graduated 60 degrees, or the Tangent of
^ twice 30 degrees, whofe center, is thecen-
■ter-hole or Pm at B, on the Head or |oync
of the Sedtor,
_ Which degrees are numbred three wayes,
Firft from D to A for forward Obfer-
Vations ■ and from the middle at 30 to A
end of the Moving-leg, with 10, 20,
3° 5 and again, from D the end of the
Head-leg to A, with 40. 50,60,70,80,
for Obfervations with Thied and

Thlrily, Next to thefe degrees on the
Moving-leg, is the Line of the Sms right
Afcencion, numbred from lt;Jo]o onthede-
gi-ees, with i, 2, 3,j-, 6, toward the
Head, and then back again with 7, p,
10, II, n, amp;c. 2, 3, 4,5quot;, on the o-
thcr fide of the Line, as the Figure annexed
iheweth : The diyifions on this Line is ( for
the raofi part) whole degrees, or every four
minutes of time.

Fourthly, Next above this is the Line of
the Suns place in the Zodiack, noted with
Y j[i S toward the Head ; then back a-
gain with .^in:^ t^i over 6o|o in the de-
grees, and 12 and 24 in the Line of the
Suns right Afcentions: then toward the
end, with rn y^; then back again with
s^andKj being the Charaders of tV^e 12
Signes of the Zodiack, wherein you have
expreft every whole degree, as the num-
ber of them do fliew, there being 30 der
grees in one Sign»

Fiftly, Next above this is a Kalender of
Months and Dayes j every fingle Day being
expreft, and three or more Letters, of the
name of every Month being fet in the
Month, and alio at the beginning of each
Month, and every loth day noted with a
Prick on the top of the Line reprefenting ic,
as 15 pfual in fuph

Line to find the Hour and Azimuth in a
)articular Latitude. Put alwayes on fmal-
er Inftruments ( and very rarely on large
Triangular Quadrants for Sea Obfervations)
the loweft Margent whereof, and next the
Months, is numbred from the end toward
the Head, with lo, 20, 30,40, jo, do,
70, 80, po, 100, no, 120,130, near the
Head Center. For the Semi-diurnal Ark of
the Su»s Azimuth, and in the Margent next
above this, with 4, 5, d, 7, 8, 9,^0, ir,
12, near the end, for the Morning hours ,
then the other way, vik.. toward the Head
on the other-fide the Hour Line, with i, 2,
Sj 45 Si 7)^) for the Afternoon hours.

Seventhly, On the fame Quadrantal-
fide, and Moveable-leg on the fpare places,
beyond the Months toward the end,is fet an
Almanack; and the Names of 12 or more
Stars, to find the hour of the Night ; which
Ï2 Stars are noted with i, 2, 4, 5-, 6 ,7,
8, p, 10, II, 12. among the degrees ia
imall Figures ^ as in the Figure.

£ightly. Next of all to the in-fide, is the
Line of Natural yerfed Sines drawii to the
Center, with his correfpondent Lme on
the other, or Head-leg. Expreft fometimes
a pricked Line, for want of room.

the verfed Sines lafl: mentioned, is firft the
Line of Equal Parts, or Line of Lines: and
on the fame common Line wherein is the
Center, is the Line of Natural Sines, whofe
length is equal to the meafure from the cen-
ter at C to dojo on the moveable-leg; fo
that the Line of degrees is a Tangent, and
the meafure from C to any Tangent, a Se-
cant, to the fame Radius of the Natural
Lines of Sines, and Lines : Alfo beyond
the Center C on the fame common middle
Line is another fmaller Line of Natural
Sines, whofe length is equal to the meafure
fromC to lt;Jo on the loofe-piece; then if
you count from the Center pin at lt;5o, on
the loofe-piece, toward the end of the mo-
vable-leg,they fhall be Tangents to the fame
Radius, and the meafure from the Center
C to thofe Tangents, fhall be Secants to the
fame Radius, which may be well to be or-
dered, to a third, or fourth part of the for-
mer, from the Center downwards: Thefe
twoLines of Sines are beft figur'd with their
Sines; andCofines, the other way with a
fmaller figure, and the Line of Lines from
the Center downward from i to lo where
90 is, which Lines of Sines may be called
a general Scale for all Latitudes.

Ttnth^ Next to this toward the outers
edge is a^ier Line of Natural Sines, fitted

to the particular Linequot; of Hour and Azi-
muths, for one particular Latitude, noted,
*Tert. Scale of Altltudts ; or Sixes.

Eleventhly, Next to this is the Line of
29', for fo many dayes of the Moon's age,
in ihoxt Rules of the whole length, but in
longer not; being eafily known by the
fingle ftrokes, and Figures annexed to thofe
ftiokes.

Tmlfthly, Next the outer-edge is a Line
of 24 hours, 360 degrees, or 12 Signs, or
in moft Rules inches alfo, ufed together
with the former Line of 29 i,' and as a
Theory of the Sun and Moon, and ready
\vayof findnig the Hour by the Moon or
fixed Stars.

Thirteentblj, To this luftrument alfo be-
longs a Thred and Plummet, and Sights, as
to other Quadrants ; and a pair of Com-
^afles as to other Se61:ors • a Staff and Ball
focket alfo, if you will be curious and ac-
curate.

And for large Liftruments for Sea, 3
Square and an Index, which makes it a per-
feft finical Quadrant, and two Aiding fights
aWo, which makes it a fore and bact-ftafF,
and bow, as will appear more at large after-

IN the firft place it will not be amifs to
hint a few words, as to the reading the
Line^, or (more properly) Numeration on
the Lines ^ wherein take notice, That all
Lines of Equal Parts, or Lines applicable
to Arithmetick, as the Line of Lines, the
Line of Numbers, the Line of Foot-meafure,
and the like ; wherein Fradlions o f Num-
bers are requifite : they are moft commonly
accounted in a Decimal way, and as much
as may be, the fmall divilions are numbred,
and counted accordingly.

: t 1
But in the Lines of Sines, Tangents, Se-
cants, and Chords ; being Lines belonging
properly to a Circle: in regard that the
Sexagenary Fraction is ftill in ufe, the inter-
mediate Divifions are, as much as may
be, fitted to that way of account, viz,, by-
whole degrees, where they come clbfe toge-
ther, (or the Line of no great ufe.) And if
more room is, to half degrees or 5 o minuts,
and fometimes to quarters of degrees or i f
minuts i but toward the beginning of the
Line of Natural Sines, or the end of the
Natural Tangents and Secants: where the
degrees are largeft, they are divided to e-
very loth minute in all large Rules, as by
confidering and accounting you may plain-
ly perce've.

Take two or three Examples of
each kind.
I. Firft, On the Line of Lines, to find
the Point that reprefents ij. In the doing
of this, or any the like, you muft confider
your whole Scale, Radius, or length of the
Line, may be accounted as i, as 10, as lOo,
as looo, or as 10000 ; and no further can
oe applicable to any ordinary Inllrument.
Wherein obferve, That if the w hole Line
one, then the long flroke by every Figure
doth reprefent one tenth of that Integer :
^»d the jie.xt fliorter without Figures, are

hundredth parts of that one Integer; and a
loooth part is eftimated in fmaller Inftru-
ments, and fometimes expreft in larger : But
the hundredth thoufand part is alwayes to
be eftimated by the , eye m all Inftruments
whatfoever.

2. But if the wholeline of Lines fliall
reprefent lo, as it ufually doth, and as it is
figured, then the long flroke at every Fi-
gure« I, and the next longer are tenths,
and the fhorteft are hundred parts, and the
thoufand parts as near as can be eftimated.
. 3. But if the whole Line reprefents a
hundred, as here in our prefent Example,
then the long ftroke by every Figure repre-
fents 10, and every fhorter ftroke is oney
and the fhorteft ftrokes are tenths, and the
hundredth parts as much as can be eftima-
ted.

4. But if the whole Line fhall reprefertt
a 1000, then the long ftroke by the Figure
lhall reprefent a hundred, and every fhor-
ter ID, and every of the fhorteft ftrokes is
one Integer, and a loth part as near as can
be eftimated.

y. Butlafily, ifthe whole Line reprefent
10000, then every Jong ftroke is 100, and
every fhorteft cut is ten, and every fingle
Integer is as near as can be eftimatedhj any
ordinary Inftrumenc^

^ oHr prefent Example will properly
come under the third Rule, by conceiving
the whole Line to reprefent loo j then the
fitftlongftro'ke by i is lo, then the next
ftorter is for li, the next 12, amp;c. to 15 ;
which is cut up a little above the Line, for
the more ready reckoning without telling
the parts : which 15: is the Point required
to be found.

This will come under the Notion of the
îth Rule, wherein the whole Lme is con-
ceived to reprefent looco; then the firft
I IS for the ithoufand, then the fifth longer
ftroke next is for the yoo ; and laftly, the
'niddle between the yoo ftroke and the,
lt;îooo ftroke is for the yo, being a little be.
yond the Point for ly in the fiiil Example.

This third Example may fuffice for this
^ork, being fo plain after a little due con-
quot;deration : For firft, the whole Line is con-
ceived to reprefent 10000, then the long
ftroke by j is for 5000, then there is no
hundreds, therefore the Point required
ïîiuft be fliort of the next longer ftroke,
^nich fignifîcs hundreds, and being it is juft
pJj which is 1 of an hundred, the true-
«.quot;int readily fhewech it felf : If you re-
quire

quire a more plainer and larger wording of
this raacter, I refer you to the third Chapter
of Mr. ivi»dgates Rule of Proportion ; or
the firft Chapter of the Carpenters Rale, by
y. 'Brown.

Laftly, In nameing of any Point found
out on the Line, great care and refpeft muft
Jbe had as to the tfue value of the Number,
according to the rate of the queftion pro-
pounded : ^ for the fame Point that repre-
fents ly, doth reprefent lyo; and alfo
lyoo, or lyoOQ, ( increafing above the
bounds before mentioned ) alfo it fignifies
one and a half, or ij of one hundred,
which is ufually expreft thus in a Decimal
Fiaftion'Jj, or more readily, o. ly.

Alfo if It fhould be a Number with a di-
git, two ciphers and another digit, as
2.005-, quot;^his Number would be found clofc
to the long ftroke, by the figure 2 : and may
reprefent either two thouiand, and y of
1000more 5 or 20 and 5 of a hundred;
or 2 hundred and y of another 10 more,
or plainly as it is fet down, two thoufand,
110 hundred, but five: Thus you fee the
manner of expreffing whole Numbers, or
whole Numbers and Decimal Fradions,
which on the Lines is one and the fame
thing 5 and thus'all Decimal Scales are to
be accouuted, and in die fame manner is

the Line of Numbers co be reac., as yorf
may fee more at large in the two Books be.,
fore mentioned.

But for Numerateon on all Circular
Lines, it is much ealier : For firft, very few
Inftruments, unlefs at one part of the Line^
can exprels nearer than minutes of a degnee.

Secondly, The whole Radius or Line of
tines is but J) o degrees, or but 4f of the
Tangents, or 60 of the Chords, or Secants:
So that in Inftruments of 12 or 18 inches
Radius, you may exprefs very well every
tenth minute, to lt;56 on the Line of Sines:
and every half degree to 7f, and whole de-
grees to 90. And on the Tar^ents or
Chords, every loth minute quite through :
^d the Secants as the Sines.

So that any clegree or minixte being na-
med, to find the fame on the rdpcótive Line,
Count thus;

Firft, every loth degree is noted with a
long ftroke, and figures fet thereunto. Se-
condly, every whole degree is cut between
two, or three tines, and fometimes with a
Point or Mark on the end of thé ftroke; and
every y th degree cut up higher than the reftj
*nd lometimes with three Points, on the enïi
Cnbsp;of

of the Eine, or fome other convenient df-
Hindhori, for readinefs 'fake : and every
lotli, I jth, or 30th minute, is cut only be-
tween two Lines and no more ; as will ap-
pear very plain with a ■liule' pra(5lice.

ExJmpltf to fini the She of the Latitude
being at London, jl degreesy
32 mimtts.

1.nbsp;iFirft^ look on any Line of SineS) on
the Quadrantal, or Se6tor fide, according as
you have occafion, till you fee yo, which is
yo degrees; then one degree forward, to-
ward 60 is JI degrees, then count three
loths of minutes more for 30 minutes, and
then for the odde two minutes, eftimate one
fifth part of the next 10 minutes forwarder,
and that istheprecife Point for the Sine of
51 degrees 3 2 minutes, the latitude of Lo»-
ion, where fometimes is fet a Brafs Center-
Pin.

2.-Tonbsp;. find the Cofine of the Latitude,
there are two wayes to count the Comple-
ment of any Ark or Angle.

Firfl, by fubfi:rafting the Ark or Angle
out of po by the Pen, and count the refidue
fi om the beginning of the Idne of Sines? and

ft taken from po, the remainder is
28, now if you count fo much from the
beginning of the Line of Sines, according
to the laft Rule, that fhall be the Point for
the Sine of 38 28, the Complement of
Jï 32, or the Sine Complement of the
Latitude.

Or Secondly, If you count ^132 from
50, calling 80, ID J and 70, 20; and do,
30i 5-0,40; 40, yo, amp;c. whereloever thé
vJumber whofe Complement you would
have fhall end, that is the Sine Comple-
quot;lent required, which will be at 38 28,
ftoiTfi the Center or beginning, for the Co-
fine of fi 32; The like work ferves for
®«y other Number, or on any other Line,
•js on thé Degrees, Tangents, or Secantsj
•Natural or Artificial, as by practice will
•^ore plainly appear, to the willing Practi-
tioner.

To find the verfed Sine of an Ark or
Angle, or the Sine of an Ark or Angle a-
bove 90 degrees, or the Chord above ï8o
^«grees^ obferve chele Rules.

; I. Firft, arightSincj is the meafureott /
the Line of Sines, from the center or begin- |
ning of that Line, to the Point that doth ,
reprefent the Ark or Angle required.

2. The right Sine of an Ark or Angle
above 90 degrees, is equal, to the right Sine
of the Complement thereof to 180 degrees,
being readily accounted, thus • Count the
excels above 90 backwards, from jgt;o toward
the Center; then the meafure or diftance
from the end of the account to the Center, is
the Sine of the Ark above po required:
Example. Let the Sine of 130 be required,
firft, if you take 130 from i8o, there-
mainder is fO ; then I fay that the right Sine
of fo, is alio the right Sine of 130 j for if
you count backwards from 90, calling 80,
joc J and 70, no ; and 60,120 ; and
JO, 130; the meafure from thence to oo,or
the Center, is the right Sine of 130 de-
grees,

3. The verfed Sine of an Ark or Angle,
is the meafure on the Line of Sines from 99
toward the Center, counted backwards, as
the fmall figures for Complements fliew,
counting 90 for 00, and the Center for 90,
(as the Azimuth Line is figured) opening
llie Line of Sines to a ftrait Line, and theo
counting beyond 90 for the verfed Sines a-
bove 90, as on the verfed Sines is plainly

Halve the Ark or Angle required, and
tike the right Sine thereof, and that (hall be
thfClwrd thereof.

I would have the Chord of 40, the half
of 40 is 20 J then I fay the right Sine of

is the Chord of 40, to that I^dius that
IS equal to the right Sine of 30 degrees, at
the Radius the Rule (lands at.

5.nbsp;To find a Chord to an Ark or Angle
above 180 degrees, you muft count as yovt
did the right Sines; for note, the Chord of

i| jsqtial to the right Sine of 90 dou-
ole4, which is the fijll Diameter of a Circle :
and a longer right Line than the Diameter
cannot be taken in a Circle j therefore it
quot;»uft needs follow that Chords of above
are (horter than the Diameter which
is the biggeft Chord j therefope the Chord
of 2(50, is equal to the Chord of locde-
Rrees, or right Sin? of 50, the Sine of 30
being Radius.

Ö. In ufing the Artificial Sines and Tan-
8?nts, or Secants i if you are to ufe a Sine
above 90, then count 80 for loQ, 70 for
?ïogt; 60 for 120, 8fc« But for Secants,
C Jnbsp;thea

then count after thé mdiineï of vMéd'Sirtes-é
Thus the Secant óf 6amp;- amp; as far-^yoAd ^q,
as it is from jo topp; fo that whoY^^oij
iiavè ot'ci^oH'fo ufe m ArtifidiV-Sfecaht,
MHhich isnotdften- -Thén fet th'amp;^nylbfthc
Rule againft a Table, and- ccftintifigl-feack-i-
wards from 90 to the number of the Secant
required, turn tkat-diïferice beyond 90 on
the Board or Table, aftd that lhall gt;be youy
Secaifc rHüil'éd, ds Will fë afceï^atd hini
tedjèischeycomeinufê. . ■

Tj GR the betth' -underftanditig'of the
JT followirtg dffc6urfe,ifislleedfl^^t6un-

■derftand thefe Eletnfents or Principles, as
the Letters are neceffary to be known before
■feadihg,

I. A Circle is a'figure enclofed in one cir-
•plaf Line,called the Circumference ; in the
puddle whereof is aPoint called theCen-

iter : From which Point all right Linej
drawn to the Circuiiifc;t:nce are equal one
to another; as in the Circle ABCD, Eis
the Center, ABCD the Circumiei^Qce^
!the Lines E A, E B, E C, E are equal,

3. Any right Line crolTing the CircanrFer
fence, and p^fing through the Center of a
Circfe, is called the Diaineicr 9 end it di-
vides the whole Circle.iiito two equal parts,
called Semi-circles ( or half-Circles.) And
the half of chat Line is called th£ Semi-
diameter or Radius to that Circle. As the
Line A C is the Diameter, and E C the half-
Diameter or Radius.

3.nbsp;Any other Right-line eroding the Cir-
tumferente is called a Chord, or Subtpnce,
««the Line FG, which divides the Circle
intoxwo unequal fayts t And note,that this
Subtence belongs both to the ieffer, and alfo
to che greater part of the Circumference;
ihat istofay, the Cho»d of 90 deg. is alfo
the Chord-of 270dcg. fo that FG is
Chord to the Ark F B G 90 deg, ^nd al-
fo to the Ark F D G be^ng^/o deg« mych
more than half ihe Circle. , gt;

4.nbsp;Half the Chord of any Ark, is the
right Sine of half that Ark ; thus the right
Line H G, the half of F G, is the right
fiine of the Ark B G the half of F. S G.'

anyArkisthcnearcftdift
cumfercnce to the Diameter: Perpendicu.,
ler to that Diameter from whence you coun-
ted the degrees and minutes of the Ark or
Angle. As thus, GI istheCofineof the
Ark BG, and the Right Sine of any Ark
is the neareft diftance from the Circumfe?
Knee to the Diameter you counted the de-
grees from, as G H is the Right Sine of
BG.

The verfed Sine of any Aik or Angle,
is the Segment of the Diameter between
ehe right Sine of the fame Ark and the Cirr
cumfercnce.

Thus H B is the verfed Sine of the Ark
BG, and H D the verfed Sine of G D. So
alfo is GH the right Sine of the Ark GCD
or the Angle GED 4y degrees abovt jgt;o,
W«,. 13 f degrees.

7.nbsp;A Tttngtm is a right Line drawn per?
pendiculer to the Diameter, beginning at
one extreme of the given-Ark, and termi-
nated by a right Line drawn from the Cen-
ter to the other extreme, of the given-Ark in
the Circumference, rill it inter-fett the
perpendiculer ; Thus C K is the Tangent of
the Ark CG, or the Angle C E G, 45 do-
prees.

8.nbsp;A Stcant is a right Line drawn from
the iDenter thorpw one extreme of the given-

Ark,till it meet with the Tangent rais'd per-
pendicularly from the Diameter» drawn to
the other Extreme of the faid Ark ; Thus the
Line E K is the Secant of the Ark C G, or
the Angle GEC.

J). Note, as in a (Natural) Sine, the
neareft diftance from the Ark to one Diame-
ter, from whence you counted the degrees
of the Ark or Angle, was the Right Sine ;
and the nearcft diftance from the fame Point
to the other perpendiculer Diameter, is the
Cofine of that Ark or Angle.
* So likewife the neareft diftance from the
Point where the Tangent and Secant meets,
to one of the Diameters aforefaid, is the
Tangent of the Ark or Angle; fo the nea-
reft diftance from the meeting Point of the
fame Secant-line is the other Tangent-line ta
the other Diameter abovefaid, is the Co-
Tangent of the Ark or Angle abovefaid.

Thus the Right-line KC is the Tangent of
4y, and the Right Line KB the Co-Tangent
of 4 J ; Alfo the Line LC is the Tangent
of 53,304 and the Line MB is the cp-
Tangent thereof, m*. the Tangent of 36,

^ Alfo the neareft diftance from L to E B,
is the Tangent of 36,30, to the Radius
LC.

dcd into |(So degrees J the Serai-circle iiitlt;*
»80, the Quadrant or Quarter into 9©.

11.nbsp;hvcry Degree isfuppofed to be di-
vided into do minutes, and every minute
into 60 Seconds, and every Second into 60
Thirds, amp;e, ■ '

12.nbsp;A Rsditu, or SemUiameter, is in
our Inftrumfaital Praftice^ fuppofed to.be
divided into 10000 parts, and every
Chord, Sitie, Tangent, or Secant, is to be
divided by the Parts of the fame Radius, of
Radius and Parts more.

13.nbsp;hn^agle is the raeering of two
Right Lines, as G E, and EC, meeting at
E, doconftitute the Angle GEC, called
a Right-lined Angle j or when two Circles
crofs one another, it is called a Spherical
Angle, the Ai%uler Point being noted al-
wayes by the middle Letter of t^iree thac
fliew the Triangle.

14.nbsp;K TUin Triangle is the meeting of
three Ri^ht Lines crolhng one another j and
a Spherical Trirngk is coaftituted by the
eroding of^three Circles, as in the two Fi-
gures noted H and III, you may plainly
fee.

17.nbsp;A Right Angle IS altvayes juft po
degrees, as you may lee in the Figures II,
and IIIgt; fey, the Angles A in bath of
diem.

18.nbsp;An OhtHce Angle is alwayes more
than 90 degrees,as the Angles at D in both
Figures fhew.-

quot;19. A Parahl Line, is any Line drawn
by another Line in fuch a way, that though
it were infinitely produced, yet they would
never méet oï Aofs one another, as the
Lines A B, lt;Zamp;.

2Ö. A P^rpenHculer Line^ is when one
Line fo falfctli'on ^mother Line, that the
■Angles on each fide are equal, as C A falls
onthelirtc-B A, Figure VL

2t. -AU Tfiitngtes are either with three
equal fides, as Figure 101, Or tv/o equal
fides, as FigurèV, or all Unequal fides, as
Figure VI; the firft of -which'is called Eqtti-
Uterki, tbeiécófe^ f/a/c-f/jj^ thé third Sca-
icntim.

■ 22. Againj^they may be fometimes na-»
'med from their Angles; thuS-: Orthigonium,
with one Right Angle, and two Acute
Angles» jimHigonium, with one Obtuce
Angle, and two Acute Angles. Oxigoni-
Hm, with three Acute Angles only.

24. All Faur-fidti Figures are cithe*
amp;][uares, with four fides, and four right
Angles all equal; or long Squares ( or Ob.
longs) with the two oppofite fides equal, or
the fame crulhed together, or not Ri?ht-
Angled, as the Romim, and %omymA^ot
elfe with four unequal Sides, called Trth
fe?:.iaes,

2f. Laftly, many fided-Figures, are
fome Regular, having every fide alike, as
5quot;) 7gt; 8, 9, 10, amp;e. Or elfe unlike, as
Fields, and Woods, and Meadows, which
being mfinite, cannot be comprehended un-
der any Regular Order or Rule.

z6. MHltifUcatoTy is a term ufed in Mul-
tiplication, by which any Number is to be
multiplied, as in laying y nwr/y is the
Multipliearor of 6.

27.nbsp;^ultiflicani, is the Number to b?
multiplied, as 6 by y, as above named.

28.nbsp;The ProduSlf is the Iffue or Refulc
of two Numbers multiplied one by the 0»
ther, asjoistheProduil of 5 multiplied
by J-} for tf times y is 30.

?9. Divifor^ is a term ufed inDlvifion,
and is the Number by which another Num-
ber is to be divided; as to fay. How many
times y in 30 ? j- here is the Divifor.
tnbsp;30. Dmitndf

31 • Qjngt;tttntj i s the Anfwer to the How
njany times ( as m the abovefaid) j is in JQ ?
lt;S times : 6 then is the Quotient.

32« Square, is the ProduÖ: of two
Numbers multiplied together, as the Square
of 6 multiplied by lt;S, is 36.

33. Sq»*re.r§ot of any Number, is that
number, which being multiplied by it fcif,
lhall have a Produâ or Square equal to the
given Number ; thus the Square-root of 36
is 6 ; for (5 mulriplied by 6, is 3 (5,equal to
the ftrft given Number.

But if it be a Number that cannot be
fquared, as 72, the content of half a Foot
of Board ; whole near Square-root is 8 :
48J2811 of 10000000, then is the Square-
root to be expreft as near as you may (or
care for ) as here the Square-root of 72,
which is called a Surd Number^ that will
not be fquared.

34' t K a fécond Produit, of power
of two Numbers increafing or muhiplied
together, as thus ; the Square of 6 is 3d,
the firft power : and the Cube of 6 is 215,
that is to fay 6 times the fécond power.

35. Cuhe-roet of an; Number, is a fé-
cond Quotient decreafing between thé Num-
ber given to becubsaand ï, as thus ; the

Cube-root of a 16, is to be fbundj out by
the Line of Numbers, the third part of the
diftance between 2ilt;J and i, is at 2 repeti-
tions found to ftay at lt;J • for if you fliould
have 216 Cubes or Dies, which is a proper
Cube, you fhali find that 36 laid together
one by another, making a Square repeated
lt;J times, will ufe or take up2id,thejuft
number, and make one great fquareDie;
and no other Number whatfoever, except
will do the like j therefore 6 is the Cube-
root of 2Ilt;J.

In Mr. mnigatt's Book of Arhhmetlck ,
is the way of doing it by Numbers or Fi-
gures, being one of the hardeft Leflbns m
Arithmeticki

htteftInftrumcm)to try your Rule,do thus r
apply one end to one Point as to A and
the other end to the other Point atB and
clofe to the edge draw the Line required :

the laft Point' ( yet keeping the fame fide of
the Rule toward the Paper ) and draw ths
Lmeapin, and if the two Lmes appear as
one the Rule is ftreight, or elfe not. Note
the Figure I.

On the Point E on the Line AB, I would
raife the Perpendiculer Line C E, fet one
jjoint of the Compafles in E, and open them
to dhy diftance, as E B and E A, and note
the Points A and B, then opfen the Com-
palfes wider, and fetting one Point in A,
make the pare of the Arch by C upwards 5
and if you have room do the like down-
wards, near D : Then the Compafles not
ftirrmg, fet one Point in B, and with the
other, crofsthe former Arks, nCarCamp;D:
a Rule laid, and a Line drawn, by thofe
twocroffings, fliallcutthe Linè AB per-
pendiculerly juft in the Point E, which was
required.

But if the Point C had been given from
whence to let fall the Perpendiculer to the
Line A B, do thus: Firft, fet one Point of
Compafles in C ; open the other to any di-
ftance, as fuppofe to A and B; and then (if
yovl have room upon A and B, ftrike both
the Arks by D, which finds the Point D, if

not) the middle between A and B givfi;
the Point E; by which todra. CED,
the Perpendiculer froni C defired. Note
Figure 2.

, Note, That if you can conie to find the
Point D, by the crofling, it doth readily
and exadtly divide the Line A B in two
qual pares, by the Point E.

On the end of the Line A B, at Bj
1 would raife a Perpaidiculer: Firft, fee
one Point of the Compafles in B; open
them to any diftance, as fuppofe to C 5 and
fet the other Point any where about the
ftiiddle, between D and E, asfuppofe at C,
then keep that Point fixed there ; turn the
other till it cut the Line, asacD, and keep
both Points fixed there, and lay a ftreight
Rule clofe to both Foiiits,. and there keep
p then keep the middle-Point ftill fixed at
C, and turn the other neatly clofe to the
other end and ed^e of the Rule, to find the
Point E; then a Rule laid to the Points E
indnbsp;draiw the Perpendiculer re-

Or elfe, when you have fet the Com-
pares in the Point C,, prLck the Point D in
Dnbsp;ehs

tlie Line, and make the touch of an Arfc
. near to E ; then a Rule laid to D C cuts
the Ark laft made, at or by E, in the Point
E : There are other wayes, but none better
than this. Note the Figure 3.

f. From A Point given^to let faB a Per-
pendiculer to the end of a Line, he-
ing the converge of the firmer.

Firft, from the Point E, draw the Line
E D, of which Line find the middle be-
tween EandD, viz,, the Point C : then
the extent C E, or C D, keeoing one Point
in C, fhall crofs the Ground-line in the
Point B, by which, and E, you may draw
the perpendiculer Line E B, which is but
the converfe of the former.

To the Line AB, I would have another
Parallel thereto ; to the diftance of A I,
take AI between your Compaffes, and
fetting one Point in one end of th^Line, as
at fwcep the Ark EIF ; then fet the
Compaflb ui the other end, as at B, and
fwcep the Ark GDH; then juft by the
Round-fide of thofe Arks, draw a Line,

\ Take B C, the meafure from the Point
quot;lat istocuttheParallel-lme, and one end
ot the given-Lme, viz,, B; with thisdi-
«ance, fet one foot in A, at the other end
^ the given-Lme, and draw the Arch at
tnentakeall AB, the given-Lme, and
mnig one Point in C, crofs the Ark atK,
then C and K fhall be Ponits to draw the
laraliel-hneby. Note the Figure 4.

The Angle BAG, being given, and I
would have another Angle equal muoits
«« one point of the CompafTes in A, and
draw the Arch C B ; then on the Line D E
Jrom the Point D, draw the like Ark E F •
hen m that Ark make EF equal toCB^
then draw the Line DF, it fhall make the
^ngle E D F, equal to the Angle BAG,
which was required,

into Efght pans: On one end, viz,. Ai
draw a Line, as AD, to any Angle ; and i
from die other end B, draw another Line
Parallel to A D, as B E 5 then open the
Compafles to any corivenierit diftance, and ;
from A and B, divide the Lines A D, and
B Ej into eight parts; then Lines drawn by
a Ruler, laid to every divifion, in the Lines
AD, and BE, fhallquot; divide the Line A B
in the parts requked. Note the Figure
marked VL

This Propofition is much eafier wrought
by the Line of Lines on the Seftor, thus 5
Take A B between your Compafles, and fit ^
•it over parrally in'8, and 8 of the Line of'
Lines j then the Parallel diftance between
1 and I, (hall divide AB into 8 parts rc^
quired.

Let A B C be three Points to be brought
into a Circle 5 firft fet one Point on A, an^
open the other above half-way to C, and,
fweep the part of a Circle above and be-f
low tie Point A, as the two Arches at V
aiui E •, not moving the Compsflcs, do thlt;
like on C, as the Arks FandG; thenfe'
ike Ccrapafe-poinf m B, and crofs thoi'

Arks ill DEFandG; then a Rule Lid
frpm D to E, and from F to G, and Li les
. drawn do inter-fed at H,the true Center,to
L bring ABC into a Circle.

10» Any two Points give» in a Cirdf, to
'nbsp;dram part of a Circle, which [hall cut

Let. A and B be two Points in a Circle,
by which two Points, I would draw an
Arch, which lhall cut the whole Circumfe-
rence into two equal parts. Firll, draw a
Line from A, the Point remoteft from c'le
; Center, through the Center,and beyond the
Circumference, as A D ; then draw another
Line from A, to a Point in the Circumfe-
! reace, perpendiculer to AD, (and cutting
• the Center C) as the Line AE: Then or»
the Point E, draw another Line perpendicu-
ler to the Line A E, till it inter-feSt A D ac
D 5 then thefe tiiree Points A B D brought
into a Circle, or Arch, by the laft Rule,
1 fliall divide the Circumference into two e-
j qual parts. Note tlie Figure 8, where the
i fiifi Circle is cut into two equal parts at F
J and G, by part of a Circle paiTing througli
the Points A and B.

îï. Any Seji^ment of a Circle givt», tt
fnd the Diameter and (Renter of
the Circle belonging to it.

tet A B C be the Segment of a Circle, to
Vvhich I would find a Center; anywhere
about the middeft of the Segment, fet one
point of the Compaflb àc pleafiire, as at B ;
on the point B (at any meet diftance) de-
fcribe a Circle, and 1iote v/here the Circle
doth crofs the Segment, as at D and E, then
( not ftirring the Compafles ) fet one point
in D, aind crofs the Circle twice, as at F
and I; and again, fet one point in E, and
crofs the Circle twice in G and H : Laftly,
by the Points G H, and F I, draw two
Lines, which will meet in the point Oj the
center required.

Multiply the Chord (or flat-fide) of the
half-Segment, viz, AK,i2 by it felf (which
is.called Squaring ) which makes 144;
then divide that Produft 144 by 8, the'
tine KB, called â Sine, the Quotientj
which comes out will be found to be 18 ;(
then if you adde 8 the Sine, andiSthej
Quotient together, it (hall make z6 fof.

Lay the Chord of the whole Segment,
and twice the Chord of half the Segment,
from one Point feverally 5 and to the grea-
teft extent, adde one third part of the dif-
ference between the Extents, and that futn
of Extents fhall be equal to the Arch.

Let ABC reprefent the Segment of a
Circle j the length of whofe Arch I would
know, or have a Line equal thereuntoquot;:
Take all the Chord A C, and lay it on any
Line, as from D to E; alfo take the Chora
of half the Arch, as A B, or B C, and lay
it twice from D to F ; then E and F are the
two Extents, whofe third part F G is to be
added to D F the greatefl: Extent to make up
DG, a Line equal to the Arch ABC,
which was required to be done. Wliich
.operation, may very well be performed on
a Line of Lines, or inches on your Rule ,
or by Numbers in Figures, thus; Suppofe
AC be u inches and 6 tenths i and twicc
0 4nbsp;AB

A B be 4? inches, 7 tenths: The difference
between thcin you may coynt, on the Rule,
to be 7 incks, and i tenth ; a third part of
which is % inches 4 tenths; which added
to 42, 7, makes 45, 2. the meafureofthe
Arch ABC, \vhich was required.

14. Tq draw 4 HtHcal Line from lt;»»ƒ
Three Polnti, to feveral Radiujfes withquot;
eat much glhkiofity ; ufiful for Archi-
ttB.'^ ShifwrightSyand others.

Let A B C D E be five Points, to be
brought into a Hehcal-Line, fmoothly, and
even ■ivit|iout gibbiofity or bunches, as the
under-fide qf an Arch, or the bending of
a Ship, or the hke.

Firfl, between the two remote Points of
3, as A and C, draw the Line A C, then
let fall a Perpendiculer from B, to cut the
Line AC at Right Angles, and produce it
to F : draw the like perpendiculer-Line
from the point D, to cut the Line C E at
Right-An^Ies produced to F. I fay, the
Center both for the Arches A B the lefler,
and BC the greater, will be found to be in
the Line B F ; th? like on the other-fide
for D E and C D, the Helical-Circle, or
Atch required»

But if you divide the Arch A B C D E
into 24 or more parts, the fevcral Centers
of the fplay-Lines are thus found; Take the
meafure A G, andlay k fromB, or D, or
C, on the Line G F j a;id thofe Points on
GF, fliall be the feveral Points to draw-
the fplay-Lines of the Arch, and Key-ftone
by.

Ja. 4f, or Secant of 00, are all one
and the fame thing, yet taken refpeaively
in their proper places, and is the whola
I^ine of Sines, or Tangents, to 4^ ; or
more particulary that point at the end of the

ooand4f on the edge of the Rule for the
Artificial Sines and Tangents, or 10 on the
Line of Numbers, and lo and 9.0 on the
Line of Lines, and Sines, on the Quadrant
tal-fide of the Inftrumcnt;.

Lme of Nacural-Sxnes. to that loinc oa
that Line of Sines, which reprefents the de-
grees and minutes contained in that Ark or
Angle required. But on the Artificial-Sines
we^refpea not any meafure but the Point

Tansem: 3 The fame account is ufed both for the
Kight-Tangent, and Secant alfo; the Na-
tural-Tangent taken from the beginning to
the degree and minute required ; the Arti-
Secant'nbsp;the Point only.

being in eflea the Right-Sine of the Cofine
of the Ark or Angle required; As for Ex^
^mple i I would take out the Cofine of the
Latitude of London, which is ci 52-
Count yi 32 from 90 toward the begin-
aung, and you fhall find your account to
.endatthe Right-Sineof 38 28, which is
theCompIement ofnbsp;for both put

watd to the Point required, without mind-
ing any diftance or meafuregt; till you coine
to Proportion.

6.nbsp;A Lateral Sine, Tangent, or Secant, LateraU,
or Scale of Equal Parts, is any Sine Tan-

gent, or Secant, taken along the length of
any Line, from the beginning onwards, be-
ing a term ufed only in operation with a
Sedfor, or one Line and a Thred, and op-
pofed to a parallel-Sine, Tangent, or Secant,
the thing next to be explained.

7.nbsp;A Parahl Sine, Tangent, or Secant, Tartnd;
is any Sine, Tangent, or Secant, taken a-

crofs from one Leg to the other of a SeiStor,;
or from any degree and minute on one Line
to a Thred dtawii ftreight with the other
hand, or any other fixed Line whatfoever,
at the neareft diftance.

8.nbsp;The Neartfl Diftance to any Line, is Nearepr
thus taken j When one Point of the Com-^'fiquot;quot;quot;'
-pafles ftands in any one Point, and the Line

being laid, I open or clofe my Compafles
till the other moyeable-Foot, being turned
about, will but juft touch or cleave the
Thred. But if you are to lay the Thred to
■ the neareft diftance, then one Point of the
Compafles being fet faft, the other is to be
turned about, and the Thred alfo flipped to
and fro, till the Compafs-point fiiall juft
cleave the Thred in the middeft.

Sgt;. To aide one Sine or Tangent,to a Sine
or Tangejit, is to take the Right-Sine, or
Tangent of any Ark or Angle between
your Compafles, and fetting one Point of
the Compailes in the Point of the other
Number, and then to fee how far the other
Point will extend Laterally. Example. To
adde the Sine of 20, to the Sine of 30, take
the Sine of 20 between your Compafles,
and then putting one Point in 50, the other
lhall reach to the Sine of yi 21 ; therefore
the diftance from the beginning to yi 21,
is the fum of the Sine of 30 and 20 added
together. The like way is to ad4 Taiv-
gents.

10. To SuhflraB a Sine from a Sine, or
a Tangent from a Tangent, is but to take
the Lateral leaft Right-Sine or Tangent be-
tween your Compalfes.and fetting one Point
in the term of the greateftturn, the other
toward the beginning, and note the degree
and minute that the other Point Ttayes in,
for that is the difference or remainder.

Suppofe I would take the Sine of 10 de-
grees from 25-; Take the diftance-lo bet
tween your CompafjTes, and fetting one
foot in and the other turned toward
....... 'nbsp;the

Or, you may fometimes take the diftancc
between the greater and the lefs, and lay
this from the beginning, fhall give the re-
mainder in diftance on the Sines as be-

The RcBifyini-Pam, is a Point ox^^^^iffm
Hole on the Head of the rrianguUr Qua-
drant in the inter-fefting of the hour and
Azimuth-line, and the common Line to the
Lines and Sines on the Head-leg ; in which
Point you are, when the Rule is open, to
ftick a fmall Pin to look to the objeft whofe
Altitude above the Horizon you would
have in degrees and minutes.

Plain, is that Board, Glafs, or flatSu-p/^/s;
perficies you intend to draw the Dial
upon, either fingle of it felf, or pyned to
fome other.

qgt;ole of the PUifif is an imaginary Point pole of ^
k the Horizon ( for all upright Dials) di- thi PtaiA

teaiy oppofite to the Plain, or in all Plains,
a Point every way 90 degrees from the
Plam.

Declination of a Plain, is only the num-
ber of degrees and minutes, that the Pole-
point of the Plain is diftant from the North
and South-points of the Horizon.

The PerdeniicHltr-Lineon the Plain, is a
Line Square to a Horizontal-line, being
part Of a Circle pafling through the Zenith,
and Nadir, and Pole-point of the Plain.

The HorUontal-llne, is a Line drawn
on any Plain, exadfly parallel to the true
Horizon of the place you dwell in.

Rtcllnatlon, K when a Plain beholdeth
the Zenith-point over our heads : But ƒ«-
clinatlon, is, when a Plain beholdeth the
Nadier j as in a Roof of a Houfe, the
Tiled-part rethnes, and the Celid-part in-
clines.

The Meridian, line, on all Plains is the
HourJine .of 12; but the Meridian of the
Plain, is the great Circle of Azimuth perpen-
diculer to the Plain, being the fame with
thePerpendiculer-line on the Plain, paflin''
.through the Points of Declination.

The Snhfllle-llne on all Dials, is that Line
\vherein the fitile. Gnomon, or Cock of
the Dial dothftand, ufually counted fiom

The Stile of a Dial, is the Angle,between
the common Axis of the World and the
Plain, upon the Subftile-line on the Plain,
on all Dials.

The Angle between \iani 6, is onely fe-
the number of degrees and minutes contain- ^^
ed between the Hour-line of iz, and thequot;quot;''^*
Hour-line of 6 a clock, on any kind of
Plain j efpecially thofe having Centers.

The Inclination of MeridianSyh thenum- Tnclha.
ber of degrees and minutes, counted on the
^quinodia!, between the Meridian
Hour-line of 12and the Subftile being the
diftance, between the Meridian of the place,
mz. 12 a clock, and the Meridian of the
Plain, but counted on the ;Equino6haI;
and doth ferve to make the Table of Hour-
Arks at iie Pole, and to prove your work.

The Linfj Tarallel to 12, are two Lines TaraMil
peculiar to this way of Dialling bytheSe-
and are only two Lines drawn equi-
diftanc from, and parallel to the Hour-line
of 12.

The Contingent Or Touch-Hne in this way ccnu»^
of Dialling with Centers, is a Line drawn
parallel to the Hour-line of 6 ; but in thofe
without Centers, it is drawn alwayes per-
pendiculer to the Subftile, and fo may it be
alfo, if you pleafe, in thofe with Centers alfo.

The Vertical Line on the Plain, is thi
fame with the Perpendiculer-line on the
Plain, being petpendiculer to the Horizon-
tal-line.nbsp;.

By the word i\7i7lt;/«f, is meant a Knot or
Ball, 6n the Axis or Stile of the Dial, to
make a black-fhaddow on the Dial, to
trace out the Suns motion in the Heavens j
or i'oiiietimes an open or hollow-place m
the Stile, to leave a light-place to do the
fame office.

But by Apex is meant the fame thing,
when the Top-end, or Point of an upright
Stile fhall fliew the Hour and Suns place,
as the Spot doth iii Celing-Dials, where the
Hours and Quarters are all of one length,
and diftingüifhed by their tullours of great-
nefsonly.nbsp;_nbsp;. ^

Ths Perpendlctiler height of the Stile, is
ilothing elie but thé neareft diftance from
the NUies or Apex to the Plain.

The Foot of the Stile is properly right un-
der the Nodféi or ApeX at the neareft di-
ftance.

The P^ertieaLPoint, is a Point only ufed
in Reciiners and Incliners, being a Point
right over, or under the Apex j and yet in
the Meridian, being let fall from the Zenith,
by or through the Apex or Naiw, to the
Plain in the Meridian-Hne.quot;

The Axis of the Horizon, is only the rnea- AxU of
fure from the Apex to the Veitical-point
laft fpoken to, being the Sccant of the com-
plement of the Reciination to the Radius of
the Perpendiculer height of the Stile.

Direü, is when the Dial-plain beholdeth
One of the Four Cardinal Points oF the
Horizon, as South or North, Eaflor Well,
that is to fay, when the Pole of the Plain,
being po degrees everyway from the Plain,
doth lieprccifely in one of thofe Four Car-
dinal Azimuths: Which in znEreä and
DireB-Plain will be in the Horizon.

declining, and �Uning, or Inclining' Declining
Plaints, are as the upper or under-fide of^ecZ/n/n^
Roofs at any Oblique Scituation from the^quot;'fquot;quot;-''
Cardinal Points of the Horizon.nbsp;-^r^g-flmns

Oblique, is only a wry, flanting, crook- Obliqne.
ed i contrary to diredl, right, plain, flat, or
perpendiculer; and applied varioufly, as to
the Sphear, to Triangles, to Dial-plains,
to Difcourle and Converfation.

Circles of Pofitlon, or rather Semi-circles /^t of
making 12 Houfes, are Circles, whofe Pole
, or Meeting-point is in the Meridian and Ho-
rizon of every Country, dividing the jEqui-
quot;oftial into 12 equal parts, being then cal-
«ed Houfes, when ufed in Aßrologle^ and
Enbsp;foijje.

But when they are ufed in A^ronm^^
they require a more near account, as to de-
grees and minutes.

Of certain Term mJßronomy ,
and Spherical Definitions of
Points and Lines in the S^hear.

N'Oc to be ctirious in this matter, ai
Sphear may be underftood to be a
nnited Spherical Superficies, or round Bo-
dy, contained under one Surface,; in the
middle whereof is a Point or Center j from
■whence all Lines drawn to th| .CirjCumlt;-
ference are equal: Or you iiia^.conceive a
Sphear to be anlnftrumentj^ cjonfifting of
feveral Rings or Circles, ys^ereby, the fenr
fible motion of theÄavenly Bodies arc;
conveniently reprefented.nbsp;j

For the better Explanation whereof, A-
ftronomershaVe contrived thereon, vix,. on
the Spheaf, ten imaginary Points, ar^d tea
Circles, which are ufually drawn on Globes
and Sphears; befides others not ufually
drawn, but apprehei^ed in the fancy, for

Totes. n-^He two Poles of the World, are the
JL two Points P and P in the Analem-
ma, being direftly oppofite one to another j
about which two Points, the whple frame ot
the Heavens moveth from Eaft to Weft; one
of which Poles may alwayes be feen by us,
called the Artickor North-Pole;reprefcnted
in the particular Scheam by the Point P.
The other being not feen, is not reprefented
in the particular Scheam j but the Line
PEP, in the general Scheam, drawn from
Pole to Pole, is'called the Axis, or Axeltree
of the World, becaufe the whole Sphear ap-'
pears to move round about it,

The Poles of the Zodiack are two Points
diametrically oppofite alfo, upon which
Points the Heavens moveftowly tromWefl:
to Eaft, reprefented by the two Points, I
andK, 23 degrees and 31 minutes diftant
from the two former Poles, in the Analem-
ma, and by the Point PZ in the Horizon-
tal projection j but the other Pole of the
Zodiack cannot be reprefented in that par-
ticular Schesm.
Eqntnom- The Equinoftial Points, are the Points
AMts.^^nbsp;and Lil?ra 5 to which two Points,

^hen the Sm cometlValoiig the Eclipticfc ,
K makechthe Dayes and Nights £qual in all
places; at-ft/^njj March loth or 11 th ; to
'^'^»•-«about the 13 th of Septemlftr, Kheje
the Spring, and Autumn begins; beingfre-
prefented in the Analemma 'by the Point-E,
and in the particular Scheam by the Points
t.andW.nbsp;. 7 .

The two Solflicial Points, are reprefeii-
ted one by the Point s,' ahd i:he other by
the Point •vy, in both Scheams; to which
Points when the 5«« cometh along the E-
chptick, it makes the Dayes in Career
longeft; in Capricora yp, fhorteft ; ©be-
ing about the nth of Jmc, and vy about

The Zenith is an imaginjry Point riglit
over our heads, being every way 90 degrees
Qiitant form the Horizon; in which Point
au Azimuth Lines do meet, reprefented by
the Points Z, in both Scheams.

- The Nadir is an imaginary Point under
our feet, direftly oppofice to the Zenith,
■reprefented by the Point N in the Analeml
ma, but not in the particular Scheam, be-
,lt;aufe it is not feen at any time.

THe HorihoM iS twofold, f/i. Rational,
and Senfthlt : TheRâtional'Horizon,
js ah imaginary great Citcleof the Sphear,
every where 90 degrees diflant from the
Zenith, and Nadir ; Points cuttihg, or divi-
ding the whole Sphear into two equal parts,
the'ówf called, The upper or viiilDle Hemi-
' fphear ; the w^rr the lower or ilivifible He-
quot;inifpheai;.

This Rational Horizon, is diftingüifhed
alfo into Right, Oblique, and Parallel-
Horizon;

I. The Right Horizon is when thetwö
Pelés óf the World lie in the Horizon, and
'the Equinoôliàl at Right Angles to it; which
quot;Horizon is pccuMar to thofe that live under
Vhe Equiiioét:aîi who have their Dayes and
Nights alwayes eqiial, and all the Stars to
iRife'aud Set, and the Çuii to paf^twice in
àquot; year by their Zenith-point, .thereby ma-
■feihg two Winters,and tvifo Summers ;Thei£
ÎVintérs being in June znàDccimher, and
their Summers, in (JMarch and Sepm-
kir.

• ' 2. The Oblique Horizon is when one
Pole-point is vifibie, and (the other not)
having E.evacion above, ag^d depreilion be-
low

low the North or South part of the Hori-
zon, according to the Latitude of the place :
w which Horizon when the Sun cdfrieth to
the Equinoitial, the Dayes and Nights are
only then equal; and the ncarèr the San
comes to the vifible Pole, the Dayes are the
»onger, and the contrary; alfo fome Stars
lever fer, and fome never rife in that Horir
Zon : And all Horizons-bat two, are in a
firiét fenfe Oblique Horizons, Thé
Right Horizon already fpoken ta: And
, The Parallel Horizon, is that Horizon
which hath the Equiho61:ial for its Horizon,
and one of the Pole-points for its Zenith ;
peculiar only to thofe Inhabitants uiider
the Pole, ( if any be there.) In which Ho-
rizon, one half of the Sphear doth only al-
wayes appear, and the other half alwayes is
hid ; aiid rhe Sun, for one half year, doth
go round about like a Skrevv, making ic
continual Day, and the other.haff year is
continual Night, and cold enough ; which
Circle in the Analenima is reprefcnted by
the Line H E S, but hi the,particular Scheam
by the Circle NES W.

The vifible or fenfible Horizon, is that
Circle where the Heavens and the Eardi
feem to touch, where the fight of the Sun
and Moon dorh feem fo begin,or ccafc to ap-
pear in our fights, being not much diflcnn^
E 4nbsp;'

in Obfervation from the true Horizon : an d
from, thence hath been called by BUgravt^
and others. The Finitor, or cnder of our
fight of the Heavenly Bodies,
Afe, Wwn. 'The Meridtan is a great Circle w hich paf-
2. feth through the two Pole-points, the Zenith
and Nadir, and the North and South-
points of the Horizon, and is called Meri-
dian, becaufe when the Sun (or Stars) co-
meth to that Circle, it maketh Mid-day,
or Mid-night, which is twice in every 24
hours: Alfo all places. North and South,
have the fame Meridian; but places that lie
Eaftwaid, or Weil wards, have feveral Me-
ridians. Alfo, when the Sun or Stars come
to the South, or ^lorth-part of the Meri-
dian, their Altitudes are then higheft, and
lowefl:.. And the difierence of Meridians is
the difference of Longitudes of Places, no-
ted by the Circle Z H N S in the Analem-
ma; and N Z ffi S in Horizontal-projeflion.

MquinoBi- The E^mnoElial is a great Circle, every
where po degfces. diftant from the two
5- Poles of the World, dividing the Spbeaj;
into two halfs, called the North and South
Hemifpheari and is called alfo the ^Equa-
tor, becaufe when the Sun paiTeth by it;
twice a year, it makes the day and nights
equal in all places ; noted by W ^ E, and,
1quot; both.

, The Zeiiack, or Signifer, is another ZodiacJ^
great Circle that divides the Sphear and E- 4.
quinoilial into two equal parts, whofc
' Poles are the Poles of the Zodiack, beinsj
degrees from it; and it inter-fedlsthe
Equinoctial in the two Points of jiries and
Liha ; and one part of it doth decline
Northward, and the other Southward,
23 degrees 31 minutes, as the Poles of the
Zodiack decline from the^orth and South-
Poles of the World : The breadth of this .
Zodiack, or Girdle, is counted 14 or 16 de-
grees, to allow for the wandring of Luna,
Ol^ars, and Fentu ; the middle of which
breadth is the Ecliptick-Line, becaufe all
Eclipfes are in, or very near in this Line.
And this Circle is divided into 12 Signs,ancl
each Sign into 30 degrees, according to,
dieif Names and Chara6ters, t Aries^
^ Taurui, m Gemm, 25 Canctty LiO\
^nbsp;^ Libra, m Scorpio, f Sagit-

tariHi, y? Capricornifu, j» Aijuarim^
X Pifces. 6 being Northern, and the lt;J
latter Southern.

The two Colurts are only two Meridians^ f^oi^^^^j
or great Circles, croffmg one another at '
Right Angles ; the one Co lure paffing ^
through the Poles of the World, and the (j.
Points of Arits and Libra^ there cutting the
Equinoctial and Ecliptick : And the other

Colure pafletli by the Poles of the World
alfo^ aha clifs the Echptick in ss, and yy,
making the Four Seafon^ of the year ^ that
is, the equal Dayes, called the Equinoi^iat-
Colure; and the unequal Dayes, in June
and Dtcfmhtr^calhd the Solfticial-Colures,
reprcfeiited in the Aflalemma by ZP 03
N S, and PEP; and in the particular
Scheam by WP^ and N P S, the Sol.
fticial-Coltire.

The leffer Circles are the Troftckj of
and -vy ; being the Lines of the Suns moti-
on in the longeft and fhdrtcft dayes, noted
in the Scheams by ©gt; ©, 2o, and d s E,
and-vy, yp; andW'VjxJi to which two
Circles when the Sun cometh, it is on the-
I ith of Jme, and the 11 th of December^
making the Summer and Winter Solftice.

The Pilar Circles, are two Circles drawii
jlbout the Poles of the World, as far off as
the Poles, of'the Zodiack ate, 23 de-
grees, 31 minutes; That about the North-
Pole is called the Artlck^, and that about
the South the Antartick,, being oppofite
thereunto, fhewed.in the Analemma by II,
and KK; and by the fmall Circle about P
in the particular Scheam.

X X through the two Poles, and cut-
ting the Equinoéb'al in 24 equal parts, as
thèUnes Pi,P2,P3,8e:c» in the Particular;
and P© H in theAnalemma; fuch alfo
arè degrees of Longitude, and Meridians ;
the Meridian being the hour of 12.

2.nbsp;Aümmhs are great Circles,-paffïng •^^''«Kffcx
through j or meeting in the Zenith and Na-
dir, points, numbred and counted on the
Hoiizon, from the Four Cardinal Points of
North and South, Eaft and Weft, accor-
ding to Four pories, br 180 degrees, or
.according to the 32 Rombs or Points of the
Compafs, as Z 0 A, and Z E, the Azi-

arc lefler Circles, all pardllel to the Horizon, nrs,'
coiuited on any Azimuth From the Horizon
to the quot;Zenith, to meafure the Altitude of
the Sunj Moon, or Stars above the Horizon,
being the portion-of fome Azimuth, between
ti:e Center of the Sun, or Star; and the

Horizon, commonly called its Altitude a-
bove the Horizon, (howedby A © in the
Analemma, and in the particular
Seheam.

iion. to the Equinoctial, as the Almicanters were
parallel to the Horizon, as s 0 ffi, the
greatefl: Declination or Circle of © : Thefe
parallels have the 2 Poles of the World for
their Centers, and m refpe6\ of the Sun or
!Stars, are called degrees of Decimation |
■ but in rcfpeCt of the Earth, degrees of La-
titude i being the Arch on the Meridian of
any place, between the Pole and Horizon,
as 4 s 4 in the Particular,.and H P in the
Analemma.

the Stars, are Lmesdrawn parallel to the
Ecliptick, as the Almicanters were parallel
to the Horizon ; fothat the Latitude of a
Star is counted from the Ecliptick toward
the Poles of the Zodiack ; but the Sun be-
ing alwayes in the Ecliptick, • is faid to have
110 Latitude.

LomitH^. à. Degrees of Longitude, in ref^a of
the Heavens, are meafured b^ the degrees
on the Ecliptick, from the firfl point of
forward, according to the fuccelTion

But Loitgltuit on the Earth, is counted
on the Equmoftial Eaftwards, from fome
principal Meridian on the Earth , as the
Ifles of Ax^ores, or the Peak of Tenntrif,
or the like,

7.nbsp;Right Afcention is an Arch, of the E- Right Af-.
quinofliai (counted from the firft Point ofcennon.
Aries ) that cometh to the Meridian with

the Sun, Moon, or Stars, at any day, or
time of the year, being much ufed in the
following difcourfe, noted in the Analem-
ma by E H, or the like; but counted asaf-
terward is fhewed.

8.nbsp;Oblique Afcention is an Arch of the' oblique-
Equinoitia , between the beginning of A- afcention.
ries, and t lat part of the Equino6i:ial that

rifeth with the Center of a Star, or any
portion of the Ecliptick in an Oblique-
Sphear.

5). Afcentional Diference,\s the difference Afcentio-
between the Right and Oblique Afcention, nal Differ
to find the Sun or Stars rifing beJfore or af-
terd.

zon, between the Center ot the Sun and the tude.
true Eaft-point, at the very moment of Ri-
fing, reprefented by 5S F, in the particular
Scheam, and G E, and F E in the Analem-
ma : ufeful at Sea,

Circles ii. AC/^f/* lt;ƒ Po/sfiV« is orie of the 12
An^n fAftronomy or Aftrology.
Fofition. 12. An Angle of Pofttm,- is the Angle-
made in the Center of the Sun, between his
Meridian, or Hour, and fome Azimuth, as
the Prime, VeMcal, or the Meridian, or any
other Azimuth, being ufefiit in Aftronomy,
and fometimes in Calculation, reprefented
bp P 0 Z in the Analemma.
»nbsp;Thus much for Aftronoraical terms.

Flift open the Rule, and put in the loofe
piece into the two Mortice-holes,
(which putting together makes it a Trittn-
gtiler Quadrant) but if you do not ufe thet
Toofe-piece, then open it quot;to an Angle of 6o
degrees, which is thus exaClly done : Mca-
fure frpm the ReCtifying-point, to any Num-
ber

ber on the S'nes or Lines; then keepine
the Point of theCompaflb ftill fixed in the
R9aify!ns[.point, turn the other to th?
Common-Line of the Hour and Azimuth-
■Lwe, that cuts the Reaifying.point, and
there keep it; then removing'the Point of
the Gompafles from the Redifying-point
^en or clofe the Rule till the other Point
(nail touch the diftance firft mcafufed in the
Line of Sines or Lines, then lhall you fee
the Lines on the Head, and Moveable-leg
to meet; and alfo fee quite through the'
Keaifying-point, to thruft a Pin quite
through ; and thus is it fet to an Angle of
fo degrees, xlt;rithout the help of the loofe-
an Angle of 4j, or whatfoever
tile the Rule is made for.

, Put a Pin in the Caiter. hole on tjie Head-
Leg, and another in the Reaifying-point,
and a third (if you pleafe )'in the end
of the Hour-line on the Moving-leg.
fhenonthe Pm in the Leg-center, hang'a
Thrd and Plummet; then if the objedt be
low, viz. under ay degrees high; Look
along by the two Pins in the Re6lifying-lt;

point, and the Moving-leg, and fee that the
Plummet playeth evenly and fteady, then
the degrees cut by the Thred, fhall be tfe
Altitude required, counting from 60 jo to-
ward the Head , as the fmaller Figures
Ihew.

But if the Objea be above zy degrees
high, then look by the Pm in theRedify-
ing-point, and that on which the Plummet
hangech ; and obferve as before, and the
Thred fhall (hew the Altitude required, aS
the Figures before the Line fheweth ; If you
have Sights,u(e them inftead ofPins^and by
Pradlice learn to be accurate in this Work,
the ground and foundation in every Obfer-
vation; and according to your exaftnefs
herein, is the following Work alfo.

Note alfo, that this^ looking up toward
the Sun, is only then when the Sun is in a
cloitd, and may be feen in the Abifs, but
will not give a clear fhadow : Or elfe you
muft ufe a piece of Red, or a Blue, or Green
Glafs, to darken the lufter that it oftend
not the eyes.

But if the Sun be clear and bright, then
you need not look up toward it, but hold
the TrianiHler Quadrant fo, that the fha-
dow of the Pin in the Center may fall juft
on the fhadow of the Pin in the Redlifying-
poinc» and both thofe fliadows on your
^ *nbsp;finger

finger beyond them, and the Plummet fc
ingfomewhat heavy, and the Thred fmall
and playuig-evenly by the Rule, then is the
Obfervation fo madei likely to be near the
Very truth.

Note alfo further. That the fhakinj^ of
the hand, yoU fhall find will'hinder exact-
nefs ; therefore, when you may, find fome
gt;Iaee to lean your Body, or Arm, or the
nffrument againft, that you may be the
more fleady.

But the fureft arid bett way i^ with a
Bail-focket, andaTliree-Ieg-ftaft; fuch as
Land Surveighers ufe to fupport their Li-
ftriiments withal, then youw^ll beat liber-
ty to move and removequot; it, to and fro, till
the Sights or Pins, and Plummet and Thred
play to exaftnefs ; without which care and
exadtuefs, you cannot certainly and know-
ingly attain the Sun's or a Star's Altitude to
a mmute, cither by this nor any other In-
ftrument whatfoever, though they be never
fo truly made : Yet I dare affirm to do it,
or It may be done as well by this, as by any
other graduated Inftrument whatfoever:
The Line of degrees on this, being only two
thirty degrees of a Tangent laidwgecher;
of which, that on the in-fide of the loofe-
piece is the largeft,and confèquently the beft,'
diftinguifh the minuts of a degree wichah
Fnbsp;life llh

Set the Moving-leg of. the Trlanguler
Quadrant on the thing you would have to
be Level; then if the Thred play juft on do
degrees, ortheftroke by 60Jo, then isic

Bat to try if a thing be upright or not 9
apply the Head-leg to the Wa 1 or Poft, and
if it be upright, the Thied will play juft on
the common Line between the Lines and
Sines on the Head-leg, and cut the ftroke
by po on the Head of the Inftrument, or
elfe not.

Firft, on the Seftor fide, about the Head,
is 180 degrees, or twice po graduated to
every two degrees; fo that opening the
Rule to any Angle, the in-fide of the Mo-
ving-le.^, paTing aboui: the femi-circle ot
the^Head, fhewcth the Angle of opening
to one degree. But to do it more exadly,

The two Lines of Sines that iffue from
the Center in Rules of a Foot^ Auc, are
drawn ufually juft y degrees affunder: or
rather the two innermoft Lines, on each
Leg, are always juft onejdegree from the in-

l^ine of Tangents, juft againft the Sine of
30, It makes the two innermoft Lines that
come fi-om the Center, juft 2 degrees af-
Innder, which is ealie to remember either
wadding or fubftradhng asfblWeth, two
Wayes.nbsp;'

I. Take the LatteralSine of 30,
^e meafure from the Center to 30 : the
^-ompaiTes fo fet, fet one Point in the Cen-
ter-pin m the Tangents juft againft 20;
and turn the other till it cut tlie common
Line in the Line of Sines on the other Le^.
and there it fhall fliew what Angle the two
innermoft-Lmes make, counting fr3in the
end toward the Head, and two degrees
lefs IS the Angle theSeCbr ftands at, both
on theni-fide and out-fide, the Legs being
parallel ; which Number muli nearly agree
with what the in-fide of the Leg cuts on
the Head-fenncjrcle, or there is a miftake.

Suppofe I open the Rule at all adventures^
and taking the Latteral Sine of 30 from the
Sines on the Se6tor-fide, and putting one
Point of the Compafs in the Center on the
Tangents, right againft the Sine of 30 011
the other Leg ( or the beginning of the Se-
cants on the fame Leg ) and turning the 0-
ther Point to the Line of Sines on the other
Leg, it cuts the Sine of 60 on the inner-
moil Line that comes from the Center ; then
I fay, that the Lines of Sines and Tan.»
gents are juft 30 degrees afiunder , and the
in-fide or out-fide of the Legs but 28, vtx.i
two degrees lefs, as a glance of your eye to
the Head will plainly fhew.

2. This way will'ferve very well for all
Angles above 20, and under 80: But for
all under 20, and above 80, to 120, this
is a better way;

Open the Rule to any Angle at pleafure^
and take the diftance parallelly (that is, a-
crofs from one Leg to the other) between the
Center-pin at 30 in the Sines, and that in
the Tangents right againft it, and meafure
it latterally from the' Center, and it fhall
fhew the Sine of half the Angle the Sines
and Tangentsftand at; and one degree lefs

Suppofe that opening the Seftor at ad-
ventures, or to the Levrf of any thing, I
Would know the Angle it ftands at: I take
the parallel Diftance between the two Cen-
ters ; and meafuring it latterally from the
Center, I find it gives me the Sine of j i de-
grees, viz,, the half Angle the Lines ftand
at; or yo, the Angle tlw Rule ftands at;
which doubled, is 102 for the Lines, or
100 for the Legs of the Senior, as a glance
of the eye prefently refblves by the inner-
edge of the Moving-leg, aud the divided
femi-circle.

3 . On the contrary. Would you fet the
Legs or Lines to any Angle, take the half
thereof latterally, or one degree lefs in the
half for the Legs, and make it a Parallel its
the two Centers, and the Sector is fo fee
accordingly.

I would fet the Legs, to 90 degrees, or a
juft Square : take out the Latteral Sine of
44, one degree lefs than 4y, the'half of 90,
and make it a Parallel in the two Centers
abovcfaid, and you fhall find the Lejs fee

iuft to a Square, or Right-Angle, as by
looking to the Head you may nearly fee. .

At the faille time if you take Latteral
50, and lay it from the Center, according
to the firft Rule, you fliall fee a great de-
ficiency therein, a? above is hinted.

Tbetiay of the Month btlngglveuy to fini
the SmsDecliKAtioKy true place in thi
Zodiack., Right Afctntion, AfcentiB--
nal Difference, or Rifing and Setting.

I. Lay the Thred to the Day of the
Month (m the upper Line of Monchs,wheTc
the length of theDayes are increafing ; or
in the lower-Line, when the Dayes are de-
creafing, according to the time of the year )
then in the Line of degrees, you have his De-
clination J wherein note, that if the Thred
lie on the right hand of 60)0, then the Suns
Declination is Northwards; the jcontrary.-i
way is Southwards: Alfo on the Line of the
Sun's Right Afcention, you have his Right
Afcention, in degrees and hours, ( counting
one Hour for ly degrees ) as the Months
proceed from/^/«rcA the loth, orEquino-
6\ial, the Right Afcention being then 00,
and fo forward to 24 hours, or 3 lt;Jo de-
grees^ as the Months and Dayes proceed.

AgarÀ, onahe Line of.;' the Sun's tru
place, yoahave the fign and degree of h;
place jn the Ecliptick^-^nVx, or the Equi-
noâial-point being the place to begin, and
then proceeding forward as the Months and
Dayes go.

Laftly, on the Hour-line you have the
Afcentional-difterence, in degrees and mi-
nutes, counting from or the Suns-Ri-
fing, counting as the morninghours proceed;
or his Setting, counting as che afternoon
hours proceed.

1.nbsp;For March the lay the Thred to
theDayj and extend itflreighc ; then en
the Line of degrees, - it iheweth near i de-
gree,,or 5-4 minutes Northward. ^ ' '

2.nbsp;Suns%lght'AfciHtio»i is intime
8 minutes and better, or in degrees, 2 deg.
J minutes. :

4. The Afcentional Dijferetsce, is t dcquot;
gree and 10 minutes; or the Sun rifeth
4 minutes before, and lets 4 minutes al-
ter 6.nbsp;^

Again, for iMay the loth, the Thred
laid thereon, cuts in the degrees, 2odeg.
J) tnin. for Northern DecUnation; and
57 deg. 24 min. or 3 hours 52 min. Right
Afcention; and 29 37 in -uTmrui for
his true place; and 27 12 for difference
of Afcentions, or rifeth 11 minutes after 4,
and fets 49 minutes after

Again, on the laß ef Oaohr, or the 21
of Janttary, near the Dechnation, is 17
5J2 Southwards, the Right Afccntion for
Oäoher is 225, 5-3, for January 21,
314 21: The true place for Offa^fr 31,
is tn Scorfla, JS deg. 22 min; hut for
January 21, luj Acjuariut u, f ^ gt; accor-
tiing as the Months go to the end at V?, and
then back again; but the Afcentional diffe-
rence, and Rifing and Setting, is very near
the fame at both times, 23 10, and
Rifeth 32 minutes, and more, after 7 ; and
Sets 28 minutes lefs after

Take the Bedlnatian, being counted on
the particular Scale of Altitudes, between
your Compajfles; and with this diftance, fet
gAefoot in 90 on jhe Aämuth-Ljne, the

other Point applied to the fame Line, fhall
give the AmpUtude, counting from 90,

The DeclfKatloK being laNorth.the Am-
plitude is ip deg. ly min. Northwards. Or
the 'Declination being 20 South, the Am-
plitude is 34 deg. 10 min. Southwards.

The Right Afcention and Afeentional-dif-
ference being given, to fi»d bis Oblique-
Afcention,

When the Declination is North, then
the difference between the Right Afcention,
and the Afcentional-difference, is the Ob-
lique-Afcention.

But in Southern declinations, the fum of
the Right Afcention, and difference of Af-
fcentions, is the Oblique Afcention.

On or between the 25: and id of fuly,
the Oblique-Afcention is by Subftradtion
SI 2, ij : On the 30th of OMer, the Ob-
lique-Afcention is gas 45 hy Addition,

, Ufe VIII. ,
The TDay of the i^onth, or Sttn's DecH-
ttatioH and Altitude heinggive», to fini
the Hour bf the Day.

• Take the Suns Altitude, from the parti-
cular Scale of Altitudes, fetting one Point
of the Compafles in the Center, at the be-
ginning of that Line ; and opening the 0-
ther to the degree and minute of the Sun's
Altitude,counted on that Line ; then lay the
Thred on the Day of the Month ( or De-
clination) and there keep it : Then carry the
Compafles ( fet at the former diftance ) a-
long the Line of Hours, perpeiidiculer to
the Thred, till the other Point, being tur-
ned about, will but juft touch the Thred ;
the CompafTes ftanding between the Thred
and the Hour 12, then the fixed Point in
the-Hour-Line fhall fhew the hour and mi-
nute required; but whether it be the Fore
or Afternoon, your judgment, or a fécond
obfervation muftdetermine.

On thz flrji ofAuguft in the morning, at
20 degrees of Altit^ide, you fhall find it to
he juft' j 2 minutes paft 6 ; but at the fame

Altitude in the afternoon, it is jS?minuteS
paft y at night, in the Latitude of
for London,

The Suns VecUnanon and Altitude glven^
to find the Suns Asjimuth from the
South'^ part of the Horiz^on,

Firft, by the 4th Ufe, find the Suns De-
clination, count the fame on the particular
Scale, and take the diftance between your
Compaffes; then lay the Thred to the Suns
Altitude^ counted the fame way as the
Southern-Declination is from do I o, toward
the loofe-piece j and when need requires on
the loofe-piece, then carry the CompafTes
along the Azimuth-line, on the rif^ht-fide
of the Thred, that is, between the Thred
and the Head, when the Declination is
Northward 5 and on the left-fide of the
Thred, that is, between the Thred and the
tnd, when the Declination is Southward.
So as the CompafTes fet to the Declination,
as before, and one Point ftaying on the
' Azimuth.lme, and the other turned about,
lhall but juft touch the Thred at the
neareft diftance ^ then, I fay, the fixed-
^oint fhall, in the Azimuth-line, fhew the
Suns-Azimuth required»

The Sun being in the Equinoftial, and
having no Dedination, you have nothing
to take with your Compaffes, but only lay
the Thred to the Altitude counted from
6o|o toward the loofe-piece, and in the
Azimuth-line it cuts the Azimuth required.

Example. At ay degrees high, you fhall
find the Suns Azimuth to be lo ; at
3 2 degrees high, you fhall find 38,20, the
Azimuth.

Again, At 20 degrees of Declination,
take 20 from the particular Scale, and at
10 degrees of Altitude, lay the Thred to
JO counted as before ; then if you carry the
CompalTes on the right-fide for North-De-
clination, you fhall find lop, 30, from
South; but if you carry them on the left-
fide for South-Declination, you lhall find
38, 30, from South.

The life of the Line of- Numberj oti
the Edge, and the Line of Lints
on the ^adrantal-fide, or on the
Seilor-fde, being all as dnei

Alfo to add or fubftraft one Line or
Number to or from one another, as in Chap-
ter 4th, Explanation the pth. I ccme now
to work the Rules of Multiplication and
Divifion, and the Rule of Three, dire6i: and
reverfe, both by the Artificial and Natural-
Lines; and firft by the Artificial, being the
moft cafie; and then by the Natural-Hnes
both on the Seftor and Trianguler Qua-
drant, being alike: and I work them toge-
ther ; Firft, becaufe I would avoid tau-
tology : Secondly, becaufe thereby is bet-
ter feen the harmony between them, and
which is beft and fpeedieft. Thirdly, be-
caufe it is a way not yec, as I know of, gone

by any other. And laft of all, becaufe one
may explain the other ; the Geometrical Fi-
gure being the fame with the luftrumenta^
work by the Natural way,

Extend the Compafles from i to the Mul-
tiplicator ; the fame extent applied the lame
way from the Multiplicand^ will caufe the
other Point to fall on the Produ6\ required.

Let 8 be given to be multiplied by lt;f ; If
you fet one Point of the Compaffes in i,
( either at the beginning, or at the middle,
or at the end, it matters not which ; y et the
middle i on the Head-leg, is for rhe moft
part the moft convenient) and open the o-
ther to 6, ( or 8, it matters not which, for
6 times 8, and 8 times 6, are alike ; ( but
yet you may mind the Precept if you will )
the fame Extent, laid the fame way from 8,
fhall reach to 48, the Product required ;

The Extent from i to fliall*
reach the fame way from 8
to 48. Or,
The Extent from i to 8, Ihallrt
reach the fame way from
to 48. .

Bj tht Natural-Lines on the StElor-fde,
er Triangultr Quadrant with a Threi
and Compajfety the T(fork.iithtu;

I. For the moft part it is wrought by
changing the terms from the Artificial way,
as thus;

The formet way was, as i to 6, fo is 8 to
4^ ; or as I to 8, fo is 6 to 48; but by
theSeftorit isthus : As theLatteral 6 taken
from the Center toward the end, is to the
Parallel 10 amp; lO, fet over from 10 to 10,
at the end counted as i ; fo is theParallel-
diftance between 8 amp; 8, on the Line of
Lines taken a-crofs from one Leg to the o-
ther, to the Latteral-diftance from the Cen-
ter to 48, the Produdt required.

Or porter thm.
As the Latteral 8, to the Parallel lo;
So is che Parallel 6, to the Latteral 48.

Another way niay you work withoiie
'altering the terms from the Artificial way,
as thus, by a double Radius; Take the Lat-
teral-Extent from the Center to 1 j (or from
1 o to 9, if the beginning be defe^ive) make
this a Parallel in 6 then theLatteral-
Extent from the Center to 8 of the lo parts
between Figure and Figure, fhall reach a-
crofs from 48 to 48, as before. See Plg'Tl.

The fame work as was done by the Se-
ftor, is done by the Line of Lines, and
Thred on the Quadrant-fide, that if youi-
Sedlor be put together as a Trianguler (^a-
drant, you may work any thing by it, as
•well as by the Seftor, in this manner; ( of
by the Scale and Compafs, as in the Fi-
gure I.) and firft,as above^ Seftor-wife.

Take the Extent from the Center to 6 latte^
rally,between your Compafles? fet one Point
in 10, and with the other lay the Thred
in the neareft diftance, turning the Com-
pafs-Point about, till it will but juft touch
the Thred, then there keep it j then fet one
Point of the Compafles in 8, and take the
neafeft diftance to the Thred ; this diftance
laid latterally from the Center, fhall reach
to 48 the Produ£l:*

Or as Latteral 6, to Parallel 10 j fo is
the Parallel 8, to the Latteral 48, the Pro-
du6t required.

Ortlielaftway by a double Radius, ort
greater and a fmaller Scale, as the Latteral
Extent from the Center to i, is to the Paral-
lel é, paying the Thred to the neareft di-
ftance ; fo is ^e latteral Extent from tbc
Center to 8 parts, lefs than the i before ta-
ken, carriea parallelly along the common-
line, till the other Point will but jufi touch
the Thred, it fhall on thofe conditions .ftay
only at 48, the Produ^ required. Obferve
and note the Figure, by protraótion, with
Scale and Compafs.

5. But if you have an Index and a
Square, as is ufed in the Demonftrative
Work of Plain Sayting, as you fliall have
afterwards, then the reprefentation of this
Natural-way will moll evidently appear, as
thus:

Set the edge of thé Square to i- on the
Line of Lines, counted from the Center
downwards, where the Figure i is, then
move the Index till the edge cut 8 of the
fmall parts, counted on the Square, from
the Line of Lines toward the end of the
Square, and then there keep the Index j
then remove the Square downward to 6, on
the Line of Lines j then there holding it
Square, you fliall lee the Index to cut 48 on
the Square, countin:» after the fame rate
ïhac the 8 pares was accountcd, Note the
Unbsp;Figu I

From hence you may obferve, That thé
■firft and third Numbers muft alwayes be ac-
counted alike, and on like Scales 5 and the
fécond and the fourth in like manner on like
Scales and counting ; and the Latteral-firfl
Number, muft alwayes be lefe than the
Parallel.fécond, in length or quantity, ot
you cannot work it ; which you muft make
to, either by changing the terms, or ufing
•a lefs Scale, to begin and end upon.

Here you muft except a Decimal grada-
tion, as thus ; fometimes the fame place
which is called 10 in the firft, may be coun-
ted 100 in the third, and the contrary ; or
more or lefs differing in a Decimal Ac-
count.

But if you would fee a Figure of the
Se61or-way of operation, then it is thus;
let the Line C reprefent one Leg of the
Seétor ; and the other Line C d, reprefent
the other Leg of the Sedior ; then take i
but of any Scale, as i inch, or one tenth
part of a foot, or what you pleafe : or the
diftance from the Center to i, or 2, on the
Line of Lines between your Compafles : put
this diftance over in 6 8c 6, of the Line of
Lines, Then is the Sedor fet to its due
//o/ffc/« Angle.

farts of the former I, from the Scale from
^'hence you took the firfl Latteral diftance,
and carry it parallel between the Line af
Lines till it ftay in like parts, which yoii
Ijialffind to beat 48, the Produarequi-,

_ ^Or to get the Anfwer inaLatteral-Iine,
is generally moft convenient, 6y changing
the terms 5 work thus : Take che L.irtei al-
diftance from che Center to 8,on ihj Line of
Lines, make it a Parallel in 10 amp; 10 ; then
the Seélor being fo let, take the Parallel-
diftance between 6 Sc Ö, and lay it lateral-
ly from the Center, and it fhall reach to
48, the Produa required. See Fig. 1IÏI.

Thus you fee that the way of the Seétor-
fide, and of the Quadrant-fide, is ina man-
ner all one; and the laying of the Thred, of
Index to the neareft diftance, is the fame
with fetting the Legs of the Seftor to their
Angle; and the taking the neareft diftance
from any Point or Number to the Thred,
Js the fame with taking parallelly front
Point to Point, or from Number to Number:
So that having thus fully explained the Lat-
teral and Parallel-Extent, and laying of thé
Thred, and fetting of the Seilor, the foL
lowing Propofitions will be more eafie, and.
ready ; and to that purpofe, thefe brief
Maries for Latteral, Parallel, and Neareft-
Ö 2nbsp;diftance^

Idiftance, will frequently be ufed ; as thiisj
for Latteral,thus—; for Parallel,thus = ;
forNeareft-diftance,thus|, or ND,thus:
for the Sine ofjgt;o, or Radius, or Taiigenc
of 4^,thusRj amp;c.

In all which wayes you may fee, that for
the want of feveral Radiuflb, which do
properly exprefs the unites, tens, hundreds,
and thoufands, and ten thoufands of Num-
bers, there muft a due and rational account,
OT confideration, go along with this Inftru-
mental manner of work, elfe you may give
an erroneous anfwer to the queftion pro-
pounded J to prevent which, obferve, that
in Multiplication there muft be for the moft
part, as many figures in the Produft, as is
ill the Multiplicator and Multiplicand, put
together i except when the firft figures of
tW Produft, be greater than any of the firft
figures of the Multiplicator, or the Multi-
plicand, and thai there is one lefs ;

2 times 2 makes 4, being only one fi-
gure, becaufe 4 is greater than 2 j but
2 times y is 10, being two figures; where-
in I, the firft figure, is lefs than j. Again,
inabigger fumj multiplied by 23, makea
1^00, coiififting of four figures, as many as

IS in the Multiplier and Multiplicand put
together j but if you multiply 4a by 22, it
makes but 924, which is but three figures ;
becaufe the firft figure 9 is greater than 2
or 4, as in the former, thp firft figure i was
lefs than y or 2 : And this Rule is general,
as to the number of places, or figures, in
any Multiplication whatfoever 5 but note,
that no Inftrunjent extant, and in ordinary
life,is capable to exprefs above y or 6 places:
Yet with this help you may come true to y
piaces, with a good Lin? of Numbers.

■ Suppofe I was by a Line of Numbers to
multiply ilt;J8 by 249; the extent from i to
l68, will reach the fame way from 249, to
41832 : Now by the Line of Numbers,
you can only fee but the 418, and eftimate
at the 3 i but the laft figure 2, I cannot fee
by any Line ufually put on two foot Rules,
therefore the ilt;J8, and 249 being before
you, fay ( according to the vulgar Rules of
Multiplication) 9 times 8 is 72; therefore
a muft needs be the laft figure; and if you
can fee the former 4, you have the Produ6l
infallibly true : if not, multiply a figure
more; Bat by this help you fliall be furc

to come right alwayes to 4 figures, or pkji'
CCS, in any Multiplication whatfoever. ' quot;
4. Aifo, by thefe operations, youraay
plainly fee, That the tine of Numbers, or •
gmters-Lint, as it is ufually called,..is i:he'
cafieft and exaaeft for Arithmetical opera-
tions, being perforrned with an Exte,nt of
the Compafles only, without any opening'
or fliutting of the Rule, or laying a Thred
or Index ; But in Queliions of Gemetry,
where a lively draught or reprefenratioh is
required, as' to the reafon of the. work,
there the Natural-Lines are'more demon^

In which Natural Work, Note, That
the Parallel-diffance muft alwayes be the
greateft, or you cannot work it, unkls
you make ufe of a greater, and a lefler
Scale ; to which purpofe, this Inftrument
is well furniftied, with, three or four Ra-
diufles, bigger and lefs, both ofSin?s,
Tangents, and Secants, and Equal-Part?,
as in their due places fhall be obferved ;
and taken notice of, in the f^rtroKomical-,
IVork. ■ . '

And note alfo. That if the Line of
Lines were repeated to 4 RadiufTes, o?
400 iuftead of 10, you might work
4 J but then the Ra-
diuflfs

Therefore this of lo, being the moft .
ufual, I lhall make ufe of, arid work every
Queftion the moft convenient way;, that by-
a frequent Practice, the young Beg-nners,.
for whom only I wrire, may fee theReafon
and Nature of che Work, and the fooner
underftand it.

For aConclufion of this Rule nf Mulci-
)lication, take three or four Examples more,
)oth by.theLiue of Numbers, and Equal-
Parts alfo;nbsp;^

I. Firft ly foot, 8 tenths, multiplied ^^
jgt; foot and 7 tenths, by the Line of Nu.a- ^
bers; the Extent of the Compafles from 1 to
9 foot, 7 tenths, fhall reach the fame way^
from ij—8, to IJ3, 2lt;J..

As the Latteral — ly-S, to the Paral-»
lel — 10, at the Énd counted as i ;

Where you may obferve, that the firft
and fourth, are meafui ed on like Scales ;
and the fécond and third, alfo on like
Spies.

But note, that asyoudiminiflicd the ac-
count in the third r= work, counting 5—7
lefs than 10 reckoned as i :

So likewife in the — fourth, you count
if3—2lt;J, which is leCs in Extent than the
ly—8, fiift taken Latterally j yet is to be
read as before, viz,. 153-25, beeaufa
$gt; times ij-, muft needs be above 100.

2» As I, to 9 foot 10 inches; So is
(o foot 9 inches, to 105 foot 6 inches 4.
To work this properly by the Line of
lumbers, you are firft by the inches and
3ot-meafure on the in-fide of the Rule, to
educe the inches into decimals of a foot ; as
hus: Right againft loinches, m the Line
tf Foot-meafure, you ftiallfind 83 Alfo,
ight againft 9 inches, on the Foot-meafure,
jou fhall find 7y j this being done, which
is with a glance of your eye only, on thofc
:wo Lines, then the work is thus:

As I, to 10 7y ; fois 9 8j, to lOj-i ,
®r loj foot, 6 inches: Now for the odd
6 fquare inches, you cannpt fee them on the
Rule, but muft find them by the help before
J—10nbsp;mentioned, as thus,

p is 90; for which you muft fet down 7
foot; and 6 inches, which is the 6 inches
you could not fee on the Line of Numbers,
and there muft needs be 105 foot, and not
10 foot and an half, and better, which is as
to the right Number of Figures.

Büt by the Liiie of Lines as the — 10—
75-, from any Scale, as the Line of Lines
doubled, or foot-meafure, or the like, is to
the^=^ie, foisthe =9—83, totheLat-
teralnbsp;as before, though not fo

As I, to ipS, fo is 3f22, to
yi^tfitf, the true anfwer; whichindeed
is to more places than poflible the Rule can
coma to without the help laft mentioned:
But if the Queftion had been thus, with the
fame Figures, 15 foot parts, by 35 foot
il* parts, as fo many feet, or yards, and
hundred parts : Then the anfwer would be
as before, yjtf foot; and cutting off four
figures for the four figures of Fra^ions,
both in the Mulciplicator and Multiplicand,
the i6 i(J, which makes near 2 inches
of a foot more, or parts of a yard more,
which in ordinary meafuring is i^ot con-
fiderabki

To multiply 3 pound, lt;S killings, and
3 pence, by it lelf j the Product is,
to./.—19.X.—5.(/.—!.ƒ.—: For the Ex-
tent from I, to 3—3125-, the Decimal num-
ber for 3./.—(J./.—3.lt;i. fhall reach from
thence to 10—9 72(5, which reduced again,
js as before, 10—19—^—2 as followeth,

Notey That Inthu way of (Multiplication
hj the Pen^ works thm ; Ton mufl fir(l
^aliif lj Pounds hy Pounds, one over the
other-, at I by ^ : Then the Shillings by
the Pounds crofs-wife both-wajes, as the
hlackzlinc jheweth. Then Pounds bf
Pence, as the long Prick-lines fhewech
hoth-wayts alfo. Then SUMngs by Shll-

, lings, M the 6 by Then ShiSings
hy Pence, both-wayes, as the fhsrt Prickp
lines (heweth. Then laftly, the Pence by
the Pence, as ^ by 3 ; whofe true power,
cfr denowinatton, is fomewhat hard tai
fonesiv' j t^hich «thus:

Sixtly, 6 times 6 is 36, every 20 where-
of is I Shilling; and every j dicreor is
3 Pence ; and every i is 2 Fa; things ai d
v^ths of a Farthing : So that 36 make
1 Shilling, 9 Pence, 2Faithings, -,Jths of a
Farthing.

Seventhly 6 times 3 is 18 ; every y
whereof is a Farthing, and every i is two
tenths of a Farthing, as the fhoit Prick-
line iTieweth.

Nmthly and laftiy ; to the right-hand,
3 times 3 is 9 ; where note, that there
goes lt;5o to make I Farthing ; therefore ó
makes one tenth of a Farthing: So that here
is I tenth and { : Confider the Scheam and
the Decimal-work, to prove it cxactly to
the hundreds of millions of a Pound, and
you will fiiid it to be very near,

Which Sum, being brought to Shilling?»
Pence, and Farthings, and tenths of Farj
things, is juft asaforefaid, vix..

Or elfe find the Square of the leaftDrt
nomination in 20 s. and divide the Pro-
duct of the Sums bemg brought to that
leaft Denomination thereby, and the Quo-

5) (Jo, the Farthings in aoi. fquared, is
921 lt;Soo. The fum of ^ I.6 s. — 3 lt;i. in
Farthings, is 3180; multiplied by it felf

I is lox 12400: This Produft divided by
921000, the fquare of the Farthings in
20 makes rol. in the Quotient,
which reduced, is 1 o/.—ip/.—J lt;i.—I tl.

To find this Decimal Fraétion is very
eafie thus, by the Line of Numbers; for if
20 fliillings be 1000, what fhall 6 (hillings

*nbsp;and 3 pence be ? Set one Point in 2, repre-
fenting ao ; and thé other in i, reprelen-
ting 1000: then the fame Extent laid the
fame way from 6 and h A^all reach to
3.12J, the decimal fraaionfor 6 {hillings
and 2 pence,nbsp;Qf

Or by inches and foot-meafure; for if
you account every 8th of an inch a farthing,
then every inch is a d. and 6 inches is i2 d.
right againft. which, in the Foot-meafure, is
the Decimal Fraftion requijed : So that
right againft it farthings,or linchandlon
the Line of Foot-meafure, is the De-
cimal Fradtion fought for.

Or if one Pound be the Liteger, or whole
Number, then every loth part is afhil-
lings 9 and every yrh is i {hilling: and
the inter-mediace peace and fahhings is Very
near the jch part j for if you conceive a jth
part, or yo of an hundred to contain one
ftiillnig, or 48 'quot;arthings; then one of yo
is very near one farJim^, for 12 and A is
juftjJ. and zs is juftdlt;f.nbsp;is juft

lt;)d. and 50 juft izd. So that to fet the
Cc.mpafs-pomt ro 3 /. 6 s. and 5 d. is
to fet the Point on 3.3125, as before^
which a httle praftice will make eafie.

As the Latteral 3,3125 is to the Parallel
io, So IS the Parallel 3.3125 to the Lat-
teral 10/. -Ip t.- J farthings ftriy
or 10. P73.

Extend the Compaffes from the Divifor to
1, then the fame extent of the Compafles,
applied the fame way from the Dividend,
fhall reach the Quotient required.

Or the Extent from the Divifor to the
Dividend, fhall reach the fame way from ij
Co the quotient required.

Let 40 be a Number given, to be divided
by 5 j here 40 is the Dividend, and j the
Divifor ; and the anfwer to how many, viz-
8 is the Quotient»

Extend the Compaffes from 5 to i, the
fame Extent fhall reach the fame way from
40 to 8 the Quotient required i or the Ex-
tent from y to 40, ihall reach the fame way
from I to 8 the quotient required.

As the—Latteral 40, to the = ParaHef y ;
So is the == 10 counted as i to the —-8.

Obferve tht Figurt with tht Line A B.
Or, as the — y, to the := 10 •
So is the — 40, to the = 8 i As the Line
CB in the Figure doth demonftrate, being
the manner of working by the Triangulerquot;
Quadrant, the way of tht Senior being the

Or, If of ip, as before.
For the Extent from jp to itfdS, fhafl
reach the fame way from i to87-] Js - the
work by the Lin es is as before.

In this vvork of Divifion, for moft ordi-
nary queftions, where there is not abote
four figures in the quotient, you may coiiie
very near with a good Line of Numbers, as
»hat on Serpeutinc-iines, and the like;

, but the difficulty is, to know the Number of
Figures, which is thus moft certainly done :
Write down the Dividend, and fet the Di-
vifor under it, as in the vulgar way of Di-
vifion j and there muft alwayes be as many
Figures as the Dividend hath more than the
Divifor ; and one more alfogt; when the
firft figure of the Dividend is greater than
the firft figure of the Divifor ; as if
Were to be divided by j^y, then there
Would be 3 figures in the Quotient; for the
Divifor would be written 3 times under the
Dividend, in the ufual way of Divifion; and
thofe figures be 417 almoft : But if
9172318, is divided'by 8231, you will
have 4 figures, viz.. iiif, being one fi-
gure more than 3, the difterence of places,
In this Rule alfo you may fee the excellency
of the Artificial-Lines of Numbers, before
aii,d above the Natural-Lines.

Td two Lines or ]lt;!umbers given, to
find a third in continual Proportion
Geometrical.

The Extent from one Number to the o-
«her, fhall reach the fame way from that
fecond to a third, amp;:c.

As the Lateral firft Number to the Paral-
lel fecond; laying the thred to the neareft;
diftance, there keep it:

quot;Any 0ne fide of a Geometrical Figure
■ bemg given, to find an the refi, or

ThU PropfnioK ts mjl proper to the Line of
Lines, and. not to the Line of Numbers ;
and done thus :

Take the Line given, and make it a Pa-
Talfel in its refpeaive Numbers ; the Tlired
fo laid to the neareft diftance, or the Se-
ctor fo fet, there keep it: then take out all
the reft feverally, and carry the Compafles
^rallelly till they ftay in like parts, which

Notcajfo, Tl^at the Line of Sines, and
the Thred will readily lie on all the Angles,
and be removed from Radius to Radius
more nimbly than any Seélor whatfoever,
only by drawing the Thred ftreight, and
obferving on what degree and part it cuts
being fo laid.

let ABCDEFG be the Plot of à
tield, whofe fide E D is only given to be
H 2nbsp;^ Chains 5

I would know all the refi:
Take E D, make it a r= in 9 ; lay the
Thred to ND, or fet the Seaor to that
gage, and there keep it: then meafure every
iide feverally, and you fhall find what e-
very one is in the lame proportional parts,
by carrying the Compafles parallelly, till it
ftay in like parts by the Sedtor, or N D, by
the Quadrant.

Jo lay down any Number of parts in
a Line, to any Scale lefs than the
the Line of Lines,

Take 10, or any other Number, out of
your given Scale, or defign any diftance to
be fo ixiuch as you pleafe, and fet one Point
in the fame Number, on the Line of Lines;
and with the other, lay the Thred to the
neareft diftance, and there keep it, by no-
ting the degree cut by ; then take out any 0-
ther Number that you would have, fettmg
one Point in that Number on the Line of
. Lines; and opening the Compafles, till the
ther Point will but juft touch the middle
gt;f the Thred? at N D ? and that fliall be the
---------- other

other fart required; or the length of fo
much, according to the firft Scale given.

Let AB reprefent a Line which is loo
parts, and I would lay down 65, 30,42,
83 parts of that 100.

Fiift, take all AB between your Com-
paffes, and fet one Point in 100 at 10 with
the other, lay the Thred to N D, then take
out , 30, 42, 83, amp;:c. parallelly, and lay
them down for the parts required^ as here
you fee.

The like work is by theSeaor, making
AB a = in 100 amp; 100; thentakeouc
= 30, 42, dj, 83, or anything elfe for the
parts required.

But note. If the Line be too large for
your Scale, or Line of Lines, then take
half, or one third, or fourth part of the
given Line; then if you take half, you muft
at lafl turn the Compafles two times: If
you take one third, then turn the Compaflb
three times; which may prove a very conr-
venient help in many cafes, in Surveying
and Dialling.

Note, That by this Rule you-may add
to, or take from, any given Line, or Num-
ber, any imraber of Parts or Lines required;

which is called the increaiing, or diminifh-'
ing a Line, to any Proportion required.

Take the whole length oF the Line be-
tween your Compaffes, and fetting one
Point in the Number of Parts, you would
have the Line divided into; with the other,
lay the Thred to N D, and there keep it;
then take the N D from i to the Thred,
and that fhall divide the Line into the parts
required.

Let AB be to be divided into 7 parts:
Take A B, make it a Parallel in 7, laying
the Thred to the N D, there keep it; then
the = I fhall divide the Line into 7 parts.
But if the Line were to be divided into ma-
ny parts, as'fuppofe 73 : Then firft, fit the
whole Line in = 73; then take out the
=r 72, 71, amp; 70, for the odd 3 ; then the
= 10 J, for every 10quot;' divifion, then the
= I for the fmaller parts ; or elfe you fhall
find it almoft an iqipoffible thing, to take at

oiice any diftance, which, being turned a-
bove JO times over, fhall not at laft happen
to be more or lefs than the defired Number
required.

Note, That if the given Number hap-
pen to be fuch, that the Part will fall too
near the Center; as fuppofe 11, 12, or any
Number under 30; then you may double,
treble, or quadruple the Number, and then
count 2, 3, or 4, for one of the Numbers
required.

Suppofe I would divide a Line into 15
parts; multiply 15 by and it makes 90 ;
Now ;n regard you have multiplied ly by

you muft take the = inftead of the
~ I, to divide the Line into ly parts, be-
tween your Compafles, becaule the whole
Line is fet in = 90, inflead of = jy; which
is 6 times as much as ly.

Note alfo, if the Line be too big for your
amp;ale, then take half, or a third, and make
it a = in the given Linethen take out the
■= I, and turn two or three times, to divide
the Line according to your mind, when it
is too large for your Scale.

Thefe two laft are not to be done by ths
l-ine of Numbers, but proper for the Line
of Sines only; unlefs you turn your Lines
H 4nbsp;to

the whole Line, in any other parts;
So is I, to as many of thofe Parts as be-
longs to J.

A mean Proportion between two Lines or
Numbers, is t lat Number, which being
multiplied by it felf, fhall produce a Num-
ber equal to the Produft of the two Num-
bers given, when they are multiplied the
one by the other.

So alfo multiplied by it felf, is 36:
Therefore 6 is a mean Proportional between
4 and 9. To find this by Arithmetick, is
by finding the Square-root of

Divide the diftanfe between 4 and p in-
to two equal parts, and the middle-point
will be found to be (J, the Geometrical mean
proportional required.

Firft, joyn the Lines, or Numbers, toge-
ther, to get the fum of them, and alfo the
half fum; and fubftra6\ one from the c-
thertoget the difference, and half the dif-
ference ; then count the half difference
from the Center down-wards; and note
where it ends: then taking the half fum
between your Compaffes, lay your Thred
to CO on the loofe-piece ; then, fetting one
Point in the half-difference, on the Line of
Lines.: See where, on the loofe-piece, the o-
ther Point fhall touch the Thred; and
mark the place, with a Bead on the Thred,
or a fpeck of Ink, or otherwife : for the mea-
fure from thence to the Center is the mean
Proportional required.

of Equal Parts, take off 4 and 9, and lay
them from C, to A and B; then find out
the true middle between A and B, as at E ;
and draw the half Circle ADB; then on
C ere6^ a perpendiculer Line, as C d ; then
if you take C d between the Compafles, and
,meafure it on the fame Scale that you took
4 and 9 from, and you fliall find it to be
6, the true mean proportional required:
being only the way by the Line of Lines, as
by confidering the Triangle C D E will ap^
pear.

To do this by the Senior, open the Line
of Lines to a Right-Ans^le ( by 3,4, amp;: j,
or 6, 8, amp; 10. thits: Take 10 Latterally
between your Compafles, make it aParal-r
lei in 6 and 8, then is the Line of Lines
opened to a Right-Angle; or if your Rule
be large, and your Compafles fmall, then
take Latteral y, the halt of 10, and make
it a Parallel in 3 and 4, the half of lt;J and
8, and it is rediangle alfo: ) Then fet half
the dift'erence on one Leg from the Center,
then having half the fum between your
Compaffes, fet one Point in the half-difrerquot;
ence lafl: counted, and turn the other Point
to the other Leg, and there it fhall fhew
the mean proportional Number required.

Find a mean proportion between the
length and the breadth of the Oblong, and
that fhall be the fide of a Square equal to
the Oblong.

Let the breadth of the Oblong be 4, and
the length p, the mean proportion will be
found to be 6 ; Therefore a Square, whofe
fide is lt;S, is equal to an Oblong, whofe
breadth is 4, and length p, of the fame
parts.

Find a mean proportion between the half
Bafe, and the whole Perpendiculer; and
that fhall be the fide of a Square equal to the
Triangle.

If the half-Bafe of a Gable-end be 10,
and the whole Perpendiculer II-18 ; the
mean proportion be:wecn 10and it-i8, is
10-575 J the fide of the Square equal to
that Tna.:gi£, or Gable-end required. '

3. Ta find a Proportto» htwet» the Sm'
perfecies, though unlike to one
another.

Firft, to every Superfecies, find the fide
of his equal Square, whether it be Circle,
Oblong, Romboides, or Triangle; then
the proportion between the fides of thofe
Squares, fliall be the Proportion one to ano-
ther.

Suppofe I have a Triangle, and a Circle,
and the fide of the Square, equal to the
Circle, is 10 inches; and the fide of the
Square, equal to the Triangle, is ly inches :
The Proportion between thefe two Squares,
as they are Lines, is as 10 to ly ; but as
Superfecies, as 100 to 4y ; being thus found
out, Take the Extent between ly and 10,
on che Line of Numbers, and repeat it two
times the fame way from 100, and it fhall
reach to 4y, the Proportion as Supcrfecies,
between that Cncle and Triangle, whofe
Squares equal were ly and lo.

4. To make one Sttperfeciet, equal to ano-
ther Suptrftcies, of another fhape : but

Fitft find a mean proportion between the
unequal fides of the given Superfecies, thtt
you are to make one hke ; and find the mean
proportion alfo between the unequal fides
of the Figure that you are to make one e-*-
qual to.

I have a Romboides,.whofe bafe is y, and
perpendiculer is 3, (and fide is 3-J5 ) the
mean proportion betw een is 3 - 8 6 (SAlfo,
I have a Triangle, whofe half-bafe is 8, and
the perpendiculer 4, the mean proportional
is j-6jj2 J and I would make another
Romboides as big as ,thc Triangle given,
whofe Area is 3 2 : Then by the Line of
Numbers, fay. As 3-866, the one mean
proportion, is to j-(5jy2, the other mean
proportion ; fo is Ae Sides of the Rom-
boides, whofe like I am to make,quot; to the
fides and perpendiculer of the Romboides
required, to make a Romboides equal to a
Triangle given, and like to another Rom-

of Che Rotiiboides given, to 7-30, the bafe
of the Rotnboides required. And fois 3,
the given perpendiculer of the Romboides,
to 4.3 8,. the perpendiculer of the Rom-
boides required : So alfo is 3-55, the fide of
the Romboides given, to 5-19, the fide of
the Romboides required: for, if youmul-
tiply 7-30, the bafe thus found, by 4-3 8,
the perpendiculer now found, it will make
a Romboides, whofe Area is equal to 32
the Area of the Triangle,that I was to make
the Romboides equal to ; and makin« the
fide to be 3-yy, it will be like the ftrft Rom-
boides propounded.

If it hadlt;bcen a Trapefi^, or other form-
fd Figure, ic might have been refolved into
Triangles, and hen brought into Squares as
before; Then all them Squares added into
one fum, whole Square-root is the mean
proportional or fide of a Square, equal to
that many-fided Figure, whofe like o'n equal
IS defired to be made and produced.

given, to find the Content of another
Circle, hy having the Diameter thereof
only given.

The Extent from one Diameter to the o-
ther, being twice repeated the right-way
from the given Area, lhall reach to the
Area required.

If the Area's of two Circles be given,and
the Diameter required; then the half-
diftance on the Numbers, between the two
Area's, lhall reach from the one Diameter
to the other.

To do this by the Line of Numbers, you
muft firft confider, whether the Figures,
whereby the Number, whofe Root you
would have, is exprelTed, be even or odd fi-
gures , that is, confifi of 2,4, 6, 8, or 10 ;

For if it be of even figures, then yo«
*huft count the 10 at the end for the unite •
^nd the Root and Square are backwards to-

But if itconfilTiof odd Figures, then the
I, in the middle of the Line, is the unite;
and the Root and Square is forwards to-
wards 10 : for the Square-root of any Num-
ber, is ahvaycs the mean proportional, or
middle fpace between i, and the Number
propounded; counting the unite according
to the Rule abovefaid: So that the Square-
Root of i728,confiftingof four figures, it
is at 41 and f.,., counting 10 for the unite ;
for the Number 42 8c , is juft in the
middeft between 1728 and 10,

And to find the Square-root of 144,
confifting of three figures ; divide the fpace
between the middle i and 144, counted
forwards, into two equal parts y and the
Point fhall reft at 12, the Square-root re-
quired.

Firft, find out a Number, that may part
the Number given evenly, or as even as
may be; then the Divifor fhall be one ex-
tream, and the Quotient another extream ;
the mean proportional between which two,
Hiall be the Square^root required, working
by dielaftRu'e.

To find the Square-root of 144. If you
divide 144. by 9, you (hall find ilt;5 in the
Quotient : Now a mean proportion be-
tween 9 die Divifor, and 16 the Quotient,
is 12 the Root, found by the laft Rule, viz.*
the 7th.

The Cubick -root of a Number, is al-
wayes the firft of two mean proportionals
between r, and the Number given ; coun-
ting the unite with the followmg cautions :
Set the Number down, and put a Point un-
der the ift, the 4th, the 7th, and the loth
figure ; and look how many Points you
have, fo many figures fhall youhaveinthe
Root.

Then if the laft Point, fall on the laft Fi-
gure, then the middle i muft be the unite,
and the Root, the Square, and Cube will
fall forwards toward 10.

But if the laft Point fall on the laft:
t^t oi.e, then the unite may be placed at ei-
ther end, viz,, at i at the beginning, or at
^Oat the end i and'thea the Cube will be
Radius beyond the unite, either for-
^^ards or backwards,

But if it fall on the laft but two, ^theS
10 at the end of the Line muft be the unitpi'«
and the Root, the Squave, and Cube will
alwayesbe in the fame Radius, that is be-
tween 10 at the end, and the middle i.

So that by thefe Rules, the Cubick-root
of 8 is 2; for putting a Point under 8, ;
being but one figure it hath but one Point,
therefore but one figure in the Root: Se-
condly, the Point being under the laft fi-
gure, the middle i is the unite; then di-
viding the fpace between i and 8, into
three equal parts; the firft - part ends at a,
the Root^required.

So likewife inij 31, there is two Points,
therefore two figures in the Root ; and the
laft Point being under the laft Figure, the
middle 1 is the unite ; and the fpace be-
tween 1 and I? 31, being divided into three
equal parts, the firft part doth end at il, the
quot;Cubick-root of 1331.

Again, for there is one Point, and ic
falls on the laft figure but one; therefore the
Root contains but one figure, and i at the
beginning, or 10 at the end, which yoU
pleafe, may be the unite.nbsp;;

But yet with this Caution, That the Cub«
muft be in the next Radius beyond that
which belongs to the unite; fo that dividing
quot;the fpace between *oandlt;J4.,

^ddlc J, towards the beginning, into three
Si'quot;'' the firft part falls on4, the

the laft but 2 ; therefore lo at the end is
the unite; and between 10 and the middle
I backward., you fhall have both Root,
Squa.^, and Cube,for the Number required'
jvhich will be at p ; For if you divide the
fpace between 10, and 720, into three equal
parts, the firft part will ftay at chcCu-
bick-root required.

Note, if ic be a furd Number that can-
not be cubed exaddy; yet the Number of
Jgures to be accounted as Integers is asbe-
^«re; and the refidue difcovcrable byths
IS a Decimal Fraftion.

fhis by theNaCural-lines,
com ^^ ^^ troublefome.aiid cannot
. CO no fuch exadlnefs, as by the Line of
I anbsp;Nura-

For Application or life of this laft Rule
of finding the Cube-Root, obferve
with me as followeth :

;. I. Between two Numbtrs, or Linesglven^
to find two mean proportional Numbers,
or Lines required.

Divide the fpace on the Line of Numbers,
between the two Numbers given, into three
equal parts and the Numbers where the
Points of the Compafles flay at each repeti-
tion, ( or tiirning ) fhall be the two mean
propordonal Numbers required.

Let 4 and 32, be two extream Numbers
(or the meafure of two extream Lines) be-
tween which I would have, two mean pro-
portional Numbers (or Lines) required.

In dividing the fpace on the Line of
Numbers, between 4 and 3 2, into three e-
qual parts, you fliall find the CompafTes
to ftay firft at 8, fecondly at ilt;J ; the two
mean proportionals, between 8 and 32, the
two extream Numbers firft given.

For the Square or Produil of 4 and 3 2»
thc'two extreams, 128, is equal to the Square

or Produa of 8 and 16, the two means
»nulfiplied together, being 128 alfo.
Tffi. To apply it then thus iov Example
■Ir 1 have the folid content of a Cube to be
5728 Cubick inches, and the fide thereof
be 12 inches ; I would know what lhall
the fide of the Cube be, whofe folid content
the double of 1728 ? Divide the
ipac6 between 1728, and 345*5, into three
equal parts; then, lay the fame diftance
the Compafles ftand at fVom 12, the fide
of the Cube given, and it firall reach to
the fide of the Cube required,
whofe fohd content Is 34^6 inches.

A'^o, If I have a Shoe of Iron, whofe
weight is 3 pound,and the diameter there-
of 2 inches, and 780 parts of an inch in a
?ooo J what fhall the diameter of a Shoe
be, whofe weight is 71 pound ? One third
gt;art of the diftance, on the Line of Num-
bers between 3 pound, and 71 pound,
lhall reach from 2 inches, 780 parts, the gi-
ven diameter, to 8 inches, the true diameter
of a caft Iron Bullet, whofe weight is 71
pound.

2 Secondly, on the contrary, if the Dia-*
meter and Content of one Globe, or Cube
be given, and the Diameter of another
Globe or Cube, to find the content thereof.

As the diameter of the Globe, whofe con-
tent is alfo given, is to the diameter of the
Globe whofe content is required ; fo is the
content given, to the content required ; by
repeating the Extent the fame way three
times. Example. Suppofe the capaffity or
content of a Globe, whofe diameter is lo
inches, be 5'23 inches foHd, and 80 parts ;
what fliall the coûtent of that Globe be whofe
diameter is 20 inches ? the Extent from 10
to 20, being turned three times from ^23-8
the content of a Globe of loinches diame-
ter,fhall reach to 4ipc'2, the Cubick inchcs
contained in a Globe ot 20 inches diameter,
being % times as much as the former.

3. TTie Proportion between the weights and
magnitudes of fever d Metals,are a»fol-
Im ath, according to Marinus Ghetaldi.

If 7 pieces of the 7 Metals, are all of
cne fhape. and bi^ncfs, either Sphears, or
Cuucï or Ciilcnders, or Parallelepipedons 5
then their weights are in proportion as fol-
Ipwcthj according to Mmnns Qbetaldi.

So that if a Cillender of Tlun, whofe
quot;dc IS one inch, weigh 1824 grains; Wllac
inall aCiilcnder of Gold weigh, the height
and diameter being juft one inch,

The Shapesaud Welghts ef the ^leces of the
feven feveral Metals beitig etjital, tbtn
the (Jl^agHitttdesofthe Jidesare as fol-
lovfethj aecording to Mr. Gunter.

So that if I have a Sphear of Iron,
weigheth 9 pound, whofe Diameter is
4 inches; What muft the Diameter of a
Leaden Sphear, or Bullet, be of the fame
weight ?

One third part of the fpace between
p2jo, andd43j, fhall reach from 4 the
Diameter of the Iron Bullet, to 3 inches
54 parts, the diameter of the Leaden Bul-
let, that weighs 9 pound.

4. So that if I bavt the weight and mag-
nitude, of A body ef one kind of Metal,
and would know the magnitude of a body
of another Metal, having the fame
Weight: work^ thus;

The firft of two mean proportionals, be-
tween the two Points, on the Line of Num- •

ters, repiefeuting the Numbers in the laft
Table, for the two Metals, fliall reach the
right way, from the Magnitude given, to
the Magnitude required.

. Suppofe a Cube of Gold, whofe fide is 2 '
inches, weigh 29000 grains; What Ihaft
the fide of a Cube of Tinn be, having the
lame weight ? Divide the fpace on the Line
ot Numbers, between 3895, the Point on
theNpbers for Gold; and looco, the
I oint for Tinn ; and this extent parted into
3 equal parts, and that diftance laid from
a, the fide of the Cube of Gold, fliall reach
to 2-74 j the fide of the Cube of Tmu re-
quired.

y. The Magnitudes of two bodies of feveral
Metals being given, and the weight of
the ont, to find the weight of the other.

Take the Extent between their Points,
oil the Line of Numbers, according to the
laft Table, for eachfeveral Metal; and this
Extent laid from the given weight, (hall
reach to the enquired weight, of the other
Metal propounded.

If a Bullet of Iron, of 4 inches Diameter,
will weigh 9 pound ; a Bullet of Lead, of
the fame Diameter, will weigh 13 pound.

6, embody of one Metal being given, to,
make another body like unta it »ƒ another
Metal, and any other weight, to find
the Diameters and Magnitudes thereof.

•■ Firft, by the 4th laft paft, find the Mag-
nitude of the fide , or Diameter of thé
Sphear, having equal weight; and note
that down, or keep it,

Xhen find out two mean proportions, be-
tween the weights given; and fetting this di-.
ftance, on the Line of Numbers, the right
way, either increafing, or diminiftiing from

Firft, fquare the given Number ( that is,
^Ultiply it by it felf) then multiply the
i rodua by y, and divide this Produa by
4 5 then find the Square-root of the Quoti-
^»t, and from it take half the given Num-
ber, the refidue is the greater portion, then
greater part taken from the whole, leaves
the leffer-part.

Open the Seaor to a Right AnglCj in the
Line of Lmes (making Latteral lo, a Paral-
«Im 8 and 6) or elfemake the Latteral 90,
=J arallel Sine of (or the Latteral Sine
a Parallel Sine of 30 ) then upon
Legs count the given Number j then
Jake the Parallel Extent from the whole

J ^quot;quot;'^''^^^^■hole Number, muft
^e added to the half Number, to makeup
, ^ greater Number ; or taken from the half
make the lefier.

then the Se6\or itanding at Right Angles,''
takctheParallel-diftance from 3, the half
of lt;J ( counted as 12 ) on one Leg, to 6 on
the other Leg ; and you fhall' find it reach
to 6-71J which doubled is 13-42; from
which if you take lt;J, the half fum, reft 7-4^
for the greater part : and if you take 7-42
from 12, there remains 4-58, the lefTer-
part.

Extend the Compaffes from i to 12, and
that extent fhall reach from 12 to 144;
then next the extent from i to 144, fhall
reach from f to 720 ; then the extent from
4 to I, fliall reach from 720 to 180 : then
to find the Square-root of 180 ; the ^half--
diftance between 180 and i, you wiil find
to be at 13-42, as before ; which ufed as
abovefaid, gives the extream, and the mean
proportional parts of 12 required.

Extend the CompafTes from i to 2lt;J, and
repeat the fame agani forward from 2(S,llialI
reach to ■ 67C).

, Lafily, the Extent from 4 to i, fliall
«ach from 3380, to 84^ ; and the Square-
*oot of 84^, is 29-07: from which Num-
or Root, if you take half the given
dumber, viz.. 13 ; then there will remain
^lt;5-07, one extieam: then 16-07 taken
worn 26.0, reft 9-93, the other extream
«quired.

The befl and quickeft way is by the Lm
ot !gt;mes,thus; Make the given Line a Paral-
Jcl-Sineof 90 J then take our theparallel-
^ne of 3 8 degrees, 10 minutes, and th^t
inall be the greater part.

Alfo, takeout the Parallel-Sine of 22 dc-
gtees, 27 mrnutes, and that lhall be the
quot;er extream required: Or, accordmgto

Alfo by confequence, having the mean,
or greater part, make it a parallel-Sine of
3« degrees, 10 mmutes; then Parallel 90
quot;lall be the whole Line, and Parallel 24
Pegrees, 27 minutes, lhall be the lefler part

it a Parallel in aa degrees, minute3;
then Parallel podeg. lomin. fhall be the
whole Line; and Parallel 38-io,the greater
part.

The Ufe whereof, you fhall have after-
wards in the tith Chapter, about thecut^
ting ofl' the Platonical Bodies.

Three Lines or Numbers given, to
find a Fourth y in Geometrical Pro-
portion s or, the J^le of Three di-
reB.

X. In all Queftion s of the Rule oîThrti,
there be three terras propounded, viz. two
of Suppofition, and one of Demand.

2,nbsp;Alfo note, that two of the terms pro-
pounded, are of one denomination, ( or at
eaft to be reduced to one denomination) and
one of another denomination.

3.nbsp;Of the three termes propounded, (in
diredl proportion ) that of Demand is al-
wayes the third term, and one of the terms
of Suppofition, viz,, that of the fame Deno^
mination, with the term of Demand, is al-
wayes the firft ; then the Other of Suppofiti-
on left, muft needs be the fécond term in the
Queftion..

5quot;. Having difcovered which be the firft,
^€cond, and third terms; If the firft and
third term be of divers Denominations^
they nmft be reduced to one Denomination,
quot; it cûïuiot be done on the Line in the ope-
I'stion, as many times it may j At ihu6 for

Here you fee that the term of D-.Tand,
30 ouncesgt;,i. the third term^is not direét-
ly of the fame Denoiniiiation with one
pound, the firft term ; but is thus to be re-
duced to ounces : Saying;

Thus the firft and third terms, are
brought to one Denomination : Alfo you
fee t ut the Demand or Qilt;eftion,t/;*,. What
aiall 30 ounccs coft? is joynedtothe third
term ; and alfo that 16 ounc« the firft
term, is of the fame Denomination ; there-
fore the 3 X. muft needs be the fecond
and the Anfwer to the Queftion is
tl'cfouith.

Having thus difcovered, which arc
^ ^ firft, fecond, and third tei-ms, and re-
Knbsp;daccjJ

duced the firft and third to the like Den(gt;
minacions ; then the work by the Line of
Numbers is alwayes thus ;

Or the Extent of the Compaffes upon the
tine of Numbers, from the firft, to the fé-
cond ; fhall reach the fame way, from the
third, to the fourth required.

Kit 6 ounces is to 24 ; fo is 30 ounces
to 45 i ; which is ^s. — ^d. as be-
fore.

But by the' Line of Lines or SetSlor,
if you will work on one Scale only, you
muft confider which term of the firft or fé-
cond is biggeft ; for you muft alwayes or-
der it fo, quot;that the Parallel work muft be
the largeft, (or at leaft fo as it may be
wrought) and as much as may be, that the
fourth-term may be a Latteral Extent, as
the firft alwayes is ; for then it is wrought
the fooneft, and alfo the exaóleft,

. Yet by this Inftrument, you need noS
much care for thefe Cautions, having feveral
Scales of Equal Parts, to begin and end the
vork on, you are freed frOra that trouble.

Whtn the fécond terra is greater then the firft, then the Work is welî
performed thus, two wayes.
As the Latteral firft x6 ' C As Latterâl third 5 a from a Icflef Scale.
To the Parallel fécond 02 (^or J To Parallel firft id
So is the Latteral third 30 ( elfe j So Parallel fécond 2
To the Parallel fourth J ^ To Latteral fourth j?-!^ by the fame Scak;
Or, As —fécond, to quot; firft ; So .z^ third, to —— fourth j by a lj;fs
Scale alfo, if need be.

But when the fécond term is lefs than the firft, then die work is per-
formed thus :

8. Thus you fee feveral wayes of work-
ing: but for Beginners, I would advifc
thus, briefly.'

Firft, cither to oMerve this Rule of chang-
ing the terms, from the firft to the fecondgt;
Wx..To take the fecond Latterally, and make
it a Parallel ift the firft; then the Parallel
third gives you a Latteral Anfwer. Or elfe
CO work direólly, as the firft to the fecond,
and fo be content with a Parallel Anfwer,
which you may alwayes do with thé help
of a fmaller Scale, when need requires it.

The Rule of Three inverfed, or the
back-Rule of Three, is, when the term re-
cruited, or fourth term, ought topoceed
from the fecond term, according to the fame
)roportion, that the firft teri^i proceeds
jpmthethir^

If 20 Carts carry 60 Square yards of
Earch in 60 hours, how many Square yards
of Earth fhall Io Carts carry in id hours?

Here it is apparent that fewer Carts muft
have a longer time to carry the like quanti-
ty ; therefore to the fame time muft lefs
Work be allotted, as in the work doth fol-
low.

For if yoy extend the CompafTes from 10
to 20, terms of like Denomination, vU.
that of Carts'; the fame extent applyed the
^«^ntrary way, from x6, the time' required,
20 Carts, fhall reach to 32, the time
'acquired by 10 Carts, to carry 60 Load.

Look how much .the third term is lefs
than the firft, by fo much is the fourth term
lefs than the fecond.

As 2 is to 4, fo is 6 to 12 ^ for as 6 the
third term, is thrice as much as 2, the firft
term ; fo is 12, the fourth term, thrice as
much as 4, the fecond.

As 12 is to 6, fo is 4 to 2 ; For as 4 is
one third part of 12, fo is 2 one third part
bf 6.

2, But now in this Rule of Three in-
verfcd, or the back-RuIe of three ; it iscon-
crarily ordered, tfcaij ;
. Look how much the third term is greater
(or leflêr) than the firft, by fo much is the
fcairth term lefTer ( or greater ) tlian the fé-
cond.

As y is to 60, fo is 30 to 2that is,
If y is 60 d. How many fliillmgs is
- ■nbsp;fence?

If i 8 Pioneers make a Trench in 40 days,
how many Pioneers is needful to perform the
fame in if dayes?

As 40 to iy, fo is 18 to 48 : Here, as
the third is leffer than the firft j fo is the
fourth greater than the fécond.

A^ain, if 12 Horfes eat 20 bufhels of
Provender, in 30 dayes; how foon will
24 Horfes eat up the like quantity of Pro-
Vender ? The Anfwer is in ij dayes.

Extend the Compafles from one term to'
Uie oth:r of like Penominatiqn ; the fame
K4 quot; extent;

extent laid the contrary way froni the othefc
term, fhall reach to the Anfwer required.

As in the lafl Example ; the e-.tent from
12 to 24, the terms under the denominati-
on of Horfes, fhall reach the contrary way
from 30 to ij-, the'number of day es re-
quired.

4. Note, That by due confideration, this
back-Rule may be wrought by the Precepts,
for the direct-Rule,

InallQueflions of this nature, there be
tVr:e terms given to find a fourth; of which
three terms, two are of one Denomination,
and one of a different Denomination ; of
which, the fourth muft alwayes be • which
in the firft Rule of the tenth Seftion before
going, are called two termes of Suppofition,
and ontf of Demand. Nowhere you are
to confider. That

y. When the fourth term required, ought
to be greater than that of Demand ; which
by realon you may certainly know ;
Then fay,

If 80 Men do a Work ini2 dayes, how
foon may 40 Men do the hkc Work ? Here
Reafon tells me, that fewer Men muft have
'ongertime; therefore the fourth term re-
quired nmft be greater. Therefore,

Suppofition 40, to the greater 80 ;
Sois 12, the term of Demand, to 24, the
Anfwer required.

But if the required term ought to be
'eller, which Reafon will difcover in like
manner; Thenthfu:

As the greater term of Suppofition, is
to the leffer ; fo is the term of Demand to
the fourth term required.

As 80 to 4.0, fo is 24 to 12 ; extendina;
the CompaOes the fame way from the third
to the fourth, as from the firft to the fé-
cond.

But Note here, That you are not tyed to
^oferve which is the firft, fecond, or third
; but to confider only the nature of

the Queftion, thac you may Anfwer accor-
dingly j and indeed this way will, general-
ly, taice in the direél Rule alfo. For alwayes
in Direft Proportion, you may as well fay.
As the third term is to the firft, fo is the
fecond to the fourth ; as to fay, As the firft
to the fecond, fo is the third to the fourth.

Alfo backwards, or inverfly ; As the
third to the firft, fo is the fecond to the
fourth; extending theCompaiTesthe con-
trary way.

8. To perform this by the Seftor, or ge-
neral Scale and Thred, on the Quadrantal-
fide, you may generally obferve this Rule ;
Enter the fecond term taken Latterally, Pa-
rallelly in the firft, keeping the Secfor, or
Thred, at thac Angle ; then the Parallel-
third, fhall give the Lacceralrfourth, La t-

p. Or elfe. As the Latteral-firft, to the
Parallel-fccond ; fo is the Latteral-third, to
the Parallel-fourth, Parallelly.

And if the fecond be lefs than the firft,
make ufe of a fmaller Scale ; ot change clie
terras, as is ftiewed before.-

Having premifcd the way to bring the
■back (or inverfed) Rule of Three, to beper-
- formed by the Rules for theDireft; and
confidering that the Double and Compound
Rirfes of Three are alike by the Line of Num-
bers ; I have therefore joyned them toge-
ther in one Sedlion.

1.nbsp;The Compound, or Double (Golden)
Rule of Three j is, when more than three
terms are propounded, or given.

2.nbsp;The Double Rule of Three, is when
five terms are propounded, and a fixt terraquot;
proportional unto them is demanded»

• If 6 Menfpend i8 /. in three months;
How much will ferve I2 Men for 6
. months ?

. If two Barrels of Beer ferve 12 Men for
14 dayes j How many dayes will 4 Barrels
ferve 24 Men ?

3.nbsp;Tl^e five ^erms d/cn c.onfi.l: oftwo
• -nbsp;quot;nbsp;quot;nbsp;parts,

Thf? Suppofition lies in thefe three Num-
bers firft propounded, viz,. If 6 Men fpend
in 3 months j and the Demand lies
in the two remaining, viz,. How much will
ferveia Mcn6 months ?

Or in the other Example, viz,. If z Bar-
rels of Beer ferve iz Men 14 dayes, are the
terms of Suppofition ; and, how many
dayes will 4 Barrels ferve 24 Men, are the

4. The next work is to rank the three
terms of Suppofition, and the two of De-
mand, in their dtie and proper order, for
convenience of Operation j which may be
thus :

Of the three terms of Suppofition, that
which hath the fame Denomination with the
term required, place in the fecond place ;
and the other two, one above another in the
firft place :

And then place the two terms of Demand
one above another in the third place. Only
pblerving to keep the Numbers of like Igt;£-

. . [ 1
nomination in the fame ranks ; as lt;S Msn,
and 12 Men in the upper rank ; and 3
Months, and 6 Months in the lower rank j
as in the Work is expreft.

5. When Queftions of this nature art
tefolved by two fingle Rules, then the Ana-
logy, or Proportion, is thm ;

the term required.
As in the firft Example j As d to 18 j fo
' is 12 to 36 a fourth.
Again, as 5 to 3 d ; fo is ^ to 72, the term
required.

f^hkh by the Line ef Numbers, is
thm wrought ;
Extend the Compafles from d to 18 j tftfe
fame extent applyed the fame way from
ïa, fhall reach to 3d. Then again, extend
*hc Compafles flom 3 to 365 the fame ex-
tent applied the fame way from lhalî
'ifach to^j^t.^ term ttxj^iiîd,

Or elfe work it Parallelly, obferving the
fame order, as by the Line of Numbers,
thus ;

7. In the other Example, is cot^rehend-
cd the double Rule of Three inverle ■ which

If two Barrels of Beer, ferve 12 Men 14
dayes; How many dayes will 4Barrels
ferve 24 Men ?

. If you Rank the terms, according to the
fornxer Precept, they will ftand thus :

like Denomination, viz,, of Barrels j the
fame Extent applied the contrary way from
^4, fhall reach to 7, for a fourth Propor^
tional.,

Again, Extend the CompafTes froni 12 to
7, the fourth laft found j the fame Extent
fliall reach the contrary ^v«y, from 24 tg
14, the number of dayes required.

9. But if you would reduce this, to bequot;
wrought by two firtgle diredl: Rules j you
«auft confider the Preeept Rule, the jth and
•lt;5th, of the Eleventh Seftion • and the terms
of Suppofition and Demand • and the in-
creafing, or decreafing of the fourth terra,
5vhich is required.

Again, If 1 Barrels laft 24 Men 7 dayes,
'4 Barrels will laft them 14 dayes; the An*
fwer to the Queftion required.

Here by the lt;Sth Rule, where the Number
fought is to be lefs; As 24, the greater term
of Men, is to 12 the lefs of che fame Deuo-
nomination; So is 14 to 7, the fourth.
jigai».

As 2 the lefTer terra, is to 4 the greater of I
the fame Denomination ; fo is 7 to 14, the ,
Anfwer required, by the jth Rule of the
iithSedfion.

As 2 to 7, fo is 4 to 14; that is, the Ex-
tent from 2 to 7, fhall reach the fame way
from 4 to 14, the term required.

To work this by the Trianguler Qua-
driaiit, or Sedor, the general Rule in this
Sedfion, Ruled and 7, giveth fuflicientdi-
rc6fion.

quot;ID. The Rule of Three, compounded of
' five Numbers, is no other than the double
Rule of Three ; and is,or for the moft part,
may be wrought by one Operation, having
prepared the Numbers by Multiplication,
for that purpofe : Which two Multiplicati-
ons by the Line of Numbers, though they ,

are prefently wroughc, yet the two Rules
of Three are done as foon j fo that the
Compoupd Rule, ii here qf no advantage
at all, therefore ! might wave it ; yetBe-
caufe the only difficulty lies in the order-
hig the (^eftioii, I fhall propound it, for
the addition fâké of another ExampleyVihich
JSthls;

if the Carriage of 2 hundred weighty
30 niiles, eoft. «f. s. What will the Carriage
of y hundred weight colt for looriiilesK
The Numbers Ranked, according to the
firfl Precept, will ftand thus; as followeth.

ri. Then for thé Operation, multiply
the two firft Numbers one by the other •
Js 2 times 30 is 60, which is the firll term ;
and let the middle Numbet be the fecond
term Î and the Produft of the two laft
C multiply^ed tojiether ) for, the third term :
1 hen the Numbers being fo prepared, Ar,
As 60, thé Produa of the two firlt
Numbers, is to 4, the middle Num-
gt; ber ;

By the tine o^Numbers, the Extent .frQoi
60 to 4, gt;vili reach the fairte way from 590
to 331, or, tliirty three fliilHngs and four
pencc/the price of 5 hundred .weight,- ca|-
ried.iqq miles..nbsp;. r

Noté, This Rule ferves when it is pei-for- |
hied by the Compound Rule gf Three di-
red.

12. But if the inverfe, or backer Ru\e of
Thi;€^,be u;f£d in theworjc» then Qpcraw
thus:nbsp;, f ■

A Merchant liath received io/. lO/. ,
for the Intereft of gt; certain fum of Money
for fixManths.; and he rcceiïid after the
rate of 61. for . the ufe of an hundred
pound in a year ; the Queftion is, how
much Money Was Principal to io /.—rip s.
for 6 Months ?

Fu-ft, 1 range the Numbeis, according to
the order firft propounded,'; in the 4tH:I\4l«
of thg 12th Section, as folloj^eth;

^ Then I obftrvc diligently, whether the
inverfe Proportion be in the firft or fecond
Operation, or Line, as thus in this Quefti-
on it is in the lower Line j therefore after
jlie Crofs Ivlultiplicadón, it is to be wroughc
by the fingle inverfed Ruk of Three 5 but
gt;vhen the Inverfe Proportion is in the uppeic
••ine, it is wrought by the fiugle Rule di-»

Then I multiply the double terms a-
^ofs i that is, the loweft on the right-hand
by the uppermoft on the left ^ a«d the up-
gt;ermoft on the right, by the loweft on the
; As thus :
f by 6, which makes'3lt;f, tobefet.under
o , and 12 by 10-y, or 10/. ( which is
and 10 s,) and fet it under 10 : then
lay by the inverfed Rule, thus ; *nbsp;,quot; ƒ« boih

As 125 t0 3(J, fojs ioo to 550, t\iQ thefe the
Anfwer demanded ; So that lyo/.
gal will^Md ro - .0 JJ .'Months , f/^S
J/ti the Extent from i2lt;J to 3 (S, fhall reach lower ivii
jfe contrary .way from, 100 to jyOjthe
»quot;incipal Money required,nbsp;. »

13. If 3/. be thelntereft of lool. in 6
Months, to how much Money fhall 10 /.
10 s. be intereft in 6 Months ? uork, thta ;
. The Extent of the Cdmpafles from 3 to
100, lhall reach the fame way from 10/.
HO/, to 350, the Principal Money an-
fwering to 101. — lOs. the Anfwer re-
quired.

As — 3 to = io, ibunted 100;
So is-ID at the firft i ne.xt the Cen-
ter, to = 3 Jo.

J. Rules of Plural Proportion are thofc,
by which thole Queftions arc refolved,
which require inore Golden Rules than one ;
aud yet cannot be Refolved by the double
( Rule of Three, .or ) Golden Rule, which
^vjis laft mentioned.

Of thefe Rules there be divers kinds
aad varieties, according to the nature of the
^^eftion propounded ; for here the terms
given, are fometimes four, five, or fix, or
more; and the terms required alfo more
than one, two, or three.

3.nbsp;The Rule of Feilowfhip, is to difco-
^er the Gain or L06 of every Partner in the
Stock, by their feveral Stocks, and the
whole gain or lofs of the whole Stock.

Alfo obferve, That the Rule of Fcllow-
fliip may be either ftnglt or ionhle ; of both
vvhich in order.

4.nbsp;The fingle Rule of Fellowlliip is, when
the Stocks propounded are fingle Numbers,

, ABC and D, reprefenting the Names of
'4 Men, pur into one common'Stock 1001.
to trade withal: A puts in 10/. B puts in
20/ C 30/. and D 40/; and with this
Stock, in a certain time, they gained 10/.
or 200 s ; Now the Queftion is,what ought
each man to have of the 200 s. that may
proportionable to his particular Stock ?
y. The Rule of Operation is, firft, by
Addition find the total of all the particular
^tocks, for the iirft term ; the whole gain
V lofs ) , for the fecond term ; and each,
L 3nbsp;paiticuquot;

^articular Stock for a third term ; and re-
lating the Rule of Three as often as there
3e particular Stocks in the Queftion, you
ftiall bring forth , or find out, as many
fourths for the particular gains ( or lofte^)
of each particular Man required.

For the Extent from 100 to 200, fhall
yeach from 10 to 20, and from 20 to 40,
and from 30 to 60, and from 40 to 80^
the particular gains due to A B C D, which
was required.

6. For proof whereof, if you add 20,
40, (Jo, and 80 together, they make up
300 ƒ, or 10 / J the w^lc gain of the

7. The iouble Rule of Fellowfhip \s;
when the Stock? propounded are-double
Numbers.

A B and C, holds a Field in csmmon,
for which thef pay 50/. a year; and iu
this Field, A had ay Oxen went 30 dayes;
B had ty Oxen there 40 dayes ; andC
had 20 Oxen went there 40 dayes : What
ought each man to pay for his part of the
Rent, wïi. yo /? Here you fee the Stocky
propounded are double Numbers, as of
Oxen, and their dayes, or time of feeding ;
3S2y amp; 30,: ij amp;40, 20 amp; 40, being
double Numbers.

Multiply.the double Numbers, fevcrally
pne by thé Other, one after another, and
take the fum óf their feveral Produ6ls, fot
the firft term; and the whole gain or lofs,
for the fecond term; and the particular Pror
duifts of every double Number, for the
third term, one after another: This done,
repeating the Rule of Three, as often a$
there be double Numbers, the 4th term
produced from thofc Operations, fliall be
Anfwers to the Queftions required, viz^,
% quantity of each liians gain or lofs.

9* To wojk bytheLineofîS[umbers,the
Extent of the Compaffes from, i to iy, fhalj
ï,each the fame way from 30 to 7 jo, the firft
Produél: of a A's double Number,or Stock.

And^as I to :ço, fo is 40 to 800, the
Produdlof C's double Number, aiid Stock
Which three Produis added , make
21 JO, the firft term ; and f o rs the fécond
term j and 7yo, 600, 2nd 8oo,,the three
Produas feverally, the third term. Then,
^ The Extent from zrjo to jo, lhall reach
the fame way from 75-0 to 17-4.5', or 17 /.
9 f. And from |oo tcgt;:i^-pej or i j l.-jçf^
'nbsp;' ' ^ ■ Aitd'

And from 800 to 18-60, or 18 I. - 12 squot;,
the feverai Anfwers required ; which being
added together, make up jo/. the' whok
Rent to be paid among them.

There be other Rules of Arithmetick, as
the Rule called t^lltgation^ Medial, and
Alteruate, and thé Rule of Poßtio» or Falfe-
fgt;9od j 'iii tjie working of which,arc fo many
Cautions in ordering the Numbers, before
you come to the proportional work, that it
Would make the Book more bulky than ufe-
ful; therefore I fliall wave it, and refer you
to the particular Books of Arithmetick, as
that of Mr. iJ^ear*/, D^f, znd Meilis-, or
that of Mr. fvingate Natural and Artificial,
having in it plenty of Examples i and o-
thers alfo, as Johnfous, J aggers, ox Moora
Arithmetick, any of which exceed the
bounds I intend for this whole difcourfe;
I fhall therefore pafs on to the Rules of
Praftice, in feveral kinds, as meafuring Su-
perfecies, andSohds, and Rules of double
and. treble Proportion and Queftions of
Intereft, which are tedious by the Pen,
Without the help of particular Tables, and
very eafie by the Line of Numbers, as will
fiilly appear in the next Chapters.nbsp;'

CHAP. VII,
quot;The ufi of the Line of Numbers in
meafuring of any k^md of Super-
ficial Meajw?.

THc Meafure chat is commonly ufed in
this Work, is a Foot-Rule, divided
into lOO parts; or elfe into 12 inches, and
thofe iirches into halves, and quarters, or
Spares; or inches and loparts: but in
regard that the Numbers do n^aft fitly a^
gree to the IOC parts of a Foot, it will be
convenient here to fhew how to reduce
then, or any other Frailion, from izs.to
■JO X. or any other whatfoever, from one
Fraftion to the other, which by the Line of
umbers is quickly done; as thusr, from i z*
to 10'.

Extend the Compafles from one Denomi-
nator to the other , the fame Extent fhall
reachthe fame way from, one Niiraerator to.
the other.

Which two Lines of Inches f and Foot-
Meafure,areufuallY fet together on Rules,
for the ready way of Redult;Shon 6y Occular
iufpecficn, only in this manner, m in tht Fi-
ciirei And the like may be for any thing
Whatfoever, as Mr. Edmoni Winigate hath
largely fhewed in his Arithmetick. Which
Line being next to the Line of Numbers. on
your Rule, will be very plain and ready in
the ufe of the Line of Numbers fot feCt and
inches, or fliillings and pence; and the fame
Rule of Redudtion, ferves for all manner of
Fraaions : For as the Denoitiinator of
one FradHon is to the Denominator of the
other, ( which in the Decimal work is al-
wayes a unite, with one, two, or more
Cyphers) fo 15 the Numerator of one, to
the Numerator of the other.

And Nott, That the operation of Decimal
lumbers, and their Fraftions, is no other
than whole Numbers, except only the cut-
^^quot;g off fo many Figures as there is Frafti-
'nbsp;ons

pns m the Multiplicator and Multiplicand,
after any Multiplication as in the follow-
ing EAraiwp/?/ will appear.

The breadth of an Oblong Superficies gi-^
ven in Foot (^teafure, to find how muck
in length ma^es one Foot.

The Extent of the Compaffes from the
brkdth to I, fhall reach the fame way froig
J, to the length required.

As the breadth in inches to 12, fois 12
to the length of a Foot in inches, and 10
parts.

'At 8 inches broad, you muft have 18
, inches to make a Foot; for the Extent from
Sto 12, fhall reach the fame way from-x2
10 18. ■

f As —t to =7, fo is =i to -1-43
the length in Foot- meafure j
By JAs —fo is =12 to
Inches.^ — 18 j Or elfe,

Hxvifg t^he hreaiib of m Oblong Super'
fides given in Foot-weafure, tofind hojt
/AHchfsinaFoot-iotg.

This isfoon wrought ; for in every Foot
long there is juft as much as the btearfth is,
either in .Foot-meafure or inches j for i
piece of Boavd half a Foot broad, and a
Poot long, isjuft half« Foot.

- Problem III. .
Having the length and breadth in Foot-
Melt;^{iirey to find the Content in Feet,

The Extent from i to tht length, lhall
'each the fame way from the length to the
Content in Feet.

As t to I foot 50, thc-breadcli; fo is i i
itoo:, Iaparts,, thé ie^th, to 16 foot and
éf patts, the Content irequired.

, As 12 to the brcadth in inches, fo is the
kngthin feèt, to the Content in fèet requi-
ïèd.

The Extent from' 12 to p, fhall reach the
iame.waj from ix to S fbot, 3 inches^or A.

quot;Büt Noté, That in workirig this,and ma^'
Jiy fuch-hke, it will be convenient to doublé
your Scale in account, calhng io at the
end 20, and every fingle figure as mucli
more, as to call i 2, and 2 4, amp;c,

Thus you may many times vary the man-
tier of work to\ get ^he Aniwer latterally,
and as large as may be on the Scale a^Lines,
by dpublmgor halfing the Numbers, or ia-
king the whole Number of quarters, or
[wg a l?fs or a bigger Scale, as bath been
fainted, and fliall be more in places coh^
venieot, in the .following Difcourfe, to at-
tain exaanefs and cafe, as much 35may be, at
tune and practice will demonftrate to the
^Villing Pra^itioner in thefe Operations.

Pfoblem iv.
Having the length and breadth £ivih 1»
Inches to find the Content in SitferfidaU

. As X inch, tcf disWadtH in inches • fo
IS the length in inches, to the Content in Su-
jperfidal inches.

Example^ 20 ihches broad, and ji
inches long.
The Extent of the CompalTes froni i on
the Line of Numbers to 20, lb all reach the
fame way from 3Ö t0 72j the true Number
of Superficial Square inchcs in that Oblong,

For Note^ The reafon that the Latteral
72 and 3 are from the fame Scale in ac-
count; and the Parallel i and 20 counted
Decimally, are from the fame Scale alfo, or
elfe according to the Proportion by thé Line
ofNunibersj

As—I torrao, Sois —to =72.
Mere alfo is the fame advance Decimally
from I to 20, as before.

Having the length and hreaitbgivtn i»
Inches, to find the Content in Feet Sttfcr^
ficial f

As 144 to thé breadth in inches, fo is tha
length in inches, to the Content in Feet Su-
perficial. •

For the Extent on the Line of Numbers
from 144, the number of inches in one foot,
to 40 the breadth, fliall reach from 60
Wlches the length, to x6 foot ( and 26 inches.
To count fo many inches on the Line, ob-
ferve with me this way of Reduétion, the
foot and 4 is very plainly feen. Ani
Nott^ That there is 10 cuts in this place be-
tween 16 and 17 i and 10 times 14 is 140,
which is near 144, the inches in a foot; fo
the Point of the Compafles ftaying at near 2
lodis beyojid the half-foot, t count almoft
twice 14, which is 26 inches for die Fraction
above IÖ foot and

As -— 144 ( found between i and z
'^^ar the Center) is to = 40 (at the figure 4)
Mnbsp;So

[ 1 ^ ,
So is half — 60, or the meafure from the
Center to to = S-jf, which is the half
6f i6 foot and 26 inches : For if you had
taken all 60, it would have exceeded the
whole Parallel Radius, where the Anfwer
would have been right itfi and better, but
taking the half, it gives the half alfo.

• Note, That irt both thefe two laft worb
Jngs, the 144 is at 72,nbsp;is the half of

144; to make the work the larger. By
thefe the excellen^ of the Line of Numbers^
over the Line of Lines, is evident in thefe
kind of Proportions. And for difcovering
the Reafon of thefe Proportions, .read the
beginning of the (5th Chapter, Seftion 3d,

the length and breadth of an Oblong Süi
perfictts hing giveny to find the fide of »
Square eftal to it, bphe LineéfNums.
hers,

_ Divide the fpace between the length and
breadth into t\\ o equal parts, and the
middle Point ihall be thehde of the Square
equal to the Oblong given, in quantity.

if a long Square, or Oblong, be 18 foot
one way, and 12 foot the other way j the
middle Point between i8,and iz is 14 and
hftre-^ for 18 multiplied by 12, makes
and 14-7 multiplied by 14-7, isnear
2llt;S alfo.

the Diameter, or Cïrcsimferencé
of a Urcle,iofind the Circ(tmfertnce,of
Diameter or Squares equal, or Infcri-
ted, atd Content.

the Ptrlfera, or Circumfcreace, is 31 41;
the fide of the Square equal tó the Circle, is
8-862, the fide of thé Square infcribed is
7-071 , and the Superncial Content is
78-^4 ; fo that any one of thefe being gi-
ven, you may find out any of-thereftby
theLineof Numbers.

As 10 to 30, fo is 78-f 4 to the Square of
theArea-of that Circle, whofe Dia-
meter is 3 0. Or, To the Diameter
turning the Compaffes twice,

Diameter ^lo oo
Circumference Vl i
Square equal 8
Square within / 7 071'
AreaorContentL78 54

Note, If the Circumference be firft given, then fay. As 31 42 is to
the Number 10 for the Diameter, or 8-862 for the Square equal; or to
7-071 for the infcribed Square; fois the given Circumference to the rell.

But to find the Area fay. As the fixed Diameter 10, is to the given
Diameter 30; fois the fixed Area for 10, vi«,» /S'H» ^^ 707 — by
turning the Compafliss txvo times.

Or if the CWcumfertnce be given, ani I muU find the Area.
As 3 i-42,the fixed Circumference,is to 94-2«? the given Circumference;fo
is 7 8-54, the fixed Area,anfwering the fixed Circumference 7C'7,the Area

Thus by having five Centers at the five
fixed Numbers; or four Centers anfwering
to the four fixed Numbers; for a Circle
•whofe Diameter is lo, having any one of
thofe y given, you may find any of the o-
ther required.

Thus you have eight Problems couched
in one; therefore be the more diligent to
underftand it. To work thefe by the Line
of Lines, obferve the former direftions,
which for brevity fake I now omic.

Divide the diftance on the Line of Num-
bers, between the fixed Concent, or the
Point 78-^4, and the given Content into
two equal parts; that diftance laid the fame
way, from the fixed Diameter, fhall reach
to the required D ameter.

Divide tlie diftance between the fixed and
the given Contents or Area's into two equal
parts, that diftance-laid from the fixed
Circumference, fliall reach to the required
Circumfereoce.

A Circle, whofe Area is 707, fliall be
54-26 about. For the half diftance between
78-^4, and 707-00, fliall reach from j 1-42
the fixed Circumference, to 94-26, theen-
quired Circuinference.

And from 8-862, ths fixed Square, e-
-lt;3«al to 26-j8 the inquired Square E-
qual.

And from ,7-071, the fixed Square in-
fcribed, to2i_2i,' the inquired Square In-
-fcribed.

For the Square, the long Square and
Circle, hatlibeenfpokento juft before; All
other Figurts arc to be reduced to a Square,
^ to a long Squa?^^ and then meafured by
^M^tiplication^ as bsfcre.

Multiply the Diameter by it felf; and
then that Produft by 11: then laftly, di-
vide this laft Produd by 141, and the pa-
tient fliall be the Area, or Content, of the
Circle required.
Chcle. For a Cirek (othcrwife) thusMultiply
half the Diameter, by half the Gircumfe-
renccj and the Produdl fhall be the Content
required.nbsp;.

half- For a Hdf CltcleMultiply half the
Circle. Diameter of the whole Circle, and a quar-
ter of the whole Circumference togedier,
and the Produft (hall be the Content.
(Xuadrant, Por a Quadrant, or a quarter of a Circle;
erthe ' Multiply half the Arch, by the half Dia-
qtimer. „lecgr, or Radius of the Circle, and the Pro-

du(ft fhall be the Superficial Content.
Lejfer- The like Rule holds for any ieffer fortiott
T^'quot;- of a Circle, whofe Point goeth to the Cen-
ter, vix.. to take half the Arch, and the
whole Radius, and multiply them together,
and che Product fliall be the Content.

Square half the Chord, and divide the
Produ6t by the Sine, then add the Quotient
and Sine together; the fum is the Dinneter^

Chord is 24 J12. Squared is 144; divided
by 8, gives 18 in the Quotient, which ad-
ded to 8, makes 26 for the Diameter.

For any other Segmtnt of a Circle, find Segments;
the true femi-Diamcter, and meafure it a»
before; then take out the Triangle, and
the remainder is the true Content of the
Segment. SetCbdptr xj, iz.

Or elfe thus, hy tht Line tf Stgmnts ■
jaynti to a Ltnt of Numhtrs, in
this manner.

To the Segment given, find the true Dia-
meter, by Chap.Wh n, 12. Then having
the Diameter, find out the Area, or Content
of the Circle, by any of the former Rules,
then the Proportion or Analogy is thus ;

As the whole Diameer is to 100 on the
Segments; So is the Altitude of the Seg-
ment,whofe Area is required to a 4quot;quot; Num-
ber on the Line of Segments, which you
muft keep.nbsp;®

As I, to the whole Content of the whole
Circle given ; So is the 4''' Number, kept.
Counted on the Numbers, to the Area of the
Segment required.

If the Line of Segments is not on your
Hule, then this Tabic annexed, will fupply
the defea, reafonably well thus:

70	I
112	Z
147	3
17»,	4
206	y
	6
258	7
282	8
307	9
	I
350	^
371	2
sr-	3
	4
	J
451	6
469	7
437.	8
507	9
5^4	2
558	2
592	4
626	6
6S7	8
	J
718	2
749	4
779	6
S08	S
0836	J
0864	2
o3gt	4
0918	6
0948	8
0970	_5
1000	2
1017	4
io;i	6
iP77	8
II02	6

The1gt;lamrterif the Circle, attfwerir£
to the Segment give», being found out, Say,
As the whole Diameter to loo; fo is
the Ahitudeof the Segment to a4thNunH
her, which fought in the Table of Segments,
or the neareft to it^ gives in the parts the
Number to be keptv Then again,
^ As the whole Content of the Circle fixed,
loo, is to the whole Content of the
»^ew Circle; fo is the Number kept, being
tne Content, or Area, of the fixed Segment,
the Area of the Segment required*

Let the Segment of a Circle, whofe whole
Area is 314-2, and whofe Diameter is 20,
®«d let the Altitude of the Segment bey,
4th part of the w hole Diameter.

to 10000:
So is the Altitude of the Segment 5, to
2 JOG, the 4th ; which fought in the Table
f u^w'^u' Parts, s.vesi9-yo for

Then agai»^
As I, to 314-2, the whole Area J So is
jY-yo, to lt;Ji-30, the Ai-ea, or Content of

TrhngUtl For all manner of TriangliSyXOakx^l^ the
longeft fide (being properly called the baft)
- by half the perpendiculer, and the Produ*^:
fhall be the Content of the Triangle ; or a«
3 to the bafe, fo is the perpendiculer to the

Ahmbm For z RbombtUy being a Figure like«
Quarry of Glafs, containing 4 equal fides»
and two pan: of equal Angles: And afii
Figure having his oppofite-fides Parallel ofl^
to another j then the length of one fide an''
the neareft diftance between the other tW
oppofite-fides multplied together, fhall W
the true Area required.
For all other four-iided-figures,call'd Tquot;rd'
being irregular Figures; draw a LiP^'
from one corner to the other, which rpakc^l
it two Triangles; then multiply that Lineal
being the whole bafe of both the Triangle«»
by the-half fum of both the Perpendicylet'quot;
and the Product fhall be the Content f^'
quired.

For all Regular P9%fl«.f, or Figures,wH'
equal fides, the meafure from the Center t
the middle of one fide, and the half fui^'
of the meafure of all the fides multipUe'Ji
together, ftiall be the true Area, or Con^^''
thereof.

All othet Figures whatfoever, of how
many fides foever they be, may be reduced
to Triangles,or toTrabeziaes, and meafured
as before ; which kind of Figure j Surveyors
and Builders oftentimes meet withal, in
their Operations,

Tcr the m*[tiring of ok Oval, tbe heft my Ovals,
•is to reiftceit ta d Circle thw J

Divide th^ diftance on the Line of Num-
bers, between the length and the breadth of
the Oval into two equal parts; and tbe
middle Point where the Compafs ftayeth
on, fhall be the Diameter of a Cirde equal
in Area to the Oval given.

quot;Suppofe an Oval be lofoot long(tranf-
verfe) and 8 foot broad (conjugate) ; the
mean proportion , between lo and 8, is
8-9 J : I fay, that a Circle whofe Diameter
is 8-9 f, is equal to an Oval of 8 broad, and
10 long; And how to meafure the Circle,is
ihewed before.

if the Content be lOO, then the fides of
. JhefeRegul ar Figures are as followeth, and

Ptrfeniic$Uer-Triangley 13^ 123.'
TriaKgaltr-fiie, 15. 2.
Square, its Side, 16,
Pant agon offive Sides, 7.
Hexagon of fix,nbsp;6.

The half diftance on the Numbers be-
tween 100 and 200, fhall reach from 15-2
to 21-5, the fide required.

A^ from 13.123 the feed perpendiculer
for a Tmngle,wh6fe Area is 100, to i8-lt;J,

The Extent from the fixed-fide, to the gi-
VCii-fide, fliall reach at two turnings, froni
the fixed Area, to the Area required.

The Extent from iy-2, to 21-5-, fliall
*each, at twice repeating, from lOo to 200.

To m^e Oft Oval equal to a Cmlu havittg
the Diameter of the Circle, aud thé
length or breadth of tht Oval given.

one Point of the Compafles in the
diameter of the Circle found out on the
Ijioe of Numbers, and the other Point to
the Ovals length ; then turn that diftance
^he contrary way from the fame Diameter-
point, and it fhall reach to the breadth of
the Oval required.

I would have an Oval to contain as much as ■
the Circle, and be 12 foot long j the Qpe-

Set one Point in 1 o, and the other in 12,
^at Extent turned the other Way from 10,
«iall reach to 8-54, the breadth of the Oval
'«quired. ~

Jf you'pleafe to alter the breadth or
length, yoii lhall foori find the length or
breadth accordingly.

T} ^ork. this hy the Line of Lints, you mufi
work^by the DirtSiionsin the ytb StUiort
tf the 6th Chapter, as thus;

Firft, To find the Content of the Oval,
joyn the length and breadth in one fum, to
get the fura, the half fum, and difference,
and half difference; then open the Seftor,
( or lay the Thred on 6o|o) to a Right
Angle ; Then count half the diflerencefrom
the Center downwards, and note the place ;
then take half the fum between your Com-
pafles, and fetting orte Point in the lialf-
difFerence, and extending the other to the
other Leg, ( or perpendiculer Line ) and ic
lhall fliew a Point, whofe diftance from the
Center is the mean proportional required ;
which is the Diameter of a Circle, equal in
Area, to the Oval, or Ellpjis given to be
meafured j as before is fhewed.

Take the guefled half.fum of the length
and breadth of the Oval, and fetting one
Point in the Diameter of the Circle- and
on the other Leg, fet at a Right Angle, the
other Point fhall fliew half the difference,

fcetweén the length and breadth of the O-
''^al ; then if the mean proportional between
them be equal to the Diameter, you have
^vrought right ; if not, then refolvingupon
length or breadth of the Oval, take more
lefs, for the breadth or length according-
ly : Herein alfo is feen the excellency of the
Line of Numbers, in many operations.

Tie leafth a«d hreetitb of any' Oblong Su2
ftrficies given in F get, to Çnd the Con-
tent in Tards.

The Extent of the Compaffes from p to
13 Ï the length, fl^all reach the fame way
»romzl the breadth, to 11 yards and a
^'larter, the Content.

■ ^ Nate^ That if you meafure by feet and
quot;Undred parts, you fhall find this way ex-
^^eding ready ; the Anfwer being given in
^^•^ds, and hundred parts of a yard. Biit if
^^^ have a yard divided into a 100 parts,
■ Nnbsp;to

As I to the length or breadth, fo is the
breadth or length to the Content in yards.

The Extent of the Compafles on the Line
of Numbers, from I to 3-72, fhall reach
the fame way from y-8 2, to 21 yards 6$
parts, the Content in fquare yards,and 100
parts.

Jht le»frth uni ktadth of an) Wall, htl»^
^ivtn in feet and i go partj, to find how
thanj Rffds of H^aUiKa thertfhaUbt at a
Brick. lt;fni aH half thick.

Pirft you muff Note, That foot and
7 quarter, makes one Rod, ( or fo many fccc
'^'naRod).nbsp;^

Wndly, Thar fet the Walls be half a
of'ck, one Brick, t-.vo Bricks, two and a
jalf, or three Bricks thick; it is to be re-

^ Thirdly Note, That this reducing to a
quot;CK and half thick, may be at the mea-
t'lng, or after the cafting-up, as you pleafe,
3ppequot;nbsp;following will plainly

A In-ont, or fide-Wall of a Houfe is to
rneafured, wherein the Cellet-ftory Wall
firft and a half thick j The Shop and
Chamher-ftory is two bricks thicks;
le other Scones i Brick and a half thick •
the Gable-ends i Brick thick.

1.nbsp;The Cellar-ftory is lo foot high, but
being 2 bricks and a half thick, i make it
16 foot 8 inches high, by adding two
thirds of 10 foot, to the 10 foot high?
which is 6 foot 8 inches, in all 16 foot
8 inches.

2.nbsp;The other two Stories, are fuppofed
22 foot i but in regard they are two bricks
thick, I add one third part of 22 foot,which
is 7 foot 4 inches, to 22 ; and it makes 29
foot and 4 inches, the height of the Shop
and next Story above.

'The other two Stories being a brick
and half thick, need no aheration, which
fuppofe may be 19 foot.

4. The Gable-end, or Garret-ftory, i»
any be, being but one brick thick ; you muft
take away one third part to bring it to ^
brick and a half. Alfo if it be a Gable-endgt;
Note, it is a Triangle, and you muft mea'
fure but half the height, and the whol«
breadth, to find the Content j whi?hhei^
may bey foot.

f. Add all thefe fums of feet high toge-
^Her, and they make 70 i then meafure the
t»rcadch, which is common to every Room,
'he out-fide going upright, which in a
double Houfe may be 3 6 or 40 foot.

As 272 i ( the feet in one Rod ) is to 40
foot, the breadth of the Houfe j fo is 70
foot, the whole height of every feveral
Story, (reduced) to 10 Rod and 29 parts ;
which 29 parts you may call a quarter of a
Hod, and 10 foot and a half.

Tht reafon vehereof is apparent thut:
As 100 is to 272 -; fo is 29 to near 79 ;
of which 79^68 is a quarter of a Rod, or
of 100 is a quarter likcwife, which by
Line of Numbers is apparently feen j
P^quot;^*quot; ^^nbsp;72 of a

Rod, there is 4 hundred parts more in 19 '
Then thm ; the double of 4 is 8, or, twic^
4 is 8, aijd four times three quarters is thre^
foot more J of \\;hich you muft abate fome'
what (becaufe 72 ^ is not 7$, which is juft
three quarters) and all put together, mak«
ten rod, one quarter, ten foot and a half'
fcr if you ftjall divide the Produdi: of 40,
niuhipHed by 70, which is 2800 by 272 j»
you fhall find the Quotient to be 10 rod,
78 I, which is, as before, 10 rod, i quartef
and 10 foot and a half.

But note alfo by the way, That when
you come to take out the deduftions for the
doors and w^indows, if any happen in ? j
Wall of two Bricfcs and a half, or in twquot; 1
Bricks; you muft add two thirds, or onf
third more to the Ipngth or bredth dne way; i
and then cafting them up feverally, whe»'
they be of feveral lengths or breadths, yoii
lhall do no wrong to the Work-mafter not
Work-man : For true Anthmitick^ an4
Geometry will he for no man, or ufe an/
kind of partiality.

This I conceive is as near a way, as aft/
fuch bufinefs can be performed. But if
will meafure every Story feverally, taki^f?
account of each Story feverally in their thicl^
rifles; then, after it is eaft up, the befgt;
way, by the Rule^ to redijcc itj is thm j .

As 3 half bricks, for a brick and a half,is
to any other number of half bricks thick,
over or under 3 i So is the Content at that
rate accordingly, to his Concent, at a brick
and a half required.

ded to latfp, makes 21 if ; For the Extent
On the Line of Numbers, from 3 to f, fhall
«■each the fame w^ay from 1269 to2ij
the Number required to be found out,
Oihirtvife tbw.

To bring any kind of thicknefs, to one
brick and a half thick, at one operation, by
the Line of Numbers.

For this purpofe, you muft ufe feveral
Points, as fo many gage Points, as in the
fliort Table following doth appear^

tet a Wall be 30 foot long, and 10 foot
liighi andlft it be fuppofed of any of thefe
thicknefles following , from half a bricks
length, to three bricks length in thicknefs}
then thus in order, increafing, See,

And fo for any other thicknefs, as far as
you pleafe; which Points are found thus ;

The Extent, from the number of bricks,
any Wall is thick to 15 (or i and A ) fhall
reach the fame way from 10, on, to the
Gage-Point required for that Wall,or Walls
of that thicknefs.

Laftly, having the Number of Feet iii
the whole work, to find how many Rods
there is. Say,

The Extent of the Compafles from 1723,
to I, fliall reach the fame w±y,from the
Number of Feet, to the Number of Rods,
and hundred Parts, or Rods, and Quarters,
and Feet. j as by the lt;$th, laft mentioned.

The Extent from 272i, to i, . fiialJ
reach the fame way, from ^269, to ip
^od, and 36 parts of a 100 j or, ipRod
* quarter,-and 29 foot, and a quarter of a
foot. Xhe ip Rod, and a quarter, is feea

plainly on the Rule; and ay being a quar-
ter,is 11 pares more ; for wch 11 parts
more, I fay, 2 times 11 is 22 foot, and 11,
3 quarters of a foot is near 8 foot, which
put together, makes 29 foot, as before : Or,
as the Compaffes fiandj turn them the con-
trary way, from the Decimal parts, above
the even quarter, and it fha}I reach to the
odd feet above tbe quarter required.

TheExtentfrom 272i, to 100; or i,
fliall reach the contrary way from 10to
29 foot, the feet above i of a Rod.

8. Obferve, That the Tyling, the Roof,
the Floors, and Partitions, are meafured by
the Square 5 which is 10 foot Square every
way, or 10Ö foot in Area. The Chimneys
are ufually done by a certain rate for 9
Chimney; or if to be meafured, thus arq
the height and bïeadths taken, gcc.

Jf a Chimney ftand lingly and alone, not
leaning againft, or in a Wall, the ufual way
is to girt it about j and if the Jauntes are
but a brick thick, and wrought upright over
the Mantle-tree to the Floor- then iVay,girt
ic about for a length, and the height of the
Story is the breadth, at a brick thick,becaufc
of the gathering together, to malte room for
the next Hearth above in th? next Story.

But if the Chimney-back 'be a Party^
Wall, the Wall being hrft meafuKd, then
the breft and the depth of the two Jautnes
is one fide, and the height of the Story ano«»
ther fide, to be multiplied together, at a
bricked a half thick, or a brick thick, ac-
cording as the Jauwies be, and nothing to be
abated for the want between the Hearth and
the Mantle-tree, becaufe of the Withs^nd
thickning for the next Hearth.

Girt with a Line, round about the Teflft
place ofthem, for one fide i and the height
for the other fide, at a brick thick, in confi-
deration of the Withs, Pargitting, and
Scaftblds.

In meafuring of Ceiling a foot broad,and
the length of the Vallies is alwayes to be al-
lowed,more than the whole Roof; Alfo the

Wbeii Rafters have their ufual pitch,
which IS; when the breadth of the Houfe is
12 foot, the Rafter is 9 foot long, which
3 quarters of the Floors breadth, be it more,
or lefs i then, 1 fay, that the Content of
Pne Floor, and half fo much, is the Area o£
ihe whole Roof in Squares; to which is to

For the Extent, from lt;îlt;î(îlt;î, to 20, fhall
reach the fame way from 30.10 900.

Alfo in meafuring of the Roof, as to Car-
penters work, by the Square, there is to be
allowance for thofe Rafters in the Dormers,
and Gable-ends, on which no Tiles are laid,
as over-work for a particular ufe and con-
irenience, more than need be in a bare Co-
vering, or Roof.

■ Alfo in meafuring of Plafterers work in
Partitions and Walls -, the Timbers and
Charters, are not to be deduced out of the
rendring for Work only, except when the
Workman finds the Work and Stuff alfo,
then fubftraft a lt;îth part for the (garters in
the rendring Work : But in Ceilings, the

Summers uliidi are feen,are alwayes abated}
and Doors and Windows alfo, unlefs by a
due coaifiderate (or-an unconfiderate) bar-
gain of ruigt;fiing meafure.

Thus you have a brief account of the u-
fual order,quot;ufed among Workmen, in taking
the Dimentions of a Houfe^, viz,. Brick-
Work by the Rod j Tileing and Carpenters-
work by the Square ; Chim_neys ufually by
the Fire ; And Plaflerers and Painters-
Work by the Yard j Glafiers, by the Foot.

There are many other things to be taken
notice of in the Carpenters Bill, as Lintels,
Mantle-trees, and Taflels; Luthern Lights,
and other Lights, both Architrave and PlaiiV
Lights, Sky-lights, or Cubiloes, Modillean
Cornifli, and guttering Penthoufe Cormili,
Timber-Front-StoryCellar-doors, and
Door-cafes; the Plank and Curb at the
Cellar-flairs, Dogleg-ftairs , and Open-
New el-flairs, Canted-llairs, counted either
by the fiep or pair ; together with the half
Spaces on the Corners of the open Newel-
ftairs, theRaylesandBallafters, fmall and
great Cornilh, Outfide-work and Partiti-
ons, Ceihng Joyffs, and the Afhlering,
Boarded Partitions, and Chequer-work ;
back-Doors, and Door-cafes; Window-
boards, and Wall-timber; Planks in the
foundation, Paking, Penthoufe-floors, and

j^nthou{e-rc»f j furring the Platfofm,
Centerings for the Chimney, Trimmers^
Gtrdets-ends, Ends of Breft-fummew, and
Plate 5 and more the Hke, which will come
in Adcompc to be rem^mbrtd and fee down
«ccordirtg as the Building is.

Alfo, with due allowaiice into the Wair
that way the ends of the Joyfts are enti ed or'
laid in the Wall, as thus t,

If it be Framing Work is only mea-
fured, then 9 Inches ought to be allowed
into each VVall, that way the Joyfts ends
are laid J becaufe every Joyft, if welllaid,
fliould have 9 inches, at leaft, hold on thé
Wall.

But if it be timber, and Boarding, both
to be meafured, then 6 inches only is a com-
petent allowance J becaufe the Tnnber is
ufually Vallued atone third part msre than
the Boarding is.

Alfo, As the Workman doth think on
this^ the Work-mafter may not forget to
dedtiét for Stairs,and Chimneys alfo,where
Work and Stufi'ate both meafured ; though
for Work only it may be very well allowed^
Unlefs the better Price make an allowance
for it.

Natt alfo, that by the Line of Num-
bers, you may readily find the length of the
Hips and Rafters, in a Roof of any large^

Tht Breadth of the Hottfe being 40 foot}
and tkt Ends Square^ tht Length ani
jingles are, lt;« in tht Tablty 4t m ufud
trne pitch.

The Extent of the CompaiTes from 40, the
^J^eadthin the Table, to 18 the breadth gi-
fhall reach the lame way from 30, the
*^afterinthe Table, to 13-50, the Rafter

ïequired* And from 3 the Hib in the
Table, to 16-22, the Hip required. And
from 22-36, the Perpendiculer in the Table,
to io-06j the Perpendiculer required. And
from f 6-5-7, the Diagonal in the Table, to
if-48, the Diagonal required. The Angles
are alwayes the fame in all Roofs, fmall or
great, as in the Table, being Square and true
pitch.

If you would have Direftidns for Bevel
or Taper Frames, to find the Lengths and
Angles of Rafters and Hips, you may have
it at large, in annbsp;to the Mirtour

of t^rchiteEiftre ; or, Vincent Stamz,z,iy
Printed for mllUm Vifher, at the Poftern'
Gate, i66p.

By which Directions, and the Se£lor,yoU ,
may find any thing that is there fee down.
As alfo, by the Trianguler Quadrant, Thred
and Compafles.

Note aifoi That having Inches-] and Foot-
meafure together, you may prefently, by
infpedtion, find the price of one Foot, ha-
ving the price of the Square, and the con-
trary. Alfo,. having the 12 Inchcs on theo-
ther Foot, divided into 8f parts (near), and
figured, at every 8 with 1,2,

ihall reprefent pence and half,
farthingsthen at any price the Rod, yoil
have the price of one'Eogt, amp; the contrary. quot;
■nbsp;Ji

Let every Incb,Teprefeac one pound; and
^ery 8th part, 2 flullings and 6 pence; 6t
^very loch part, 2 fliillings 5 becaufe 8
|^aIt-cro\vns,or 10 two fliillings, is 20 fliil-
Imgs.

. Examflt.
Right againft 6 Inches and a half, fqr 61.
Jo s J on this other Line, I find y pence ^
rarthingsjthe price of one Foot, at 6/. lo/,
?*rRod: And at 7 farthings p*rFoot,I
nnd near 40 fliillings, or 2 pound per Rod,
Alfo, at 40 fliillings ptr Square, found
oa Foot-meafure, is 4pence } far-

N the Anfwering of this Queftion, iti^
_ not amifs, but very needful to premife
how many Square Inches, Feet, YardSf
Perches, or Chains (I mean a Chain of
Foot long) is contained in a Square Acre
Land; for which purpofe, have recourfe ^
the Table annexed, which is drawn
great care and exadtnefs for that purpofe.

By which Table yoU may perceive, Tb»^
6272640 Square inches are contained quot;
one Square Acre.nbsp;^

?eet, Yards, Paccs, Perches, orGhajfis, is
to the number of Square Feet, Yards_,Paces,
Perches) or Ghams in'a Square Acrt^; So is
i to the hreadth of the'Lind-fin thal-mea-
fiire the length was given) to make a^ uare
Acre : Seethi; '■ExMmfte's ^«f all thefe nica-
furts in their order, viz,, of Feery Yards,
^aces, Perches, and Chains.
■■'Suppofe'a Vi^ce of Land' 66o.fctt
Jong, or 22.0 Yards, or 13 2 Races, ^r 40
I^crches, IjI'- iC'Chaim in kngth - which
Several meafures are all of the tame qAanti.
ty ; I would know how much in brpdth I

43J6d, Vot ¥tamp;or ta'4840^for Yar^s;
Or to 1742, for Paces; or to tfio', %Per-
dies; or to 10, for Chains ; To the Num-
ber ill thtrT«ii^fcr chatiiicafr?rc4n aSqamp;are
Acrej .jhe fame Extent applied thcfam«
j^ay frofn-k,-fKall rea^htb cHe'Fect, Y^s,
^acpsj Perches, or Chains requir^ ■ *

4. As 40, the length in Perches,to i6o j ,
So is Ï, to 4, the breadth in Perches.

, f. As 10, the length in Chains, to lO 5
Sois I, to I, thç breadth in Chains r^^.
quired.

3. As— 132 doubled, is to= 1742
likewife doubled, becaufe it falls near
the Center ;
So is 1-quadrupled , uix;. 4, to
=: i3-2quadruplccl, vix.. J2-4.

If you would know how much breadth,
at any length, fhall make 2, 3, or 4 Acres j

Examffe at ^oPereb 1» l,„gth,
, TheEctent from 30 to ilt;jo, lhall reach
the fame way from 4 to 21 Perch, and 34
100 (or J Foot, 06 Inches) the breadth
^5 4 Acres, at 30 Perches in length.

Problem II.
The length and headih '^tven in Verchth
to find the extent in FercheSy of an}
pfceofLaad. . .

_ The Extent from ly .to the breadth in
Perches, lhall reach the fanje way, from the
length io Perchey^ to the true Content in
SquarePetches. ■■vi:.- ,
Example,
► ■ - • f.
As I to 5:0,.fo is 179 to ow the Gon^

The Ektcnt from j6o to the breadth irt
1 crches, ILalbcach the iameAvay, fVom the
fcngth in Pcrches, to the Content m §quarf

'^ht'ltngtb4niigt;uadt¥cf apuceef Lini
being given inChnfts, tofind tie C9»-
^^t in Acres.

The Extent from i, to the breadth in
^hains, and loo parts, which arc Links,
J^all reach the lame way; from the length'in
^hains and Links, to the Content in Square
Acres., ;nbsp;■ ;nbsp;-

Hnvwg the Bafe and Perpend'icuter of a
Tnsngle ^ven in Chains or Perches, to
f'»d tht Content in \Acres.

The Extent from if you ufe Chains •
t^r irom 320, if yourticafure by Perches
to the whole Bafe, fhall reach the fame way
'rom the whole Perpendiculcr, to the whole
^oiitent of the Triangle ; or if it be a Tra-
P'^jn, joyn both tltePerpendicalers in brie

So is 47-20, the length, or bafe Line, in
Perches, to Aas, 24 Links, the
Content in Acres.

Problem VI.
■ The tArea, or Comtnt of a fleet of LaA
given, that wof meafured by Statute'
fyjbes; t9.find the Content of the fit»'
peee of Land in mod-land Hftafurej ef
Cttfttmarj ActtSy pr Jrifh Acres.

jpor.the better underfl^nding of this Pro-
blan, i^is neceflary to defcnbe the feveral
kinds and quantities of Perches, which arC
fpoken of by Authors, and ufed in feveral
places; together with their proportion tlt;?
the Statute Pcrch of ilt;5 Foot and a half
fquare, London meafure.nbsp;' '

The kinds of Perches, are fiift Statute-
meafure of foot ^ to the Perch, according

«0 the Standard at GHild.Hall, or .the King's
Majcfties Excbtqfter. Secondly, Woodlani-
nicafure, a Perch whereof contains i8 Foot
Square of the fame London meafure. Third-
Irifh Af^es, of 21 Foot to the Perch or
Pole. And laftly, Three forts of Cufiomary^
ufed in feveral places of England, of 20,
24) Chtfhirc meafure, and 28 Foot fquare
to the Perch.

As for the Proportions pne to another,
that is, as 16 i, to 18, 20, 24, 28, or
any the like whatfoever.

But to find their diflerencp in Squares or
Scales, the Work is thus i. By tht Line of
Numhers, Firft appoint what Numberjn an
Inch ftiall be the Scale for Statute me^ufe,
InS^ 1 appoint a Scale of join au

iVoodUnd meafure, fliall reach the contrary
twice repeated, to
^ n^l' that a Scale made to 2f-2

Again, For irj/^Acrcs, which are met-
lured by a Pole of 21 Foot to a Perch, the
cxten? on the Line of Numbers from 21 to
1, ftall rcach ( being turned twice the

fame way gt;from to ilt;J the quantity
of the Scale lot Irifh Acres, to be in propor-
tion to a Scale of 30 iii an Inch for Statute-
meafurcj and fo for the reft, or any other
wsfeatfocver, as in the following Table.

Takfe the Acres, meafured by Statute-
quot;vcafure, out oi tlie Scale of 30 in an Inch
appointed for Statute meafure, and meafure
« in the Scale of 2^-22 in an Inch fot
Woodland; or by the Scalc of iS-^y for
Irifh Acres; or by the Scalle of 16-89 for
Cuftomary; and you fhall have the quantu
ty of Woodland, Infh, or Cuftoinarx,
Acres required.

Suppofe I have 30 Acres of Statute-mes-
fure, how many Acres of Woodland,^^rifh,
or Cuftomary meafure will they mafee ?
Take 30 from the Scalc of 30 in an Inch,
and on the Scale of 2J-22,. it ftiall give
2 j-22, for fo many Woodland Acres.; and
w the Scale of 18-5 y, for Irifh Acres, it
Ihallvgive. i8-yj for fo many IHfh Acres ;
and on the Scale of 16-89 in agt;nlnch for

theFeet in a Scatute Pcrchj to l8 the
Feer iij a Woodland Perch (or to 21 the
F et ill an Irifh Perch, pr to 22,24,28 the
Fcer in a Cuftomary Perch ) lhall reach
frctn vO, the Acres in Statute meafure, be-
ns twice repeated, to 25'-22, the Acres ill
-Wooilaad meafurerequired, Src. tt being a
larg?i:,Acre muft needs be lefsin qtiantity'»

Having the Plot or DraHght of a Vieliy
and tts CoHteMt in AcrUy to fi»d hy what
Scale it wa* Plotted 5 that by what
farts in an Inch.

Supplofe a Triangle, or a Parallellagtam,
or long Square, do contairt 4 Acres and a
Iialf, which is fet down in figures thus,
4-y o ; which if I ihould meafure by a Scale
of li irt an Inch, might happen to be 2-2^
Chains one way, and i Chain, 2^ Links
the other way; whith two fums being mul-
tiplied together, make 2-y200, whereas it
Ihould be 4-5 000 ;

Note, That if the Scale I gueffedat,
8'ves more than Ilhould:have, then I have
[00 many man Inch ; bm if Hs, j muft
jiave more in an Inch, as here, which infal^
quot;bly fiieweth which way, which is alwayes
^ne fame way asypu divided the fpace, from
Jhcjucffed Sum or Produdl^to the true Pro-

fo this Rule may he referred the way to
difcoverthe true fee of gujters Quarries ;
the method whereof is thus: They are ufu-
aUy cut to, and called by 8s, ids, 12$, 15s,
«s, and 20s m a Foot, or any other what

Glafeof 8s, make a Superficial Foot; and
10 quayies of los, make a Foot Superfici-
i2ofthei2s,amp;c. ^

This béingthe ftanding Rule er Method,
and thofe two fizes being known, I wouW
find out any other,, as 13 s, or 14«, or i/S;
and the like. , '
t)o th$u ;

Divide the diftance ön the Linè of Nmn-quot;
hers, between the Content of fome knowH
fize, and the Content of the inquired fize, j
into two equal parts; and that diftance laid |
the rif^ht way from the fides of the knowfl
fize (increafing for a bigger, and decreafing ;
for a lefs j fhall give the reciprocal fides o(
the fize required,

Tht Sidts.Rangis, Lengths, aniJlriAdtk
ef Square lox, are at in the Table fel*
lowing; and I would have tht RangeSt
Stdes, Lengthy and Brtadtb ef 14J, ;
unufual Six^f, '
The Content of a Sqüaré quarry of Gla^
talléd los, is a juft loth part of a Foot»
which is I inch and 20 parts; or one lOth
J)art óf a Superficial Foot, containing
lortg inches«nbsp;'

And the Content of the fize called 14s»
muft be one i4.th part of the fame mcafurO
Or Foot Superficial, which is o-85'7i4, th^'
IS o-gyy parts of one long inch in a ioo'^
parts.

.'[209]
* ^11, by the Line of Numbers, divide
f jctpace between I-2000, the Content o£
. ^ aado-Sfz the Content of the 14s
equal partsj tliat Extent, I fay,
fam the fame way from 3-76, the Ranges of
pS»iarc I OS, fhall reach to 3-18, the Ranges
14s: And from 3.84, the fides of
llare los, to 3-2 j the fides of Square 14s ;
And from 4-80 the breadth of Square las,
J0 4-Of the breadth of Square 14s: And
Tom 600, the length of Square los, to
y-07 the length of Square 14s, the requifites
Jgt;f the unknown Size required. And the
«ke for any other whatfoever.

The ufe of the Line of Numbers
in meafuring of Solid meafure,
as Timber, Stone, or the like
Solid bodies^

Ptctsf Timbtr hittg brtaitr eat vaj
than the other, to f,d tbs fiit of a
Square that {haU be eqnal thmmto, be-
ing called, Squaring tbt Picct.

Suppofe a piece of Timber is i2 inches in
^cpth, and \6 inches in breadth ( and iq
foot m length.)

, to 12 and 1(5, the depth and thicknefs of the
piece of Timber propounded. For if you
fhaU multiply ij-Sj'p by i3-85'9, you
lhall find 192-071881, the neareflRoor,
you can exprefs in 5 figures, and an in-
different true mean proportion, between
12 and the depth and breadth j fo that
in fine, 13-86, is the fide of a Square,
nearly equal to 12 and 16, whereas the
doubling and halfing, the old falfe wayj
gives full 14. '

Divide the difiance on the Line of Nuffl''
hers, between 12 and 16, into two equal
parts, and you fhall find the Point to ftay
at 13, .and near 86 parts, the Anfwer re-
quired.

The way of doing it, bj the Line ef
Lines, is fhewed in the VI Chapter, and
7th Propofition, either by the Se6i:or, or Tri-
anguler, Quadrant, and therefore needs
lepeijtjon jn this pla«e.

'At My quot;Bre/dtb ani Depth, or Square}
Ktfs, to find hw much makfs a Foot (if
Timber.

I. If the Timber be fquare (or fquared)
*henthe way by the Line of Numbers, is
thtti^

Extend the Conipaffes from the fide of che
fquare to the midd e i, the fame Extent ap-
plyed, or turned twice the fatae way from
I, fhall reach to the length that makes a
Foot of Timber, at that fquarenefs.
Example,

Suppofe a piece of Timber be yo of a lOO
(or «inches) or half a Foot Square (which
is all as one) Extend the Conipafres from j

I (fonvards) the fame Extent being turn-
ed two times, the fame way from i, fhall
reach to 4, being 4 Foot, or 40 fuch part?»
whereof the fide of the Square was j. ■

The Extent frbm 6 to 12, fhall reach, be-
ing turned two times the fame way from i z,
to 48, the number of inches in length that,
quot;lake? a Foot, at that Squareivcfs j being 48
P 3nbsp;fuch

As the fide of the Square, in inches, xs to
12: fo IS 12 to a 4th, and fo is that 4th
to the length of a Foot required, turning
the Compaffes,twice, the fame way as .you
, turned from the fide of the Square in inches
to 12.

The Extent /rom.i to the depth,, fhafl
reach the f^m? way fram^the breadth tpa
jfth Number.nbsp;' ^

The Extent from that 4th Number to t,
lhall reach, being turned once, the fame
way from I,to the length of a Foot iu Foot-
meafure required.

. Suppole a piece of Timber be 0-7 j 3 Jeep,
and o-7yo broad in Foot-meafure ; or 4
inches deep, and 9 inchcs broad, as with,a
glance of your eye on inches and foot-mea-
lure, you may fee how thefe Numbers agree.
Tne Extent, I fay, from i to 0-,,,, 'fhaH
jeach the fame way from o-zyo to 2-50.

fhall reach the fame way, from i to 40,
^ 4 Foot, the length required, to make a
at that breadth and depth.

As 12 to the breadth in inches, fo is the
in inches to a 4th j then as that 4th
to 12, fois 12 to the length in inches re.
Quired.

The Extentquot; from 12 to 9 the breadth,
quot;lall reach the fame way from 4 the depth,
^03fora4th.

Then thamp;Extbnt from 3 the 4th to 12,
^all reach the f^ime.way frop 12 to 48,
We inches in length required,to make a Foo t.

' quot;f. Thebrettithanddeptifghiii'ifflnchtsy
to find tht length of a Fm of Timber,
i» Feet and Parts.

The Extent from 36 to i a, fhall rcach the
fame way from ijto4f6ot,thelengtl'
in feet required.

■ ', ■ ' '
The reafon of this Order and Method, i
.you confider, you will find thus;

. ■ As 12, the inches in a foot, is to th'
breadth in inches;
So is the'depth to 3 Foot.

As I foot »the depth in inches;
So is the breadth to 3Ö hichjes, ■ whi^ ^^
3 foot alfo.nbsp;, J ^

But altering the Order in th'ebeginninf?»
alters it m the ifiue, thoughthe fame trtJthgt;
yet in or under divers denominations;

48 inches, and 4 foot, are the lame; f^
.(anermcsonc way is njore convenient thaH

any Squartne^, or Breadth and Depth
given »» Foot-meafurty er In(:hti\ to
find how much Timber it in a Foot
long, in Foot-meafurey or ftttiani ioo
pans or incbts'e ■ •

1. If the piece of'Timber be Sijuart (or'
Squared) the» worl^ thfu for Foot*
meafure.nbsp;'

As I, to the fide of the Square, fo is the
«de of the Square to theipiantity ofTiihba
jn one Foot long j which multiplied by the
^^ngth, gives the whole Content required.

is in a Foot long i
Extend the Compaffes from i to. f, the
lame Extent turned the fame wa.y from c,
reaAe^toay or a^quatterofaFcit^ diea
It the Tree be lafpotlpug, ij quarter*

a. The Side of the S^egiven in Inches;
t? fiai the Quantity, or Contest^ in a
Foot.nbsp;. J

, 'As I, to the fide. of the Square, {q is the
fid? of riieSquare tpjtf, 144th parts of a 1
SoptSoIid.nbsp;quot; V ^ I

Examfit^.- .- I
The Extent from 12 to tf, the inches
Sc|uare, ihajl reacb^ttie^faiBe way ftom.d to
j i^hes in afc^ iougi.wh'ichis a 12th
parts of a Foot Solid.nbsp;-

s^JJ^c-Extent ftona i to 6, the inchcs fqilate,
flai} f^biheAane wiaf from 6 «lt;gt; id - the
nhmbBr of k,«g .hcHes ift afoot Wie ; oi
pu:ccs.^of .t inch Jijuare', and a fOot^long,
«44 Of which makes one foot of Timber.

iftbtPimk mtfqudrgfirfq04'
, rtdj then to fini fitiv much u in a Foot',
'J . pug, vtrk. PhfU;' y • ^ /T

,3- .Tljie Extent from i to p, fhall rc«h
f'le fame way from 4to5lt;J, the long inches
'^a.footV'g i for Inch-meafwe.
3- The Extent from 12 104, ihail reach
fame way from p to 3 inches, or 3 i fths,
a qu^r of a Toot j for Inch, racafnre,

Tbt pit of the Square, er tbt brtaitb and
dtfthgiven in Inehes, or Foot-mafHre^
maiibt length, in. Fettj to find the
titfy ^rCenttm of the while Twe, i»
feet 0nd parts, ■ j'nbsp;5;. _ .

foi^V^ to the fide of the Squarf,in inches,;
^ ^ the length jn feet, to a 4th j and the«

The Extent of the Compaffes, on the Vfl^;
of Numbers, from 12, to 10 inches SquaJ'^'
ftall reach the fanje way from 10 foot i,
3 inches, to for a 4th ; and fiO^.
thence to 7-11, or 7 foot i inch, and a tSif''
part, the Content required. As by look'
ing for II on the Line of Foot-meafuf^'
right a^inft which, on the inches, is 1 in'''
and a quarter, and fomewhat more.

3. But if the piece of Timber be
Cquare, and you woiild meature k'withquot;quot;
fguaring, by the ^r/i Probltm ;

As I to the 4th, fo is the length in feet'■
the true Content, in feet and parts.

Let a Timber-tree of one foot ay,
quarter one way, and one foot jo the och^
way, and 12 foot long be meafured.

The Extent of the CompaiTes from
1-25, fhall reach the fame way from 15 '
to 18.74, for a 4tb.

Then the Extent from i to 18-74, fbaU
«ach the fame way from 12 foot, the let^tb,
for the Content; viz.. 22 foot
a half, the whole Content required.

4' Whtn tht hreadtb and depth it give»
in Inches., and the length in Feet, tefind
the Content mthoHt fqnaring.

As 12 to that 4th, fo is the length ill
feet and parts, to the Content in feet
and parts required,

Extend the Compaffes on the Line of
Numbers from 12 to ly the depth ; the
lame Extent applied the fame way from 18
2nbsp;'hall reach to 22.jo, for a

Then the Extent, from 12 to 22-fo,thc
Vh, fhall reach the fame way from ij
the length, to 24 foot 38 parts, or 4
J^ches and a half, as a glance of vour eye
Inches and Foot-meafure will plamly

thus'you havc thc Solution ofany
ßfM that may coiktrn proper Meafuri^f
quot;Fooe-mea(ure, and Inches i ufing oulyth«
Centef a£ lo for Foot^eafare, and at i*
for Inch-meafure, wkhout troubling jo^
or 1728,' or 41-y7, ör tht Ito
as inthe little Bookbf the Carpenttrs Ruth
may be feen.

To work thefe Qatßions by the Line oi
Lines, thoiigh it mäy be done feveral \yaysgt;
yet no way fb foon, nor fo exad, as by th«
Line of Num.bers: Yet I fhall fhew
in this place, to|ether by themfelves, tb«
Three princtpal ^eftions, viz.. How mttd
makjs a Faot in qttamity ; And, How mach
u in a Foot long-. And, By tht Ungth
ireadth and deptb, tht Content in Feet Iquot;
the doing whereof, you mufl: conceive tb®'
I o principal parts to be doubled, and th^'^
10 iscalled 20 5, and confequently «J iscal'
led 12, the Point fo oftenufed; and f
callcd 10, the Point ufed for Foot-meafut^'

12, to that = 4th Number i
is,=: 12, to t^ -^.J^unjber of Inchci
that goes to make a Foot of Tjpiber,

Take the diftaHce from the Center to ^quot;
®«counted as 8 5 ?nd make it a Parallel

counted as 12 j or lay the Thred to the
quot;eareft diftance, and there keep it. Then

^ Then take the Latteral diftance from'the
Renter to 12, according to the ufual ac-
laft?'nbsp;it a Parallel in the 4th

' the Thred to thenea?eft
theakeep it j then take the
counted as 12, to
Zitfrnbsp;^^ Latter'ally

Jormtrlj u doHt,
As — 4, counted as 8, to = d, count-
ed for 125
So is =8, to a — 4th.

a. If yoM mnli ufe Feot-meafurty counf
the f in the midft for 10, or i Foot;
and workman tht reji 4* before : As thff*
for Example.

As — I, to = 22'} Sois = y count'
edfor I, to2ay, which is 2 Foot J)
as by the Foot-meafure and Inched
you may fee.

As—xt, to = i-8oî
« = 12, to 4 Foot, the length in
j^eet, that goes to make i Foot cf
Tiiuber.

y. The fide of the Square given in Inch*'»
ani the length in Feet, to find the O^
tent in Fett.

As — 4th, to = J2 ; fo is = fide
the Square to — Content required, '
feet and parts.

So is == 9, the fquare to 9, the true .5
tent of fuch a Piece in feet and P®
required.

Tht Lengtb pz/tn i» Pm^ and tht
Brtadth and Defth in IncheS) to fini
tht Cffnttnt in fett and farti,

Thus you fee the way and raanner df
• °tKgt;ng py thç Line oi Lines, either on the
^•»drant, or 5lt;rffffr-fidc,For the ufual Que-
'ç'quot;«^ f for ï have neglefted to give the
/^■^teiit of Pieces m Cuhe Incha^ for two
^ifons : Firftj Bccaufe it is very feldom
0^2nbsp;te-

required. Secondly, Becaufe the Lirifi O',
Numbers at moft will {hew bus 4 figurée»
which is not fufficient for any Piece abovj
lt;îFoot, therefore not fit quot;for Inftrument^'
Wôrk.,

And withal you may obferve, Thatajj^
wayes the Latteral Extent firft\aken, fflUl'^
be lefs than the diftance from the Center
the parallel Point of Entrance, which
thefe Examples is remedied by calling d if
And alfo, there are fo many Cautions
doubling; and halfingof Numbersjto mi^f
it applicable, that without due confidera«'
on, you may foon err 5 Alfo, the openi»'?
and ihutting the Rule, and ufing of IScvef^,
Scales, makes it far inferior to the LinC^
Numbers, which may be eafily enlarged.

Htivlng tbtDiamtUrof ACihnitr^imn
in Inchis, or Foot-mafure, tf fini the
lengthofone Foot.

reach to 21 inches, 8 loths, forthe
Y^'^Sth of a Foor, a,t that Diameter, in
■'^«hcs.

Or, t'^'i.;-
The fame Extent being turned two tiifls^
the fame way froip i, jQiallreach to 1-85 J»
which is.the.Decimai for 22 Inches, as
and Foot meaftire, you

The Extent from 0-83 3 (the Decimal'''
10 Inches) to r.i28j being turned twice ^
fame way fVom i, lhall reach to 1-85»
which is almoft 2zln9hes, as by compari^'^
Inches and-lfctot-lneafure together, isplai»il5'

Problem II.
^^ivlng tht Diameter given In Inches^ tr
Foot-meafitrey to find hort much is in *
Foot long.

As i3-f4 (the Inches Diameter that
gt;nake a Foot of Timber, at one Foot
to the Diameter in Inches j
So is 12 to a 4th, and fo is that 4th, to
the quantity in a Foot long.

: ~ The Extent from 13-^4 to 10, being re-
lated two times the fame way from 12,
Jhall reach to 6 Inches or, 54 of 106,,
t^nig fomewhat more than a half Foot, for
the true Content pf one Fopt long.

2. But if theTimber is great, then It is
more convenient to have the quantity of a
Foot, mfecc and parts.

As 13-J4, is to the Diameter in Inches 5
So IS I, to a 4th, and that 4th to the
quantity in a,Foot, in feet and parts.

fnetcrtn Inches, fhall reach, being turncQ
twice the fame way from to 0-^45', the
Content of a Foot l^ig.

The Extent from ij-^, to 30, bein?
turned two times the fame way from if
fhall reach to 4 foot, 93 parts; which 4^9?
muldplied by the length in feet, fhall give
the whole Content of the Tree.

The Extent of the Compafles from x-iiSt
(the feet and loths Dian^eter that majce a
Foot, at one foot in length) to the DiametcJ
in Foot-meafure, fhall reach, being turned
twice the fame way from j, to the quantity
in a Foot lon^.

The Extent from 1-128, to i-jo, lhall
reach , being turned twice the fame way
from 1, to 1-77, the true quantity in onC
Foot long.

I. The Diameter of any Cihttder given in
Inches, and the length in Feet, to fini
tht Content in Feet.

The Extent from i3'-5:4, to 8, being
turned twicc the fame way from 20, the
length, fhall ftay at 6-94, or near 7 foot.

Î. The Diameter and length of a CiUender
given in IncheSy to find the Content in
Cube-inches.

The Extent from 1-128, to the Diameter
»n Inches, being turned twice the fame way
from the length in Inches, fhall reach to
the Content in Inches.

Thus the Extent from 1-128 to 10 inches
liiametcr, fliall reach from 34 inches, the

The Extent from i-i28,to the Diameter»
fliall reach from the length, being twice re-
peated the fame way»to the Content in feet
required.

Thus the Extent from 1-128, to i-yo,
lhall reach,being turned twice the famp way»
from f-30, to 9-37, the Content in feet
reqirired.

Having the Circftmference of a Cillender
pven in Inches, or Foot-meafure, to
find the length that taakfs ont Foot tf
SoUd-meafure.

As the Circumference in Inches, is to
134-50, (becaufe at fo many inches a-
bout,pneof a Foot in length, is a Foot) fo
is I a to a 4th, and fo is that 4th to the
length of a Foot in inches.

TheExcenc from 30 to 134-50, .beinj
turned twice the fame way from 12, fhal
reach to 24 inches, 13 parts; the inches and
parts that make one Foot Sohd.

So is I to a 4th, and that 4th . to the
length in feet and parts, that makes
iFqot,

-Eca: the; Extent of theCcppafles fromjo
to 134-yp, being turned twice from I, the
fame way, fhall reach to two foot, and one
tenth, the length that makes one Foot Solid.

So is that Extent twicc repeated the ferae
w^y from i, to the length that makes
^FooiSoUd^

The Extent from 2-yOj to 3-54, beiri?
famed two times the fame way from i, doth
ircadi to 2 foot, 001, the length in Foot-
meafure.

Tftf ClreumfertKct gtvquot;* «« Inches, «r
Fott-meafHre, tdfind how much is in 4
Foot long.

So is t foot to a 4th, and that 4th to the
folid Content in one foot long.
ExampU.

The Extent of »he Compaffes from S'Hff
to 2-50, the feet about, lhall reach, ^mg
turnpd twic? the fame way from i,to 0-497,
the quanatyin a foot long, vix,. near half a
foot..

2.nbsp;The Cireumftrence given in Inches, to.
find the Content of one Foot in length,
§olid.meafure, in Incbes.

Th jBxtent from 134-j', to 30, being
turned two times from 12, ftiali reach to
near 6 inches for the Content of one foot
long, at 30 inches about.

3. TbeClrcumftrenee of tt CiStudtt givt»
in IncbeSi to find the qnantity of ont
Foot long in fctt and incbts.

So is I to a 4th, and that 4th to the
quantity of one foot long in Feet and
Inches.

The Extent from 134-5, to 30, being
twice repeated the fame way from i, fhall
»each to 0-497, or near half a foot, the
Content of one foot long, at that Circum-
ference , which being multiplied by the
length in feet, gives thc.tiue Content of any
Cillender whatfoever.

Tht Cireumftrence, ani length of any Cil'nbsp;j
leader give» inches, or Feet and In' j

t. The Circumference give» in Inches, and
the length in Feet, to find the Content in i
fett ani parts.

As 42-5'4 (the CifcumferenCe in Inches^
that makes i foot long, a Foot) is to
die Inches ill Circurafereuce;

The Extent from 42-54, to 48 the inches
about, being twice repeated from 12 foot
the length, fhall reach to ij-2 8, the Con-
tent in feet required.

2. The CircHinference and length given
in Feet, te find the Content in feet ani
parts.

As 3-J45'gt; (becaufe at 3 foot and a half
about, and a foot in length, is a Foot)
is to the Circumference j

The Extent from 3-545',' to 4-6, the
Circumference, being turned two times from
»2 foot the length, fhall reach to 15-28,
the Content in feet required.

3. The CircMmference and length glveti
in Inches, te find the Content in Inches^

The precife Extent on a true Line of Num-
bers, from 3.545, to 48, bemg turned two
times from 144, the length in IncheS: fhall
teach to 26383, the itumber of Inches in
a Tree 48 inches about, and x44mchesin
length.

This is fufficient for the Menfuration ttf
any folid body in a fquare, or Cillender-
hke form, as Timber or Stone ufually is,
after the true quantity of a foot, or 1728
Cubical inchesbut there is a cuftome ufed
in buying of Oaken-Timber, and Elm-Tim-
t»er, when it is round and uiifquared, to take

a line, and girt about the midft of tl^
Piece; and then to double the Line 4 times^
and account that 4ch part of the Circum-
ference, to be the fide of the Square, equsl
to that Circle; but this is well known to be
lefs than the true meafure, by a fifth part of
the true Content, be it more or lefs,

Alfo in meafuring Elm, and Beech, and
Alh, whofe bark is not peeled off, as Oak
tifually is} to caft away i inch out of thc
4th part of the Circumference, which may
well be allowed when the Bark is 3 quar-
ters of an inch, or more in thicknefs, and
the Tree about 40 inches about, or the 4ch
-part, 10 inches j but if thc Bark is thinner^
and the Tree lefs, then 8 inches-fquare;
then aitinch is too much to be allowed.

Alfo, if the Tree ispeaterthan a foot-
fquare, and the Bark thick, an inch is too
little to be allowed, as by this Rule you may
plainly fee, by the 7th Problem of Superfi-
cial-meafure in the 7tb Chapter.

Suppofe a Tree be 48 inches about, the
Diameter will be iy i, the 4th of 48, for
the fqUare is 12.

Now if I take away I inch i from the
Diameter, then the Tree will be but 43
inches and i aboUt, whofe 4th part is
under 11; fo that here I may very well a-

Confequently, if the Rind be thinner,and
the Treeiels,a lefs allowance will ferve; and
'f the Rind be thicker, and the Tree large,
*here ought to be naore, as by cutting the
Rind away, and then taking the true diame-
you may plainly fee.

This meafuring by the 4th part of the
^'tcumference, for the fide of the Square,
^quot;d allowance for the Bark being allowed
as before, I fay will prove to be juft one
yth part over-meafure.

. Efpecially confidering this. That when k
hewed,and large wanes left,then the Tree
^ marked for more meafure, fometimes by
Jo foot ni 60, than there was befot': 1. .vsi
ncwed; the reafon IS, becaufe when this
^ree is round and unhewn, the girting ir,
^nd counting the 4th part for the fide.of quot;the
fquare, is but very little raor^ than the lii-^
■^tibed Square i and then being h jwen.and
that fcarce to an eight Square, and meafur-
ing with a pair of Callipers, to the extremity
'^f that, doth not then allow the Square e-
^Ual to the Circle for the fide of tlie Square,
as by the working by thofe feveral Squares,
very plainly appear, which being forc-
and warned of, let thofc whom it con-
•^^tnslooktoit.

But this being premifed, and the Parties
®§teeing, the diftererice being as 4 to y, che
Rnbsp;bef^

bell: way to meafure round Timber, Icoffi
ceive, is by tire Diameter taken with a pair ;
of Callipers, and the length j which for the
juft and true uieafure is largely handled al-
ready.

As 1-^26, to the Diameter- -
So is thé length to a 4th, and lb is thst
4th to the Content in feet.

reach, being twice repeated from 10 foot,
the length, to lo foot the Content required,
being all at one Point.

The Extent from i-sad^ to 20 inches
the Diameter, being jwke repeated the
fame way from ic foot, the length, fhall

Then the Extent from 48, to the incheJ
aoout being turned twice the fame wa/
from the length in feet, fhall reach to ihe
Content required.

The Extent from 48, to 62, the inches a-
oo«t, bang turned twice from 10^ the fame

Thus you have full and compleat Di-
i^ftions for the meafuring of any found
V^nber by the Line of Numbers, by ha-
^^quot;g the Djameter and length given, after
'quot;^y ufuil manner, there remains only
One general and natural way, by find-
ing the bafe of the middle, or one end, by
7ih Problem of Superficial meafure j
^'id then to multiply that bafe by the length,
^^'ill give the true Content in feet or inches.
ThWy

, Having found the Bafe of theCilleiider
the 7th or loth Problem of Superficial-
'^Wfurei tben if you multiply that Bafe be-
Î'^S found in fquare inches, by the length
!quot; inches, you lhall have the vt hole CöntiaiE
Cube Inches.

Suppofe a Qllender have 10 inches for
Jts Diameter, then by the 7th or loth above-
aid, you lhall find the Bafe to be 78-54.5
%nii you multiply 78-j4.by 80, the fup-
P'^fed length m inches , you (haU find
^356-20 Cube Inches, which divided by
the inches in a Cube Foot, iheWeth
many feet there is, amp;c. And as to the
^^mber of figures, and the fradciorls cutting
you ha'N e ample Dire6tion$ in the firfi
R 2nbsp;Froblewi

Multiply half the Diameter of the Ba^'^'
A B, by k felf; then meafure the fide A
'aiid murdply that by it felf ; then taket^ '
lefl'er Square out of the greater, andtj'■
..{kjuare root of the refidue is the Pcrpen^''
*culer Altitude required, viz.. D B.

Suppofe the 'half Diameter of the B^''
AC,\vcrc lo-zy, and the fide DA
A B 10-2 f, and 10-2 j multiplied togeth^^'
called Squaring, makes loy, 062$ ;
. ipo, multiplied by'iob, called Squaring''
makes 10000 ; then the Ieffer Square
062^^ taken out of 10000, the grea''
Square, the remainder is 9894,
whofe fquare Root found by the %th
hlem of the ftxt Chapter, is 99-47J,
true length of the Luie D B, the length quot;
height of the Cone.

^thor loth of Superficial meafure is found
^obei(Jo-o8, by 33-ijS, a third part of
^5-475, the whole height makes 5308,.
^^quot;ing off the Fra6fions for the true Con-
^«nc of the Cone, whefe length is 99 inches,
near a half, and whofe Bafe is 20 inches
a half piameter.

2. Then Secondly, for the Segment or
SeBion of aConCy the fkape or form of
all round taper Timber^ the trusfl rvaj
is thus i

By the length and difference of Diasne-
find the whole length, of the Cone,
^^hich for all manner of Timber as it grows
^his way is near enough.

A-S the difference of the Diameters at the
the two ends, is to the lcn;;th between
the two ends 9

So is the Diameter at the Bafe, to the
wholelengthof theCons.
Example, .
The difference between the Diameter,?

Ac, and EF, 1513-70, the length, AE
IS 66-^2. then the Extent on the Line of
j!^uaibers from 13-70, the difference of the
^iameters; to 66.^1 y the length between,
reach the fame way from 20-50, the

greater Diameter to 99 and better,tbe leng' ^
that maiicsnp the Cone,-ac that Angle 0'.
Tapering in the Timber j then if by tli®
laiV Rule yoii meafure it as a Cone of tn»
length, and alfo meafure the little end .
point at his length and diameter j and tli^
la'ftly, this little Cone taken out of the gre^'^
Cone, there remains the true Content of iquot;®
Taper-piece that was to be meafured, f»^ ,
^246-71, when di-gO) the Content of
fmall Cone at the end,is taken out of 5 1
the Content of the whole Pyramid.

3.. /ƒ tJjis way feew too troublefme for tf''
eommon ufe, then afe this, being
brief:

To the Content that is found out, by tb^
Diameter inthemidft of the Timber, aO^
the length, add the Content of a
found out, by half the difference of Diame^
ters, and onethird part of the length of tn
whole Piece, and the fum of them two fh^'
be the whole.Content required.

Divide the length of the Tree into 4 oquot;quot; jj.
parts, and meafure the middle of each pf*^
feverally, and that cafl up by his prop^
length, fliall give the Content of eac^.
Piece ; then the fum of the Contents ^^
ail the Pieces put together, is the true Cojj^;

^ote. That this curiofity fhall never need
to be i;fed, but when you meet with Timber
^^ch Taper, and Die-fquare, or on a Con-
elt or Wager ; for according to the ufual
^^y (and meafiire) of fquaring the Timber,
fwell, if the meafure of the Square, ta-
with Callipers from fide to fide, in the
•Middle of the length of the Piece, will make
^piends for half the Timber \vhich is wan- .
'quot;g in the wany edges of your fquared
*'mber,and the knots, or fwellings, amp;• hol-
j^Ws of mofl round Timber, may well bal-
*ance this over-meafure found bytbeDia-
quot;Ijeter taken in the middle of the length of
Piece. But indeed for Mafts of Ships
''^d Yards, being wrought true and fmoot i,
^bere the price of a Foot is confiderable,
^fte exadnefs is requifite, and neceflary to
^ ufed J and thus much for Solid-meafure
Squares and Cilleaders.

. T. To meafure a Sphear or Globe by A-
fithmetick, the ancient way, is to multiply
Diameter by it felf, and then chat Pro-
to multiply by the Diameter again j
^^hich t\vQ multiplications is called Cubing '

of the Diameter; then multiply thisCubf^
by 11, and then divide this laft Produét by'
21, and thc Quotient fhall be the Solid Cori'
tent of the Sphear, in fuch njeafure as th«
Diameter was,

Let a Sphear be to be meafured, who!«
Diameter is lo inches: Firft, lo times
is lOo; and lo times lOo, is looo ; di^
Cube of I o, that multiplied by 11, makegt;
I looo ; which being divided by 21,make» |
523-81, for the Solid Content.nbsp;;

2. The Extent from i, to the Diameter»
fliall reach the fame way from the Diametc^
to the Square of the Diameter.

Thc Extent from 1, to the Square of thi:
Diameter, fhall rcach the fame way frofquot;
the Diameter, to the Cube of the Diameter*
Then,

The Extent from i, to the Cube of th^
Diameter, lhall reach the fame way fi'oquot;'
11, to the ProduiS: of the Cube of the |
meter, multiplied by 11.nbsp;^ 1

Lafily, This E^itent from 21, to this la»^ '
Produét, lhall reach the fame way from i'
to the Solid Content of the Sphear require'

3. The Extent from i, to the Diameter^
heing turned three times the fame way from
fhall flay at the Solid Content of
^he Sphear, or Globe, required.

The Extent from i to 12, being turned
three times the fame way from 0-523 8,fhall
teach to' 90^-143, the SoHd Content re;;;
lt;iuired.

Square the Diameter, and multiply that
hy 3-1415, and the Produdl is the Superfi-
cial Con-ent.

The Extent from i, to the Diameter, be-r
turned twice the fame way from
3-1416, fliall reach to the Superficial Con-
tent, of the out-fide round about the Gobe,
viz.. at 12 Diameter, 45 2-44.

The Extent from ? to 0-3183, fliall reach
the fame way from the Superficial Content,
the Square of the Piameter, whofe

The Extent from i to i-popS«?, lhall
reach from the Solid Content to the Cube of
the Diameter, as at 90y-i4j Solidity givegt;
1718, the Cube of 12.

The Extent from i, to the Chord of the
half Segment, ftiall reach, being twice re-
peated, from 3-i4ilt;J, to the Superficial
Content of the round part of the Se'^ment,

Let the Segment be the half Sphear,
ABC; AC being 12, then BC which
is the Chord of the Peripheria, B C is
8-485', whofe^quare is 72.

The Extent of the CompaiTes from i, to
8-485-, being turned twice the fame way
from 3-i4i(?, ihall reach to 22^-22, the
Superficial Content of the round part of the
Segnknt, or half Sphear or Globe, to which
if you add the Content of the Circle 01'
Bafe, you have the whole Superficies round
Sbouc»nbsp;7. To

Fiift, you muft find the Diameter of that
Sphear, of whith the given Segment to be
meafured is part.

Add theSquare of the Altitude, and the
Square of the Diameter of the Segment to-
gether, and the fum divide by the Altitude
of the Segment, the Quotient ftail be the
whole Sphears Diameter.

Taking the Ahitude of the Segment gi-
ven, from the whole Diameter, there re-
mains the Altitude of the other Segment.

Extend the CompalTes from the whole
l-gt;iametcr of the Sphear, to i ; the fame Ex-
tent applied the fame way from the Altitude
of the given Segment, fhall reach to a 4th
TNumber, on a Line of Artificial Solid Seg-
ments joyned to the Line of Numbers,
which 4th Number keep.

The Extent from i, to the whole Content:
of the Sphear, fliall reach the fame way from
the 4th Number on the Line of Numbers,
to the Sohd Content of the Segment required.

Let the whole Diameter of a Sphear be
14, then the whole Solid Content by the
former Rules, you will find to be 1437 ',
a Segment of that Sphear whofe Altitude cr
Depth is 4, the Sohdity is required.

Extend the CompafTes from 14, the whole
Sphears Diameter,to i • that Extent applied
the fame way from 4, the Altitude of the
Segment, fhall reach to z-2,6 on the Num-
bers, or to 19-88, on the Line of Solid Seg-
ments joyn'd to the Line of Numbers,which
ip-88, is the 4th Number to be kept.

The Extent from i 101437, the whole
Content of the whole Sphear, lhall reach
the fame way from 19-88, ^0284.^, the
Content of the Segment required to be
found.

If you want the Ll»e of Sef^ments, the
Table arinexed will fupply that defeél :
Thw,

Look for the 4th Number, found on the
Line of Nftrnhers, among the figures on the
Table, and the number anfwering it in the
prft Column, is the Solid Segment, or 4i:h to
be kept ; as here, on the Numbers, you find
2-8(Sj feek 2-86 in the Table annexed,

8. To perform the fame bj e/fritbmeticl^
after the vaj fet ftrth by Mr, Thomas
Diggs, if74-

Multiply the Diameter by thé Circum-
ference, theProdu6t fhall be the iSupirfi-
cialContent round about the Globe.

Multiply the Content of a Circle, having;
like,Dia«ieter, by 4, the Prodult;S: fhall be
theSuperficial Content.

If you multiply the .Superficial quot;Content,;
by a 6th part of the.Diameter, the Produdt
ihall be the Solid Content of the Sphear.

ti. For the Segment work^thttil
Multiply the whole Superficial Content
the whole Globe, by the Altitude of the
°^ment, and divide the Produa by ths
, Phears whole Diameter, the Quotient lhall
^ the Superficial Content of the Convexity
^ ^ound part of the Segment.

But for the Solid Content, workthtu •
. Pirft, find the difference between the
J'^ight of the Segment, and the half Diame-
of the Sphear 5 then multiply thisdif-
j jnce ( being found by fubflradling t!Ue
from the greater) by the Superficial Con-
the Bafe of the Segment, and. the
^J^odudtfubftradt from the Produa of the
Pneats femi-Diameter, and che Convex Su-
J^rhcies of the Segmeiu 5 then a third pare
the remainder fhall be the Sohd Content
the Segment required.

Hgt; the Segments
^^iutude is 4, the Segments Altitude taken
7, the half Diameter, the remainder is
Which multiplied by 13d, the Superfiaal
^°Jtent of the Bafe of the Segment, makes
th ^nbsp;multiplied 7, the Sphears

the Segment 176, the Produa is 1332^
from which number take 378, the Prody^^^
laft found, and the remainder is 854, whofe
third part 284', is the Sohdity of the Seg-
ment required.

There are other fragments of Sphears, aS
Multiformed and Irregular, Cones or Pyra-
mids, aid Solid Angles; but the Menfura-
tion of thefe I lhall i^ot trouble my felf, not
theL^iarner with, for whom I only write,
intendmg the Menfuration of things that
may come in ufe only.

113 . But yet to conclude this Chapter, tak?
thefe O.yfervatians along with you, cor,'
cerning the Proportion of a Cuhe , 4
frifma, and a Pyramid, a Ctllender,
. • Sphear, and Cone j whofe Shapes and

If a Cube be made or conceived, whofe
fide is 12 inches, theli - tte iolidityr. thereof
is 1728 Cube inches,; and a Pnfma, having '
the fame Bafe and Altitude, contains S64
Cubetnchesi and a fquare Pyramis, of the ;
fame Bafe and Altitude, contains 576 Cube
inches; and a Trianguler Pyramid, as be-
fore, contains 249-6 Cube inches ; A Cil-
lender contains 1357'^, being the fafflC
Height and Diameter of 12 inches : A
Sphear, whofe Axis is 12 inches, con-
tains

( ;i .
tams 90 y ^ Cube inches; and a Coiie, df
^he fame Diameter and Altitude, contains

The Superficies of the Cillender about,
wcepting the top and bottom ) is equal toi
Superficial Content óf a Globe.

By the foregoing Proportions, it is evij
dent that a Cube is double the Pnfma, and
^^ J ^he Square Pyramis of equal Bafe
and Alturude, or as 3,2, i ; for 3 times
576 is 1728, and 2 times 864 is 1728.

Alfo a calender is of a Cube; and a
Wobe is of a Cube, or ^ of a Cillender,
^^hofe Sides and Diameters are eqnal; and
^Cone is ^ of a Cillender; lb that tfi(|
proportion between the Cone, Sphear, and
Cillender, is as i, 2, 8c 3 ; for 3 times
makesnbsp;i and two thirds

of 472 I makes 90J the Content of *
Sphçar. The Trianguler Pyramid is hrt'^
more than y of a Cube; fo that if any oTfi
bave frequent occafions for thefe proporti'
ons, let Centers be put in the Line ofNui«'
bcrs, at thefe proportional Numbers, an»
then work with thofe Points from the Cube
and Cillender, as isdireéfed before, for the
Circumference, and Diameter, and Squaresj
equaland infcribedin Chap.S. Prob. 6.

So much for the meafuring of régula*
ordinary Solids ; for the extraordinary and
irregular, the beft Mechanick way is by
Weights or Waters to meafure their Craifi-
tudes or Solidities, cither by Weight oï
Meafure.

further irhprovernem of the Tri-
anguler Quadrant, as I have made
U feveral times, with a fliding Co'
•ver m the in-jide, when made hoi-
lovpi to carry Ink, Pens, and Cm'
pajfes 1 then on the Jliditig Coveri
md Edges, is put the Line of Num-
hm, according to Mr. White's firfi
Contrivance for manner of Opera-
tion; but much augmented, and
made eafie, by John Btown,

THe defaipcion thereof fot out Jide^
being the fame with the Line of
Numbers on the outter-Edge, except that thé
hrlt part is fometimes ( when required for
that particular purpofe ) divided into xz
parts, fot inches^ inltead of lo* that is to
»ayi The fpace between the firft i, and the
fiddle I, on thc Rule 5 ( the fpace I fay )
oetween every Figure, on the firft half part,
cut into 12 pares, infiead of 10, to an-
'^jl^er to t he 12 inches in a Foot; and the 6-
half, as the Line of Numbers on the
A,nd in the fame manner are fvhttt'i
S 2nbsp;fliding

2. On the other Jide, is the Line of Nu®'
hers drawiT^dcjife the one ^.ine to the O'
rherj /foV^t^e'TeAdy rnealuring of foilquot;
Meafure. at, one Operatfon ^ the defcriptioi'
\Vhereof in- brief* ft thu^'; '

quot; Firfty The divifions on the fliding-piece
ill. hollow^RuIes, or f)it the^ tight-fide iquot;
lliding-Rules; when the figures of the Tim'
ber-fide fl'and fit to read, I call the right-

■■Thedrnflons therefore on the frxed^dgc
of the Rulcymull heeds be the left-fide, and
is alfo divided to a double Radius, oi on^
Radius twice repeated.

.. Soalfo inflidiiig-Rules, the double Ra-
dius is on the left-fide alfo. Set the Figurt
thereof, withjight and Itft-Jide expreft ttfo»
U.

The Figures on the right or fingle-fide»
do ufually begin at , 3 or 4,. and fo proceed
with 4, y, 6,7, 83 p, 10,11, for fo many
inches of a'Foot. ;

^«fs of inches • And the fmall DivilionS be-»
the Figures, that reprefent Feet j- ar^
Only every whole. Inch; The .i^alfe, and
I^T^ners of Peer, alfo, noted by a .lon^c
lyoke-as in foch^wbsk is necefT^ry and ufuals
'' V.' On thefame nghtrfide atfo,-. for mop«
readiuefs in the ufe, aresiiofed fe^
Gage-points'(as it were) ; Jsy i'
Plrfl^ At I Foot is the wotdrfquarr.: ■
^ Steondty^ At ^ l incli and .V;, is a
^i- 'and'clofe to it is fet t.dlt;fQr-,«ue Dia-
''^«er of a round folid Cillender,, 3 '

Thirdly, Ac i foot 3 inrlxfsji
T^'- and neiir ttxir is fet Dj for'W Piaoie-»
•^^df a rough fieceif TimbfiUjafc-ftcding; t:q
'^quot;«^afual ^ib^mice for uirl^wd J.imljer:,
^•^ording to the- fourth pars.iof' a Line .gift
«feutWd counted f(?r the fidc; of tjie,fquare,
Fmrthlj^ At 3 3ooc 6 inches and ftn^t
near to ic is fet t: r: for the true Circum-
ference of a round Cillender.

Tiftlh At-4 foot juft isffen K, for the
pii'ciumfereii«e, according tQ the former al-
»^^Wance.

Sixclj, At I foot 5lnchcs;..ftrf,, IS a
[potj and clofe to it the letter W, -as the
^^^ge-poinr for:« Wine-'gallogt;ii: .J aI .
quot;quot; ■ quot; S 3 : quot;h'S^^^^^hi

Seventhly, At near 19 inches, or i fr®^
7 inches, is auotlier fpot; and clofe to i
the letter a, as the Gage-point of an AU' ;
gallon.

Eiabtly, At 2 foot 8 inches is a fpoj
and clofe to it is fet B, .for the Gage-poipc ^
a Beer Barrel 5 and at 2 foot 7 inches is
A, for an-Ale Barrel.- TheMfes where^
order'follow,nbsp;.

The Figures on tht left-fUt^ or fixej'
edges, are read and counted as thofe cret''
right : For the fmall, 1,2,
I o, 1 J, are to reprefent inches,, apd the cu»
between, quarters of inches; Then th« i
3,4,7,8,9,XO Figures next, fome^^hat
bigger,'as to reprefent fo many feet, - ain^tflf
cuts between,-are whole inchesThen 20,
30,40,y0,60,70,80,90,100,'i JO, for ten« ,

pf feet, and the parts between, for fijiglfj
feet, for the moft part j or elfe whole anlt;»
half feet, as is ufual. The Ufes,follo\^^.

piece of Timber being not Square^ (
having Its breadth and depth unequal )
to makf igt;, Si^uare, or find the Squ0*
equal'j

Set the breadth of the Piece, counted oj
the right-fide, to the fame breadth counts^

on the left-fide; then right againft the depth
raund on the left-fide,on the right or fingle-
is the inches and quarters fquare re-
lt;iuired.nbsp;^

Set 9 inches on the right-fide, to 9 on the
^ft gt; then right againft i y inches, or 1 foot
3 inches, found our on the double or left-
quot;dc, on the right or fingle-fide., is 11 inches
^'^d {the Square equal required.

Alfo, if you fet ij to if, then right a-
Sainft 9, found out on the left-fide, on the
quot;^'ght-fide is 11 inches J, the Square equal

Set the inches (or feet amp; inches) found out
^n the right-fide, to i foot on thc left; tlien
.'■Jght againft i foot on the right, is the in-
ches, or the feet and inches required,to make
^Foot of Timber.

Set 9 inches, found out on the nght-fide,
to I foot on thc left-fide ; then right againfl

If the'Square be fo big, that the i on the
right falls beyond the End at the beginning,
then right againft lO foot on the right-fide,
is on the left, the hundredth part of .3 foot,
that makes a Foot of Timber.

Set 4 foot, fouiid out on the right, to I
foot on the left then right againft i-o foot
onthe right-fide,0-063 on the left-fide,
or agajnft i2foptyouhave 9,. 12 parts ol:
I inch, the length that goes to mak-e j foot ot
Tiinber required.

At any (blg»ej?(ir} lnçhes, or feet ani
inches fquare, t9 find how much is i^
1 Foot long.nbsp;■, :

Juft as the Rule ftands even, that is î
foot on the right, againft i foot on -the left,
fpek die inches, or feet and inches, the Piece
is Iquare on the right or finglcrfide j and
juft againft it on the left or double-fide is
the Anfwer required ; in inches, or feet
and inches.

Example at ip inches fquare.
• Juft againit i foot 7 inchesjcr 19 inches,
is all one) found out on thç right-

fide, on the lefc.fide is 2 foot lt;J inches, the
^tiantityof Timber in i foot long, at 19
inches fquare; which Number of 4 foot
inches, multiphed by the length infett,^
gives the true Content of the whole Piecc '
Timber required.

I^ote, That this is a moft excellent way
great Wood, and very exaft.
Mfo NotCy That here, by infpeftion, yoii
■^ay icjuare a fmall Number, or find the
%Uare-root of a fmall Number.nbsp;,

Tie Jlde of the Square, and the length of
any 'piece being given, to find the Con-
tent in feet and parts.

Set the word fqttare, or i foot, alwayes to
the length, found out on the left-fide; then
right againft, the inches, or feet aud inches
iquare counted on tlie right, on the left is thq
Content required.

Set I foot pn the right, to- 20 on the leff^i
'hen right againft i foot 3 inches on the

iVT'rff, That if the Piece be very finally
call the feet on the left-fide, inches; and
thepiirts between 12' of inched; then the
Anfwer will be found on the left-fide in
I44'. of a Foot.

Set 1 foot on the right, to 30 foot on the
left; then right againft 2 foot on the right,
counted as a inches, is 120 parts of a foot
divided into i44 parts, being juft 10 in-
ches^ for 10 times 12 is 120.

Set I foot, or the word fquare, to the
length on the left, counting the fingle feet
IO' of feet; then right againft the feet and
inches fquare ate the 100' of 'feet re-
quired.

Set I foot to 4 foot, counted as 40, on
the left; then right againft 4 foot on the
right, islt;J40, the true Content, increafing
the 10' to 100', Thus mnch for fquare
Timber.

Though there be many other wayes and
manner of workings, fome whereqf you
jnay in a Book Ictforth under the .i^ame
of Thi;Clt;irpent(rs J^ult,
3ud well known abroad already..

Set the fpot by-p,d' to x foot on the left,
then juft againft the inches, or feet and in-
ches Diameter, found on the right, is the
quantity of Timber in i foot long on thc
left-lidc required. '

Suppofe a piece of Stone-Piilar,- or Gar-
den-Rpul, be two foot p inches Diameter,
fet the fpot by t.d. jitft againft i foot, then
right againft 2 foot sgt; inches, found on the
right, on the left is 6 foot, the quantity of
folid hieafure in one foot'long ^ which be-
ing multiplied by the kngth in feet, gives
the true Content of the whole Piece.

Ufe'VI. ■
Tht VUmettr of etnj Piece of TimF^r- gi'
vheky to fini •hroW''Wft6h in length'
mak.e'onePnh^^'^'- 'nbsp;— quot;

Set I foot on the lEfr,~to tKe inches Dia-
met^s,counted on th«-J!ighl.v then.^i^i^ a-
% triie flj^iirf' or Dipr the
ufual allowance, is the ^nfwer.: required ,
found on the left-fide.

Set I pn'phe left'tOjP oiiihe right ^' theii
juft againl\ t.d.^s^ ^ fopy oh tlie.left}
andquot;frght againfl; D,'. is i f^'oc 11 inches pn
the left, the length requite'd to make a foot
folid at true meafur^' w- ufual ajllow-
ance, when the 4th part pf-thegirt abput,
i; counted the fide.ot\ the ^uare, equal, to
the round piece of Timber.;nbsp;(, 1 •

Note, That for great :Timbcf, you muft
fet theleiFt i foot, to the fe,e_t'and inches j^ji-r
merer as' before * but count the t.d. or.D,
as farb^ond 12 foot, as it is placed beyond
12 inches, and you (iialt hav^ the Anfwei;
in 144® of afoot.nbsp;,; ,

If you fet ion the left to y foot on the
jighr, and count fo much beyond 12 foot

On the right, as t.d. is beyond 12 inches,'
yott iiiaji find 7 ^, ^tliat is, 7 and a
quarter, to make t foot true measure, and

But for fmall Timber,,:fet i foot, 2 foot,
car theright, counTed as jt, 2, 3^4,and
quot;iches, to I on the left; then right againft
^•d. orD, is a Number,, that multiplied by
is the Number of-feet required.

Set 2 foot on the right, counted as i inch^
^0 I on the left j then right againft t.d. is
3 foot 10 inches,quot; calling the inches feet,
the feet los of feet; which 3 foot lo
^ches, multiplied; by 12, make 46 foot,
'or the length of i foot of Timber at 2 in-
ches Diameter, the'thing required for true
'tieafure.

Set t.d. orD, for true meafure, orufual
allowance, alwayes to the length counted
on the left ; then right againft .the inches
diameter counted on the right, on the lefc-
^^de is the Content required.

Set t,d. to 30 gt; then right againft 6 in-
ches, counted on the right • on the left is
y foot li inches, the Content required.

Nott, If thé Piece be fmail, thbi coont
every foot on the right as inches, and you
have the Anfwer in 144s of a foot^ whidi
is eafily counted by having i fet at 12,
a at 14, 3 dt 3*5, 4 at 48, f at do,amp;c. to
12 at 144, which little fmall figures are
counted as inches of ias of a foOt.

But for great Pieces, fet t.d. or D^ to the
length, counting i foot, or the left for 10
foot, then have you the Anfwer in lOoS of
feet.

Set t.d. to 3 inches, counting 30 foot for
the length ; then right againft y foot on the
right, on the left is Jp2 foot, the Content
reouired.

Ufe VIII.
meafure. round Timber, hj having thi
Girt, or Circumference about,and length
given.

This being the fame in operation with
Diameter, I fhall pafs it over more
^n'efly 5 which way of wording, may ferve
the Square and Diameter alfo i only I
'abour to be plain and brief.

Set t.r, or R, for true round, or allbw-
to I foot on the left; then againft the
^quot;ches about, on the left is the Anfwer re-
quired.

At 2 foot about, you will find 3 inches
in a foot long true meafure ; or juft
3 inches at the ufual allowance.

As fingle 18, to double i ;
S6 is t.r. to y foot, 7 inchcs' Or,
So is R, to 7 foot 2 inches, fbr the ufual
' allowance.

As t.r, or R, to the length ■
So is the feet and inches about, to the
Content.

Set the fpot by VV, for Wine-gallons, al-
wayes to the length of the Veflei, given in
in'ehes, counted on the left-fide ; then right
againft the inches, or feet and inches Dia--

For greater Veflels, count the feet on the
I«ft for ID' of inches in the lengthj and you
wve your defire.

As W, to 6 foot on the left for 60 inches;
So is 3 foot 2 inches, or 38 inches to
apj Gallons, the Anfwer required.

The Diameter and length heinggtven i»
feet and Inches, to ^nd the Content
in Betr-Barrelsy at one O-
peratlon.

Tun, counted on the left, in Feet aquot;^
inches 5 then right againft the mean Diainf
ter, found out on the right, on the leit
the Content in Barrels required.

Thus much for the Timber-fide, theuf^
of the other,or board-fide, is the fame wi^j?
that by the Compaffes before treated of, a«''
therefore needs here no repetition, unlefs ^
to the bare manner of working with it.

The fliding-Rule is only twQ Rules,
Pieces fitted together, with a fhort GrovC;
and Tenon, and two Braces at each end,
keep It from falling affunder ; and even»''
alfo is the fliding-Cover, and two edges
the infide of the Trianguler Quadrant; 1
l?he Numbers graduated thereon, are cut
crofs the middle Joynt, having the fame d''
vifions on both fides; that is to fay, on ea«^quot;
Rule, or on the Cover and Edge of the f'
fide of the Rule.

That fide or part of the Rule, on which
.^QU count the firft term in the Queftion, is
ahvayes tire firft-fide i then the other
.«luftneeds be the fecond.

So is the Mulciplicand, on the firft-fide,
where lAvas, to the Produól on thc
fcçond.'' ,,nbsp;quot;

As the firft term on the firft-fide, to thà
fécond on the other }
■ Sa is the thîrd term found on the firft-
fide, to the 4th term on the fecond-fide*

For any thii^ elfe, the fame Rules and
■precepts you find in VII. will give
you ample and plain direftions.

The Lines being fitted, as much as may
te, to fpeak out the Anfwer to the Queftion»
as by well confidering the f igwty you may
(ee.

To ma\e and meafure the Five
Regular Flatomcal iBodiesy
•»ith their Declinations ani
declinations,

IT is a Square Solid Body, every way aJ
like, and fpoken of largely before, as to
^he Meiifuration thereof,and obvious enough
to every indifferent Workman, as to the
®iaking thereof, and needs no repetition in
^his place.

It is a Figure, comprehended of 4 equi-
latteral plain Triangles, or a Trianguler Py-
ramid, aft mentioned, the beft and neareft
^ay, as I conceive of making, is thus. Ac-
cording to due^Uons of Mr. Jthn Leak.

On any rough Piece, make one fide
and flat, fo large, as to con tain J:he Triangle
which you intend ftiall be one fide of the
Tetrahedron -, then fet a Bevel to 73 degrees
31 minutes and 42 féconds ; and plain ano-
ther fide, to fit the forraeir fide, and the Be'
vel^ fecmdum Art em) y then mark this laft
plained fide, according to the former, and
cut away the refidue, plaining them away
juft to the ftrokes, and fit to the Bevel for-
merly fet, and you fhallcoiiftitute the 7f-
trahedron required,nbsp;. '

The Superficial Content is the Area of the
4 equilatteral Triangles mentioned before,
and the folid Content is found by multiply^
jng the Area of one Triangle by one third
p.art of the Altitude of the Pyramid, or Te^
trahedron, from the midft of one Plain to
the Apex, or top of the oppofite Solid
Angle.nbsp;!

If the meafuring the fides, Perpendic«'
1er, aud Altitude of the Tetrahedron with
Compaiîes, Callipers, and Scale, ferve
to exaétnefs ; then proceed thus j

being fqmiatteral.
Mukiply one fide given by 13, and di-
vide the Pipduft by ly; the Quotient
fhe Perpendiculer,

Example.
the fide of the Tetrahedron be i2,
multiplied by 13, gives which
prided by ly, leaves 10.4, for the length
■J, . the Perpendiculer in the equilatteral
^nangle^ whofe fide is 12.

Then for the Perpendiculer Jlcitude,work^
thui, by the ^Artificial Numbers and
Sines.

Or by the SeBor, work, tb^ \
P Take, 12, the fide of the Tetrahedron,
(any Scale, or) the Line of Lines, and
l^t the Se£lor to 60 degrees, by making the
^tteral 12, a Parallel 12, then thenearefl;
diftance from 12, to the Line of Lines, is
the true Perpendiculer; which meafured
the fame Line of Lines, will be found to
10-4, as before ; then make this 10-4 a
^®tallel Sine of 90, and 90 theSeftor fo
take out the Parallel-fine of 70-31-42»
meafure it on the fame Scale, a jd it
y^llbe 0-8, as before.

Bût if you ufe the Quadrant-fide, thequot;
firft lay the Thred to 60 degrees, counted
from the Head ; then take the neareft di-
ftance from 12, on the Line of Lines, to the
Thred, and it fhall be the Perpendiculer of
the Triangle, 10-4 j then fet thisPerpetiquot;
diculer in po, and lay the Thred toth®
neareft diftance, and there keep it; then
take the neareft diftance from 70-31 in the
Sines to the Thred, and that fhall be 9-80»
for the Perpendiculer Altitude of the Tri-r
anguler Pyramid, or Tetrahedron.

Laftly, This perpendiculer Altitude be»
ing multiplied by the Area of the Bafe,gives
a Number, whofe third part is the Solid^
Content of the Tetrahedron required.

For 12 the fide, and yz the half Per-
pendiculer, makes ^2-4, the Superficial'
Content of one Triangle, or Bale -, tbei^
lt;52-4, the Bafe, muhiplied by 9-8, the per-
pendiculer Altitude, gives 611-52, and3
third part of d11-5 2 is 203-80, the folid
Content required.

The three Triangles recline from the Per-
pendiculer upright, 19 degr. 28 min. and
18 fee. and decline when the edge is South
to. South Eaft, and South Weft, and the
Pppofite Plain a juft North j but if y®'-'
■ quot; ' ' ...... makç

3. For the OBabedro»,
Which is a fohd body, comprehended
^nder 8 equilattcral Triangles: The way,
making which, is thm ;
Make a plain Parallelepipedon, or long-
Cube', if the breadth both wayes be 1000,
^«t the length be 1-414; or if the length be
Sooooo, the breadth both wayes muft be
3-J3ff3 ; then to thefeMeafures fquare it
exaftly ; then divide the length and breadth
juft in the midft,and draw Lines both wayes
on all the 6 fides; then draw the Diagonal'
Lines from the midft of the length, to the
Riidft of the breadth; and cut by thefe
Diagonal-Lints, and the OElahedran will
appear to be truly made.

For the Menfuration thereof, it is the
fame as inthe7'«rlt;ifcfiro»; For, fuppofing
the fide of one of the Triangles 12; the
Bafe is 144 in Content, and the Triangles
Perpendiculer is 10-4, as before: But the
Perpendiculer Altitude is juft half the
length, viz.. 8-49 ; for if the breadth be
J2,'then the length muft be i6-j)8, whofe
halfsarelt;Jand 8-49 '■gt; Then if you multi-
ply 144 the Bafe, by 8-49 the perpendiculer

theTttrahedron, and 815-04 is the whole
folid Content of the Tetrahedron required,
as near as we can fee by Inftrumental Ope-
ration ; but if you work to a Figure more,
you' fliall find the total Area to be but
814-tf jlt;5 more exad.

Firft, The Triangles Perpendiculer being
30-4, as before; Take the Latteral 10.4,
from the Line of Lines, make it a Parallel
in s»o, lay the Thred esadlly to the neareft
diftance, and there keep it; then the Paral-
lel-diftance from the Sine of 54 deg.44 min.
45 fee. the Reclination ftiall be 8-486, the
true'PerpendicuIer Altitude required.

Then if the OEtahedron ftand on one
Triangle, you have one Horizontal Plain,
and one South and North Reclining and In-
clining ip deg. 28 mi*. 18 fee. as the Tc
trahedron was; and two South, and two
North, declining 60, and reclining and in-
clining 19 deg. 28 min. 18 fee. as afore.

But if it ftand on a Point, then you have
4dire61: or declining 45, and reclining
54,44,45 J and 4 incliners, iaciiningas

Which is a regular folid body, contained
*inder (or made up of ) 12 Pentagonal Py-
ramids, or Pyramids whofe Bafe hath y e-»
qual fides, and the perpendiculer Altitudes
^f thofe 12 Pyramids equal to half the Do-
^decahedrons Altitude, Handing on one fide^
equal to the femi-diameter of the infcri-
l^ed %hear.

To cut this Body, take any round Piece,
5nd if the Diameter be 100000, the length
»ftuftbe o-Siooy, or as 4-995'to 3-973gt;
^hen the Piece being turned round, and the
•^^vo Ends flat to the former meafures o£
l^ength and Diameters ( which are near ac-
cording to thc Sphear infcribed, and to the
Circle circumfcribing ) being meafured by
Compafles, Callipers',and Line of Lines very
carefully and exadly. Then divide the Cir-
cumference of the two Ends of the Cillen-
der into 10 equal parts, and dra-,'^ Lines
Perpendiculer from end to c iJ, an'i p^ain
all away between ti-e Lir.cs 9af a^'.d i'uj''oth,
fo that'the two Plains on bc-b e.^ds will be-^.
come a regular ten-'quot;ie ' F'gine.

Then making rl-. v i. : Diau.aer-.bcve-
taidjioooo w the Ui»; of Lines, fake out

0-309, and with this meafure ( as a RadiuS
on the Center ) at both ends defcribe a
Circle ; and if you draw Lines,from every
oppofite Line of the 10 firft drawn, you
Inall have Points in the laft defcribed circle,
to draw a Tentagon by ; which is the Bafe
of one of the 12 Pentagonal Pyramids, con-
tained in the body. This Work is to be
done at both Ends; but be fure that the
Angle of the Pentagon at one end, be op'
pofite to a fide of the Pentagon at the othet
end; then thefe Lines drawn, the twoend$
are fully marked.

Count the firft length 1000, viz.. the
meafure from the top to the bottom, or from
Center to Center i and fit this length in lO
and 10, of the Line of Lines; the Sc6tor fo
fet, take out 0-3821, and lay it from the
two ends, and either draw, or gage Lines
round about from each end ; and iu the
midft between the two Lines will remain
0-233:8; then Lines drawn Diagonally on
theio fides, will guide to the true cutting
of the Dodecahedron.

If you fet a Bevel to iilt;S deg. 33 min.
54 fee. and apply it from the two ends,you
may try the truth of your Work,

- The Dtcl'watiott and, Rtclinatlou of aB the
I o Pant agonal Plains^re as foUomth.

Firft, You have i Norch, reclining 26
34 min i and i South, inclining as
quot;lUch.

Secondly, You have 2 North declining
and reclining 2lt;J, 34; ^nd 2 South,
lt;^eclimng 72, and inclining 26, 34.

Thirdly, You have 2 North, declinii^
3lt;5, and incHning 3lt;J, 34 ; and a South,
declining 36, and reclining 2(S, 34; And
* Horizontal Plain, and his oppofite Bafe to
«and on.

Firft, find the Superficial Content of the
Bafe of one of the Pentagons, by multiply-
ing the meafure from the Center to the
middle of one of the Sides, ( which is the
Contained Circles femi-diameter ) and half
ihe fum of the meafure of all the fides put
together ; and then to multiply this Produ6t
fcy one third part of half the Altitude of
'he body, and the Produa fhall be the
Content of one Pentagonal Pyramid, being
One twelft part of the Dodecahedron ; and
^his laft, multiplied by iz, gives the folid

Content of the Dodecahedron • or 12 times
: fhe Superficial Content of one 'fide, is the
Superficial Content thereof.

Example,
Suppofe the fide of a Dodecahedron
then the fum of the fides meafured is 3

the contained Circles femi-diameter is
then 15-the half of 30, and 4-12 multi'
plied together, makedi-80; andi2timc5
this, makes 741-50, for the Superficial Cofl'
tent of the Dodecahedron.

Then for the Solid Content, multiply
lt;Si-:8q, the Superficial Content of one hd^
by 2-233,,one dth part of 13-392, the
whole Altitude of the body ; the Prodixiil
is IS7-PPP40 : Again, this multiplied b/
12, the number.of Pyramids, makes t^Sff
9P28, the Solid Cdntent, as near as may begt;
in fuch a Decimal quot;way of Computatioii.

, 'S'Forthelcofaheirony
Which is a regular folid body, made up
of, or contained under zo. Trianguler Pyra'
mids, whnfeBafe (or one of whofe Sides)
is an equilatteral Triangle; and the per-
pendiculer Altitude of one of thefe 20 Py-
ramids,, is: ;equal to half the pcrpendiculci'
Altitude of the Icofahedron, from any one
fide, to his 6ppofite fide, or equal to the
femi-di^eter of the uifcribed Sphear.

To cut this body, take any round Piecc^
and if the Diameter thereof be looopj lec
'^he length thereof be turned flat and even to
; or if the true. Round and Cillen-
drical Form in Diameter be 4910, let the
true length, when the ends are plain and
flat, be 3964 ; then divide the Cillejidrica!
part into 36 equal parts,- and plain away all
to the Lines, fo that the two ends may be
two d-fided-figures; then making jooo,
the former femi-diameter, looo in the Line
®f LineSj take out (Silt;J, and on the Center,
®t each end, defcribe a Circle; and by
drawing Lines to each oppofite Point, make
® Triangle, whofe circumfcribing Circle
'tiay be theQrcle drawn at each end; but
he fure to mark the fide of one Triangle op-
pofite to the Point of the other Triangle ac
the other end, as before in the Dodecahe-
dron ; thus both the ends fhall be fully and
truly marked.

Then making the length a Parallel'in
looo, of the Line of Lines, takeout -579,
and -cjgt;5, and prick thofe two meafures
from each end, and by thofe Points (dr^w
Or gage) Lines round about, on the 6 fides.
Then Diagonal Lines drawn from Point to
I-ine, and from Line to Point round about,
fliews how to cut the Body at X2 cuts.

if you fee a Bevel to ^-i 3 8-1 lt; | ^
33, and apply it from each end, it
guide you in thc true plaining of the fid?'
of the Icofahedron, And a Bevel fet to lOquot; ^
degrees, will fit, being applied from t^e j
midft of one fide, to the meeting of tWquot;
lides.

The Reclination of the three Triangles?
whofe upper fides are adjacent (or next) tquot;
the three fides of the upper Horizontal' ^
Triangle is 48 11 23, from thePerpendi- i
culer, or 41 48 from the Horizon'
tal, and when one corner ftands South,th«
Declination of one of thefe 3, vix.. that of
pofite to the South-corncr a direct North i
th'other two decline 60 degrees, one South'
call, the other South-weft ; theother tf, a-
bout the corners of the Horizontal-plain»
do all recline ijgt; deg. 28 min. ilt;5 fee. th«
two that behold the South, decline 22 dep'
14 min. 29 fee. and thofe two that beholii
the North, decline 37 deg* 4f min. 51 fee«
toward the Baft and Weft J the other
remaining, recline as before, and declio«
one North-eaft, and the other North-weft
82 deg. 14 min. 19 fee.

The other Nine under-Plains, oppofitet«?
every one of thefe, decline and incline, ^
much as the oppofitedid recline and dechnc,
as by due confideration will plainly ap'
pear.nbsp;quot;nbsp;;nbsp;fot

Forthéineafuring of this body-, do as
Vou did fay the Dodecahedron, find the
Area of one Triangle, and multiply it by
^Oj gives the Superficial Content; and the
quot;^ea of one Triangle, multiplied by one
part of the Altitude of the body., giveS
the folid Content of one Trianguler Pyra-
mid J and that Product multiplied by ao,
number of Pyramids, gives the whole:
Content of the Icofahedron.

quot;Suppofe the fide of an Icofahedron be 12,'
^'tft fquare oiie fide ( viz,. 12, which makes
^44) ; then multiply that Square by 13,
then divide the Produft by 30, the
^orient and his remainder is the Superfi-
''■'al Content of the Equilatteral Triangle^
quot;^hofe fide is 12 ; namely, 62-400; or
'»lore eyaaiy, xhe Square-root of 3888,
^hich is neat 62- 3 54 ; 20 times this, is the
superficial Content, namely, 1247-08.

Then for the Solid Capacity or Content,'
^ukiply 3-023, the fixt part of the bodies
Altitude, or one third of the Pyramids
;^ltitude, by lt;J2.35'4, die Area of one
Itianguler Bale, and the Produft will be
*88-49322p, Laftly, this multiplied by
the number of Pyramids in the Body,
ynbsp;the

Thus you have the way of cutting, aquot;®
the Declinations and Reclinations and Me®' ^
furesj Superficial and Sohd, of the f Re' j
gular Bodies, as near as by Decimal AC'
compt to 100 part of an Integer mayb?)
the exaft meafuring whereof, requires the
help of Algebra, whereof I am ignorant.

The Meafures of the Containing, and
Contained Sphears, Circles, and DiamC'
ters, Sides and Axis's, Diagonal-lines and
Altitudes of the five Regular Bodies, gS'
thered in a Table to a Containing Sphear,
whofe Diameter was 10 inches (or Integers)
found out by Geometry, according to this
Scheam, taken from Mr. Tho. Diggs.

Let the Line A B be 10 of fome Diagoquot;
nal Scale, reprefenting the Diameter of the
Containing Sphear. Which Line A B, yoquot;
muft divide into two parts at C, and inC !
three parts at E ; A E being one third part»
and on the Points C and E, raife two Lin^s
Perpendiculer to A B ; and with y of your
Diagonal Scale, on tbe Center C, defcrib®
the femi-Circle A F D B, and note tfac
Points F and D, in the femi-Circle, with r*
andD, drawing Lines from either of theii'
to A, ar.d from F to B.

[ ipi ]
Divide A F by cxtream andmcM
Proportion ; the greater Segment being aG,
(by the loth Problem of the dth Chapter)
then extend the Line AF to Ff, making F fl
equal to F G, and draw the Line H B; and
from F, draw another Line Parallel to H B,
^tting the Diameter in I, and from I, draw
^ Line Parallel to CD, as IK; then make
1L a third part of IB, and draw M L
I^arallel to I K ; alfo, draw the Lme M B,
and divide it into two parts at N, and into
4 parts at f; then divide the 4th part, M S,
by extream and mean Proportion , whofe
greater Segment (or part) let be S V ; then
^vide F B iii 4 parts, making F O the
quot;»If, and FR thc quarter ; divide like-
Wife FE in two parts, and at the middle
The being thus made, then
^vith your Compaffes, and Diagonal Scale,
you may meafure all the Diameters, Sides,
atid Altitudes, of all the j Regular Bodies.

Jf JOH mgt;»U have the llke^ for any ethery
then fay by, the Line of Nambers, or
Line of LiaeSj^ tr EfUe..of Threef.
thtu;nbsp;» . i r ,

As thefide (,Diameter or Altitude.) for;
IO, as in the Ti^/«, is-to the given
Side, Diameter, or Altitude ;

So is any other Number, in the Table, for
. Diameter, Side, or Altitude, to bis
Proportional Meafure retjuired.

ïhave a Dodecahedron, Vfbofe Side « 6,
' fVhat jhall aB his other Sphears, or
Circles, Diametersiir Altitttde be?

The Extent of the CompafTes from
3-570, -the Dodecahedrons-fide in the
Table, to 6 the fide given, lht4l teach iram
10, the Cftntaining Sphears Djameter in the .
Table, to 16, theContaining Sphears Dia-
meter, for a Dodecahedron, whofe fide is
And from, 7-P70 the Contaiüed
Sphears Diameter, to 12-643, the Con-
tained Sphears Diameter. And fo for an/
other whatfoever.

CnnWiningSpheir.
The cJOta ning bpheirs Diame- AB
ccr.thstcompiclwndithebody lO.ooQ
in it,is foterec one of them.—

contained Spheii.
Tht contained SphewS Diime.
tet that u contiineii in the bo-
dy, ctlled alfo Axis, is —— EC
The half thereof, lt;»

ContsiningCitcIc.
The cbntjming Cirdes Diamc
ter, ( or the Diameter of that 9.419

The half thereof, jgt; • ■ -
Contained Circles.
The conrained Circles Diam!-
ter, Comprehended in the Baft

This TaUt was gathered from this (7/9-
^'trical Figure, drawn on a Slate, by a
good Diagonal Scale of lt;S parts in a Foot,
^^hereby I could very well copie to the
part of an Integer ; and is true
Enough for any Mechanick Operation, for
J^hofe ufe I only do it, and I hope it may
^^ as kindly accepted, as it was carefully
Calculated, and offered to Publick view.

GAging of VeJels, is no other than the
Meafuring of Solid Bodies; and the
former dire6l:ions for folid Meafure, con-
veniently and aptly applied, is fully fuffi-
^'ent; only obferving this difference. That
the refult or ifl'ae of the Queftion is to be
^eudred in proper terms, according to the •
V 4nbsp;demand

demand of the Queftion, as thus; in
fur ng of Timber or Stone, the ^JueftioH
is, Hovf many Feet, or Inches, is there In thi
Sola Body? But in Gaging, theQiieftioii
is. Hew many Gallons, Kilderkins,or quot;Barrels
«there in the Vejfel to be meafured f Fof
which purpofe there are fit Numbers, o^
Gage-points, requifite to be known, for tbe
more fpeedy attaining the Anfwer to the
Queftion, of which in their order, as fcgt;l'
loweth;

Firft, You are to remember. That the
fohd capacity of a Wine-Gallon, is
Cube Inches ; a Corn-Gallon 272 Cube
inches J an Ale or Beer-Gallon, is 282 \
Cube inches; or as fome fay, 288 Cube i»'
ches; So that when you have found the |
Content of any Veflel in Cube inches,
you divide that fum in inches, by the re'
ipe£tive Number for the Gallons you woul»
have, the Quotient fhall be the Content ii^
Gallons required.

From hence it follows, That to meafure
any Square or Oblong Veflel, you
Hiultiply the length and breadth taken
mches, and tenth, parts, together; that is

'ay, The one by the other ; and the Pro-
duct fliall be the Content of the Bafe ii) in-
'^hes, fuperficially : Then mukiply this Su-
P'^tficial Content of the Bafe, by the inches
^f^d tenth parts deep, and the Prodnft fhall
the folid Content in Cube inches ; then
divide this Produôl: by 282, gives the Con-
tent ini Ale-Gallons in the Quotient, and the
'■f'tiainder, if any be, are Cube inches.

But if you divide by 10161, the Cube
inches in a Beer Barrel ; or, by 9032 the
Cube inches in an Ale Barrel ; the Quotient
^eweth the Number of Beer or Ale Bar-
'^elsj (and the remainder Cube inches.)

The length let be 78 inches and i tenth,
the breadth let be 320 inches and y tenths,
and the depth 9 inches and y tenths, or
half an inch ; by multiplying and dividing,
as above, you will find 843 Gallons, and 68
Cube inches, to be the folid Content of thac
Cooler ; which work is very readily done
by the Line oi Numbers, in this manner ;

Extend the Compafles from i, to the
brcadth or length ; anlt;l the fame Extent fhall
leach from the length or breadth to a 4th,
Which is the Superficial Content of the
^afe, or bottom, In Superficial mches.

The Extent from 282 ', to the laft Nuffl-
ber found, ftiall reach the fame way froffl
the inches, and tenths deep, to the Content i»
Gallons.

The Extent from i, to 78-1, fhall reach
the fame way from 320-5-, to 25031; then
the Extent from 282 ^ , to 25031, IhaD
reach the fame way from 9-5, to 842-68,
the Solid Content in Gallons required.

Indeed, you muft Note, You cannot fee
fo many Figures on the Line, as the Produ(5t
of 4 figures multiplied by 3 j yet by the
Rules linChap.6. SeU.i.) you have di-
re»aions as to the number of Figures, which
here is 7 ; the two laft (next the right hand)
being Fradhons, or parts of an Inch, and is
therefore negledfed.

Individing the Produft of, 25031, and
9-5, multiplied together, which makes
Figures befide the Fra6\ion by 282, there
muft needs be three Figures in the Quotient,
which arc the Gallons': This artificial help
you have, behde the prefent view of the
VefTel, which will dircdt you not to call
842 Gallons, only 841, nor 8420, as you
muft needs do, if you miftake as to the de-
nomination.

, Vou need not to trouble your fcif, to
what the 4th Number is 5 but having
^Und the Point reprefenting it, keep the
^ompafs-point fixed there, and open the o-
Jhfr to 282 -1, where you may have a Brafs
C^ntcr-pin for more readinefs; but let your
j'^'^ount go as 282 5 to the 4th, for methods

muft fay, fo is the depth the contrary
J^ay to the Content in Gallons. All this is,
'lilted for plainnefs and caution fake, in be-
■'^efit to young Learners.
- Alfo Note, That if you would havFhad
^e Anfwer in Ale or Beer Barrels; then in
ileadof 282!, you muft ufe the Point at
for Ale Barrels; or the Point aC
61 for Beer Barrels, being the number
Cube inches in thofe Barrels, as 2821 is
^he number of arches, in a Gallon of Ale or
Beer.

Example for the fame Cooler.
The Extent from i to 78-1, fhall reach
from 320-j, the fame way to 2 J031; the»,
^he Extent from 10161 to 2J031, fhall
gt;^each the fame way from 9-j to 23 i, the
true number of Beer Barrels required. Or,
The Extent from PO32, to 2J031, lhall
'■each to 26 BarrelSjand near i third: which

The pUjin and natural way for meafuri)'^
of a round Tun, is this ; Meafure theDi^quot;
meter in inches and tenths, and fet
♦ half thereof; Meafure alfo, the CompS''
round about the infide, and fet down tl^^
ha^of that alfo, in inches and tenth parfS)
apd.multiply thofe two Numbers togeth«*quot;»
the Product fhall be the Content of tb^
Bafe, or bottom, in Superficial inches; thequot;
this Produft multiplied by the depth inii^'
ches, gives thequot; f^olid Content in inches» i
thqn laftly,' this Product divided by
orbyioifji, or by 9032, gives the foli'^
Content in Gallons, or Beer, or Ale Barrel'»
as before.

For, half the Diameter, and half th^
Circumference, doth reduce the round Ve'j'
fel to an Oblong Veflbl, equal to that rouiiquot; 1
Y^flel.nbsp;i

«Ut when the Véfletis Tipery thac is to
the-bottoitJand topcif difterent ttamei^quot;
as generally tlwy all are; then the
^ll^f cai-p is co come' by the true Diameters^
to? ^^^nbsp;^ flidi!i|;Rule applied

iiifide, whofe regular equal com^ca^'
°^»sthBS to be ordered , - '
p WhentheVeffelis taper,'^'and the'Sideï
thnbsp;like the Segment df a

you nray add the Diameters 4k tb^an^
f^ttom together, and cotBit the half TtniS-'

^ ' the mean Diameter of fhat taper yefifel,
j^*}^ triultiply half that Dia-meter, and. half
an!lnbsp;Circumferencè,quot;as-6éf9pe ;

multiply andquot; divide, to get tWfoIid
in Gallons,or Barrels. ;
But when the Staves are'bending, ^moft
J your clofe Cask are. then the rêadieft
^ay to come to a mean DianSetei-, is thus';

' ihall-find moft trite fór feveralCask:
So is the difference of Diameters- to a 4th
Number, which is to be added to the
leaft of the Diameters, to make up a

If the Sides be round or arching, ^^^
the lefs Diameter be 30 inches, andcb«
greater 40 uiches ; then. As 10 to 7 i
is 10, the difference to 7 inches ; whid'
makes ( being added to 30, the leaft Di^'
meter) 3 7, for a mean Diameter.

But Nott, It is hinted by
Thar Veflels, ufually, are between a Sp^^'
roid and aParaboIickSpmdIe; then, if
10 to 7, be too much to add to the
Diameter j Tou may fay.

' So is the difference of Diameters to a '
Number, which you muft add toth«
leaft Diameter, to make a meanpi^'
meter.

Having thus gained a mean Diamete^^
you may work as before; or rather ù^t
more readily and eafily, by the Line
Numbers, thtu;

The Gage-point for Wme, and Oyl-
gallons, at 231 Cube inches in a Gal-«
Ion, is-----17-1 r

The Extent of the Compaffes, on the
Line of Numbers, from the Gage-point to
^quot;e mean Diameter of a VelTel 5 bemg turn-
ed two times the feme way from the length
of a Veflel, (hall jeach to the Content of the
VelTel, in Gallons or Barrels, accordmg to
the nature of the Gage-pouit.

A mean Diameter tei^ 30, and the
■^fngth 40, the Content is ii Wice-gallonj
near.

The Gage-point of any Solid Meafure^ is
o!nl;f the EHameter of a Circle, whofe Su'
perficial Content is equal to the Solid Con-
lent the fame Meafure.

. i/fs that more pialnly ^
TheSolid Content of a Wine Galloir is'
2131 Cube iirches: Now if you have a
Circle that contains 231 Superficial inches,
the Diameter thereof will be found to be
i7.inches, and if of a hundred ; as by the
■ir^?roamp;,em of the 'jth Chapter, is well
feen.

Thefe Dire61:ions may ferve for any round
Vefiel, either clofe or open; yet Mr. Ough'
trei, a very a6Ie Mathematition, hath »

»Way accounted fomewhac more exa£l, and
«onfequently more tedious «nd troublefomc
either by the Pen or Compaffes.which

You muft meafure the Diameters at head
«nd bung, or the tcp and bottom in inches
loths, the length alfo by the fame mea-
gre J then find out the Superficial Con-
j^i^t of the Circles, anfwerable to thofe two
diameters, and take two thirds of the grea-
and one third of the leaft, and add
^quot;etn together in one fum ; which fum you
•^uft niukiply by the length in inches and
f^is, and the Produ6t (hall be the Cotitenc
^ Cube inches j which Produ6t divided by
gives Ale Gallons j or by 231, gives
Wine Gallons, as before.

The Extent from i, to o. ƒ 2 jtf, a Num-
ber fit for 2 thirds, of the Circle at
the bung •

.quot;a X, too-2lt;Ji8, the half of the fof-
mer Namber, and fit for onc third
theCircle athead;
So is the Square of the Diameter at hcad
toa4th.

Asaji ( forVVine, or 282 ^ for Alc-
Gallons ) , is tp the fum of the tw»
4ths added together;
So is the length to the Content in WinC
Gallons.

The Extent of the' Compaffes from
«-J236, fhall reach from 1024, the Square
of 32, 10536-4, tvvathirds of the bung'

The Extent from i,to 0-2Ó18, fhall read'
from 324, the Square of 18, the Diamet^
athead, to 84-9, the fum of 53^-4,
84-9, is Ö21-3.

'f^chfrom4o the length, to 107-J8, or
Gallons and a half, and better, the
J-ontent in Wine-gallons, as briefly aj can
done this way/

But if you take the Diameters at head
quot;a bmig, with a Line called Oughtrcd'j
1 ^ge-line i and fet the meafure found at the
by that Line, down twice; and the
^''eafurefdundatthehead, found by the
^^'Ue Line, once, and bring them into one
i then nmltiply that fum by the length
the Veffel in inches, and 10 parts, and.
y the Pi odua fhall be the Content in
^ine-gallons required.

this way fliould prove the
' ^elf ift the Book of the Carpmtrs'Kult,
y have iTdlt toreaifie this difference,'
^Jiich you will very feldom have occafion

Notedfoy That this Line, called
trti'% Gage-Line, is very excellently iraprf
ved to find the Concent of Great VelTels»
either in the whole, or inch by inch ; which
you will find at large in the Book before
mentioned.

tAlfo, The ufe of the Lines called Dia'
gonal-Lines, and Lines to find the empti-
nefs of Cask, and to meafure Corn-mca'
fures by, t6 which 1 lliall, for the prefenc»
refer you.

Ejttcnd the Compaffes from the Diameter
to the Gage-point, the fame Extent twi'®
repeated from the Content, fhall give ih«
length requiredt

If the Content be 60, and the Diamcte'
24, then extend the Compaffes from
to J7-iy, the Gagc-point for Wme; tlquot;^
Extent turned twice the fame way, from
the Content, fhall reach to 3o mches,anlt;'
P tenths, and a half, the length required.

, Divide the fpace on the Line of Num-
between the Length and the Content,
'quot;to two equal parts j the Compafles fo fet,
reach die lame way from the Gage-poinc
the Diameter of the Veflel.

length, and do the Content, fhall reach the
Wway from 17-if the Gage-pmt, to
H the Diameter required.

Thefe two laft Problems may be ufeful for
^oopns, to make Cask of any length, diame-
and quantity.

Problem VI.
Tefini vbat it wanting in any cloft Cétkf
at any number tf incbis and forts, (tbt
Cénk, lying after the ufual manner, mtb
the bung-bolt ufftrmoïb ) from the
bung-bale to tht fuferficies tf tbt Liqiuij
^I'vf», two wayes.nbsp;:

This Problem I fhail refolve tm v^ye),
either of which i$ experimti\ted to c hiic
X Jnbsp;near

Firp^hy the Line of Numbers, ani
(ial Line of Segments, to find t he qutr^jquot;
ef Galons that any Vefftl wants of bst4
fuU, at any number of Inches, from
injideoftbe bung-bole, tothe fuperf
of the Liquor, which « ufually Ci
Inches dry.

Extend the Compaffes, on the Line quot;j
Numbers, from thp inches and tenths diaif''
ter at the bung, to looon the Line of
ments, the fame extent applied the fame vv'^^
ifrom the inches and parts dry, fhall rea^j
to a 4ch Number, on the Line of Artific'^
Segments; which 4th Number you
Iceep. ( Or, if you will, you may ufe
inches wet, laying the fame extent from ^
iflchcs wet, and that alio will on the SeT
'..........mefiquot;

As the Extent from i,to the whole Cdn-
tent of the Veflel in Wine or Ale-gallons j
So is the 4th Number kept to the Gallons of
«niptinefs,or fullnefs, that it wants of being
fuH, or the quantity of Gallons in the Vef-

meter ai hung, is 28 Inches and 7, 4»»
fun Content is Gallons ii6 \, at 12
inches dry, or 16 inches, and 7 tenths
wet.

The Extent of the Compafles from 28-7,
to 100, (at the end of the Line of
ments) fliall reach the fame \yay from; 12,
the inches dry, to 39 on the Line of Seg-
mentsfor a 4th; or from ilt;S-7 wet, to
lt;5oon the Segments, for his 4th alp,
which two 4ths keep. .. *nbsp;•

The Extent from i, to 1161, the whole
Content in Gallons, fliall reach from 39,
the dry 4th, on the Line of Numbers, to
4lt;J 2_, for the gallons dry or wanting : or
the lame extent fliall reach the fame way, 011
- the Line of Numbers, from 60 , the 4th
Number for wet, to 70 gallons,and a tenths
quot; X 4nbsp;in

in the Vcflel, at \6 inches and 7 tenths wet;
which t4o Numbers put together, makes
up 116 galloiis and a half, the fall Con-
tent.

The like manner of working ferves for
»ny Cask whatfoever, and the nearer the
Veflel wants of being half empty, the more
«ear to the truth will your work be, and
the moft errour in very round and fwelling
Cask, when the emptinefs is not above one
or two inches; but in Veflels near to Cil-
lenders, it will give the Anfwer very true,
and as readily as any way whatfoev.er.

Ohfcrv« aife. That if you ufe the Seg»
ments in taking the wants, you muft abate
cf the gallons found, till you come to the
t thirds of the half diameter; that is to fay,
the Rule fayes, there is more wanting than
indeed there is; and that fomewhat con-
fiderable about the firft 6 inches in a veflel of
|o inches diameter: So that 1 RndsLTabl*
mad? as a mean between the Superficial and
folid Segments, would do the work the
trueft and beft of any other ; Or elfe, ufe
the mean diameter and mean parts of empti*
ncfs; found thus.

Take the eqnaded diameter, from the
idiameter at the bung ; and note the diffc
Tcnce : then half this difference taken frc«n
the inches and parts empty gives the mea»

^uptincfs i then ufe the tncattdjaoieter, and
emptinefs, inftcadof the other, and
'oe work is more exadt,

tion it thm ;
Firft, you are to fill an ordinary Cask,
a competent magnitude, as do or 109
^•ons, of a mean form, between a Spheri-
(or rouudifh form ) and a Cillenderi-
^ form; or elfe fill two Casks of each form,
learn the true Content^ and Diameter of
^^Jat mean Vefltl, or rather of both thofe
Vtflck J and the Veflel being full, draw oft^
^'tha true gallon-meafure, and on the
tlrawing off every gallon, take the exadl
Quantity of inches and loth parts, that the
^rawii^ (M of every gallon makes in the
cnptinefs or drinefs of that mean Veflel, or
»athtr both thofe Veflels, at leaft until you
have drawn off the half quantity of the
Veffel, which number of gallons drawn off,
and the ihches and tenth parts of emptinefs.
Or fulncfs, or drinefs or wietnefe, you muft
draw into a rab/e, or infert them on a
Hule, making the inches as equal parts, and
the gallons, and his proportional part of a
Rallon, the unequal parts.; then with the
tine of Numbers, and this racanTiWf, or
^«her two TUbUt or Seättf which you may

put on a Rule, as Mr. Sennit hath donejycn^
may find out the wants of any Cask what-
foever • cither Spherioid, or Ciilender-like,
as followeth.

This meafured Cask on the Scale, o'^
Table, for meithods fake, and avoiding taU'
tologie, I fhall call the firft Cask, and the
Veflel or Cask,-whofe wants you woul«!
know, I (hall call the fecond Cask 9 theo
the proportion is thus.

As the Diameter at the bung of the fé-
cond Cask, is to the bung diameter of
the firft Cask (which is always fixed) gt;

So is the inches dry of the fecond Cask f
a 4th (on thc Line of Numbers) which
4th Number fought on thé inches ot
your Table, or Scale, on the oppofitc
part of your Scalc or Table, gives a Jtb
Number, which yau mull^eep.

As the whole Content of the firft Cask,
is to the whole Content of the fecond
Cask;

f So is the firft Number kept, to the Nu^*'
ber of Gallons the VelTel wants of be
ing full, at fo many inches dry.

There is fuch a Scale made en purpofe fo^
Viduallers ufe, to meafure what they wao^
of a Barrel of Ale, being put into a Beer

Suppofe, as before, a Canary Pipe want
li inches of being full, and the Content
gallons, and a8 inches and 7 tenths
diameter at bung; The Extent on the Line

Numbers from 28-7, to aa-y, fhall reach
from 12, to 9-4 J then juft againft p inches
and 4 tenths, on that Barrel Scale, I find
'4 gallons of Beer, which is 17 gallons and
a half of Wine, being the yth Number to
he kept.

Then the Extent from 44, the Content of
»Barrel in Wine-galions, to iilt;Sj, the
Content of a Canary-Pipe in the fame gal-
lons, fhall reach the fame way from 17.^ the
Number kept, 1046, and near a half, the
gallons wanting at 12 inches dry, in the
Canary Pipe, and 46 gallons, and 5 quarts,
is the Number Mr.B*»«»t finds in a Canary-
Pipe, by meafuring at 12 inches dry.

Thus you have an account of the two
cafie Mechanick wayes,tq difcov^r the wants
of Cask, very; applicable, and ready, and
experimented to be Prope verum.

iAtany ratt of Imirtfi per annum ff*'
hundred founds, to^ nbat tht httrV
quot;fnbsp;mrleStrfHm comes tt i*

Extend the Compafles from xootothe
increafc of loo /. in one year, the faflO«
Extent fhall rejch from the fum propund'
ed, to its increafc fbr one year, at that ra^®
propounded.

ffhat is the increafe or profit of 124 /. \01.
for one jearj at 6 ftr cent, per annm ?

t* (»vhlch Is It ^124-j ) ftall reach r®
^-47, which is 7 /.- p - 4 d, the pro-
ot 124-10 s in one year.

Problem II.
K) fum of Monty, and the ratt ef
ttrefl propoMnded^to pnd what it will in-
ertafe to, at any nnmber ef years, count-
ing I atereft upon Jnttrefi.

quot;The Extent of theComraffes from 100,
ro the ncreafe of 100, beif| turned as ma-
quot;y times from the fum propounded the fame
I there be yeafsjjropounded, lhall at
ait Itay at the Principal and Intereft rc-
•JUired.

To what fum jhaU 143 pounds 10 fbiUings,
amount to tn ictytars, counting Intereft
»Plt;gt;»I'gt;tertft,at6ptrctnt} -

Nott, That in doing this, you ought to
I . oc very precife, in taking the firft Extent
J jrom 100, to to6; but to cure the uncer-
ainty thereof, you have this very good rs-
quot;^Wy: If you have a Diagonal Scale, equal
Ynbsp;to

to the Radius of the Line of Numbers, tb^
ufe that ; if not, ufe the Line of Lines on the
Se6i:ot-fide, which (hould be made fit ^
(or the double, or the half of ) the Radius
of the Line of Numbers.

Take the Extent from the Line of Nu^'
bers, between loo, and io6 ; this Extefli^
meafured on the Line of Lines, will ^
O2J30j8, could you fee fomany Figures,
but 02f3i, will ferve your turn very well»
which Number you muft uote, is the Lo'
garithmof io(J, negle6i:ing the Cara^ter»'
ftick; then this Number multiphcd by ï'''
the Number of years, is 2^310; this B*'
tent taken from thc Center, on the Line
Lines, and laid increafing from 143 i,
reach to 2^7/. oj. od. the true Numb^'
of the Ufe and Principal of 143/. iOf,
put out, or forborn for ten years.

Problem IIL
'AfumofiMoney bting dut At any tlt»t^^
come, to fnd what it is worth i»
Monty to be paid prefentlyt at any f^

This Problem is the contrary tothela^J
for if you fhall turn thc Extent betwc
100 and iolt;J, ten times backward

Multiply 025'30j8, the Logarithm of
by 10 ; then this Extent taken and laid
'hedecreafing way from 257, flwll reach to

For Note, That the Line of Lines is the
^cale of equal parts, that makes the Line
Qf Numbers, and lo, or 7,or if, or any
other Number multiplied by the Logarithni
of iolt;J, taken from that Scale of Lines all at
°quot;ce, is equal to fo many repetitions 5 and
confequently more exa£l:, becaufe of the dif-
ficulty of taking the 10, 12^ or ijth part
of any Number whatfoever ; and obferve^
That fo much as you err in the firft, it will
he 10, 12, oris, or 20 times fo much ac
laft,

A yearly Rtnt, or Amaitj belvgforborn a
certain number of years j to find what thi
Arrears thereof will ammnf ttntq, ac-,
carding to anj rate frofounit^^ . ■

Firft, you muft find out'thé Prihcipal-
Money, that anfwers to the R,eiit; ' or An-
«uity in queftion; then' find' the fum of
Y2 ■ ■■ ■■ thaf

that Principal and Ufe, at the end df tW
term given, at the rate propounded; the'
the Principal taken out of this fum, both
Arrears and Principal, the Arrears do r«' j
main, which is the fum you look for. '
Examplt.

Suppofe a Landlord live far from h)^
Tennant, and yet judging his Tennant hf
neft, and able, is content to take his Requot;'
once in every fourth year, which ihould |
paid every year, or every quarter of tb'' :
year j and fuppofe the Rent hz\o I. ptr
mm, and the rate of profit, for the fo^'
bearance, be 8 ptr cent.

Firft, to find the Principal for lo /. K
Annumy at the rate of 8 /. per eint. S4jf
If 8/. have ico for his Principal, wha'
fhall 10 /, have ? The Anfwer will be 12? '
for the Extent from 8 to loo, fhall read'
from lo, the fame way, to j

the 2d Problem of this Chapter, 12j /. fcr'
born for four years, -will come to
which is 170 l.os. o d. from which fum»^
youfubftraa: 12j/. there remains 45/. t^J -
Arrears for lo I. ptr annum for born fo''^
years, at the rate of 8 ptr cent.nbsp;.

But if you would have the profit of thcf^
Arrearages, fuppofing a/.-^io/. the-^''
part of 10 /. per annum to be paid quarterly»
and to coyint Ufe upon Ufe at the rate abolt;

Line of Lines, and laid from 120,on the
J of Numbers, fliall reach to 171 j, or
[j^ ^10 s. being 30 s. more than the for-
I k ^ when I JO 1. the Principal is ta-
^ away, the refidue Arreares is 451, i o s.

j^s between 100 and 102, 16 times from
T' which you may help thus; turn firft
tj dien take chem 4 times in one Ex-
^ and turn 3 times more, and you will
V at 2711, the Anfwer required.

Firft, by thc 4th Problem, find thcArJ
•^ars that fliall be due at the end of thc
and at the rate propounded ; then by
3d Problem, find what thofe Arrears
Worth in ready money, which ihall be
^ Worth of the Annuity,or Rent required.

î Example.
7here Ù a Leafe ef a Hottfe or Lani mrth
121, fir annum, and there is ^^yeor^
yet to come j which Leafe a man vm*'*
buj^ provided he may lay out his mone)''
gain after the rate of lo /. per cent : tUf
qneflion is, What is it mrth ?

Firft, by the laft, if ipl. have lOO fof
his Principal, What fhaU 12 ? the Anfwe'quot;
is 120 ; Then by the fécond part of th^
fecond, 120 L-fbtborn \6 years, comes
yjil. the Principal and Intereft : froquot;'
which fum, taking 1. the Principal,th£r^
remains 431 the Arrears» Then by the thir''
Problem find what 431 due 16 years
come, is worth in ready money j and rij
Anfwer will be at 10 in the 100, ^y'
14 s.

Alfo herein obferve, that if there be 2»y
Reverfiou of a Leafe to be expired, before
may be in joyed j then you are to findtP^

worth of 4311. after fo many years mor«»
as fuppofe ic be j years before the Annu^
begin -, then find the worth of 431,
born 21 yciurSjwhichwrillbefSl. 45»

Problem VI.
A fum ef Money is fropouniei, and the
rate whereby a man intends to Punhafe,
to find what Annuity, and how many
years to continue, that fum of money mil
buy.

Take any known Annuity at pleafUrc,
find by the laft, the value of that in
'^eady money, then this proportion holds;

ffhat Annuity, to continue x6 years, wiS
JOO /, Purchafe, whereby a man may
gain after the rate of lo /. per cent f

By the lall Problem I find, That 93 1.
14s. will purchafe 121, a year, for iiJ
years, at 10 per cent.

Therefore,
The Extent of the Compaffes from 93 1.
to 12 I. per annum, fhall reach the fame
from yoo, to 641. per annum. For fuch an
Annuity, to continue i6 years, will 5:001.
purchafe, to gain 101. per annum, percent,
f^ot your Money.

tands or Houfet, fold at any certain ndf^
her of years Par-cbafe ^ to find what thf
value of the whole will he !

The ufual way of valuing Land of
Houfes, is by the years Purchafe, and Laoquot;
Fee-fimple is ufually vallued at 20 yearj
Purchafe; Coppy-hold-Land, at if or»®
years Purchafe ; and good, ftrong,and ne^^
Houfes, at 12, 13, or 14 years Purchalquot;^
for Fee-fimple.

But a Leafe of a Houfe of 21 years about
7 years Purchafe; and a Leafe of 31 years,
about 8 years Purchafe, rather lefs tha«^
more; and a Leafe of do, or 100,
worth above 81 years Purchafe.

The ufual profic allowed for inFcC'
fimple, is not above j 1. in the 100 f
fium, becaufe of the certainty thereof; fquot;'
Coppy-hold Land, full 61. in the 100 ff^
annum; for the beft Houfes, 7 and 81«
the xoo Fee-fimple.

But in laying out Money on Leafes, fith^'^
of Land or Houfes, Men lhall hardly bef»'
vers, if they gain not 8, p, or 10 in the
ftr anrntntf fer theu Money • The yeafo^

and demonftration whereof, you may read
large in Mr. PhiSlfs his Pttrehaftrs Pat'

Thus the number of years Purchafe a-
Si^eed on, ( which ought to be deer, from
^it-rent, and Taxes, and the like; the
^ent is ufually various, according to the
fUce, and time where, and wherein, the
^Urchafe fhall happen to be ) then to find
the quantity of the whole Purchafe,

As r, to ao,i8,i5,i4,i2,io,or8,thc
number of years Purchafe, for Fee-
fimple, or Coppy-hold Land, or
Houfes Fee-fimple, or Coppy-hold;
For Leafes of 60, yo, 40,0r 30 years,
or 21 years;
So is the yearly Rent to the whole value,
Examplt.

A Parcel of Land worth 10 /. ptr aMttum
Fee-fimple, valued at 20 years Purchafe,
will amount to 2001.

The Extent from i, to 20, will reach the
fame way from 10 to 200, the whole price
of 20 years Purchafe, at 10 /. ptr anmm.

'Any Ntimher of Sonldiers being profound'
ed, to order them Into a Square Battel'
ef Me»; that is, at many in Rank''*
in File.

Find die Square-root of the Number of
Souldiers, and that fhall be the Num-
ber of Men in Rank and File required.

As fuppofe it were required to oidei
1770 Men, in the order abovefaid, yoU
fhall, by the Probl. of the Chapt.
find, that the Square-root of 1770 is
and 6 oyer, which here is not confider-
able.

Problem II.
Any number of Souldiers propounded, tv
ordtr thm into »double Battel of Men;
that is to fay, twice as figt;(t»y i» Rankjt
File,

Pind the Square-root of half the Num-
ber of Men, and that is the Number of
^en inEile, and die double che Number m

As for Example, , ,
If 2603, were fo to be placed, the halt
2603,151301; whole Square-root by

the 8th of 6th, is 36; the number of Men
'nFile : and 72, the double thereof, is the
«umber in Rank. For if you (hall multiply
7i by 36, the Produft is 2592, almoft the
number of Men propounded.

. Any Numbtr of Souldiers being propound-
ed, to order them into a Quadruple
Battel of Men-, viz. times as manyt»
Kansas Filf.

Find the Square-root of a 4th part of the
Number of Men, and that fliall be the
Number in File; and 4 times fo many the

So the 4th part of 2603, is ^yo; whofc
Square-root is 2y}., and 4 times 2c is 100,

'AuyNambtrof SoHliitrs bting givm, W
getber witb tbelr Diflance one from ano-
tbtr tn Rank, and File, to ordtr tbtm
into A Square Batttl of Ground.

As fuppofe I would order 3000 Men fo,
that being 7 foot afunder in File, and 3 foot
apart in Rank, the Ground whereon they
flood Ihould be Square.

Extend the Compafles from 7 foot, the di-
ftance in File; to 3 foot, the diftance in
Rank; then that Extent applied thc fame
way from 3000, the Number of Souldiers,
reaches to 12U, whofe greateft Square-
rootis 3y-7 J that is, 35, the Number of
Men to be placed in File.

If you divide 3000, the whole Number,
by 3y-7, thcQuodcntis 84, the Number
in Rank, to ufe and imploy a Square plat of
ground to ftand in.

Any Number of SouUiert propoufided, to
order them tnto Rank File, accor-
ding to tht rafio of any two Numbers
giveh.

As the Number given for the diftance in
File, is to that for the diftance in
Rank;

So is the whole Number of Souldiers to
a 4th, whofe Square-root is the Num-
ber of Men in Rank.

Suppofe 3000 Souldiers were to be or-
dered in Rank and File : As 5 is to io,or as
y is to 9 ; that is to fay, that the Men in
Hank, might be in Proportion to them in
File, asjjistoj.

Thtrt are 8100 Men to h ordered
Square Body of Men, andtobaver
many Pil{es, as to arm the main Square^
Body round about, with 6 Rankj quot;J
Pikes; the Queßion is. How
Ranks muß be in the whole Square Bat'
tel ? And, How many Pikes and Mfj'
quets ?

Firft, the Square-root of 8100, is
the Number of Men in, and Number oi
Ranks and Files; now in regard that tbe^^
muft be (J Ranks of Pikes round about tb^
Mufqueciers, there will be 12 Ranks lefsquot;'
them, both in Front and Flank, than in tb^
whole Body; therefore fubftra^ing of
from 90, reft 78, whofe Square is do84»
the number of Mufquetiers; which take''
from 8100, there remains 2061, thenu«'
ber of Pikes,

, Por as much as like Squares, are iti
double the Proportion of their anfwerable
'^«lesi therefore you muft work by their
Squares, and Square-root.

If a Fathom of Rope, of 6 inches com-
Pafs about, weigh 6 pound, 2 ounces, ( or
what fhall a Fathom of Rope of
inches compafs weigh ?
Here Note dxvajes. That when the two
•^umbers of like denominauon, which are
Riven, are of Lines, or fides of Squares, or
liiameters of Circles; then the Extent of
the CompafTes upon the Line of Numbers,
from one Line to the other, or from one
fide to the other fide; that Extent turned
twice the fame way from the given Area, or
Content, fliall reach to the other required:
So here,the Extent of the Compafles from
® to 12, being turned- two times the fame
^vay from 6-125, ^hall reach to 24-f o, for
24 pound and a halfjthe weight required.

But if thc two terms given of one defquot;'
taination, arc of Squares, or Superficies,
Areas; then the half diftance, on the
of Numbers, between one Area and the^
ther, being turned the fame way oDtPj
Line, from che given Line or Side, it ih»quot;
reach to the Side, or Line, required.

For the half-diftance, between
and 6-12J, lhall reach firom 12 totfj
the contrary, from 6 to 12.

An Example whereof, you have in tbfi
4th and jth Problems of the i2thChapte''''
Alfo, in the dch and 7ch Problems of
8th Chapter, which treates of Superfici«''
meafure, in meafuring of Land.

Note alfo. That if you have three Lin^'
of Numbers, viz,, a Great, a Mean, ind^
Lefs; after Mr. tylndmes way ; then the»''
Qgellions are wrought without doubling»
or halving, and very neatly and fpcedily*

The Extent on the mean Line ,
24-yc, to 6-125', the weight of thc t«'quot;
Ropes, fhall reach on thc great Line, frquot;'*'
12 to 6 i or from 6 to 12, the inches in coiH'
pafs about of each Rope.

Problem VIII.
To t hret Numbers give», to find it fourth
in a trifled Proportion.

For as much as like Solids, are in a tripled
proportion to their anfwerable fide/the
^ubesof their fides are proportional one to
another ; therefore, to work thefe Que-
étions by the Line of Numbers, do thus •

When the two given terms, of like deno-
mination in the Queftion, are of Sides,
or Diameters ; then the Extênc of
^ne Compafles, on the Line cf Numbers,quot;
roai one fide to the other, that is, from the
•''e, whofe Cube or Solidity is alfo given,
the other ; the fame Extent, turned three
^'«es from the given Cube, or Solidity, fhall
^^ach to tlie inquired Cube, or Solidityj

As for Examplei
If a fide of a Cube, being 12 inches,
^°ntain in Solidity 1728 cube inches; Hpw
»any inches is thefe in a Cube, whofefide
» inches ? The Extent from 12 to 8, be-
^quot;g turned three times from 1728, fliaif
' C Knbsp;Solidity required of thé

jiga'tH, on the contrary ',
^Whenthe two terms of the fame denotui'quot;
nation, are Cubes or Solidities, then divid'^
the fpace on th?Line of Numbers, between
the two Sohdities,into three equal parts,and
lay that Extent the fame way, as the reafoi^
of the Queftion doth require, either incre^'
finger djminifhing, from the given Side
Line, and it fliall reach to the inquired
Side, or Line.

If 1728, be the Cube of 12, the Root ci
fide ^ what fhall be the Root or fide of 8d4t
the half of 1718 , being half a foot
Timber ? The Extent between 1728, and
8*54, being divided into three parts, and
that third part, laid decreafing from
lhall reach to p-5'2y, the fide or root requi-
red of half a foot of Timber, though no:
exadlly, yet very near.

If an Iron BuUer,.of 6 inches Diameter'
weigh 30 pound ; what lhall a Bullet of
7 inches Diameter weigh ? The Extent froin
lt;Jto7, fhall reach/being turned thre^
times, to 47./.

gt;5 Foot by the Keel; what fhall that Ship
he, whofequot; Ked.is.j;£)ft Foot? . The Ewcm
between 7S and 'ooj turned three times
from JOG, ihallj-each to 713 Tun Bur-
then. ■ V ;

If a Ship«f Foot and a half at the
heam, be 300 Tun Burthen, what fhall a
Ship of 713 Trt^hufthen be ? The third
part of the diftance between 300 and 713,
ftall reach from 29 » to 39-3fj its mea-
fute at the beam.

a Ship of 713 Tun be in hold f
The third part between 300 and 713, lhall
f^ach from 13 fobt, to i7-35'i the Feet
in the Hold of a Ship of 713 Tun.

If you have airebleLine, then you may
fave tlie dividiiig, by taking from th^ littlc-
^'ne, and meafuling on the great-Lincj and

^lOW, that by adding of tWelvf Centers
^•^d Points^ the Line n:ay be made to fpeak,
^^ It Were, and fo made more fit for any
'^ans more pajfticular occafions.

'A Brief Tduch of'the üfe éf the Lo-
garithms, orTabTes, of the Arti-
ficial Numbers, StTter, and Tan-
gents.

this Book, that liad rather ufe the Tables
of Logarithm, fromwheuce. thefe Lines arc
framed, than the Lines on the Rule; or out
of cufiofity to prove the truth.of their work,
for whofe faies I have adde'd thefe following
plain Precepts, without Examples.

Set the Logarithm of the Mulnplicator,
and Multiplicand, right under one another,
and add them together, and the fum is the
Logarithm of the Produft.

Set down firft the Logarithm of the Di'
vidend, and then right under it the Loga'
rithm of the Divifor, aiid then fubftraiit
the log. of the Divjfox, from ihe log. of the
quot; 'nbsp;Dividend?

Half the Logarithm of the Number gi-
^'fnj is the whole Logarithm of the Squarc-

One third part of the Logarithm of the
given Number, is the full Logarithm of the
Cubick-root of the given Number, as a
third of 14313637 j the logarithm of 27 is
^-4771212, tire Log. of 3, the Cube-root
^f 27, required.

5quot;. Toxvork. the%uleof Three direB, or
three Numbers giveiiy to find a jf^th by
the Logarithms.

Set down the Logarithms of the i, 2, amp;
3 Numbers, one right over anotlier 5, then
add the logarithms of the fecond and third
together; and from the fum, fubftraft the
logarithms of the firft, and the remainders
^^ the logarithms of the 4threquired.

6.nbsp;when in common ^rîthmetick the fif
eonà term « divided hy the firfi-, and tht
Quotient mttltiplied hy the third.

Take the Logarithm of the firfl term,
from the Logarithm of the fécond -, and add
the difference to the log. of the third, and
the fum is the log. of the 4th.

7.nbsp;when in common Arithmetic^ the ft'
cond term is divided by the firft, and tht
third by the Quotient,

Then take the log. of the fccond, from
the log. of the firft term ; and take the dif-
ference out of the log. of the third, and the
remainder is the log. of the 4th term re-
quired.

9* To mrk tht Rule of Three in the Lo-
garithms of Artificial Numbers, Sines,
and Tangents.

I. when Radius ii the firfi term.
A-dd the Logarithms of ths fecond and
third terms together, and Radius, or a
unite, in the firft place, taken from the
fum, there fhall remain the logarithm
of the 4th term required ; according
tq the jth Precept.

Then the firft term ( and fecond virtually)
taken from the third, cutting off a unite
in the firft place for Radius, is the
4th term.

Then fubftraéf the logarithm of the fecond
term, from the log. of the firft term,
cutting off a unite Vor Radius, aiid the
remainder is the 4th term.

Then add the Logarithms of the fecond
and third terms ; and from the fum,
Z 4 ■nbsp;fub^

fubftraa the logarithm of the firft term»,
and the remainder is the logarithm ot
the 4th term.

Sec down thc Arithmetical complement ot
the firft term, and the logarit ims of the
fecond and third term, and add all to-
gether, and the fuip cutting olf Radius,
is thc 4th term.

10» When the Number is not to be
found in the Canon of Logarithms of Num-
bers, Sines, or Tangents ; take the next
neareft, or for more exaôhiefs ufe the part
proportional.

11.nbsp;Though Numbers and Sines, of
Numbers and Tangents are ufed together,the
work is all one, as with Sines and T^gents,
as CO the Precept in working.nbsp;quot; •

12.nbsp;In ufing the Logarithms, great re'
gard is to be had to the Index , or Ch^f
ra6leriftick,to rule in the Number of places;
the Charaéteriftick being one unite lels
than the Number of places to exprefs that
Number ; thus the Charafteriftick oI
3-5:P328()i, the logarithm of 3920 is h
being one lefs than the Number of placés ii^
3920, which confifts of 4 figures.

The %ailHs of a Circle, or Line being gi-
ve», to find readily, any required Sine,
Tangent, or Secant, or Chord, to that
Radifu.

Firft, If your Radius happen robee-
qual to the greater Scale of ( Alti- -
tudes or ) Sines, iifuing from the Center, Sines.
then the meafure of any degree or mi-
quot;Uite, from the Center toward the head,
^all be the Sine, thquot; mpaluie from the
Center.point at '^olo, ou the degrees, to
aiy degree and rainuic nqui. d, fhall be

the Tangent to the fame Radius; and th«
meafure from the Tangent to the Center,
fhall be the Secant, to the fame Radius.
And if you have an Index, or a Bead upoi'
your Thred, and fet the Bead, when the
Ihrcd is drawn ftreight, to the Center a^
doio on the degrees or Tangents, or to
the Sine of po ; then if you lay the Thre^l
to any Number of degrees and minuits,
counted from po, and there keep it • the»
the extent from the Sine of po, to the Bead,
fliall be the Chord of the Angle the Thred
IS laid to, to the Radius of the greater
Scale of Sines, ifl'uing from the Center.

But if this happen to be too large, the«
the other leflbrLine of Sines, ifiitmg up-
wards from the Center, being about oiic
third part of the other, hath firll it felf-for
Sines j fecondly, the degrees ontheloofe-
piece for Tangents, counting from the Cen-
^rat5o; thirdly, the meafure from the
Tangent, to the Cciiter, for a Secant}
fourthly, the Bead and Thred, for «
Chord, as before j ■ all at once to one Ra-
dius, clearly and diftindlly, without any
interruption, to 7; degrees of the Tangent,
or Secant.

But if any other Radius be given, the»
they will not be had fo readily altogether,
bur th'vs in order one after another, and
........... firft

J. Take the.Radius between your Compaf-
jj^'^^conefbat: in pc, with the other lay
Thred to the neareft diftance, and there
^P if i then take the neareft diftance fronj
Sine of any Ark or Angle you would
and that lhall be the Sine of the Ark
Angle required to the given Radius.
J But by the Sector-fide work thus, be- By th'
pg near alike, fit the given Radius in the ^
f 2!-alIel-finc of po amp; po ; then take ont fide.
Parallel-fine, of the Ark or Angle re-^,-«,.
1 quot;^ed, and you have your defire.
^ 2,. Alfo the Sedor being fo fet, if you Targcnt.
skeout any Parallel Tangent under 45;,
)ou have that alfo to the fame Radius.
, ^ • Alfo, if you would have any Tangent tan.to 76.
quot;»der 76, as the Seizor ftands, rake^out
«e parallel Tangent thereof, and that ftiall
o™ 4th part of the Tangent required to
He fame Radius, and is to be turned 4
times for that greater Radius.

4. Alfo, if you want a Sccant undc^ secant.
degrees; at the fame Radius take out
parallel Secant of the Ark or Angle rc-
S^hed; and that fhall be the half of the
^eant required; for note, the Secant of
degree is more than Radius; and why

By the Artificial Numbers, Sines, ai^f
Tangents, this cannot properly be don^'
' only thus you may do by them, countJi'Sj
your given Radius (be it great or little/
10000 parts,you may by diem find out realt;l''
ly how many of them parts will go to vai^^
the Sine, Tangent^ or Secant, to any N«'^'
ber of degrees and minuts. Asthma
Sines. Take the diflance from the Sine of

on the Artificial Sines, to the Sine of an)'
degree and minuit required; and fet th*^
fame diftance, the fame way, from 10 on
Line of Nunibers, reading it as a Scale
equal parts, and that fliall be the Nacurf»
Sine of the degree and minuit required.

If you lay a Square to the Sine given,
the Numbers^ it cuts the Natural Sine
quired.

Right againft che Artificial Sine of
on the Line of Numbers, you find jooo)
which IS the Natural Number thereof.

But, if you meafure this diftance frcH'
10, in the Line of Lines, it will give di^
Logarithmal Sine thereof, 69897.

from 4y, to the co-Tangent of 45:, for
* Tangent; This extent laid the contrary
'^^yfroiTi I, in the Numbers, Ihews how
■nbsp;Radiufl'es, and alfo how much above

jT'lius, you muft have to make up the
)^atural Tangent, or Secant required, ia
^^«mbers.

The Secant of 50 degrees, and the Tan-
gent of J7 degrees, 16 minuts, being near Artificial-
is I Radius ^556; for theNaturaWo^*'^quot;^'«'
Number thereof: and this diftance mea-
j^ted On the Line of Lines, gives Radius
^caufe above 4J, and ip ip 5 more for the
•'ytificial Tangent of y7-i'5gt; or the Secant
fo degrees.

This have Ihinted, in the firft place,that
Jhereby you might fee the nature of the
^ines, and the making of the Inftrument,
^^ith its great convenience in theContri-
J'^nce of the Work on both fides, and the
hai^mony, and proportion, the Natural way
to the Artificial; alfo hereby you may
^^adily prove the truth of your Inftrument,
^^'quot;g an equilatteral Triangle, whether you
H'e the greater or the lelTcr Sines; For the

the fmf meafure from the Center, where the Thr«^!
quot;truth of quot;nbsp;to the Center-point of Brafs d'

the lnjiri. '^e moveable-leg, and loofe-piece at 60 '
the degrees, ouglit to be equal to each Uquot;^
of Smes 5 and alfo to the Tangent of
on the Tangent Line ; The meafure frquot; ''
the Center to the reftifying-point on
Head, at the meeting of the Lines for.thf
Hour and Azimuth, and the Lines fort^'^.
Sines and Lines, is equal to the Tangent

The meafure from the Centefj to tli^
rcdifymg-point on the end of the Hea'''
leg, fhall reach from thence to 30 on th^
loofe-piece i and being turned twice,reacbf'
to 0160 on the loofe-piece : Alfo, the
dius, or Tangent of turned twice frfi^ '
o]5o on the loofe-piece, fhall reach to 75'' '
as by comparing the Natural Numbers to'
gether, will moft exadfly ap^:2ar . Thougl'
perhaps without this hint, it might i
h'ave beeen obferved by an ordinary eye.

firft U/i, I fhall be, I hope, as plain, thoug'^ ,
far more brief in all the reftj for if yquot;^
look back to Chapt. VI. Prok. I. h
you fhall there fee the full explaining
Latteral and Parallel, and Neareft-dift^'^^»

given Sine ; and with the other Point
^y the Thred to the neareft-diflancej and
'quot;^te keep it; then the neareft-dilfance
'join the Sine of 90 to the Thred, lhall be
^^ Radius required.

Make the given Sine a Parallel Sine, and Sell sr.
then take out the Parallel Radius, and you
^ave yourdefire.

The Artificial Sines and Tangents, are not
proper for this work, further then to give
the Natural Number thereof, as before;
therefore I fhall only add the ufe of them
gt;vhen i: is convenient in the fit place.

The Radlw, er any know» Sine being gi-
ven, to find the quantity of any other
Mnknown Sine, to the fame Radius.

and make it a Parallel in the Sine of po
Radius, or in the Sine of the known
given, and lay the Thred to N D. Thd'»
take the unknown Sine between your Com'
palfes, and carry one Point along the Li»^
of Sines, till the other foot being, turit^''
a'bout, will but juft touch the Thred; thequot;
the place where the Compaffes ftayes, fha^
be the Sine of the unknown Angle requit'
ed, to that Radius or known Sine.

SeSor. Make the given Radius a Parallel Radius»
or the given Sine a = Sine, in the anfwefquot;
able Sine thereof: Then, takmg the uu'
known Sine, carry it parallelly along th^
Line of Sines till it ftay in like pares, which
parts lhall be the Numerator to the Sine rS'
quired.

The Raditti being giveny by the Sines ahquot;^
to find any Tangent or Secant to th»*'
Radius.

Qvadrant. ^^ke the Radius between your Compaffes
and fet one Point in the Sine complemeoquot;^
of the Tangent required, and lay the Thre^
to the N D; then the N D from the Si»!^
of the Tangent required, to the Thred»
fhall be the Tangent required: And the N P
from po, to the Thred, lhall be the Secant
required.

Make the given Radius a = in the co-Seffor]
me of the Tangent required; then the =
^quot;^e ( of the inquired Ark or Angle ) fhall
the Tangent required ; and = po Ihall
^ the Secant required to that Radius.

Any Tangent er Secant being given, to
fnd the anfwerable Radimand then
«ny other proportionable Tangenty or
Secant, bj Sines only.

j^J^i^ftj if it be a Tangent that is given, Qjlt;dlt;/r.
I'lce ic between your Compafles, and fet-
3 one foot in the Sine thereof, lay the
^hied to N D, then the = Co-fine thereof
be Radius ;
quot;UCj if it be a Secant, take it between
your CompalTes, and fet one foot alwayes
in 90, lay the Thred to the N D, then the
öearett diftance from the Co-fine to the
Thred ( or the = Co-fine ) fhall be the
ivadius required.

Take the given Tangent, make it a = inSe^?ar3
the Sine thereof; then the Co-fine there-
of fiiall be Radius.

Or, if It be a Secant given, then
Take the given Secant,make it a = in po,
A anbsp;thin

Then having gotten Radius, the 4th Uk
flrcwes now to come by any Tangent, or S^'
cant, by the Sines only.

the Sine of 30, lay the Thred to the N P*
( and for your rriore ready fetting it again,
note, what degree and minuit the Thred
doth flay at, on the degrees) and there
keep it. Then the N D from the Sine of
half the Angle you would have, ihall be
the Chord of the Angle required.

SeHor. Take the given Radius, and make it al-
wayes a = in30, and 30 of Sines; the
= Sine of half the Chord, lhall be the
Chord required.

Q^adr. Take the Radius between your Compaf-
fes, and fetting one Point in 90 of the Sines,
- ^ ^ — - jjj,

Then taking the = Sine of the Angle re-
S^hed, with it fet one Point in the Line to
J^'hich you would draw the Angle, as far
quot;tom the Center as the Radius is; then draw
tne Convexity of an Ark, and by that Con-
^'exity, and the Center, draw the Line for
tne Angle required.

Let AB be a Radius of any length, un-
^er or equal co the whole Line of Sines :
fake A B between your Compafles, and
•^tting one Point in po J lay the Thred to
N D, then take out die Sine of 38, or
a'ly bther Number you pleafe, and fetting
Point in B, the end of the Radius from
A the Center, and trace the Ark DC, by
the Convexity of which Ark, draw the Line
A C for the Angle required.
__Take the given Radius A B, niake it a Se^or.
^ m 90, and 90 of Sines; then take out
38, and fetting one foot in B, draw
the Ark DC, and draw AC for the
Angle required.

Or ilfe mrk, thm •
Take A B, the given Radius, ( having
^rawn the Ark B E ) and make it a = in
Aa 2 ~ the

the Co-fine o!^ half the Angle required; and
lay the Thred to N D, (or let the Se6tor).

Take the == N D, ft-om the right-fine ot
the An;'c required,'and ir Oiall be BE,
the Chord required to be found.

at the Point B, raife a Perpendiculer, as the
Line B C extended at length, then make
A B, the RadiuSj a = Tangent in and
4^; then take out the = Tangent erf the
Angle required, and lay ic from B to C i*^
1nbsp;the Perpendiculer, and draw the Line A ^

Alfo, If you take out the Secant of the
Angle, as the Sedfor ftands, and lay it twice
Jn the Line A E, ic will reach juft co C, the
Point required.

Alfo Note, Thac if you wane an Angl«
^ above degrees, as the Secfor ftands, take
the fame from the fmall -Tangent thac pro-
ceeds to 75, and turn that Extent 4 tjmeS
from B, and it lhall give che Poinc required

and IcX A be .die Arigulerl^oinc;
J^«gt;\uiaking AB Kmake. AB ■
j-o-fine of ;he ■ rje you would have, and
'^y the Three tc lie-^sveft diftance, thcu
^quot;eND from the Sii.e .oi: the Angle

For making A B, or A D, the fide of the
Square Radius, lay off the Tangents of
^^y Angle under 45^ from B toward C,
the complements thereof above 45 from
toward C, calling 40, jo ; amp; 30, 60 j
10, 80, 6cc.

To take out readily, any Tangent aboVl 4f'
by theTangent to 45 on tht SeBor-ftdc-

SeHor. Take the given Radius, make it a = in the
co-Tangent of the Tangent required; the''
the = Tangent of 45, fhall be the Tange»'
required.

I would have a Tangent to 80 degrees?
take the given Radius, make it a = m lO?
the complement of 80 j then the = Tan'
gent of 4J, fliall be the Tangent of 80 re-
quired.

But if your Radius be fo big, thatyo^
cannot enter it, then take the half, or a
quarter of your Radius, and then 4?
will be the half, or the quarter of theTaO'
gent required.

There are 4 Varieties in this Work, th^^
include all Proportions, yl^j.nbsp;^^^^

then the work, is thui l
Lay the Thred to the fecond term, count- ^^
on the degrees firoa^ dieHead, towa.u
the loofe-piece; and co-,;nt the third term
on the Lme of Sines, the Center down-
\vards; and taLii^ï- t^n near^^ diftance
from thence to thf -.ed, and thw diflance
»neafured from che Center downwards, on
the Line of Sines, gives the 4th term re-
quired.

Take theLatteral fecond term, make itS
a = Sine of po ; then take out the = third
term, and meafuring it from the Center, it
gives the 4th term required.

then vforkjhM
Take the — Sine, ot the fecond terni, (
from the Center downwards, and make it
a = Sine in the firft term, laying the Thred
to ND; then on the degrees, the Thred
' lhall give the 4th term required.

Bttt hy the SeBor,
SeUor, Take the — Sine of the fecond, mak^
it a = Sine of the firft term ; then take ouj
= 90, and meafure it from the Center, ao®
it fhall give the 4th term required.E;trf»»/'^'
as before.

uppermoft Sine above the Center, or the
Line of Right-Afcentions, or the Azimuth'
Scale, or the hke ; and make it a — in the
Sine of the firft, laying the Thred to N Vy
then the = third term, taken and meafure^
on the fame Scale that po was taken froiflj
ftall give the 4th term required.

% carrying the CompafiTes till it fo ffayes,
^^ that che foot turned about, will hut juft
^ouch the Thred, at the neareft dUlance.

Where the Radius being brought to the
wird place, it is wrought by the fecond
Rule, as before.

Take a fmaller — Sme of po, make it
a = in 30 • then the = Sine of 20, taken
and meafured on the fmall Sine gives
43-12, as before.

^uadr. ' Then when the firfi term is greater tha«
the fecond and third, work thm-.

Take the — fecond term, make it a
in the firft, laying the Thred to the NDi
then the neareft diftance, from the third
term, to the Thred meafured from the
Center downward, give the 4th Sice re-
quired.

I'lftly, ufe a Parallel entrance, or anfwer
Fächer, as before, which being carefully
^^fought, will do very well.

The fame manner of work, is as before by
fhe Quadrant, and the fetting the Seftor,
is all one to the laying the Thred, as will be .
largely feen in all the following Propofitions,
Wrought both by the Artificial and Natu-
'^al Lines, of Numbers, Sines, aud Tangents,
as followeth.

Lay the Thred on the day of the Month By the
in the Kalender, and in the Line of de- Q»ädr.
grees, on the Moving-leg, you have his De-
clination, either Northward,or Southward,
according to the time of the year, counting
from (So jo, toward the Head, for Noxth-
declination ; or toward the End, for South-
declination.

Extend the Compaffes from the Sine oi
po, to the Sine of 23 degrees 31 minuts,
the Suns greatefl: Dechnation: The fame Equot;'
tent applied the fame way, from the Sine of
the Suns place, or the Suns diftance from the
next Equinoftial-point, fhall caufe theMo'
ving-point to fall, on the fine of the Sun^
declination j This being the general way
workmg.

The Extent from the fine of po, to the
fine of 23-31, flrall reach from the fine o'^
30, to II deg, 31 min. the Suns decimati-
on, innbsp;30 degrees from y Arifh
the nextEquinocfial-point,' and from 60
degrees, the Suus diftance in jj Gemini
degrees, from? T 20 deg. 12 min. theSun^
declination then. This l^ing the manner pf
working by thefe Lines, by extending the
Compaflb from the firft to the fecond
term: I fhill for the reft wave this large
repetition of extending the Compafles, and
reader it only thus by the words of *lie
Cannon-general in all Books j

Lay the Thred to 23-3 r,' on the degrees Qtrnd. Ge^
the Moveable-piece, counted from the
Y^d toward the End j then count the Suns
P'ace from the next Equinoftial-poinc, on
Lme of Sines from the Center down-
Y^rds, and take the N D from thence to-the
jilted i then this diftance being meafured
hoiTi the Center downwards, fhall be the
'[tie of the Suns dedination, required for

Take ■— 23-31, hom the'Sines, make it
^ ^ in the fine of po; then the= fine of
the Suns diftance from the next Equinoffial-

The Suns Declination heingglven, to find '
his true place or diflance from Tortc:,
the two EquinoHial Points.

Lay the Thred to the Declination count- Quadr-
inthe degrees from lt;5o|o, and in the PafticH-'
Line of the Suus place, is his true place re-
Httired.

When the Suns dcchnation is iz
grees Northward, the dayes increafing, thequot;
the Sun will be 31 deg. and 23 min» froquot;'
r, or. I deg. 23 oiin. in if, his true pla'^'^
required.

Artificial. As Sine of 23-31, the Suns gteatcft de-
S. T. clination, to Sine ot 90 •

So Sine of 12-00, the Siuis prefent de-
clination, to Sine of Suns diftance frofl*
T or 31-23,

Which, by confidering the time of the
year, gives his true place, 'bjf looking on the
Months and Line of Suns p ace on the Qua'
drantal-fide.

C}uad. Ge~ . Take the — Sine of the prefent declina-.
iieralljt. tion, make it a r:^ Sine in the greateft de'
clination, laying the Thred to N D ; and
on the degrees the Thred fhall give the Sufl5
diftance from r, Or j required. Exa^fp^'
as before.

SeSor. . —Sine of the given Suns declina-
tion, a = Sine in the Suns greateft declina-
tion, then = Sine of 90, meafured from ih«
Center, is the = Sine of the Suns diftance?
fromnbsp;required ; or Count 30 deg.

1'he Suns' place, or Day of the Month,
und greatefl Declination given ; to find
bit Right Afcention from the fame E-
qtiinoUtal.

place given, and in the Line of the SunsQiquot;quot;''^«
7'ght Afcention, you have his Rignt Afcen-
in degrees, or hours and minutes, coun-
^^g 4 rainuts for every degree.

On the pth of April, near night, the Sun
oeingthenentringb'j the Suns Right Afcen-
tion will be I hour yz min. or 28 degrees

As the Sine of po, to the Sine comple-y^t■ƒ;ƒc/^j.
ment of the Suns greateft declination S. i^r T.
(orC.S.) of 23-31, counting back-
wards from 90, which wiU be at the
Sine of 6lt;S-5.p'.)

So is the Tangent of the Suns diftance
from the next Equinoftial-point 9 to
the Tangent of the Suns Right Afcen-
tion from the fame EquinoiSial.point.

Quad. Ge- Take the ■— co-fine oF the greateft
neratly. dination from the Center downwards,

ing the gt;—. fine of 66.2^'. make it a ^
fine of 90, laying the Thred to N D; an«
note what degree and minuit it cuts, for thi'
is fixed to this Proportion; Then take tli*^
Tangent of the Suus diftance from the nequot;''
Equinoftial-point, from the Center at
on the de'grees toward the End, and lay
on the fines, from the Center downwartl^»
and note the Point where it ftayeth, for tb^
N D from thence to the Thred, ftiall be tb^
Tangent of the Suns Right Alcention re-
quired.

Note, That if the Suns diftance from V»
or be above 45- degrees, then the Tanquot;
gents on the loofe-piece, are to be ufed iquot;'
fteadof the Tangents on the moveable'
Jeg.

Or, Take — Sine of the prefent
declination, make it a = in the Sine of the
Suns greateft declination, and lay the Thre'»
to N D; then take = Co-fine of the S««^
greateft declination,and make it a =: in Cfi'
fine of the Suns prefent declination , afquot;'
lay the Thred to N D, and in the degreed
it cuts the Suns Right Afcention, requi-
red.

riT '—of 23-31, niz. theSsSd»-, 1
^jSnt Sine of 66-29, a = line of 90, thennbsp;'

^^ = Tangent of the Suns diftance from
or t^j is the = Tangent of the Suns
^^ght Afcention from the fame Pcdnt of T,
j'' as at 30 from T, it is 28 degrees, or
quot;our and y2 minuts from r gt; (neer).

Ufe XV.
fiavlng the Sms Right Afcention, ani
greateft Dedination, to find the An git
of the Ecliptick^ani Meridian^

^0 IS the Co-fine of the Suns Right Af-
«nrion, to the Co-fine of the Angle
of the Ecliptick

. Lay the Thred to 23-31, Counted on
le degrees from the Head ; then count the
^o-r,„e of the Ri^ht Afcention, from the
Renter downward, or the Sine from 90
^P^vards, and take the ND from thence to
^^^ Thred, and meafure ic from theCen^
^r, and it lhall reach to che Co-fine of the
^quot;gle required.

Make the — right fine of 23-31^ a =
fine of po ; then the = co-fine of 30, vi^-
:=z60y fliall make the — fine of dp-jOj
the Angle of the Ecliptick and Meridian.

Having the Latlttide, aniVtcllnatlon 4
the Sm or Stars, to find, the Sttnstf
Stars Amplitude, at rifing or Setting'

Partie. Take the Suns declination, from the paf
ÇLifddr- ticular Scale of Sines, and lay it from

the hour or Azimuth-hne, and it fhall give
the Amplitude from South, as it is figured }
or from Eafl, or Weft, counting from 90
obferving to turn the Compafles the faine
way from 90 or 6, as the declination i^
Northward, or Southwards.

The Suns declination being 10 degree^
Northward, the Suns Amplitude, or Line»
is iod-i2gt;from theSouth, or 16-12froif
the Eaft-point.

As co-fine of the Latitude, to S. 90 ; r
So is S. of the Suns declination, to S.
the Amplitude.

Ciuadr. Take the — Sine of the Suns declinSquot;
generally, tion, make it a = in the co-fine of the Ir

^îfUde, and lay the Thred to the neareft SiSifi
diftance, and on the degrees the Thred fliall
fliew the true Amplitude required.

Make the — right Sine of the Suns de^
clination, a = in co-fine latitude, then =
50, taken and meai'ured from the Center,
gives the Amplitude or Line.

Take the — fine of the5«»j declinati- Quadri
fet one Point in the Sine of the Suns ' J
■^'ïiplitude, lay the Thred to N D, and on
the degrees it fheweth the complement of
quot;he Latitude required.

Head, toward the End.
. Make the right Sine of the Suns Déclina^ Sf^ifa/
'^on, a = fine in ths Sum Amplitude
J.hen the = fine of po, fhall be the ^ col
Moc of the Latitude required.

Ufe XVIII.
JJavlngthe LAtitude, and Suns DtelinO'
tion, to find his Altitude at EaB or
f^eft, commonly called the Vertical'
Circle • or Ax^mmh ofEafi or fVeft.

Partic.Cl^ Take the Sms Declination from the par-
ticular Line of Sines, fet one Point in po on
the Azimuth-line, and lay the Thred to the
N D, and on the degrees it Iheweth the
Altitude required; counting from 6o\o
toward the End.

6eh Quad. Take the — fine of the Sms declinati-
on, make it a = in the fine of the latitude,
and lay the Thred to N D, and on the de-
grees It lhall fliew the Suns Altitude, aC
Eaft and Weft required.

Having the Latitude, and Suns 'Declinati-
on, to find the time when the Sun will
be due Eafl or Wefi.

Having gotten the Altitude by the lafl Part. Q. •
'vUle, take it from the particular Sine j
'^neu lay the Thred to che Suns declination,
counted on the degrees ; then letting one
i oint in the Hour-line, fo as the other turn-
ed about,fhall but jufl: touch the Thred,and
^he Compafs-point fhall ftay at the hour
'quot;d mmuit of time required.

So is the Tangent of the Suns declination, S. amp; T.
to Co-fine of the hour.
Or,

latitude 5:1-32, declination 10, the Sun
^^dl be due Eaft at 6-? 2, and Weft ac
Î-28.

Take the — Tangent of die Latitude (on Gen.Qaad.
^he loofe-piece, countmg from 6u toward che
quot;loveable-leg ; or elfe from dolo, on the
B b 5nbsp;moving-

moving-leg, or degrees, according as the
Latitude is above or under degrees) and
lay it from theCenter downwards, and note
the Point where it ends. Then take from the
fame Tangent, the Tangent of the S»ns de-
clination, and fetting one foot in the Point
laft noted, lay the Thred to N D, then the
= fine of 90, fliall be the —fineof the
hour from 6,

Take che — fine of the Sms declination,
make it a = in fine of the latitude ; lay the
Thred to N D, then take N D from the Co-
fine latitude to the Thred ; then fet one
foot

in the Co-fine of the5««j declination,
lay the Thred to N D, and on the degrees
it gives che hour from noon, as it is figured,
or the hour from counting from the head,
counting 4 minuts for every degree.

Sf^sr. . Make the fmall Tangent of \heLatitude,
if above 45-, taken from the Center, a
fine of 90; then the — Tangent of the-
Suns declination, taken from the fame

Tangent, and carried Parallelly till it ftay
in like Sines, fliall be the Sine of the hour
from lt;i.

Make_fine Declination, a fine La-
titude ; then take = Co-fine Latitude, and
iSake ic a = Co-fine of the Suns Dechnati-
oil. then take = 90, and lay it from the
Center, ic gives the Sine of the hour From

Having the Latitude, and Suns Declina-
tion, to find the Afcentional Difference,
or the Suns Rlftng and Setting, and
Ohlique Afcention.

Lay the Thred to the Day of the Month, Panic.Q^
(or to the Suns Declination,quot;or true Place,or
to his Right Afcention; for che Thred be-
ing laid to any one of them, is then alio
laid to all the reft ) then in the Azimuth-
luie, it cuts the Afcencional d it^rence, it
ityoucounc from 90, orthe Suns Rifing, as
you count the morning hours; or his Setting,
counting the afternoon hours.

The Oblique Afcention is found out for oblique-
the fix Northern figns, or Summer half-^«nquot;«quot;.
year by fubftraAing the Suns difference ot
Afcentions, out of the Suns Right Afcentioji.
But for che other Wmcei-halfyear, or fix
Southern figns, ic is found by adding che
Bb 4nbsp;Sunt

Suns difference of Afcentions to his RiS^''-
Afcention; this fum in Winter, and thlt;j
remainder as above-faid in Summer, fhaU
be the Sms ObUque Afcention required.

the loofe or moveable-piece, as it is above ot
under 45 degrees, make it a= in fine poj
lay the Thred to N D, then take the
Tangent of the Sms dechnation from the
feme Tangents, and carry it = till it ifay i«
the parts, that the other foot, turned about,
will but juflr touch the Thred, which parts
fhall be the Sine of the Sms Afcentional dif-
ference required.

Make the — fine of Declination, a
■Co-fine of the Latitude; lay the Thred to
N D, then take the = fine of Latitude,
ma;ke it a = in Co-fine of the declination,
and lay the Thred to ND, and onthede-
grees it fhall cut the Suns Afcentional-dif'
Terence required; which being turned into
time^by counting 4 minuts for every degree,

Make the —■ Co-tangent Latitude, a Scffor.
fine of 90 J then take—- Tangent of the
-S««; dechnation, and carry it =: till ic
quot;ay in like parts, vix.. the Sine of the Sms
^fcemional differaice required.

So is = Tangent of 23-31, the Smt
greateft declination, to the — fine of
the Sms greateft Afcentional dif-
ference, 33 deg. and 12 min.

When the Latitude and Declination is
Doth ahke, ylg,. both North, or both South;
then fubftradt the Declination out of the La-
titude, or the lefs from the greater, and the
remainder fhall.be the complement of the
Suns Meridian Altitude,

But when they be unlike, then add them
together, and the fum lhall be the comple-
ment of the Meridian Altitude : The con-
trary work ferves when the complement of
^he Latitude and Declination is given, to
^'id the Meridian Altitude.

Lay the Thred to the Dechiiation, conquot;'
ted on the degrees from 6ojo, the rigi^'-
way, toward the Head for North, and tf
ward the End for South declination.

Take the neareft diftance, firom the Cei^'
ter-prick at 12, in the Hour-line, to tb^
Thred ; this diftance meafured on the Pat'
ticular-line of Sines, ihall ftiew the
Meridian Altitude required.

The Latitude, and Hour from tht ntl^l
night Meridian given, to find the An^'^
of the Suns Pojition, viz. the Angle
tween the Hour and ty€z.imuth-liKCS »quot;
the Center of the Sun,

S. Sc T. So is the Sine of the Hour from Mi^'
night, to the fine of thc Angle of Pquot;'
fition.

So is the Co-fine of thc Hour from inio'
night, 120, for which you muftu'^
60, to 32-34, the Angle of Poficion-

Panic, Q^ Take che diftance from the Hour to tb^
90 Azimuth on the Hour, line, and meafur^
ic in the particular fines, and it fhall fhe^^^

. Take ^— Co-fine Latitude, make it a r= Gen.QitaJ
^[J fine 5)0; then take out the Co-fine of

The Angle of the Sms Pofition may be
y®tied, and it is generally the Angle made
the Center of the 5«(», by his Meridian
Hour-circle,being a Circle pailtng thorow
the Pole of the World, and the Center of
^he Sun ; and any other principal Circle, as
the Meridian, the Horizon,or any Azimuth,
the Anguler-Point being alwayes the Center
of the Sun»

Ufe XXII.
The Sms Decli»attoi} give», to flni the
beginning and end of Tm.light,or Daj-
Ireak:

Lay the Thred to the Declination on
degrees, but counted ihe contrary way,viz..
South-declination toward the Head; and
North-declination toward the ~ iid, tnen
take 18 degrees from the particular Scale of

Sines fot Twi-ligbt,' or I3 degrees for D^r
break, or clear light; Then carry this
ftanceof 18 forTwi-Iight, on? forDaf
break, along the Line of Hours on that
of the Thred next the Eiidi till the otbef
Foot, turned about, will but jult touch ^^^
Thred, then ihall the Point Hiew the tii^t
of Twi.lighc, or Day-break, required.

The Suns Dedination being 12 degf«^^
North, the Twi-light continues, till p hotiP
24 minuts; or it begins in the morning
38 minuts after 2 ; but the Day-break
not till 22 minuts after 3 in the morni»!?»
or 38 minuts after 8 at night, and laft quot;quot;
longer.

Find the Hour that anfwers to 18 degt^'^'
of Altitude, in as much Declination th«
contrary way, and that ihall be the time of
Twi-light; or at 13 degrees for Day-break,
according to the Rules in the 2(Jch
where the way how is largely handled to tbc
.33'^ Ufe, both wayes generally.

Take the neareft diftance from the Ccn-PrfH/'e«f/lt;r
On the Head-leg, to the Azimmh-line
the moveable-leg; this diftance meafu-
on the particular Scale of Sines, fhall
jf^^w the Latitude required ; or the Extent
i^m o to 90, on the Azimuth-line, fliall
ipw the complement of the Latitude, be-
meafured as before,

Ufe XXIV.
Having the Meridian Altitude given, to
f»d the time of Sun Rifng or Settings
frnt Place, or Declination.

Take the Suns Meridian Altitude from
particular Scale, and fetting on Point
P A^gt;muth-Hne ; lay the Thred to
Je N D, and on the Hour-lme it flieweth
i^e time of Rifing or Setting ; and on the
^^grees, the Declination ; and the reft in
quot;^eirrefpe6hve Lines.

Particular I-ay the Thred to the Day of the Moflt''^
Qjtadrant. or Dechnation, then take the N D fr®'''
the Hour-point of q6, and 6 to the Thre»'^
and that diftance meafured on the parties?'
lar Scale of Sines, fliall be the Suns Alt''
tude at 6 in Summer time, or his deprefliquot;''
under the Horizon in the Winter time.

Gcn.Cluad. Count the Suns declination on the tl^'
grees from 90, toward the End, and thef^
lay the Thred ; then the leaft diftance fro'^
the fine of the Latitude to the Thred, mdquot;
fured from the Center downwards, lhall b«
the fine of the Suns Aldtudc nt lt;5.

= fine of 90; then the = fine of the L»'
titude, fliall be the — fine of the Sui^f
height at 6.

having the Latitftde, the Sms Déclina'
tioK and Altitudef to find the Hour of
the Day.

Take the Suns Altitude, from the parti-
JJlar Scale of Sines, between the Compafles ;

^onth, or Declination; then carry thc
^oiftpafles along the Line of Hours, between
^he Thred and the End, till the other Point
wing turned about) will but juft touch the
^■^.quot;ted, and then the fixed Point ftiall fliew
true hour and min. required, in the

Ore, or After-noon ; if you be in doubt
J^'hichit is, then another Obfervation pre-
eiitly after, will determine it,
Examfle.

May lothj at 30 degrees of Altitude,
the hour will be 32 minuts after 7 in the .
Morning, or 28 minuts after 4 in the After-
noon.

This Work being fomewhat inore diffi-
cult than the former, I lhall part it thus ;

JTiakc It a =: Co-fine of the Latitude ; la/
the Thred to N D, and on the degrees
fiiall give the Hour from 12, as it is figur«lgt;
counting ij-degrees for an hour, or froquot;'

Latitude f 1-50, Altitude 20, the hour is
8 amp; 12' in the forenoon, ox 3-48' in tlie
afternoon.

2. The Latitude, Declination, and Ahi'
tude given, to find the Hour at any timf'

Gen.Quad Fiift by the 25 th Ufe, find the Suns AI^
titude or deprefTion at 6; then in Summer-
time , lay this diftance from the Center
downwards ; and in Winter-time, lay it
upwards from the Center toward the End
of the Head-leg; and note that Point for
that day, or degree of Declination ; for by
taking the diftance from thence to the SunS
Akicude, on the General Scale, you have

®lt;ldcd, or fubftra^ed the Altitude at 6, to,
from the prefent Altitude.
(For by taking the diftance from that
loted Point, over, or under the Center, to
Suns prefent Altitude, yoij have in
^•Jiumer the iiffertnce between the Suns pre-
f^it Altitude, and his Altitude at 6. And
Wmter you have the fum of the prefent
Altitude, and the Altitude at 6.)

This Operation is plamly hinted at, in
4th Chapter,and pth and loth Sedhon,
^bich being underftood, take the whole
operation in fhorter terms, thtu }

Count the Suns Declination from po, toi
^ard the endgt; and thereunto lay the
^ bred; the neareft diftance fitom the fine
Ji.cne Latitude to the thred, is the Suns
Jl^ight, or depreffion at 6 : In Winter ufe
Jne/«» of, in Summer the diftrenee be-
fween, the Suns Altitude at lt;J, and his pre-
»ent Mitudej with this diftance between
your Compafles, fet one Point in the co-fine
Of the Latitude; lay the Thred toND,
then take the N D from po, to the Thred j
»hen fet one foot in the Co-fine of the Suns
dechnation, and lay the thred to N D,and
the degrees it gives the hour required j
.fooi 6 counting fiquot;om 5gt;o, or from 12, as it
^^figuxtd.

^ On. Jprll.at 30 deg. 20 tnîn. of M-
ticùàe^ Latitude, ç 1-3.2, the hour will bc
fpundt«? be jvill 2 hQurs from or juft 8.

On the lôth of iVoî/fwétfr, at 8 deg. 2.?
min. high,' it i$ juft 3 hoiurs^'om or 9 ^
dock in the forenoon, or 3 M^ernoon,

• Také the —^-fine à{%\Kfutli,ol:i'tpmch
of thé Sinis^refent A'titude^and Altitude
at and hïake it a in ^heco-fine of tft*
Latitude,' and lày the ''Thred to the nea'rS''
diftancé i:h'èfi -tâke oiit the Sécant
the decliiiâtibn btyoïid ptî^,- and make ic ®
= line of ba; and layicg the Thred t*^
thè ncareffdiftance en the ^egrees ir
fh'êw tbehour from 6 reqyired. /
By Artifi- Firft. by Ufe 2Ïiïid thé Sùns l^ight at
cral Sines^, ôr ae?^ff\tin in Wihttri theirby'^'
former 2d,-iftnd the fm tt d.fferm^^^
tXveén the Ahitude at 6, afldthe Suns pr«'
fent Altitude j but if you quot;hâve Tables ^
Naturai'Silies and Tangents; then in Wii|'
ter, add che Natural Sines of the two Alt^
tudes together ; and in Summer, fubftrâ^
the-leffer'^ur of the greater, and finddr^
Ark of différence more exaédy.

As the Co-fine of the Latitude, to thé
Secant of the Dechnation (counted be-
yond cfy as much forward as froiti'po

. Take the -4—fecant of the Suns declina-quot;
■tion, make it a iiuhe co-fine'of the'La^
titude ; then take out the fine of the ƒ»gt;»
or difference, and turn it twice from the
Center latterâlly, attd it fhall'-be the fine of
thc hpur from'required;

'April 20, the Suns Declination is if df'
grees; and the Suns Height at 6, then i^f
II deg. 42 min. now the Natural fine
.11-42, 20278j taken from the Natural
fine of 30 deg. io min. f ojo;^ tire Suns
prefent Altitude, the refidue is 30224, the
line of 17 deg. 3 5 min. and a half.
Then,

The — Secant of ly made a fine 01
38-28, and the Seétor fo fet, the = fineol:
17-3 f-', turned latterally twice from the
Center» fhall reach to 30, the fine oi
2 hours from 6, the hours required.-

Ufe XXVII.
Having the Latitude, the Suns Vecll»^'
tion, and ty^ltitHde, to find tht Sft»^
Az,imuth,

Particular Take the Declination from the particular
^'quot;''''■quot;quot;^•ScJleof Snies, for the particular Latitude

the Inftrument is made for ; Then, couo^
the given Altitude on the degrees from 6o\°
toward the loofe-piece, and fometimes on
the loofe-piece alfo ; and thereunto lay the
Thred, then carry the CompalTes, fo fet, a'
, long the Azimuth-fine on the right-fide 0»

the lefc.fide iu Southern-declinations, till
quot;e other foot, turned about, will but juft
jotich the Thred ; then the fixed-point ftiall
^ay at the Suns true Azimuth required.

Take tm or three Examples.
I. Firft, When the Sun is in the Equi-
^oftial and hath no Declination, then there
nothing to take between your Compafles,
•^iit juft to lay the Thred to the Suns Aki-
^^de, counted from lt;5o|o on the loofe-piece
|oward the End ; then on the Azimuth-
quot;ne, it cuts the Azimuth from the South re-^
lt;3Uired.

At 00 degrees high, the Azimuth is 90
quot;omSouth; and at jodegrees high, it isi
; at 20 high, the Azimuth is 62-4.^ ;
^^ 30 degrees high, it is 43-30; at 34 de-
|tees high, it is 3 2 degrees of Azimuth from
^outh ; and at 3 8-28 degrees high, it is juft

. 2.. Secondly, at 10 degrees of Declina-
tion Northward, and 20' degrees of Alti-
take 10 degrees from the particular
and lay the Thred to the Suns pre-
sent Altitude,as before, and carry the Com-
pafles on the right-fide of the Thred onthe
•'^iimuth-line, till the other foot, being
turned about^ will but juft touch it; then
C c Jnbsp;fliall

3. But if the Dcclimalion be thefai^^
to the Soutlwards, and the Altitude alW
the fame ; then carry the Compafles on th«^
left-fide of the Thred, on the Azimuth-
line, till thc other foot, turned about,
but juft couch it, and you fhall find ib^
Point to ftay at 41 deg. 10 min. the tru^
Azimuth from the South required.

Note, That any thing, as thick as tbj
Rule, laid by the Rule, and the Thre^
drawn over, ic will keep the Thred fteady»
till you get the neareft diftance ino^^
truly.

. Firft, hy the i%th Ufe, find the Sms
titude in the f^ertical Circle, or Circle O)
EaBandlVeft, thfts;

Which Vertical Altitude in Summer o
Northern Declinations, you muftfubftra
out of the Suns prefent Altitude ; or taK
the lelTer from .the greater, to find 5
. ference ^ buc in Winter, you muft a-.d tn
deprelfion-in the Vertical Cixcle, to the

^fent Altitude to get a /ww,-which-mud:
oe done on a-Lir e of Natural S mes, or
Wthe Ta.jji e of. Natural Sine^, as be-,
focjinthe Hour, by laymsit over or un-
d«r;jthe Ceutev, and takmg from,that poted
I^oint tonbsp;A'Uitude-allthac

'^ay. Th^i ©k-ö the ctilfance from; the Gen-
^t to the Tangent-ot the Suns prefent AU
^itudeonthe loofe-ptece, which isth^Se^
Cant of the Suns'prefent Altitude, and lay
it from the Center on the Line. of.Sines,
and note the place ; :theu take the diftance
from lt;Jo, on the ioofe-piece,to che co-tangcnc
of the Lacicuda ( by counting i9, 30,
^c. from 6®,upward the moveable-leg) be-
tween your Göjiipüfles; then ferting one
l^oint on the Secant of the Suns Altitude laft
found, and poted pn the Line of Sines; and
^vith die other, lay the Thred to the neareft
diftance, and thefe ke^p it, (by noting wliac
degree, day of the month,qr hour amp; niuiut,
Ot Azimuth ic cuts).

Then take che —• diftance on the Smes,
ftom the fine of the Suns Vertical Altitude,
to his prefenc Akitude, for a dlference m
Gümmer; Or,

The diftance from a Point made beyond
quot;^he Cencer, ( equal to che fine of the Suns
Vertical deprelTian) to the Suns prefent Al-

Then having this — diftance of fat» ^^
ilffertnct, for Winter or Summer, between
your CompafTes; carry one Point parallelly
on the Line of Sines, till the other, being
turned about, {hall juft touch the Thred ^^
the N D, the place where the Point ftayethi
ihall be the Azimuth from Eaft or Weft, a®
it is figured from the Center j or ffon»
North or South, counting from 90.

As •— co-tangent of the Latitude, to th«
= fecant of the Suns prefent Altitude?
laying the Thred to ND ;

So is the — fine of the fum,ot differenCti
of the Suns prefent Altitude, amp; Verti'
cal deprefiion in Winter, or the dif'
ferewe between his Verdcal and pre-
fent Altitude in Summer ; to the
fine of the Suns Azimuth, at that Alquot;-
titude and Declination.

note, That in Latitudes Unien
when the comfleoients of the Latitude
ire too large, then workthtu;
As the — co-fine of the Suns Altitude,
j?^ Tangent of the Latitude, taken from
degrees on the moveable-leg, laying the
* «red to N D, then the — fine of the fum
^!^^iference, carried parallelly, fhallftay at
me Suns Azimuth required.

If the Tangents are too fmall, on the
J'ettor-fideisalarger; and if the Sines are
great, on the Head-leg there is a lefs.
Find the Vertical Altitude by Ufe 18,
^d the/«w or diference of the prefent and A
Vertical Altitude by the Table, or Line of S»
Jl^atural Sines, as before fhewed ; then the ^^
^anon or Proportion runs thtet;

the Tangent of the Latitude;
So is the fine of the fum or difference, to
the fine of the Azimuth, from Eafl: or
Weft.

Suns Altitude;
So is the fine of the fum or diference^ to
the fine of the Awmuth.

be half the — fine of the Azimuth ;
being turned twice firoin theCenter, 'the
whoiefine,nbsp;:

Ofelfetbta-,
Make the Tangent of the Latitude,
a — Co-fine of the Suns Altitude; thenth^
= fine of the fum ox di ference, fhall be the
«— fine of the Azimuth, meafured on the
Sine, equal to the Radius of the Tangents
firft taken.

1 in Latitude fi-32, Declination North
and South 13-1 y, the Vertical Aldtude or
Depreffion being 17-01, and the prefent
Altitude 20; the Azimuth for Soiuh-dequot;
clination will be found to be 31-45, fto®
South, the Depreffion, at Eaft and Weft'be-
ing 17-01 ; and the fnm of the prefent Al'
fitude and D eprdfion 3 9 -2 y.

For North-declinauon, or Summer-'time,
the di ference between the Vertical and pre-
fent Altitude, is 2-y4; and thc Abmuth
from South, will be found to be 86 dtgrecs
and I y minuts.

Ufe XXVIII.
To met\t a Scale, whereby to perform all
thefe Propofuions, by the former Rales,
agreeable to the TriangHhr Quadrant,
being added chiefly at a Demigt;nliratio»:of
the Injlrumem, aniformer Operations.

Firfl, Draw an Equilatteral Triangle, as
■^BC, at any largeuefs you pleafe, by
drawing fi'ft the Line AB; then take the
Extent A B between your Compafles, fee
Point in A, and with the other draw a
•^ouch o£ an Ark about C, then removing
Point to B, crofs the former Arch in the
^oint C; then the drawing the Lines A C,
^d B C, will conftituce the Equilartcral-
Triangle. Then confidec whether oneRa-
dius of your Scale fliali be double, tripple,
quadruple one to the other, and accor-
divide the Line A. B into 3, 4, or $
parts, as here it is into. 3 parts, to mak^-cs^
double to the other, ( and for Sea-Inftru-
ments into j parts is beft to make the Scales
quadruple one to the other ) whereof A D
is one; Then make B Hetpal to BD, and
A G and G I equal to DA; Alfo, make
equal to DF, the neareft diftance
horn D to A C; and D po. equal to D E,
^he neareft diftance from D to CB. Alfo,|

make BE the half of BH, and A F the
half of A G. Again, make F 4;, equal to
F D, and E 45; equal to E D, at neareft d'-
ftance. Further, if you lay the Radius D r
onee from I, which is 60, it fhall reach to
lt;y9.y4 near C; and being repeated again^
it fhall reach to 75 j for i-oooo the Radiiquot;
or Tangent of 45, once added to 1-732»
the Tangent of (So, makes 2-732, the
Tangent of lt;ïp-y4 ; but if you add
twice, it makes 3-732, theTangent of
jufl.

Then making D E Radius, defcribe the
Circle 90 E I, and divide it into 180 equal
degrees ; Aifo, draw the leffer Circle pof
to the Radius D F, then a Rule laid to the
Center D, and every one of the 180 de-
grees, fhall divide the Tangent Lines A
andBC, into 180 degrees; and if yoi»
work right, you will meet with all the (ot'
mer Points, F, G, 45', I, lt;^9-54, 60-4^}
H,and E, in their true places, as firft drawn«
Alfo, Perpendiculers let fall from every
degree in the Circle 90 E I to the Line D
fhall divide the Line of Si ies, D 90, to the
the greater Radius; and the like Perpendi'
culers from the decrees in the lefTer Circle,
to the Line D A, fhall divide the leffer Line
of Sines ; Alfo, the Extent from the Center
D. to the Tangent of any Ark or Angle iquot;

I ^hcLine AC, counting from F, fiiall be
Sccant to that Ark or Angle, to the lef-
Radius; and the meafure from the Cen-
, D, to the Tangent of any Ark or Angle
JJi the Line C B (but counted from E) lhaU
the Secant to that Ark or Angle, to the
^''eater.Radius.

^quot;red faftened at D, will perform any Tr«-
f^tionhy the,Rules here inferred, and is
the very making of the Trlangultr Qjn^
quot;'■««r j, or you may put thefe Lines on a Rule
^ a plain Scale, and ufe them thus:

Firft draw a ftreightLine, as AB, re-
Ptefenting the Line A B in the Trianguler^
^airanf, then appoint in that Line any
Ponat for a Center, as C ; then for this Pro-
portion of .finding the Azimuth, the Sines
and Tangents being on a ftreight Scale,work
thtts.

. Take t]tie Sine of the Latitude, and lay
from C to J1-3 0; then cake out the Sine
' ..... of

''' Summer'pi'-J^j'ffoftitlieSouth, becaufti
thet 'refent Altitude is lefs than the Vertical.]
Eaft and Weft. . .. . ,
But when the G6-t^gent of the Lati-1
^'ide is too large for a Parallel entrance, then'
Pficlcolf firft theTii^eiit of the'fatitudc,
®nd-take the Co-fine tjf the Suns Ahitude |
^^ Work in a Parallel Nvay v which will reme-
dy the inconveniehccs V Thus you fee that
^ drawing three : tind only this work is
done ; -yet not fo foon by fa:r,as by the In-
ftrument with the Thred, which reprefents
^hofe Lines more certainly and exadly, after
'he fatiie way of Operation.nbsp;-

, As the Co-fine of the Latitude, to the By Arrfi^
Sine of the Suns prefent Altitude • S. ^ 1.

?o is the Sine of theladtude to i 4th
fine; which 4th iine is to be added
to the Suns Amplitude, for that time,
on a Line of Natural fines, and the [nm
obferved, as a jth.nbsp;. ' ;

Ufe XXIX.
Having the Latitude, Suns Altltuih
and Vertical Altitude, to find
the Azimuth,

piiftj.find the Vertical Altitude by the
former Rule, and fiud the differeiice be-
tween it and the prefent Altitude, by the
Line of Sines: then take this difference fron*
the general Sines between your Compafles,
and felting one foot in the Co-fine of th®
Latitude, lay the Thred to the N D, the»'
take the N D from the fine of the Latitude
to the Thred; having this diftance, fet one
one foot in the Co-fine of the Suns Alt''
tude, and lay the Thred to ND, ando»»
the degrees it fliall ihew the Suns true
muth at that Altitude and Declination
quired.

, The Suns Declination being 7, the Vtr-
^cal Altitude is 8-J7 ; the Suns prefetit
Altitude being 30, the,difference or refidue
Sines will be 20-13, and the Suns Azi-
'^Uth found thereby will be (So-ia'.

As Cc-S, O Alt. to the 4th fine •
So is S. 90, to S. of 0 Azimuth, ffotil
EaftorWefti

Firftj find the Suns Amplitude for that
Oeclination, thui ; take the — fine of the
t)eclination, make it a —an the C -fine of
Latitude; lay che Thred to N D, and
oti the-degrees it gives the Suns Annlicade
for that Declination, which you mull: re^
quot;member. '

, Take the —fine of the Suns prefent
* ^titude, make it a — ia the Co-line of
Ddnbsp;Lati^

Latitude, lay the Thred to the N D, thequot;
take the ND from the fine of the Latitude
to the Thred, and as the Compaffes
ftand, let one foot in the fine of the Suquot;®
Amphtude firft found, and turn the other
fcot onw ard toward po; then take froiO
thence to the Center. Thus have you added
the Amphtude, and laft found diftance tc
gether on Sines, then this added Latteral-
diftance, muft be made a Parallei in the Sun^
Co-altitude, and the Thred laid to the
neareft diftance in the degrees, gives the A'
zimuth required.

At ly degrees of Declination, and
degrees of Altitude, the Azimuth will b®
found to be 4p degrees 46 minuts from the
South, and the Amplitude «I-S o in y
of Latitude.

As co-S. of Lat. to S. of © prefent A^*
So is S. of Lat. to a 4th; whi:hyouinU^
add to the Suns Amplitude on Naturk
Sines, and keep it as a fum i
Then,

Ufe XXX.
'having thc Latitude, the Sms Pecllua-
tioK, hts Meridian and prefent Alii',
tude given, to find the Hoar,

^ in the Co-fine óf the Suns deciinacion,
''ying the Thted to N D ( and note the
Wacè) ; then take the — diftance on thé
i'nes, between the Suns Meridian and pre-
Akitude, and lay it from the Center
*°ward jgt;o- thenthe'ND from that Point
the Thred (asbelbre laid ) (hall be the
Verfed Sine of the Hour, meafured on a
^ine of verfed Sines, equal in Radius to
^^e Line of Secants firft taken, as the Sines ,

Make the -—Secant Latitude, a = Sine gy ,he
Co-dechnation ; then the — diftance SeSar*
between the 5««x Meridian and prefent Alti-
JUde, laid on both Legs from the Center
latterally, .and the = diftance between,
•Meafured on verfed Sines, equal to the Se-
^ants, fhall give the hour requir.e^j ■ as the
^reat Line of Sines oa the Seótör quot;kfe, by
turning the Coitipaftes twice, becaufe the
of Secants is half the Radius of thofe
•''les, as at firft was hinted.

r 40lt;J ] frort'
and recken the cxcclk above
Center toward poj and takenbsp;qo ' ,

But when the complements ijcc
!gt;P,then the N D from the ' d quot;
the Thred, fliall be the verfednbsp;^^

ButinWmtcr, when the
complements arc above po, ano
backwards, from the Center, fo^ i^j!»'''!
take the—diltance from thence ^
of the 5«»; declination, the Irfl'-'' jcliuquot;
greater, and fet this diftancc,
from the Center downwards; 'flif'*quot;
neareft diftancc from thencc to
ftall be the verfed Sine of thenbsp;'quot;J

But when the Latitude is 'el' , (!''
Suns Dechnation, and the 'a'quot;^
take the ~ diftancc (on the Sii^V j^^ngdJ
fum of the Suns Altitude, and o, ^^
found by Addition, whennbsp;jl

greater, and fet this diftance, or refidiJ '
from the Center downwards; then '
neareft diftance from thence to the Thre^'
fliall be the verfed Sine of the Azimuth.

But when the Latitude is lefs tban '
Suns Declination, and the fame way ; ^
take the — diftance (on the Sines) from '

found by Addition, when under pO,
counted from the Center to the declination
and lay that from the Center, as before
fiiewed.nbsp;J

Center to po, count the excefs from 90
toward the Center, and take the — diftance
from thence, to the Sine of the Suns dechna-
and lay it from the Center, as before î
then the ND from thence to the Thred,
^all give the verfed Sine of the Suns Azi-
•^tith on the fmall Smes beyond the Center.

The very fame manner of Operation that SeSor^
^=rves for the Central-Qitadrant^ ferves al-

for the SeBor, and this way being more
^toublefome than the reft, I fliall fay no
tioretoit, but proceed to others.

Suns diftance from the Elevated Pole toge-S amp; Tan.
ther, for z fttm; and find the half/«w, and
the difference between the half fum and thc
Co-altitude.

So is the Slue of the ilffertnct, to the
verfed Sine of the Hour, if you have
them on the Rule; if not, to a
Sine, whofe half-diftance on the SineS
towards po, gives a Sine, whofe com-
plement doubled, and turned into
time, is the Hour from South
quired.

Example, at deg. Ap.mln. Alt It tide f
and deg. min. Declination, La'
titude $1-1% North.

J 3-18 the Co.altitude 5 3 8-28, the Co'
latitude ; and ,66-29, added together»
makes 15 8-15-for a /«wj then the halfquot;
fupt is 79-07, and the difference between
79-07 and 53-18, is 2J-49 for a difference«
■nbsp;Then,

The Extent from fine po, to the SineCt
58-28, will reach the fame way from th^
fine of 66-ipj to the fine of 34 47j
a4ihSii.c.

The Extent from Siiie 34-47, to Sine
79-7» ihall reach the faiiie way-fioin the

of 2J.4P, the dlfftrenee, to the Sine
^ 48-34, a 7th Sine, right againft which,
On the verfed Sines, is 6o,viz.. 4. hours from
«oon.

Thehalamp;diftance, between Sine4S-34,
^d the Sine of 90, is the Sign of 60 de-
uces, whofe complement, vtz,. 30 doubled
is 60 degrees,or 4 hours in time, from noon,

To fl„d the Suns Aamp;mttth, having the
fame things given, viz. Co-latitude,
Co.altitude, and Suns dtjianct from the
Pole.nbsp;-

Add, as before, the three Nunbers toge- Artifidal-ii
ther, and thereby find the fum, and half- amp; T.
fum, and the difference bftween the half-
ƒ»»», and the 5m«/alliance from the Eleva-
ted Pole,

So is the Sine of the Jiferenct, to the
verfed Sine of the Sms Azimuth, from
South, (or to a 7 th fine, whofe half-
diftance, toward 90, gives a fine,whofe
complement doubled, is the Azimuth
from South).

The 3 Numbers, viz.. 38-28, 49-7, and
77-c, added together, makes 163-3 f}
whofe half is 81-474, and the difference
between the halfl/««,, and the ^»»i diftance
from the Pole, is 4-47 i.

As fine 27-3 6, to fine 81-47;
So fine 4.47 ^ , to ( V.S. of 13 o, the A-
Zimi^th froni the North :) the fine oi
lo-ij-, a 7th fine, whofe half-di-
flance toward po, is2j, whole com-
plement doubled, is i jo, the Azi-
muth from the North, whofe compk-
naentto 180, vis,, yo, is the Azimuth
from South,nbsp;Having

Jiavlngtle {me complements^ to find tht
Hour, and Az,tmitth-, hj the General-
Quadrant and SeSlor; and firfi for iht
Ai^imutb,

Flrftf of the complements of the Lttu By the
tvide, and Suns prefent Altitude, by fub-
ftrattion find the difference.nbsp;^teiw.

Secondly, Count this difference on the
Line of Natural Sines from 90, toward the
Center, as the fmaller figuresare counted.

Thirdly, Take the diftance on the Sines,
ftom thence to the — fine of the Suns de-
t^lination. Biit note. That when the Lati-
tude and Declination differ, Ws;,.one North,
and the other South, as ic is with us in
Winter ; you mufi: count che Suns Decli-
iiation beyond che Center, and call it thc
Suns diftance from the Elevated Pole, and
take from thence.

Fourthly^ Make this — diftance,a = in
the Co-fine of the Latitude, laying the
Thred to ND, or keeping the Seftor at
that opening. The»^

Fiftly, Take out the = fine of po, Ani
Sixtly, Make it a = fine in the Suns
Co-alcitude, fetting the Se6for, or laying '
the Thred co the (neareft diftance) N D.
Seventhly, Takeout tbe= fine of po.

'Elghtly, Meafure it irom the fine of pOgt;
towards; (andif need be beyond) the
Center, and it ftiall reach to the verfedfine
of the Suns Azimuth from North or South,
Avhenyoii count from 90; or from Eaft br
Weft, if you count from the Center, on a
Line of Sines, or middle of the Line of
veirfedSines»

Note, That if the general Sines are too
big,:you have a lefs adjoyning, whereon to
begin and end the Work ; as fcmetime the
Hour-Scale, and fometimes the Line of
Right Afcentions.

In the Latitude of 51-32, the SunsDc-
clination 18-30, the Altitude 48-12,
you fliall find the Suns Azimuth to be 130
from the North, of j o from South.

I. Firft of the comrlement of the Lati-
tude, and the. diftance from the Eleva-
ted Pole, find tne 4'fftgt;ence by Subftrawli-
pn.

3.nbsp;Take the —- diflance from theiKC, to
^to the fine of the Sms prefent Altitude.

4.nbsp;Make this diftance, a = in the
Co-fine of the Latitude, fetting the Seflor,
or laying the Thred to the N D, and there
keep it;. ' , , ' ^

lt;5. Make that ain the Co-fine of the
'^Suns Declination-,, liyiiig the Thred to N D.
' ■ A Then take oucthe = fine of po again,

; ■■■■ ' ' \ ..
quot;8. Meafure it .frtopo, toward the Cen-
ter, - and it -{hill^cWcthe' verftd fine of the
Hour from Mitf-hi^it, or the cofitrdty from
iiooh ; or from if you count from the
..Center of the Sines, or the middle'on verfed
■ Sines, .

- Latitude yi-32:. Declination North
20-14, Altitude J0-5 J, you fhall find the
Hour to be i jo from Norch, viz..''xo in the
fore-noon, or Jo'degreesftiortof South.

Ufe XXXIV.
ïïavlng the Latitude, Suns Altltudé, ani
dtflance from the Elevated Pole, tofi«^
the Hour, by the Line of verfed SineSt
en the SeSlor.

Firft y By Additien, find the fum of, and
by Subftraftion, the diferenee between the
complement of the Latitude, and the Su»'
diftance from the Elevated Pole.

Secondly, Count this fum and differ end
from the Center, or the verfed Sines on the
Seélor, ( or the beginning of the Azimuth-
Line, if you ufe that, or any other, which is
not drawn from a Center ) and with Com-
pafles take the — diftance between them.

Fourthly, Take thc — diftance between
the verfed fine of the fum, and the comple-
ment of the Suns Altitude, and carry
parallelly till it ftay in hke verfed Sines,
which fhall be the verfed Sine of the Hour
from the North Meridian, or mid-night.

If you take the — diftance from the
difference to the Co-altitude, and carry that
till ic ftay m like fines, it fliall be the
hour from noon ; counting the Center

JJdvlng the Latitude, the Sum Altitttiti
and Dtfianee from the Elevated Tolt, to
find his true Az.imuth from South er
Northy hy Natural verfed Sines.

firfi, OftheCo-altitude, andCo-lati-
Jl^e, find the fum and diference, by Ad*
^«lonandSübftraaion.

Secondly, Count the fum and diferenct
«om the Center, and take the — diftance
them with Compafles on the verfed

j ^ counting the Center tbe Eleva-
ted Pole, and 90 the Equinoaial) and
^rryit ^ till itfiay in like parts, which
lhall be the Azimuth from South.

If you take the.— diftance from thé
^fftrence, to the Suns diftance from the
^ole, and carry it as before, it fhall ftay at
^e verfed fine of the Azirauthj from the
North part of the Horizon,

S There five general wayes of finding tlt^
Hour and Azimuth, are not all needful
be learned by every one, but to delight the
ingenious, and to hold forth the ufefulnciS
of the Inftrument, and to fupply-
t^a^at fome times^.may ha^en by Excurn'
ons, and as a foiir-fold Teftimon.^, to fhe^vquot;
the harmony ib feveral way es of Operation?
the firft par ticular quot;way, and this laft by
verfed Sines, being moft eafie - and compre*
bcniive of any. other.

verfed Sines be fet but on one Eeg, then ij
is-jto belaid the neareft diftapcs inifead
9'f like parts', after the manner ofufingth©
Thred on the.,General Qmdranti,
. jilfo, if you have ic noc at aU, then the
^.zimulh-hhe fcr the parcicular' Lacicude}

and if thac^e .coolarge, che fitde Line of
Sines, beyond the Center, will fupply this

very wpllthus; ' ■
r^.tii^k Turn the Radius, or whole length
of that Line of Sines, two times from
Center downwards, ( which in Sea-InftrU'^
maus, will moft conveniently ftay ac 30 oa

large Lineof Sines, or general Scale, aS
^as hinted in the 28th Ufe, being jiift 4
quot;Hes as much one as the other). For a Point
J^Prefentingi8oof verfed Sines, to fet the
Jfmpaffesin, when you lay the Thred to
^ 0, and to take any verfed fine above po
^^grees j this being premifed, the Operati-

unt the Center po, 10 on the fmaller
3 quot;quot; * and 20 on thé fame Sines, no,

urn this diftance the other way from the
Renter downwards, and note that place, for

th/j-i-—^quot;'quot;' between this fun,, and
'«^rmf 41-32, as the fmaller figures,
^cjcoiut, being taken between your Com-
foi j'nbsp;m 180, the Point firft

«ld, and lay the Thred to N D, and there
P't (or obferve where it cuts), then ta-
E enbsp;king;

king the ~ dii;i^nce between tlie verieOj
fine , of the ilftrmHi counted as the fm»quot;
figures are reckoned, and the fine of tiiC
Sms Altitude 25-, as the greater figures are
reckoned from the Center toward the End;
and carrying this Extent parallelly aloflS
the greater Line of fines, till the other Poiflt
will but juft touch the Thred at N D»
Then, I fay, the meafure from that Point to
the Center, meafured on the fmall fineS) a'
verfed fines, ftiall be the verfed fine of tbe
Hour required, viz.. 62 from South,
7 ijo^J'ts J2 minuts from mid-night.

This Rule, or Ufe, is longer far in wor»^
ing, than the Operation need be in worquot;'
ing; for if you fhall approve of this waY'
the adding of two brafs Center-pins
fhew you the two Points moft ufed very
readily, and the Thred is fooner laid, tbaquot;^
the Legs can be opened or fhut, and the Jn'
ftrument keeps its Trianguler form; as i^'
in^ during the time of Obfervation.

. This Problem being not of fuch ufe as
the contrary» vl^t having theAltitude^^'J

Lay che Thred to thp Day, or Dechnation,
•^hen the N D from the Hour to the Thred,
quot;leafiired in the particular Scale of A!ri-
•t^des, lhall ihew the Sms Altitude re-

I. Firft, of the Co-laticude, and Sms
Q'ftance from the Pole, find the inm alt;id
'^'Jference.

Take the -— diftance berween . nem.
make it a = verfed fine or i8o oy ic:-
the Sedor, or laying the Thred ta

the s6th, and ,s nothing dfe but a oaci.--
Ward working; but the Altitude ac cny
Azimuth, is not fo to be done.

Firfl, having the Latitude, and the Sunt
Dedination, find the Suns Ahitude, or
Depreffion at lt;J; and note the Point, eithei
below, or above, or in the Center, as is large-
ly fhewed in Ufe the 2lt;5th, where the Al'
titude is given, to find the hour in any La'
titude.

Take the N D from the Co-fine of the
Suns Declination, and make it a = in the
fine of 90, laying the thred totheND?
then the N D from the fine complement 0«
the Latitude to the Thred, fhall reach froi^
the noted Point, for the Suns Altitude ot
Deprcflion at lt;J, to the Suns Altitude rC'
quired.

So is the fine of the peclination, to^^quot;quot;^'
the fine of the Suns Altitude at 6.

So IS the fine of the Suns diftance from
6 in degrees, to fine of the Suns Alti-
tude.

Which 4th Ark being taken from the
Suns diftance from the Elevated Pole, then
Jne refidue is the yth Arkbut for hours
Pefore and after add the 4th Ark, and
we diftance from the Pole together to
^'^akeajthArk.

So is the Co-fine of the refidue (or
being the jth Ark,. to the fine of ttiC
5^»iAltitüde at that hour.

Tht Latitude, Suns Jz.imutb and
clination given, to find th Altidude,or
height thereof. . lt; ,

■ Aslin? ?^} to the Co.fine;óf thc Azi-
muth from South;nbsp;'■ f
So is Co-tangent Lat. to the Tangent
the Altitude, atthat AziniWiiquot;;
theEquinoaial, which you ™ «a
• ther intoTable for every fingle ^e .

So is the Co-fine of the Suns Altitude in
Equinoaial, to the fjne of a 4th A^k,
^ht»,

When the Latitude and Declination are
®hke, as both North, or South; then add
the 4th Ark and the Altitude (in the Equa-
tor), together, and'the/«/» is the Altitude

But in'Winter-time, when the Latitude
and Declination is unlike, take the 4th
Ark. out of the reciprocal Altitude in the E-
quator, and the ■refidue is the Alti-
tude required.nbsp;' . : . „

Alfo, in all Azimuths from Eaft and Weft
NortWards, in Summer-time' alfo, you
HHift life Subftraaion alfo, and not Additi-
on i as.the Rule before-goingfuggefts.

. Take the 5«» ov SWj Declination from
the particular Scale, and fetting one Point
in the' Sms hzimvLth, on the Azimuth Line,

and with the other lay the Thred to the
K D, the right way, and on the degrees the
Thred cuts the Altitude required.

quot;quot;As the-—Co-tangcnt jLatitudCy t^equot;
from the Moving-leg, or Loofe-piccC,
to = fine of 90, layine the Thred w
ND;

So is the = Co-fine of the Azi-
muth from South, to the — Tangent
of the Suns Altitude in the Equator»
at that reciprocal Azimuth.

As the —- Co-fine of the Suns Altitude
in the Equator, to the = fine bf the
Latitude , laying the Thred to the
ND;

Which 4th Ark is to be added, or fub-
firafted, as immediately before is direftedj
and the fur» or refidue, fhall be the true Al-
titude required.

At 60 degrees of Azimuth from South,
the Equinodhal Altitude will be found to
be 21-40, for I,9»lt;/«« latitude of y 1-32 i
and the 4th Ark in s or -^yis aS-K?.

•«Kes 49-ylt;î, the Suns Altitude at 60 de-
^^es from the South in
p The fame way of working ferves for the
as is ufed for the General Qteairant,
, obferving to fet the Seftor, inftead of
gt;ng the Thred to the neareft diftance, as
^^ Ingenious will foon perceive.

Ufe XXXIX.
having the Lathtfde, Declination,
^»thy and Altitude, to find the Hour.

Ufe XL.
Having the Latitudef Declinatien, UoUT)
and Altitudef tn find the jizamuth'

Particular Firft* find the Altitude at that HoUiquot;?
Quadrant, and then the Azimuth at that Altitude, ^
before.

. So is Co-fihe Declination, to fine of ^^
Azimuthiamp;om South, or North, ast''^
Hour is counted ; ,ths «to fay, fr'quot;''

if the Hour.'.is^ bietween
moirhingj and d at night ; and fi®'^
- ' the North if the contrary '; that

So is the Tangent of the Sunt Declinati-
on, to Co.tangent of the Suns Azimuth
from the North, at the hour of lt;5.

tion,a = fine of 90, laying the Thred to
^ D, then the r= Co-fine Latitude ftiall be
the — Co-tangent of the SUns Azimuth

To find tht Amplitude, %AxAmuth,ti- .
fing. Setting, and Southing of the fixed
Stars, having the Latitude, Altitude,
and Vtclination, or time of the year
given.

Azimuth-line; and count the fame way»
and the other Point lhall ftew the Star^
Aipplitude required.

The Declination of the BuOj Eyt) being
if-48 5 if you take iy-48 from the parti-
cular Scale, and lay it from pointheAz»'
muth-line, it will reach to 26 de^teeSf
cottnring froni 90 towards either end, l'»®
Came as for the Sm in Ufe 16. But in other
Latitudes, work as yon do for the ^^
the Rules in the i(Jth life abovefaid.

The work here is the fame as for the
thtu • Take the Stars Declination ftom ch^
particular Scale of Altitudes, or Sines,
tween your CompafTes, and lay the Tbrelt;^
to the Stars Altitude, counted from
wwardthe Loofe-piece; then carry the
Compaffes, or the right-fide of the
for Northern-ftars; and on the left-fide
Southern-ftars, along the Azimuth-line, nil
the other foot, being turned about, will but
jufl touch the Thred • then the fixed Poinquot;^
on the Azimuth-line fhall fhew the Stars
Azimuth, from the South, required.

-The Ejt being 30 degrees high,
have 77 degrees and 10 minuts of A-
^inuth from the South.'

you be in other Latitudes, ufe the ge-
j^eral wayes, as for the Sh» in all refpeSs,
paving the fame Dechnation that the Star
''»th North or South.

Count the Stars Dechnation on the de-
^rces, as you count the S«»/, North, or
South, and there lay the Thred; and in the
of Hours is the Stars Rifing, or Set-
ting» when the Stars Right Afcention and
declination are equal.

ufe 14, and fet down the complement
thereof to 12 Hours, and the Stars Right-
i^cention, and the hour of Rifing the
Thred cuts, and add them into one fum,
and the fum, if under 12, is the time of his
pfing in common hours; or if you add the
of Setting that the Thred fheweth, it
give his letting.

If you lay the Thred to t5'-4?, the D^quot;
clination of the Buffs Eye, in the Hoquot;'''
line it Cuts 4 hours 5 6 min. for Rifing ;
7-24, for his Setting ; then if you wot''^
for April the 23d, the Sms Right Afcenf'
on, then is 2-44, and the complement thet^
of to 12, isnbsp;and the Stars Rigl'^;

Afcention is 4 hours and 16 minutes;
the Hour cut, is 4-3 (J for Rifing; and t^®
three Numbers, viz.. 9-15, the completne''^
of the5««i Right Afcention, and
the Scars Right Afcention, and 4-3 (J, tquot;®
Hour of Rifing the Tlued cuts, bd'^^
added, makes i 8-8; from which, taki^ i
12, reft (J-8, the time that the Balls
Rifeth on April 23 J and if you add 7*
the time of Setting that the thred cuts, ther^
comes forth viz.. one hour and
min. after the Shk.

Suhftrait che Right Afcention of the Shui
from the Right Afcention of;che Scar,
creafed by 24, when you cannot do with'
out, and che remainder, if lefs chan ''
the time between .la at noon, and

'^^ht J but if the vemauidcr be more thm
it is the time between mid-night, and
®iid-day, foUowiag*

The LyoKS-Heart, whofe Right Afcenti-
is 9-yo, will come to the South ou
^arch IO, at p-48, the Smi Right Afcen..
being then only ^ minuts.

Extend the Compaffes from the Suns
^igbt Afcention, to the Stars Right Afcen-
that diftance laid the fame way from
at the middle, or at the beginning, IhaH
»each to the time of the Stars coming to

Ark is, having the fame declination, and;
that doubled, is the whole time of continu-
ance ; Or, if you fhall add and lubftrati: it
to, or from the time of the Stars coming to.
South, you fhall find the time of Setting:
orRifing.nbsp;:-n

By laying the Thred cothe5flt;irj Decli--
*lation, it ilieweih tb« Alceucional mfterencq

in this Latitude , which added in thof«
Stars that have North dechnation, or Tub'
ftrafted in Southern to lt;J hours, gives the
femi-diurnal Ark of the Star above the He
quot;zon;nbsp;Examfh,

The BuUs Eye's Afcentional-difference,''
one hour and 24 minuts; which added tquot;
6 hours, becaufe of Northern dechnatioquot;)
makes 7-24, for the femi-diurnal-Ark,
14*' 48', for the whole time of being above
the Horizon»

Note, That to work this for other Laquot;'
tudes, the Smus Afcentional-difference is f®
be found for that Latitude you are in, and
the Operation is general for all places.

On any flat Hori^ontal-PUin^ fet up »
ftreight Wyre in the Center of a Circle ;
or hold up a Thred or Plummet, till the
lhadowof the Thred cut the Center, and
any where in the Circumference, which
two Points you muft note; then immediate-
ly take the Suns Altitude,and find the Sft»^
Azimuth, and count fo many degrees
the Circle the right way, as the Suns Azi-
muth comes to, from the Points of the fha-
dow marked in the Circumference , and
draw that Line for a true Meridian-line«

•^prning, and after two afternoon; or I'ri the
by two Plumb-lines, fet in aright-
^'quot;e with the North-Scar, at a right fcitna-*
tion.

\ Pirft, find the Stars Altitude^ by looking
I ®'ong the Fixed or Moveable-leg, to thc
'^^'ddle of the Star, letting the Thred, with
Inbsp;Plummet, play evenly by the de-

' ^'■'^es, between your Tluimb and Fore-
l^ger, to the end you may command che
^quot;fed, and know whecher'it playeth well

lake che Altitude found, from the par-
quot;cular Scale of Sines, and laying the Thred
over the Stars decimation, which for readi-
gt;iefs fake is marked with r,a,3,4,y,lt;^,7,8.

to he 12 Names of tL Stars on che
. llt;ule ; and then carrying che Compafe as
you do in hndmg che hour by the 5««, yoii
I »hall hnd how much the Scar wants, or is
I pt the Meridian, which is called the Stars-
'Hour ; And note, That if the Star be paft:
•the South, ic is an afternoon hour } if not
F fnbsp;eomö

. Jlfo, knowing the Sms Right Afcentioflj
fet one Point of the Compafles in the S«quot;*,
Right Afcention, ( counted in the Line quot;
twice 12, or 24 hours, on the outwatd-leamp;
of the fixedlpiece, next to the particu-l^^
Scale of Sines) and open the other to
.Stars Right Afcention, noting which ^^^
' you turn the Compafles; for the fame
tent, applied the fame way, from the Star
Jiour laft found, fliall Ihew the true hour
the night required.

• Suppofe on the loth of fagt;t/iaryfl(ho^
obferve the Altitude of the Bulls Eye to^
20 degrees; if you take 20 degrees,the A
titude, from the particular Scale, and
the Thred on i j-48, the Stars decli^^ioP
Northward, and meafure from the ^
fcale the neareft diftance to the Thred, yquot;
•fhal] find the Compafs-point to ftay at
onthe Eaft-fide of the Meridian j

The Extent from 8 hours 12 minuts t» ^
the Line of twice 12 hours) the Su»s K'g
Afcention, tonbsp;the Stars Right Ag'^^

quot;'ici the Moons Age, .and true Place fbr the
Ptefent time; then, by laying the Thred on
he Moons place, you may have her Right
^icention, and alfo the Suns Right Afcen-
i and by the Moons Altitude, taken
S ^^^ particular Scale, and the Thred
jj^'^'over the Moons place, you find what
e Moojj wants, or is paft coming to South,
^^ich IS called the Moons hour.

°out 40 mm after 3, there is a New Moon;
«en note. That die Suns true place, is thc
loons true place ; and confequently, their

As 8 hours 04 min. the Suns Right'A/-
cention, is to 8-04, the Moons Rigquot;
Afcention ;
So is the Moons hour at any Ahitude;
the Suns true hour.

Suppofe that on the ill Qiiarter-dayj^n
Moon being gone 90 degrees from the S^ ^
to find her place;

Set one Point in the Moons place ^
Change-day, and open the other to the
ginning or the end of the Line of 24 hoi' J
Then,nbsp;y

The fame Extent applied the contr^j^
way from 6 hours, or 7 dayes and a h» '
thé Moons Age, fhall give 28 deg- J ^
T J to which you mull add 7 degrees ^^^
30 minuts (the Suns place) between, ^
the fum lhall be the Moons true plaquot;^^
quired, viz,. 6-2S degrees in bquot;.

ter, will be on the^i y ch day later at mg^^j
then the difference between the Sun ^^
Moons Right Afcention, will be found t
near 6 hoars j for the Suns Right

J^marj i j, is 8-32 ; and the Moons Right
'cention, the fame day, being about 8 de-
btees and a half inv, is 2 hours and 28
^quot;luts ; if you talce the diflance bctween-
on the 24 hours, ic is near 6 hours ;
j^nich is the difference of time between the
^oon and the Suns hour.

Por the Full Moon ; on the 22 day, near
4 hours after noon, thc Moons Age being
H dayes I; if you add labours, or 6
^gn«, to the Moons place at the Change, you
'hall find S5 29-0; to which if you add
Mhj, the dayes between the New and
J^till, you fliall find il 13 deg. 45 min. for
lie Moons place ; the Suns Right Afcention
f 22 day is 9 hours, and the Moous the
lame day at i afternoon, is 9 hours alfo (or
father 12 difference) fo that the Suns hour

hours added, and 22 degrees alio together,
makes ,n 22 nmm. for the Moons
'lace, by help of which, to fiad the Moons
lour by her Altitude above the Horizon
tound by obfervation.

Without regarding the Sun or Mooquot;j
Right Afcention, having her true Age,
Hour,

The Extent from irin the middle, to^n
Moons Age under or over the middle,
reach the fame way, on the fame Lme, »
the Moons hour, to the true hour.

The like work ferves to find the hour
the night by any Planets, as Saturn,
or Jupiter, which are feen to ihine v
brave and bright in Winter evenings;
having learned their Place by their difta«
from the fixed Stars, or by the Ephe?ncr'»r^^
then their Altitude and Place will find
iiourfrom the Meridian, and the conip
ring their Ri|;ht Afcentions with the
gives the true hour, as before, in the
Stats.

Ufe XlV.
fi»i the Moons 'Place and Declination,
»ithout the Ephemendes, fowe-
what near.

Phft, obferve when the Moon is in the
^^quot;■idian, and then find her Altitude, and
pe the fame from the particular Scale be-
^^een your Compafles; then fet one Point
^^ the hour 12, and lay the Thred to N D,
^•^d on the degrees it fliall fliew the Moons
J^^hnation; and in the Line of the Suns
the Moons prefent Place, counting
JjerProgrefs orderly from the lafl:Change-
^^y? or New Moon, when flie was with

Obferve what Hour the Moon flieweth on
any Sun-dial, at the fame inftance by the
Hxed Stars,, or other wayes, find the true
Hour ;

The Extent from the Moons Hour, to the
^he true Hour, fliall reach the fame way from
to the Moons Age, right againfl which

her coming to South, at which time you
find her true Altitude, and fo ccme by
Declination.

ret again, for hey Age and Place, accor*
ding to Mr. Street, and /l^r.BlundeV)!'

AddtheEpacl, the Month, and Day
Month in one fum, counting the Mon^ .
from March, by calhug March the
Month, April the fecond, amp;c. then tn
fum, if under 30, is the Moons Age ; ^
if the fumhs above 30, then fubflrad^: 3 ^
and the remainder is the Moons Age, ,
the Month hath 31 dayes ; but if the M»quot;^^
hath but 30, or lefs than 30 dayes, then
flrad but 29,, and the remainder is ^
Moons Age.

Add to the Epail for the prefent ƒ ^^^
and in January o, in February 2, in
I, in April 2, in May 3, in June 4'
July'), 'n\ Anguß 6, in September'ij
(Moher 8, in November 10, in
io; and the fum, if undergo, otthcC^'^,
above 30, added to the day of thcMoiJJ '
abating 30, if need be, gives the Moons
that day; but fubftraéted from J^^^f
the day of her Change in that Month,
kom the beginning of that Month.

. îhcEp aft thac year is 2(î, andtheNiim-
/or July is y, theExcefs above 30, is i ;
^^hich added to any day of the Month as to
^Oj gives II, for the Moons Age, 7«/» 10.

Multiply the Moons Age by 4, and the
Produét divided by 10, the Quotient giveth
tnc figns ; and the remainder multiplied by
gives the degrees, whidi you muft add
^ the Suns place that day, to find out the
Moons place for that day of her Age.

On fuly 10.1668, the Moons Age is 11,
^vhich multiplied by 4, makes 44 ; and 44
divided by ID, gives 4 figns in the ^oti-
em ; and 4, the remainder, multiplied by
3,makes 12 degrees more j which added to
Lancer 29 degrees, the Suns place on the
lothday ofnbsp;^aj^çj 11 degrees in

Or rather by the Rule thus, on the Line of
34 hours by particular Scale, having
tht Moons place, to find her Age b) the
Line of 24 hours.

The Extent from the Suns true place,
the Moons true place, fhall reach the fafl^®
way, from o day, to the day of her Age.

The Extent from o day old, to the Moo''^
true Age, fhall reach the fame way from th^
Suns true Place to the.Mooiis.

Or, having the Moans true Place at the
Nerv Moon, to find her Place any
day of her Age after.

The Extent from r, to the Moons true
Place at thfe Change, fliall reach the fame
way, from the day of her true Age, t^ hequot;^
true Place, adding as many degrees to the
Number found, as the Moon is dayes old.

Having her Place, and Age, it is eafie to
find the MoOns Hour, and then her true
Hour; but I fear I fpend herein too much
time on an uncertain fub ie61:.nbsp;_

Ufe XLVI.
The Right Afcention ftni Declination of
«»7 Star, with the Suns Right Afcen-
tion, and the Hour of the Night given,
to fitii the Altitude and Ax.imuth of
that Star, and thereby to kgt;tow the Star,
if you k»ew it not before,

^ Set one Point of the Compaffes in the Stars
^'ght Afcention, found in the Line of twice
»a hours; and open the otiier to the Suns
ll^'ght Afcention, found in the fame Line;

this Extent fhall reach, in the fame
^^rie, from the true hour of the Night, to
jne Stars hour from the Meridian ; then

tne N D from the Stars hour, in the Line of
flours, to the Thred, meafured on thepar-
J of Altitudes, gives the Stars
Altitude; then by his Dechnation and Al-

And if the Inftrument be neatly fixed
to a Foot, to fet North and South, and turn
to any Azimuth and Altitude, you may
quot;nd any Star, at any time convenient and
yifible.

Ufe XLVII.
The Altitude ani Az,lmutb of any Star
beittgglvengt; tofinihtsDecltnatto».

Lav the Thred to the Altitude on the de-
grees, counted from 6o\o toward theen^
then fetting one Point on thc Stars AzimU
counted in the Azimuth Line, and take ti
N D from thence to the Thred j which oi
fiance meafured from the beginning ot tn ^
particular Scale of Altitudes, iTiall give I

If the Compaffes Ifand on the right-»»
of the Thred, then the Dedmation is NorC'S ^
if on tlie left, it is South 5 accordmg as y 0
work for the Suns Azimuth ina particul»
Latitude.

- The Altitude ani Declination of any St^Jj
with the Right Afcention of the Si* *
ani the true Hour of the Night '
to jni the Right Afcention of tquot;
Star.

'he Line of 24 hours on the Head-leg) from
'he Stars hour to the true hour, fhall reach
'he fame way from the Suns Right Afcenti-
on, to the Stars Right Afcention, on the
^ine of twice 12, or 24 hours.

Tg fi„i Vfhen any Fixed Star ccmetb to
South, by tht Line of twice 12, er 24
hours.

Subftra£lion, with its Cautions: But by
'he Line of twice 12, or 24 hours, mrk,
thus ;

Count the Suns Right Afcention on that
Line, and take the diftance from thence to
'he next 12 backward, viz,, that ac r, ac
the beginning of the Line, when the Suns
Right Afcention is under x2 hours j or, to
the next 12 iu the middle ot the Rule at
when the Suns Right Afcention is above 12
hours, (which is nothing but a rejeaing iz
for more conveniency).

The fame Extent laid the fame way^rom
'he Stars Right Afcention, fhall reach to
^he Stars coming to South. ,

the Extent from the Sun, to the ScatJ
Right Afcencion, lhall reach the fame wa/
from 12, to the Stars coming to South.

The Suns Right Afcention the 20th of
Augufir, is 10 hours 3 (J minuts ; the Right
Afcention of the Lions-Heart, is 9 hours
and 50 min.

The Extent from lO hours 35: min. tf
the beginning, fhall reach the fame way
from 9 hours jo min. ( by borrowing i®
hours) becaufe the Suns Right Afcencion^
more than the Stars) to 11 hours 13 miH'
of the next day, viz,, at a quarter paft i»i
or, at II hours and 13 min. the fame day»
where you may obferve, that the remainder
being above 12, if you add 24 hours, the
time of Southing is between mid-night,ano
mid-day next following,

To find what two dayes in the yt» an ef
equal length, and the Suns Rifing and
Setting.nbsp;,

Lay the Thred on any one day in the
''pper Lmc of Months, and Dayes, and at
^•le fame time the Thred cuts in the lower-
^•ine of Months the day that is anfwerable
^^ ic in length, rifing, letting, and declina-
tion, and other requifites.

The iR of April, and the 21 of August,
'tc dayes of equal length ; and the Suns
^fing and Setting is the fame on both thofe
^ayes; only in the upper-Line, the dayes
'quot;■e increafing in length, and in the lower-
^ine they are decreafing.

To ^nd how many degrees the Sun is un^
der the Horiz^on at any Hour, the De-
clination and Hour being given.

Count the Suns Declination on the de-
crees, die contrary way, viz,, for North De-
^lination, count from (Jr|o toward the end;

count for Southern D .clination toward
'quot;e Head, and thereunto iay the Thred ;

then take the neareft diftance from the hour
given to the Thred; this diftance mes'
furcd in the particular Scale of Altitude^
ihall ftiew the Suns Depreffion under cquot;^
Horizon at that hour.

January the jotharS at Night, hoquot;^
many degrees is the Sttn under
the JJorlx,on.

On that Day and Hour, the Suns Dccj'j
nation is about 20 degrees South; thenii
lay the thred to 20 degrees of Declinatio»
North, and take the neareft diftance ft®'quot;
8 to the thred, that diftance, I fay,
luted in the particular Scale, gives 34quot;^'
grees and 9 min. for the Suus Depreffion
der the Horizon of • 8 afternoon.nbsp;^

to do this in other Latitudes, you ^^
to find the Suns Altitude at 8 in Norths'^quot;
Declination, by Ufe 3 7.

Tie ufe of the Tmngnler
drant, in finding of Heights
iig. and Vijlances^ acceffabk or
inacceffahk.

Firft, The Trlangnltr ^adraut being
re6lified,aiid fixed to a Ball and SockcC
three-legged-ftaff, being neceflary in
thefe Operations to perform them exactly,
efpecially for Diftances; look up to the ob-
jeft as you would to a Star; and obferve
what degree and minut the Thred cuts, and
f«t it down : Alfo, obferve the place where
you ftand at the time of Obfervation, and
. the diftance from your Eye to the ground^
■and the place on the objefii that is level with
Vohr eye alfo; as the playing of the Thred
quot;Qd Pli^ramec will plamly fliew«

'Alfo, you muft have the meafure from
the place where you ftood obferving, to the
Point exadfly right under the objea, whofe
height you would have in Feet, Yards,
Perch, or what you pleafe, Cto Integers, afl'J
Fraftions in Decimals, if it may be.

Alfo Note, That in all Right-Angl^-
Triangles, one Acute Angle is alwayes the
con^Iementof the other ; fo that obferving
or finding one by Obfervation, by confc-
quence you have the other, by taking that
from po.

Thefe things being premifed, the Opera-
tion followes, by the Artificial Number^)
Sines and Tangents, and alfo by the Na-
tural.

Note alfo hy the rvajf. That in regard the
complement of the Angle obferved is fr^'
quently ufed, if you count the degrees th«
contrary way, that is to fay from thq Head»
you fhall have the complement required J ^^
hath been oftentimes hinted before.

As the fine of the Angle, oppofite to the
meafured fide, i? to the meafured fide,
counted on the Numbers j

So is the fine of the Angle found, to the
, Altitude or Height required on Num-
bers,

Standing at C, I look up to B the ob jedi, F^S-
whofe Height is reauired, and I find thc
Thred to fall on 4* degrees and 45 minuts j
but if you count, from the Head, it is 48-if,
the complement thereof, as in the Figure
you fee.

As the fine of 48-if, the Angle at B,
being die complement of the Angle at
C, is to 218 on the Line ofNiunbers;

Sois the fine of the Angle at C, 41-4^,
toipy the Altitude of AB the height
required, found on the Line of Num-
bers.

To which you muft add the height of
your eye from the ground, in the time of
Obfervation ; or on rifing grounds from a
mark on the Building, or any other objeA
that is level with your eye in time of Ob-'
fervation.nbsp;I

Fig-1. But if I were ftandingat D, 129 foot and
a half from A, and would find the height
A B, the complement of the Angle at Vf
that is to fay, the Angle at B is 33-30lt;

As the fine of 33-30, the Angle at B, to
the meafured-fide D A, 129 4 counted
on the Numbers i
So isnbsp;thefine of the Angle at

to 19y, the Altitude required, A
- and y foot more, the ufual height ot
the eye from the Level to the ground»
makes 200, the whole height iC'
quired.

to the = fine Of 33 deg. 30 min. l^Xquot;
ing the Thred to the neareft di-
ftance jnbsp;.
So is the = fine, of fy^-30, the Angle at
D, to the — meafure of ipf on che
Sc^le you took 129 ' from.

Firft, as before, find the Angle at D, or
rather the compkment thereof, t/jx,. 33-30 ;
then go further backward in a right Line
with the objedt and firft ftation, any com-
petent Number of feet, as fuppofe 88 ; to
C y there alfo obferve the Altitude or Com-
plement, viz,, the Angle A B C, 48-15.
Then,

As the fine of the diferenee M found,
viz.. the Angle C B D, 14-4^,
8 8 -î, on the Line of Numbers ;

So is the fine of the Angle at C, 41-45quot;»
to the meafure of the fide OB, 233»
on the Line of Numbers,

go is the = fine of 5 6-p, the Angle at
D, to—i9j:, on the Line of Lines,
the height required,nbsp;^

Firft obferve the complement of the Fig. 1.
Ansle at D, and alfo the complement of the
Angle atC; then count thefe two comple-
ments Oil the Line of Natural Tangents, on
the loofe-piece, or moving-leg, and take the
diftance Uveen them, and meafure it on
the fame Tangent-line from the beginmng
thereof, and note what Tangent the Com-
pafs-point ftayethat, and count that for the
firft term, in degrees and minuts.

The diftance meafured is 88 J, the two
complements 33-30, and 48-if j the di'
fiance between them makes the Tangent ot
24-34, to he ufed as a firft term.

Bttt If the d'tfianct from D or C, to A,
foot of the Ob]eBy ngt;ert required, tbtquot;
the manner of Calculation runs thm^

As the Tangent of the difference of the
Co-tangents firft found, 24-34, is tO
the diftance between D and C 881 j

So is the Secant of the Angle at B the
greater, viz.. 48-15', counted beyoO»
90, to C B 2pi.

So is the Secant of 33-30, the lefler
Angle at B, 10233 the lefler diftanc?
P Bj the Hypothenufa required.

Quadrant gt;
Firft, prick off the Tangents and Secants
be ufed parallelly, from the loofe-piece,
^n the greater general Scale ; and note
thofe Points for your prefent ufe.

The Tangent of 24-34) '^ken from the
loofe-piece from lt;Jo, counted as oowill
reach to the fine of 10-40, on the general
Scale.

TheSerantof 33-30, being the meafure
from the Tangent of 33-30, on the loofe-
piece (counting from 60) to the Center, will
reach on the general Scale from the Center,
to 28-5:0. .

The meafure from the Tangent of 48-1 f,
on the loofe-piece, to the Center, being the
Secant of 48-15:, will reach from the Cen-
ter to 3 2-y, on the general Scale,

As —diftance between the two ftati-
ons, to = Tangent, of the firft term,
at 10-40 ;

diftancc C A 218. Or,
, Soisthe—Tangent of the iefler Angl^
complement, at if-2f, to the lefle'
diftance DA, 129 Or,
So is the = Secant of the greater Angl^'
complement, at 32-f, to the greats'^
Hypothenufa C B, 291. Or,
So is the = Secant of the lefler Ang'^'
complement, at 26-fo, to the lefl^''
Hypothenufa D B, 233,

Another for Altitudes, hy the Lint 4
Sbadovfj,either acceffahle or unaccejfalflh
bjoneortwojlations.

Wic, as fuppofe to C, till looking up by
'lie two Pins put for fights, Ae T.hred falls
47 degrees on the Quadrant, or on i on
[W Line of Shadows; then, I fay, that the
quot;«ight AB, is eqvial to the diftance C A,
•^orcby the height of your eye from the
Etound.

But if you go further back ftill to D, til!
Thred falls on a on fhe Line of Sha-
dows ; that is to fay, at 2lt;S deg. 34 min. thc
Altitude will be but half the diftance from
A i but if you remove to E, the Thred fal-
jjiig on 3 on the Shadows, the Altitude will
^ but one third part of the diftance E A.

Frem hence you may obferve, that obfer-
ving at C, and at D, where the Thred falls
1, and on 2, the diftance between C and
is equal to the Altitude; fo likewife iK
and at E, and fo by confequence at i ^ and
? land 3 or any other equal parts. This
^ an excellent eafie way.

The like will be if you obferve at D and
c, looking up to F, where the Altitude
A F is twice the diftance A C.

the nkefrorrvai^y object; as fuppofe froquot;®
A toD were 2oo,f00t, and looking up
B, the Thred cuts the{lrokeby2 ontl^'
Line of Shadows.

Suppofe I meafured any other xxnes'^'^
Number, and the Thred fall between oo
the Loofe-piece, and i on the Shado^'^.^'
commonly called contrary Shadow.

So is the meafured diftance, to the heigM?
required, being lefs than the meafut^
diftance.

'But when the Thred falls between i and 9°
at the Head, called right Shadow
the Rule goes thus •

So is the meafured diftance, to the height;
being alwayes more than the me^fute
diftance froni thefwt of the obje6t,tO

ani tke $ngt;t jhining.
^'hen the Sun fhineth, find his Altitude,
^''d alfo as the Thred lies, fee what divifion
^^ the Line of Shadows is cut by the Thred,
then ftjraightway meafure the fhadows
^quot;gth on the ground; and if the Sun be
'quot;^der 4y degrees high, the fhadow is lon-^
than the length of that objed: which
J^^^feth the lhadow ; but if the Sun be a-
45. degrees high, then the objeft is
quot;^•^ger than the fhadow ; and the Operati-
is thus by the Line of Numbers, only
quot;quot;ha pair of Compafles.

' As the parts cut by the Thred on the Sha-
dows, is to I ;
J So is the Shadow meafured, to the height

Line of Shadows ;
So is the nieafure of the fhadow, to the
height in the fame parts,

Ufe vk:/
Ttft»i an inaccefahlt Jfhitudtf hjtbe
Quadrat and Shadows, «tberwift'

Obferve the Altitude at,both fiations,
count the obferved Altitudes at both ftati'
ons, on the Quidrat or Shadows, accordiquot;^
as it happens to be either above or under 4)
degrees ; and take the lefTer out of
greater, noting the remainder for the
term 5 and the Divifor to divide the
ftance between the ftations, increafed
Cyphers, if need be j and the Quotient *
the Anfwer required.

The Extent from the diference to t, ^^^
teach the fame way from the meafured ® '
ftance, to the height requited.

Let A B C D E reprefent the OW^ f .
three Stations J let the Line AC repre'^'J^
the Altitude; the Point B one ftation, f
foot from A ^ D another ftation, 1°°
from A, or jo from B j and E another fta-
tion, 73 foot from D, or 17 3 foot from A»
all which meafures you need not know W

^ngle at B, lt;^3-27, and his complement,;
hunting the other way, being the Angle at
I degrees 33 minuts; the Angle atD
I 1J, and his complement fo alfo ; the Angle
I 30, and his complement 60. Now
iiind the Operation by either of thefe, Firfl
W the Thred on 2.6.^ h andintheQua-
^tat it cuts yo ; lay the Thred on 45, and
the Shadows, or Quadrat, it cuts 100, or

As 73, the difference in Tangents be-
tween the two obfervations, is to the
diftance in feet, 73 ;
So is Radius 100, or the fide of the Qua-
drat, to 100, the hight required.

in the firft Figure, the Angles at thetquot;?
being 33-30, and 48-15: ; and the
furcd diftance 8 8 foot and a half, the dif-
ference in Tangents will be

This way is general for any Statioquot;»^
though both of right fhadpw, or both °
contrary, or mixt of right and contrary^
and done by the Line of Numbers, of
Multiplication and Divifion. .

Take the diftance between thc cotrpfe-
plement otquot; the two obfervations, on the
greater or lefler Line of Tangents, ( as is
^^loft convenient) and meafure this diftance .
i« the Line of Lines, or equal parts equal to
^hat Radius; and that fhall be thequot; diffe-
tence in Tangents required. The like for the
decants.

Alfo, By the Artificial Numbers, Sines,
'ad Tangents, you may come by this diffe-
rences in Tangents, or Secants, very well
thus-,

the Lme of Numbers, is one Number ;
and againft the Tangent of the compiemenc
of the*(Dther Angle, is the other Number j
only with this Caution, That if theTangeic
be above 45, then take the diftance from 4?
to the Tangent, as it is counted backward,
with CompaCes, and fet thc fame the in-
creafing way from I, on the Numbers, to
the other Number required then tiie leffer
faken from the greater, leaves the diifcrence
in Tangents that was required. In the fame
uianner, the Sines countcd from po, and
quot;laid the contrary way from i increafing,will
Sive thc difference in Secants, to meafure d\t
e, and Hypothenu.faby Namb?rs only.

Oigt; any plain Boards end, or Trenchef)
draw a right Angle, and in the meedng'
Point, and on one Lines-end, knock Vf^
Pijis, or fmall Nails, as near as you can up'
right; then on the Pin that ftands iftdiC
right Angle, hang a Thred and Plummet;
then lift up the Board, with the right Ang'j
toward the Objeift, whofe height you woulquot;
have, till the two Pins and the Objeft
brought to a ftreight Line, the PlumtW^''
playingeven and truly. .

Then meafure from your ftand ing, to tli^
foot of the Objedi:, and take the numb«/
of feet, or yards from any Scale, and lay
from the right Angle on the other Line, aquot;''
1-aife a Perpendiculer from thence to the
Plumb-line made by the Thred, and that
fliall be che Altitude required, being mea-
/ured on che fame Scale,

Let A B G D reprefent the Boards endgt;
fcrTrencher, and on char, let AB be one
Rrcig}« J.inej and ^ G jiijocher pexpendi-

to it 5 in rbe Point A, knock in one
^in; and in B, or anywhere toward thé
end, another; On the Pin at A, hang a
ïlired and Plummet; and ftanding at I,
^i^y convenient ftation, look up by the two
l^ins at B and A, till they bourn in a right
I-iiie with the Point H, the objeót whofe
^''eight is to be meafured ; tlien the Plummec
playing well and even, make a Point juft
^lerein, and draw the Line AD, as the
Thred fliewed.
Then,having meafured the diftance from G
tlie foot of the Objeó^, to I the ftation, take
from any firft Scale, and jay it from A to
^; then on the Point G, raife a Perpendi-
culer to A G, till it interfea the Plumb-line
A D; then, I fay, the diftance C D, mea-
'^red on the fame Scale you took A C from,
^all be equal to the Altitude GH, which
^as required.

But i£ you cannot come to meafure from
ij the firft ftation toG j then meafure from
* toK; and having obferved at I, and
drawn the Plumb-line A D, take the mea-
^'Jre between I and K, the two ftations,from
fit Scale ofcquafparts, ^nd lay it on the
Hh anbsp;Line

Line AC, from AtoC, vtz^ 7p par«)
and in the Point C, knock another Pin, and
haaig the Thred :and Plummet thereon,
obferve carefully where this laft Plumb-Iiquot;^
doth crofs the other, as fuppofe at E ; the»
from E, let fall a Perpendiculer to theLi'ie
AC, which Line AC ftiall bethfeheigll'
Gti requued;. (orthus, the neareft d''
.ftance from E to A C is the height
..quired) 120 of the fame parts tl'^J
IK is 79 i Note thenbsp;and behold

that A C F E, the fmall Figure on the Boi^^
is hke and proportional to A A, G H,

Let C reprefent a Glafs, a Bowl, or PlaJ'
of Water, wherein the Eye, at A, feestquot;^
picture or reflexion of the Objedt E-

Then, by the Line of Numbers
As CB, the meafure from your foot t''
the Glafs, is to A B, the height from
your eye, to the ground at your

to ^»d a diftance by the Square,
that is not over-long.
Let C reprefent the upper-corner of a,
^^Uare, hung on a ftaff at F; then the
part of the Square direded to E, and
other to A.

So many times as you find A F in F C ;
So many times is F C in F E, and the
hke.

, Note, That you mufl; conceive AFE to
e the Ground, or Bafe-hne in this Opera-
'on by the Square j C being the top of an
fcP»ght Staff, c foot long,' called 50 for

Firfl, I plant my trUnguleY Qaadf«^''
fet upon a three legged Staff and Ball kcke^
right over the place A j and then bring
Index with two fights in it, laid or falteii^'^
to the Center of the Trianguler Quadr f^'^
right over the Lines of Sines, and
cutting po at the Head ; the Index aquot;®
fights fo placed, hold it there, and brinB'
and the Inftrument together, till you fee ^
mark at C, through the two fights, by lif'P
of the Ball-fockct, and then there keep »
then remove the Index only to 0-60 on
loofe-piece, which makes a right Angle;
fet up a mark in that Line, at any convci'^'
em diftance ; as (uppofe at B, J02
from A ; then remove the Inftrument to^j
and laying the Index on theCenter, ^^^
o-do on the loofe-piece, direft the fig'^'^.^r.
Ajthefiiftftation, by help of a mark
there on purpofe ; Then remove the fw^.
till you lee the mark at C, and note exactly
on what degree the Index falleth, as here 0 ^
6o, counting from o | do on theloofe-piffV,
or on 30, counting from the Head, wbi^^
the Angles at B, and at C,

As the fine of 30, the Angle at C, to
102, ^ meafured diftance counted
on the Numbers;

So is the fine of 60, the Angle at B, to
117, on the Numbers, the diftance re-
quired.

As the — meafure of 102, taken from
any Scale, as the Line of Lines doub-
ling, to the = fine of 30, laying the
Index, or a Thred, to the neareft di^
ftance ;

alfo, If you obferve at B, and at D
you muft be fure to fet yourlnftru-
•^ent «oneftation, at the fame fcituation,

as at the other, as a looking back from Ra-
tion to ftation will doit, and the feme way
of work will ferve.

ipjg.IV. Let A B be two marks, as two corners
a Houfe or Wail, and let the breadth
tween them be demanded, and their difta'jf^
from C and D, the two ftations ; Flt;gt;/?,
up two marks at the t^vo ftations, then
ting up the Lillrumeut at C, fet the fiquot;^'
c'al Line on the Rule to D, the other iiiai'l''
then direct the fights exactly to B, and ^
A-, obferve the Angles DCB 4f, anquot;
DCA J13-0, as in theF/jf«rf.

. Secondly, Remove the Inftrument to
the other ftation, and fet the fiducial-Ln^^
of the Quadrant (t-/«,. the Line of Lme^
- and Sines) diredly to C ; then fix it there?
and .remove the Index and fights to A, aw

CDB icp-e; Then obferve, that the 3
Angles, of every Triangle, being equal to
degrees; having got the Angles at
C113, and the Angle at D42-30, by
tonfequence, as you take i j y, the fum of
^he Angles ac C and D, out of 180, then
'here remains 24-30, the Angle at A.

So alfo. Taking 109 and4y from iSo,
quot;^«fts 25, the Angle at B j then alfo, taking
45', the Angle BCD, out of 113, the
Angle D C A, refls 68 degrees; the Angle
^ C A, in hke manner, taking 42-30 from
^09, the Angles at D, refts 66-30, the
Angle A D B ; and let che diftanee mea-
^red, between thc two ftations,. be lOO,
C D. Thefe things thus prepared by
'he Artificial Nmnbers, Sines and Tangents
®«theedge.

As the Sine of 24-30, the Angle at A,'
to 100, on the Numbers, the mea-
fured fide CD;

So is che Sine of 113, the Angle A C D,
to 222, on the Numbers, the diftance
from C to B.

As the Sirie of 26, the Angle Camp;V,
to 100, on the Numbers, the mW
fured diftance CD;

So ii the SiHe óf 45-, to ilt;Ji, on the LiP^
of Numbers, the diftance from D t''
B;,

50 Js thé Sine óf 109, the Angle C DB;
to 2Ilt;J, on che Numbers, the diftaiic'^
from D to A.

Then, having tht Sides DB ani
A D 222, and A D B the %Angle »gt;
eluded 66-^0 , to find the t^Anglquot;
DAB, cr ABD, nft this Prof of'
tien.

So is the Tangent of half the fum oi t^®
two Angles fought, to the Tangent
of half their diferenee.

^(J.jo, taken from 180, reft 113-30,
for a [um of the Angles fought, whofe half
5lt;J-4j:, is the third Number in the propor-
tion.

As 3 83, the fttm of the two known fides,
is to 061, the dijfcrenct on the Num-
bers ;

fum of the two Angles fought, to the
Tangent of half the d!ference 13-40;
which hsli-difference, 13-40, added
to jlt;J-45'j makes 70-25, the greater
Angle required at B, viz.. A B D.

fam of the Angles inquired, reft 43-oy, the
■^quot;g'eBAB; the like may you do with
fbe other Triangle ABC, being ncedlefs
in our Propofition.

Thus having found the Angles, and one
quot;de, the Sines of the Angles, as proportional
to their oppofite fides.

As the Sine of 44-33, the Angle A B C,
is to the fide AC 146, on the Num-
bers J

So is the Sine of 68, the Angle ac Vj
ai7, the diftance between the marks
required.

So is Sine of 66-30, the Angle at
to 217, the diftance between thc mark^
required.

That if this manner of Calculation be
tedious or difficult, then on a Slate, or ihe^quot;^
of Paper, you may do it by protraif ion, W
the Line of Lines and Chords, or hal'gt;
Sines, very near the matter with care ;

Thus : Draw C D the Station-Line, of
meafured-diftance ; and make AD
from any fit Scale. Then, on C and P
draw a Circle, and on that Circle, lay
from C and D the Angles, found by obfer-
vation, and draw thofe Lines, and whei'®
they crofs one another, as at A and
the Line A B : thofe Lines and Angles niealt;

fured on the fame Scales and Chords, fhafl
be the Sides,breadth,and diftances required j
as you fee in thc Figure.

inftrument, and let E be the mark afar off;
Whofe diftance from you C would know :
^'■ft, move iu a right Line berween C and E
A, any number of Yards or Perches, as
ijjppofe jo'Perch, and fet a mark at A;
Then move in a Perpendiculer-Line to C E,
fi^oni A to B any diftance, and there fet up a
'^ark at B, as fuppole 66 Perches from A.
. Then come back again to C, and remove
a Perpendiculer-Line to C E, till you fee
'^^e mark fet up at B, and the enquired point
the diftance E in a Right-Line; and note
'quot;at place at D, getting the. exadl: diftance
quot;lereof fromC, fuppofe 7'S.

Then fubftrad the meafured diftance A B
ho'n the meafured diftance C D, and iiote
difference 10. Then, by the Line of
•^unibcrs, or by the Rule of Three, fay,

As the Difference between A B and BC, 10,
IS to the diftance between Aamp;C jo: ^
is the meafured diftance C D to the
diftance CE 380.
. . ,nbsp;Or,

^ojsthe meafured diftance A B 66, to the
diftance AE 330, the diftance required,

ATotf,. That you muft be careful and ex-
a^in meafuring the Diftances A C, A B, amp;
and the Anfwer will be the more ex-.

Fig. VlII.- ^^y^ti divide the infide-edge of the Loofe-
peice into inches, or any equal parts, fuch aS
the neareft diftance from the RedifyinS'
Point to that infide-edge may be looo,
and for this ufe two fmall Aiding fights
be convenient: Then the ufe is thus for an/
Angle under 3 o degrees •

Fix the Inftrument to the Ball-focket aflo
Staffquot;, and turn ic toward the Objeftjcaufiquot;^
the Plummet to play on 30 degrees;
then the Loofe-piece is perpendiculer. Th^
one pin or fight fet in the Refilifying-Poi'''^''
flip on a fight along the inner-edge of tb®
Loofe-piece, till you fee the Objeft at tb^
iipper part of the Altitude, and anoth^
light at the lower part of the Altitude
known ; and obferve the precife diftance
parts between the two fights, on the LoolC'
piece J Or, the feveral parts cut by the/»'
dtx at etc'.i Obfervation j
Then,

As the diftance between the two fights,
to the diftance between the remoteft figquot;
from the middle of the Loofe-piece;
So is the height of the known part, to. tl)

Let CI reprefent thc Altitude of a Pyra-
quot;^'d on thc Tower of a Steeple 30 foot high,
®P^gt;ftanding at B, I would know the height
IA from the levd of the eye upward.
Fix the Triangmitr Q»drata on the Staff
''idBaU.focketjwith the Head-Center at B,
^vith the Plummet playing on 3a degrees,
the Loofe-piece perpendiculer : Then
jJ^P two fights on the Loofe-piece, one in a
^'ght-Linc toC, the other to I; and note
parts between, and the parts the furtheft
^^bc cuts, from the middle ftroak on thc
, ^fe-piece,from whence the parts are num-
which in our Example let be 5 00, thc
Ighc of H, and thc fight at G to cut 3^9 «
J ^fi the diftance between the fights will be
,43) and the remoteftfrom thc middle of
e Loofe.piece to be yoc ; and the known
^^titude, being part of the whole, to be 30

the diftance betwixt the fights at
G 8c H,to 5-00 thc remoteft fight from the
.levelor middle,

^^30 foot, part of the Altitude known,
^ I, to loj, the whole Altitude un-
sown, AC.

parts cut at H, the rcnioteft fight :
■ So is] the height of the lower part,
loy the whole height A C.

As 143,, the diftance between the fights,
IC the part of the height known 30:
So is 357, the parts cut between F and
to 7y the height AI unknown,

Let C A be the Altitude given, and
the diftance required. Then I ftandi«?^
C, obferve the Angle CAB, by fetting th
end of the Head-leg to my eye, and tW
Head-end downwards, and fee down, as
Thread cuts, numbring both wayes, for
Angle ad C and at B his complement.

So is f? deg. 20 min. the Angle at C
the Sines, to 176 the diftance rcqt»quot;^^
on the Numbers,

'jns,as fuppofe at E and at B; ifthe Angle
D'erved at B, be found to be the half of the
^quot;§ eat E ; as here in FigurcVlll, the
gquot;gieatE, being 61-10, and the Angle at
the jufi half thereof; then, I fay,
• the diftance between the two ftations,
« ^S^al to the Hypothenufa E C, at the fiift
i^'on, EB is equal to E C; which
''^quot;Jg obferved,

As the fine of po,to i20,0n the Numbers,
is lt;Ji-2o on the Sines, to lOjjthe
height required on the Numbers.

■ ^o be approached \
J^Srees on the in-fide of
jj ^ that breadth one way,is thus;
mto jj^g two holes in the 1

(or fet the fights there in large
«ruments); then move nearer or further
till your eye, fixed at the
^^quot;fyuig Point, can but jaft fee the marks

only be at the mark C, at an Angle of
° degrees; for fo the Rule is made to that
^'^gle: then the Inftrument being ftill fixed
, C, look backward in a right Line from
middle of the loofe-piece,' and reaify-
linbsp;mg

ing Point toward D, putting up a mark ei-
ther in, or over, or beyond the Point Dj aw»
alfobe fure to leave a mark at C, the fiquot;®
place of obfervation j Then remove the figquot;'
to ly degrees, the half of 30, counting froi^
the middle;and go back in a right Line fr^j^
C, toward D, till you can juft fee the mar»'
by the two fights fet at i y degrees each way»
for then, I fay, that the meafure betwee^
quot;the two ffations, C and D, fhall be exai^'X
equal both to A B,the breadth required,aii'l
alfo to CB, orCA, the Hypothenufaes»
then, having the fides C B, and C D, aj?
the Angles B C E, and C B E, and B D 4
it is eafie to find all the other Sides aji
Angles, by the Rules before rehearfed, ^^
the Lines of Artificial Nmmbers and Si«^®'
For,

As the Sine of i y degrees, the Angle
D, viü. B D C, to ig8 on the Nu^quot;'
bers;

amp; alfo is the Angle at B, Wa. D BC iJ'
to 10 8 on the Sines and Numbers- ^
So alfo is the Sine of 150, the Ang'^ ®
C,wx,.DCB, to BD2o8i oa^^
Numbers.nbsp;1

Note Alfoj That if the Angles of a»
'30 be inconvenient, then you may make ^
of 5 2 and 26, or 48 and 24, or 40 and M
or any other, and the half thexeof j ® j,

then the meaTured diftance, and the Hypo-
thenufa BC, at the neareft ftation, will al-
wayes be equal; but not equal to the breadth
at any other Angle, except 30 and 60, as
gt;n the Ftgurf. But having the Angles, and
thofe Sides, you may foon find all the others
hy the Artificial Numbers, Sines andTaa^
gents, by the former directions.

TRianguler Quadrant j why fo called, 2
The Lines on the outter-Edg, N. T.
J^.VS,nbsp;^nbsp;2

quot;^t Line on the inner-edge, I. F* 112, 3
quot;^ht Lines on the Setior-ftde, L.S.T.Sec. 3
^*ftr SineSj Tangents, and Secants, f
Tht Lines on the Quadraat-Jide,nbsp;lt;S

The 180 degrees of a femi-Circle variiujlp
accounted, as ufe and occafion requires, /
lt;^0 Degrees on the Loofe-fiece,at afore-Staff
ftr Sea-Ohfervation,nbsp;7

what an Angle, a Triangle, Acute, Right,
or Obtufe-,PUin^or jphericalAngle is,26,27
Parallel-lines, and Perpendiculer-lines, what
they are,nbsp;, ibid.

Terms 1» Arithmetick, at tMuhlpHcitor,
Produa, Quotient, amp;c. what they mean,

Ceemetrlcal Profsfulons,
To dravp a right Line,
To raife a Perpendiculer on any line.
To letfaU a Perpendiculer anywhere,
Todraw ParaSil-llnes,
To make one Angle eqtial to another,
To divide a Line Into any number of parts,
To bring any 3 Points into a Circle,
To cut any two Points in aCircle, and tbe
Circle tntotveo equal parts,nbsp;37

A Segment of a Circle gllt;ven, to find the Cen-
ter and Diameter,nbsp;3®
A Segment of a Cirde given,to find the length

To draw a Helical-hne, and tofindthe ten-
ters, of the Sflayes,. of Ellptlcal arches,
and Key-fiones,nbsp;4^

Explanation of Terms particularly be-
longing to this Inftrnment,
Radlwyhjjw taken,nbsp;4'-

^•fht Sines, bore tak^n and counted, ' 42
Tu„geKts,Secants and Chords.how take», ib.
^Sine complement, or co-fine Tangent, comple-
ment or co-tangent, how take» »nd counted
cnthis Inftrument,nbsp;ibid.

JjOtteral Sine ani Tangent,
Parallel Sine ani Tangent,
Neare/} Diftance, what it means.
Addition on Lines,
SuhfiraUion on Line.

line df a Plain are,
Meridian-line, Stshflile-line, ani Stile-U^^^,
Angle of 12 and and tht Inclination')
Meridians, what they are,nbsp;4^

The 2 Poles of the World or EquinoUial, ibi'^'
The i Poles of tht Zodiack,nbsp;^^^

De7rlion, true pLe,
or Riftng and Setting, by tn^eUion onlj, 71
To Andthfsunt Amplitude, and d,ference of
Afcentions, and Oblique Âfctntton, 71
To find the Hour of the Day,nbsp;74

The ufe of the Line of Numbers, and the ufe
of the Lme of Lines, both on the Triangu-
ler C^iadrant and Sedfor, one after ano-
^ ther, in moll Examples.
1.0 multiply one Number by another,
A help to Multiply truly,
A crabbed Queflion of Multiplication
precepts of �u[lion.
To divide one Number by another,
A Camonm Divifion.
Toz Lines or Numbers given, to find a zd iquot;

Oeometrical proportion,
Any one fide of a Figure beinggiven,to find aH
the reft; or to find a proportion between trt»
or more Lines or Numbers,
To lay down any number of parts on a Ltne to
''fynbsp;joO

^ divide a line into any number of parts, io2
To find a Geometrical mean proportion betwit»
two Lines or Numbers, three wayes, i o4
To makea Scjuare equal to an Oblong, lO?
OrtoaTr,^nbsp;^'ibid-

To make one Superfetes likf another Superfi-
cies, 4Kd ef^nal to a fhird,nbsp;lop
The Diameter and Content of a Circle bein^
g'ven^ to find the Content if a iOthtrCircU
py having hu Diamner, j 11

^hejveight and Magnitude of a body of one
kind of Metal being given,to find the Mag-
nitude of a body of another Metal of equal
flight,nbsp;quot;I

*he magnitudes of two btdies of feveral Me-
tals, having the weight of one given^tp find
f^he Weight of the ether,nbsp;122

The Weight and magnitude of one body ofany
^ttal being given, ani another body like
the former, is to he made ofany other
fJMetal,tofind the diameters or magnitudes
ofit,nbsp;123

^ ua ani %everfe, with Exmpleh ^SP
7be Rule ofFellowjhip with Examples, 14^
Tbt ufe of tbe Line of Timbers in SHpirficilt;^
_nbsp;4ni the pans on the Rule, i f 4

the length of one Foot,
The ireith given in Inches,to find, bow mlt;tcb
_ t» length makes one Foot,nbsp;ibid.

Waving the length and bredth given in Foot'
meaf»re,t 9 fi^d the Content in Feet, ibid-
Haytng the bredth given in Inches, and length
»» Feet, to find the Content in Feet, If ^
Havtng the length amp; bredth given in Inches,

to find the content infuperfkial Inches,
Havmg the length amp; bredth given in Inches,

The length and bredth of an Oblong given, to
M the fide of a Square equal to it, l6}

^be Diameter of a Circle g, ven, to find the
Urcumference, Square, equal Square, if
fcribed and Com ent ^nbsp;1Ó4

The Content of a Circle given,to find tbe Dia-
meter or Circumference, x66, itf/
Certain Rules to meafure feveral figures, 10 8
A Segment of a Circle given, to find the trite

meafuring of Triangles, Tappdaes,Rm-
^»des, Poligons, and Ovals, 172,17 J
^ Tailzie of the Proportion between toe Sides
and Area's of regular Poligons,and the uft
thereof for any other,nbsp;174»

^be neareft way to meafure a pirty WaU, 180
multiply and reduce any length, bredth, or
thic^nefs ef a fVaBtoont Bric^and a half
at one Operation,nbsp;'83

Examples at fix feveral thitknefes, _ 184
To fini the Gage-points for this reducing, 185:
^t one opening of the Comp^fes, to f^ ho»
many Rods, garters, and Feet in any fum
tender lo Rods,nbsp;l8lt;5

Of Plaiftertrs-work.,er Painters-work, 18?
Of particulars of work, ufually mentioned in
aCarpenttrs-Bill,mth Cautions, 189,190
At any bredth of a Houfe, to find tht Rafters,
and Hip.rafters, length and Angles jhj the

7 he price of one Foot being given, to find tht
pnce of a Rod, or a Square of Brick^r»^''^'
or Flooring, hj inffeSton,nbsp;IP ?

much in bredth makes one tyicre,
•AHfefulTzhk in meafuring Land, and tht
ufe thereof in feveral Examples, ipó, I?/
1 he length and bredth given in Perches, «gt;
find the Content in Squares,Perches, Poh'*
. or Rods,nbsp;20^

The le.gth and bredth given in Chains, quot;
Ji^d tae content tn fquare Acres, Quarterh

Knowing the content of a piece of Land plotte^
out to find by what Scale ic was done, 206
1 he lame Rd, applied to the meafurng of
Giaz^iers Quarries,nbsp;I08

ibe bredth amp; depth of any folid body beinggl'
quot;quot;enitofiadjhe fide of the fquare equal,
'nbsp;The

hredth and depth, or fquare equal given,
find hotv much in length makjs one foot
/fl«r manner of wayes, according to
■J,Wording of the quejiion,nbsp;212

' hredth and depth, or the fide of the fquare
quot;f any Solid given, to find how much u in a
^'quot;t long folid meafure, three wayes,accor-
to the wording the quejiion, up,220
I jgt;redih depth and length ofany folid body
i^fen, to find the folid Content, four wayes,
i quot;'cording to the wording the quefiion, 221,
' f,nbsp;222,223

•j.®' 3 lafi Probl. wrought by the StBor, 226
'l^iameter of a CiUender given, to find
quot;«w much in length makes i foot, 4 wayes,

d,ameter of a CiUender given, to find how
WHVi is in a foot long, 3 wayes, 232
diameter of a Cillendtr with the length
f »z/f»», to find the Content 3 wayes, 2 3 5
Circumference given, to find a foot^

*he cufioms amp; allowances in meafuring round
limber^as Oak^or Elm, amp; the like, 240
Tif uft of z Points for that aUapfanct, 242

7o meafure Globes, and Segments of Globeh
»othfuferflciaUj round about, and with tkf
foUditj feveral wayes, by Arithmetith
aud the Line of Numbers, and folid Sti'
ments -, with a fmaU Tzhk of folid H'

Jhe Experimented proportions, Utween *
Cube,a CiSexder,a Sphear,a Cone,a Prif'^'
a Styuare and Trianguler Pyramid,
The ufe of thejliding (cover or) Rule, ^^^
iae defer tption,nbsp;i60

7he Gage.points, and places if them, »
lifes, to fquare a Piece, to find how mttC»
*n length wiUmake i foot of fquareTH^'

To find how much in length makes i foot, «
ViMieter and length given, to find the Content,

Gage round hj the Rule or Square,
counting 6 Foot fer a Barrel of Beer, or one
Feoffor 6 Gallons,or one Foot for 7 Gallons
and a half of Wine-meafure, ^ 273
'^he diameter and length of a Cask given, to
find the Content in fVitie-gallons, er Ale-
; gaUons,nbsp;ihid.

' ^oGage Brewers great roundTuns, and to
have the Content in Barrels at onevorkt

Tfc» ufe of the other-fide in fuperficial mea-
fure, Golden Rule, and Divifion, 27 S

To make and meafure the c regular bodies,
with the Declination and Reciination of
every fide, at any fcituation of them.
'^heCuhe,

To find the Contents of Ca»\ othertrife, ibi'^'
The Content andmean Diameter given Jo fquot;quot;

the length of the Cask., amp; contrary,
To find the wants amp; nnllage, two wayes, 3
A Table of the Wants in a Beer Barrel,
Beer and Wine GaUons, at any Inches,
or dry,nbsp;3I7

The u[e of the Lint of Numbers in Intereft
and feyeral Examples thereof, many Wif*^
ufeful, •nbsp;334

Theafeofthe Linein Military queflionS, 35^
The uf»of the Line in folid Proportions, Oi t^^
weights and meafures of Rope, and Burth'quot;
of Ships,

The way to uft the LogarithmalTdhks, 34quot;
The ufe of the Rule in Geometry (^AflronO'
my, in jp Propofitions, or Ufes, by the
ticular Scale or Quadrant, the geneer
Scale or Quadrant, the SeUor and Arti-
ficial Numbers, Sines, and Tange»quot;*

345-, to 44^
The ufe of Trianguler Quadrant,/»
tng of Heights and Diftances, accep^'^'
er inacceffable, in 14 Ufes, 449, co ,


/ ^	™ \
/	tof \
	—-.1

if, Tinn----
S Iron---;	1680
2 Copper——	iSpo
D Silver---
?? Lead	2415
V Qnickfilver—	28jO
©Gold	■

	; ^		• ■• Î
M		bJ
(J	H		H
		N
l'Ov	-t»

t. I.	'H 8 j	'O 0 O ■O	H O O O 0	c\ lo gt;'1	VJ ■ff'	bi p N!	NO ^	H	M	.iquot;
ra lt;Tgt;	00	-1- VJ Ol O'	CN		M	-O	H '	■ H M ' .•»O	w ' W	. rj
Hlt; w S- io	O -vj §	üiC O	C/C	O M	Ngt; N OC	M		c\	Os	u 5-
. p O	HH	M Nl	n NI -f	p oc	H	M • Oy		N NI	Os -.o	^ lu n
. p O	HH					Ov				■ ftt
rs i-i n	w 0 H -f' O •0	0\ O	M Ov	H	UJ 1	O	, H 0\ ^		OO	fTgt; »-t lt;-l
5	O O O	tH O	H		if	N	Ov Os 1	H O O	M	f M
tt- Ö ngt;	Ov O	H	M O	O,		K) tJ O	0\ ogt; O	M O O O	NI NO N O	gt; n i-t m
S (-6	H	CC	oc O	UJ to O	M O CN	^^ N 0\ O	M OC ö	Oo O O O	Ok SAJ Os 9	g

Co	Rang.	Si^es.	bredth	length	Content-	Content.	Co	Rang.	Sides,	br^th	lenitb\	Contem.	Content.
**	In.100	In.ioo	In.ioo	In \ 00	Inc. 100	Tent-	f ' n	In.io:gt;	/n.ioo	In-iao	In.ioo	Inc. TOO	Tens. '
8	4 20	4	9	6 70	I JOO	I 2^0	1	4 09	4 41	4 90	1 34	I 5QO	X 2JO
10	3 75	3 84	4 80	6 00	I 200	I 000	10	:3 «Îî	3 Pf	4 38	6 J7	I 200	I 000
12	i 45	3	4	S 47	I 000	0 833	12	3 34	3	4 00	i' qo	•I tOOO	b 83?
	3 07	3	3 P2,	4 po	0 800	0 667	ly	2 98	3 23	3 58	r 37	o''80o	0 '667
78	2 80	2 8d	3 Î7	4 77	0 666	0 ffy	18	2 72	2	3 2d	4 9°	0 666
\ao\2 66\2. 7A1					lo 6qq . „	\o 500	20	U sS	2. 79	[l	\ 65	.0 600	Vo 500

A Table of Segments,
Kum.Seg \Num.Segm.\Ntiim.Segm.\Num Segm-
oyp	33y	506	67p
■084	342
1041 349		y2o	^94
la?	3jd	y 27	703
5 137	30 363	yy y34	80 712
152	3/1	^40	720
I '4	378	y47	728
		5 H	737
188	392.		74Ö
io S97	Î5 399-	60	753
107	406	f74	753
2I8	413	580	772
22^	420	y 8	782
	426	594
ly 24;	40 433	tfy (ÎOI	90' 803
2^4	440	lt;îo8	812
2lt;Î3	447		82J
272	453	lt;522	836
- 280	460	629	84«
20 288		70 637
	473	644	878
3od	480		89(5
314	487	6J8	9id
321	494		941
2y 328	50 yoo	7y lt;Î72	1 100 lOCO

	httt (fall.		Wine Gall
	gal. pi- loo		gal. pi.ioo		gal	1.1000
	0	o 40	0	0 4P	0	0(Jl2
	o	I 20	0	I 47	0	184	22
	o	2 xo	0	2 S7	0	3quot;
X	0	3 10	0	3 80	0	47 y
—	p	4 33	0	s 39	0	66^
	o	6 00	0		0	p20 i	21
	o	7 tfo	!j	I 2p	I	itfi;	21
2	' I	I 80	I	4 00	I	yoo
—quot;	I	3	; X	Ö 56	I	821
	I	Ö 10	2	I 22	2	'y3	'20
	2	0 lt;50	2	4 34	2
	2	3 yo	2	7	2	pp8
	2	6 16	3	3	3	388 .
	5	0 70	3	6 20	3	772
	3	^ 80	4	2 CO	4
£	3	6 fO	4	f 3°	4	003
	4	I 80	J	I 3S	5	Itfp '
1	4	f ay	J	y do	S	700^	10
	f	0 42	6	I 4?	6	i8i ,
	f	3 po	6	y 70	6	713'
1				1

	y	7 20	7	I 70
	tf	2 80	7	(J 20
	6	lt;5 50	8	2
6	7	2 20	8	7 201
	7	5 JO	9	3 20
	8	I 10	9	7 70
	8	4 80	10	4 20
7	9	0 70	II	I 00
	9	4 50	11	5 40
	ID	0 40	12	2 20
	10	4 30	12	7 00

r3gt;81
Beer Gatl.	y^ine Gai.
gaJ.pi.iooj	gai. pi. 100	gai. 1000
17nbsp;7 90 18nbsp;y 4jgt; 19nbsp;2 00 19 6 16	22 0 00 22nbsp;6 98 23nbsp;4 31 24nbsp;I 48	21nbsp;000 22nbsp;($44 23nbsp;3P» 24nbsp;184
20 3 00 20nbsp;7 40 21nbsp;3 10 21 7 40	24 7 30 2^ 4 ÖO 26 I 3lt;î 2(î 6 38	24 9Ö1 2y 26 170 2Ä 799
22 3 CO 22nbsp;7 00 23nbsp;3 00 23 7 30	28 2 18 28nbsp;y 18 29nbsp;2 19 29 7 40 JO 3 90 31 1 00 y 80	27nbsp;130 28nbsp;174 28nbsp;648 29nbsp;27f
24 3 70 24 7 40 2y 3 do 2y 7 50		29nbsp;926 30nbsp;488 31nbsp;i2y 31 720
25nbsp;3 30 26nbsp;7 00 27nbsp;3 00 27 lt;î 40	32 2 6b 32 7 00 3 80 34 0 30	32 32J 32 87? Î3 475 34 037
28 2 20 28nbsp;y 80 2p 1 40 29nbsp;4 80	34 4 80 jy 0 80 3J y 34 3d I 80	34 600 SS 100 3y 668 36 22y

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