Shewing theNatural, Artificiai,and Inftrii- \
mental way, of making o{' Sm Dials^ on
any flat Superficies: W;th plain and eafie
Dire£tions, to difcover their Nature and
AfFed^ions, by the Honx:,o»tal ProjeBlon. j
With tilquot; way of Drawing .the ufual Orna-
ments on any Plain : Alfo, a familiar eafie
^vay to draw tiiofe Lines on the Ceiling of;
3 Room, by the Trianguhr Qnadrant. f

London, printed by John Darby, for John Wing-
field, and are to belold at his lioufe in Cmtchc'd- '
Fryas; and by John Brown at the Sfhear and |
Sun Dial in the Minories % Knd by John Seller j
tht Hermitage-ßain m Wtipfinii 1671. |

.^HHou lajlhereprefentedto thy vierv
X (Courteous Reader) in thisfécond
^^n, a plain difcourfe of Diallings
j^'atural, Artificial, ^ Inftrumental.
Natural / call it^from the plain lUufira-
^^n thereof, by the Armilary Sphear of
quot;raji lergin àefcribed, or by the Poor-
man^s Dial-Sphear, as 1 fancy to call
being only a movtngHori^ontal-Dtal,
amoving Plain ^according to the Fi-
ltre thereof in the Boohjmnexed^ where^
y all the Arl^s^ Angles,Scitnations and
^ffe^ions, are very plainly reprefented
an ordinary capacity. Artificial I

Horizontal-projeaion,^/^^fittefl in my
quot;pinion for the making plain the my fiery
Dialling. Inftrumental I quot;call it,
f^^om the applying of the Trianguler-
^adrant, to the ready refolving all the
Arithmetical and Afironomical worJ^i
A anbsp;need*

lo tne N^cmitcwvio iv-.
needful thereunto i and that to competent
exa^mj: as in thejirjl Part, and alfo ^^
this fécond Part is fufficiently feeny
[mdingthe re^uifites anddelineatingtb^
hour-hnes to (mallparts, exa
dtly by the naturcU Sines, Tangents anO'
Secants on the St^ox and Quadrant.

Jlfo^the ready way of finding the Suii^
jihiiude, Hour, J^imuth, Angle cft^^^
Plain, and any/ncb bufinejs relaiing ^^
Dyallirig, asm the fir ft Part is U^f
treated on. .

F-ircber, in this fécond Part you haf^
Tables of the Suns Dechnatton to eve^l
day uf the years, i 3, after the Bu
fxtile, as near as any extent. Jlfo' ^
Jhortt but flam direbhon, how to ufe tquot;
Triangxilcr-Quadrant, at any mann^^

and the Crofs-ftaff « ufeà ^
alfo, as Gunter's Bow îsufedbothfO'
the Sun or Stars.

To the Courteous Reader.
^quot;^king the fix ufual Problems off lain
j h the Natural and Artificial

arts and Numbers y andplaouy and
'^^nvenientlfto apply it, to the rejolvin^
^hat hard il^eflton, cf Sayl. ng by tue
■^'fch of a great Circle, to Jhew plainly
Longitudes and Latitudes alteration
tvhole Courje or Foyage.
The Prints of the Lines of IS!umbers,
^ youfee here infer! ed,are in part accor-
«^^«^roA/r. Windgates, as to a jingle
(ind broken line of Numbers: But the ad-
^^t^onofthe line of the Ffailionai parts
fi pound, and the feveral Gage-points,
^Jre never hefore ufed as I know of i but
much eafe cj expedite the Operations
quot;y the Line of Numbers,Smes and 1 a?(-
gents. Alfo, thefc Scales of l^eduUion
areconimient for the finding the Heci-
fnal-fTaction, equal to^tije other Sexage-
na/y.fraUlon, and are agreeable to thufe
Tables in Mr.Windgates Book of A-
^ithmetick^, pag. 82.

To the Courteous Reader.
(tt the beginning of the Book, pafled on ^
Board,/J- the very fame with that J^ch^
c/Chap.XV, Ufe 28. pag.B97. of^quot;^
fifjl Part, and will work^ all ^e^ionf
ivrought by the Trianguler-Quadiai^f'
to exercife them tf^at are out of the tv'^J
to have them made, and may ferlie ^
^ood direUions to the young Inftrume'^^'
Maker, though theje are made too t^O
Jhall a P^dius to arrive at exaUnefs.

The like may I fay of the Guntef^'
Lines in the Figures amiexed, yet ^^
large as the YiOoX will bear.

7hui I have given jott a hrlif account quot;l
ntyprefem Ihotights abmt this matter,
fomewhat more farticularl) in the Firfi
fUfclaimlug all botijimg or vain
knowing that at the, next Imprejjton it tn*) ^
amended in many places I [hatlrefi
pjo-in, read^ to mizke amends«« the makif0
thefe, or any other tJMathematical
went Si at my Hrntfe at the S[,hear and S^^'
Dial in the Great Minories.

Vn-Dials may be made on any Plain,
and all kind of Plains are either F/lt;if,
as HorUontalor Vpright^oi Lian-

T^^t Horiz^ontal hath two faces, the one
beholding the Zenith, called the Horizontal-
Plain ; tile other, beholding the Nadir, as
the Ceiling of a Room is.

IhtUfright^latKs, are thofc that make
right Angles with the Horizon, and do be-
hold neither the Zenith or Nadir, but arc
parallel to them.

The LeamKg Vlaim are of two forts ge-
nerally- the one called Recliners, behold,
ing che Zenith ; the other ford called In-
A 4nbsp;diners.

diners, beholdiiig the Nadir, as the out-
(Tde, and in-!ide of a Roof of a Houfe, ma/
reprefenr.

Tlie two laft ioits, viz.. Upright
t-eanlvg, may l:cDire61oT Declinins^;
beholding the South, or North, orEaft» ^^
Weft Pome of the Hociz-on j orDechnin^
therefrom,, Ws:.. Declining from South, quot;Iquot;
North,.to\yard tlSi^Eaft or Weft.

All which Plains, are hvely reprefetitc''
by a Sphear, made for that purpofe, in Braquot;
or Paftcboard, or by the Projedbon of tU^
Sphear inThw^

Equal to the Radius of the fmaller Tan-
genes, defcnbe the Circle E S W N repr^'
fencing the Horizon, crofliiig it precifefy i'j
tile the Center Z, with the' Line$ SIS aquot;^
E W, denotiija the Points of South a^d
Norih.Eaft and Weft,

Then counting the fmaller Tangent on tH^
Sedfoi-fide doubly, as riE?*«, calling y,
amp; 10, 20 • amp; 2o, 40 ; 8c 30, 60 ; amp;
80 ; amp; 4j, po 5 amp;c. Lay off from Z, to-
wards S, the complement of the Suns Me-
ridian Altitude, in in r, and -yp ^
tliofe Points on the Meridian-line, between
Z and S; and confequently the half Tan-
gent of the complement of the Suns Mequot;^!-

J. Then for the Interfeaions of all thofe
J-^nes and Parallels of Declination on the
JJorth-fide of the Meridian, Ohftrve, That
fame number of degrees and minuts, that
quot;y Point is above the Horizon on the South
of the Meridian in Summer, jufl fo
^anv degrees and minuts is his oppofite
^quot;^allel in Winter below the Horizon.

As thm for ExampUt
^ The Sun being in or 2? deg. 51 min.
r Declination North, hath for his Meridian
Ituude 62 degrees, and fo many degrees
quot; oppofite Parallel of 23-31, or vp, be-
the North part of the Horizon, at mid-

^ Lee the Center, at the beginning of the
aiTl Tangents, leprcfent'che Center Z ;
iet the Tangent of 4j, reprefenc the Hor
m the Scheam, vl^j. S. and N.

As the diftance fVora S. to Vp, is jc deg.
taken from toward o, and laid from
b. to inwards toward the Center Z,
as the diftance was taken from the
Tangent of toward the beginning
of the Line cf Tangents, that repre-
fents the Center j
So the Point C/iKffr fromN. is ijdeg.
counted beyond 4^, toward the end,
below or beyond the Horizon.

So is the other Point 62 degrees below
taken from 45:, viz.. at 76 degrees i
which being laid from N, dothove^'
reach this little page.

So that to draw the Tropick ofi^c,
Point © being his oppofitc, is 28 deg'^^
from Z, or Ö2 deg. from S ; and theoth«
Point of VP, on the North part of the Me^''
dian,is 62 degrees,counting from doubiV ^
alfo J or 28 degrees from po, the fuppo'^
end of thé Tangent, which is naturally
finite, being the Tangent of degrees,
the Semi-tangent of ip, reading the Taquot;?'
gents doubly from the Center j which ,
fiance from the Ceitter, to the Tangent ° i
76 ; or as half-tangents, ifa, laid fro® ^^
gives the Point -vji on the North-part of tP
Meridian, below the Horizon j the f«^^
between which two Points of
Souch and North part of the Meridian»
the Center to draw the Tropick of Caf^
corn.nbsp;^

Again, to illuftrate this difficuIty,to df^
the Tropick of Cancir, the Suns Mendiai'^
Altitude in his oppofite fign is iy
above the Horizoiron the South P^J^J]-quot;^

l^eridian, and ly degrees below the Hori.
on the North-part of the Meridian,
the Extent from the Center to the Tan-
Sent of j2 deg. 3o mm. or the Scmi-tan-
pnt of loy, reading it doubly j being laid
'fom Z, gives the Point © below the Horir.

Again, for the Equino£lial or Parallel of
the Meridian Altitude in T, is 38-28-
the Meridian Altitude likewife in
his oppofite Parallel is 3 8-28 alfo; fo that
quot; you count 38-28 doubly beyond 45-,
^^hich will be at the Tangent of degrees
14 minuts, and take from thence to the
^^nter 5 this diilance laid from Z, fliall
S'Ve the Point £ below the Horizon, and the
middle between the two Points M, is
quot;le Center to draw the ^Equinocaial.

Then for the Hour-Lines; firft, fet off
the bemi-tangent of 38-28 from Z to P •
and the Secant of 38-28 to the fame Fxadil
us from Z to L, and draw the Line L 4^
paralle to E W • then make P L a Tangent
ot 4 J degrees, and lay oft the Tangents of
^ y-^o, and 4;, from L both-wayes^ as you
fee in the Figure.

Lafllj, Sec one Point of theCompafTes
L, and open the other to P, and draw tb«
Line W P E for the hour of 6.

Set one Point in 7-1 j degrees fron)
and open the other to P,a)!d draw theHoi^'quot;
line J P y gt; Set the fame Extent a!fo in
y, on the other fide of L, and draw the
Hour-line 7P 7, lt;« the Figure(htwtth.

Sec one Point of the Compaffes in 8,
degrees from L, and open che other
to p, and draw che Hour-lme 8 P 8, an'»
remove it to the other fide of L, and dra^^
the Hour-lnie 4 P 4 : And fo for all the rquot;*^
in order.

Thm having irawn the Figures ; lo
Lines therein, vphich jhall truly re f'
[ent any Plain whatfoever , ohftrvi t^
following %ules.

JfprefentedbytheRigh t-Iine lt;fo Z lt;fo,whofe
are at Cand D, declining jo degrees
or N. toward E. or W.
, 4. An Eafl or mft Plain, is reprefented
°y the Meridiaivline of 12, viz,. S, amp; N.
, y. KTolar Tlalu^ is reprefaued by the
«our of (j^ the Line E.P.W.
, An EquimUlal Plain, is reprefented
the Equinoaial-line E.^.VV.
7. Any DireB Reclinihg^ or Indinlng-
between the two laft, is called, A di-
RtcUner, whofe Poles are alwayes in the
^eridian, and are reprefented by any Re-
clining Circle, as the two Circles W. E»
andE.0.w. dofhew.

.5' A Declining and 'B^tUmng^ or Jncli-
^*'gPoUr.PUlgt;7-^ that is, it fo Declines and
Reclines, or Inclines, as to lie parallel to the
i ole, as the Circle 8 P 8 doth reprefent.

A Dtclining RecUmng-PUiM, that
lo Declines and Reclines, as not to fall in the
Pole or Equinodlial, as generally they will
do, as the Circle 60 G 60 doth reprefent,
^vhich Declines from the South-eaftwards,
and Reclines 62 deg. which kind of Plains
^re various and infinite, yet confined to fix
^'arieties, as afterward»

i^ow, the my of Drawing thife SchtAtf^
to rcprefent thefe vmeties, u brtijlj
tkw, by-theSeUor.

Firft,to the Radius of the fmali Tangequot;«,
draw the Cucle N. E. S. W. obferving tb'S
Method, if it be a South Recliner, to fet t^
letter N above, and E on the right hand;
and contrarily, in North Recliners; for we
meddle not with lucliners till afterwards^
(and alwayes oblerve, that a South Inchner
is the fame with a North Recliner, and tb«
contrary ) then crofs that Circle with tw«
Diameters, precifely in the Center, as the
Letters fliew; then according to your Plains
Decimation from North or South, toward
either Eaft or Weft, fet oftquot; the Declinati-
on with a Line of Chords or Sines, as befoi^
is fliewed ; and draw that Line for the Pet'
pendiculer Line of the Plain, and laymg th«
fame diftance as much from E- and W»
draw another Line Perpendiculer to the
former, reprefenting the Plain; then, on
fhefirftLme, viz,. tHe Plains Perpendicu-
ler, lay oft' from Z, the half tangent ot tne
Plains Reclination from Z to E, and the halt
Tanoent of the complement thereof hotn
Z to Q_the contrary way ; and the whole
Tangent of the complement thereof trom^,
contrary to E, on the fame Line, extenclea

Toirtm the Scheat» for a Plat», Vtcli-
*iing from the South to the fVefl 3 J de~
guts ; ani %tclimng 20 degrteSf far
fhc Latitude of 51-30.

^itft, to the Radius of your fmall Line of
^ngents, being the Latteral diftance from
quot;c Center to 4y, ( or larger if you pleafe )
^raw the Circle N.E.S.W'. reprefentmg the
^Prizon, crolTing it in the Center with the
^quot;les N.S.amp; W.E. for the North and South,
quot;•^dEaft and Weft Lines.

J Take out the latteral Tangent of half the
19-15, for 38-30,calling the
*®ngent of 10, the half Tangent of 20 ^

Then conftder the Declination of your
Plain, and which way, as here 3 J deg. -O
from the South towards the Weft ; take out
the Chord of 3 J deg. and lay it from S to C,
and from W to A, and from N to D, ai^'J
from E to B, for the more exa6f drawing ogt;
the Lmes A B, C D ; the Lines C D repre-
fenting the Poles of the Plain, and the L'fquot;^
A B the Declining Plain it felf; then froquot;^
Z towards D, layoff the Tangent of lO deg-
( being the half Tangent of zo degrees,th^
given Decimation) to E.

Jlfo, Take out the Secant of 70 degre^'^
the complement of 20, to the lame RadiiJ® '■gt;
and that laid from the Point E, on theL'lt;^
D C produced, fhall be che Center to dfa^
the Circle AFEB, that reprefents the
dining, Rechning Plain, that decline« ? ^
degrees, and reclines 20 degrees. Alfo',^'^'^
off half the Tangent of the compleinen'^ ^^
the Reclination, viz.. 35 degrees (fort ^

Then to drarv the Line QJ', do thui ^ ^^
Obferve how many degrees yqu count rrO'
2 to the Point E, countingtroin the Center^
count fomany i,n the manner of haU

from; and the latteral diftance fronj
quot;^nce to the Center, laid from the Center
J ^^ C D, gives a third Poinc,wi.

Point I ■ which three Points, QP I,
wrought into a Circle, will cut the Circle
into two equal parts.

.The Semi-tangent of the complement of
^^ Heclination to 180 degrees, laid from Zt
^^ the Line C D, will find the Point I.

The Reclination is 20, the complement
f being taken from 180 refts 110, whofe
is 5 f, the Tangent of Z I.

. Set one Point of thcCompaffes in the
«lall Tangent of 45^, and count the Recli-quot;
®tion from thence in the way of Semi-tan-
S^'^ts, both wayes, both above and under
i and lay one, viz,, that under 45 , from
fnbsp;and the other, viz,, that above 45:

«otn D to I; then on the middle Point,
■between Q^nd I, laft found, raife aPer^
P^ndiculer to C D, and in that Line will be
Center to draw IP Q^,
Alfa,

If you count J1-3 2, the latitude on the
Line of Tangents from 4y forwards (aiS
^quot;^'lu-tangcnts) af]d lay it on the Meridian-

Prop. 10. by drawing theLine dP»
and then a perpendiculer Line to
the Point B, till it interfea CD in I pro-
duced: ThePole of which Circle QFP, J«
in the Plain at Y, found by laying a Qu^'
drantfrom HtoK, and drawing theLil®
ZK.

The Scheam thus drawn, then A G
prefentsthe diftance on the Plain, betwee'^
the Meridian and Horizon, G E the coflf'
plement thereof, or his diftance from
Plains perpendiculer ; P F the Stiles Ele^®'
tion, F G the diftance of the Subftile an'i
the Mendia.:; the Angle FPG, thelncjj'
nation o; Meridians, E F the diftance (jf ^^
Subftde from the Plains perpendiculer, P
thenbsp;on the Meridian of thepl^''®

Thus you have the Definitions of DlAi '
and the way of Drawing the general an®
particular Scheams for Dials; which ft»quot;
be further illuftrated in every patticulaj
Dial, viz:. 'mi6 forts ofnbsp;wherein!

To Draw the How-Lines on dU
Ordinary Viais - the eajiej);
in the firft place.nbsp;]

A N EqulnoBlal PU'iny as before is lliew-
ed, is thac whofe plain or flat Super-
^'^^ lieth parallel to the Equinotfial, and
^^ »quot;eprefented by the Line W jE E in the ge-
quot;era Scheam, and therefore needs no other
^cheam.to..reprefcnt it ;. In which Dial all
I J^fie Hour-lines are equal- one to the other,
; ^^ing juft JJ degrees aflunder ^ (o that to
Qraw, the Hour-lines here, defcribe a Circle
' ^^theCkcle 12. I2.amp;. and fi: the Radi-
: ^^ m the Sine of 30 degree?,(or the Chord
ƒ o ) and take out the parallel Sine of
^ degrees 30 minuts, the half of i j degrees^
B 2nbsp;and

lib] ,
and number it from 12 roundabout,
that fliall divide the whole Grele into »4
equal parts, for the 24 hours, forthetru«
Hour-lines on the Equinoflial-ptaift, ano igt;
the fame in all latitudes; only in the fettmS
of It, the Poles of it are to be fet due NorJquot;
and South J the Horizontai-line on
Plain, lyingParallel to the Eaft and
points of the Horizon, and the Scile ther^quot;'
of, only a Wyreor fharp Edge ftandioS
perpeiTdiculerly on the Center; which ,
ing fo fet, muft point direftly to the No'''^
(and South) Poles of the World. The re-
clining Dials-Stile pointing to tbeNor^
Pole, and the inclining Dials-Stile pointing
to the South Pole; then is the Dialnbsp;,

To fit a Plain (or to try whttbtr a J'l*'?
bef€t) Polar)0r ParaMt0tbtil**' .
xotlialj do thuii

Set the TriangHler'Quadrant togethejj
( by putting theTennons of the Loofc-P'^''^
into the Mortife-holes ) and on the
hang a Thred and Plummet, and apply ^
Moveable-Ieg to the Meridian-line on t
. Plain (whicfeonalU«Vfi??/^/Wisthelarp^ ^
xvith the Perpendiculer as here) ^ i
Head-end uppermoft; then ifnbsp;1

f^IIs right on the complement of the latitude*
We Plain lies parallel to the Equinoilial, or
^Ife not.

But to try the Inclining ?laln, apply the
^ofe-piece to the Plain with the Head-end
downwards ; or elfe apply the Head-leg to
^quot;e Plain with the Head-end dowiiwards,
®id the Thredfhall cut on 38-30 in Lo«-
latitude, if the Plain be fet parallel to

the next Dial, fhall be a VlreR Polar-
_ which is reprefented in the general
Jncam by the Hour-line of tf, viz,, the Line
^ P W J And here alfo the Horizontal-line
Plain, is parallel to the Eaft amp; Weft-
^°ints of the Horizon ; and the Pole ( or
°int oppofite to the Plain ) is in the Equt-

the Hour-lines, in this Plain, are all pa-
'^'lels, becaufe the Axis, or Stile-line, in all
^ «ms, is parallel to the Poles of the World ;
^nd this Plain it felf, being fo parallel, the
^fde or Axis therein makes no Angle; there-

Plain, w4urh is-dcne thus by the rrM^'
ler. ^Arnnt^ Hang a Plummer and Threo
■on the Center, and apply the Moveäble-leg
to the Plain, to and tio, till the Thred
neady on 6'olo, and draw that Line along
by the Moveable-leg, which fliall be a tr^e
Horizontal-line on any reclining '
and a Perpendiculcr-line thereunto, is
perp�iculer Line on the Plain.:
! • y-melfe-,

. WliientheSiluihineth ( the Sun beg'quot;^
:in the Pok of the Plain ) hold lt;up ä Thte^
and Plummet, till: tW'(hadow of the Thr^ ^
fall on the Plaitoi-m^knigcwo Points in'tquot;^^
-fhadow at the remoitft diftance'afuutie^^»
then a Line drawn through thofe two p^quot; ,
Aalhwia ciuc i?drpendiculer-line,(this lhal

r-^i'iiis is ^tna^fdf Ufright Plainf, '^^^ * r .
Midiners, the Sun'muft be in che Pok
the Plain^then fhe^Öiödow of the Plumb-h'- ^ j
iSthi-Pa'pen dicirie^iThisPerpendicüler-'i^i
in chefe Plains, ought to be juft in the mJ^ ;
thereof, bemg the Hour-lme for 12 ; . ;
alfi, at any convenient diftance, t^oHor

iontaUin«, Perpendiculer to tbehoun 2 ;
^hen confider what Houis fliall be the firft
^nd laft hour, which in no latitude can well
^ more than 10 hours, viz,, /rem y in the
'■norning till 7 at night, viz,, y hours on a
fide; then take the meafurc on the Plain,
from 12 m the Horizontal-hne, to the place
^liac you inrend for 7! or y,- and make u a
^Tangent ot 7y ; then is the Sector fet
a fit Radius, to fit and fill the Plain with
hours you refolve to put on.
The»,

The feweral = Tangents of 604^, 3 o
^ ly, laid both wayes from 12 on both the
Horizontal-lines, ihall give you Points
^■vfiereby to draw all the Hour-lines in their
places.nbsp;•

The =:Tangentof 4y, ftiallbe thetrde
^»■eadthof the Plate that muft be a Stile to
J^is Dial; or the length of an upright Wyre
^et any where in the Line 12.

That forthe hours under 4y, you may
from the fmail Tangents,-.and
•^akfr it a Tangent of 4y in the great
Taiigeius; Mid then take of = Tangent oF
3Qamp; 30, for 2amp; 10 ; ari'^ the r= Tangent
^f 'ty for II amp; and if you want them
3tgt;ove .4y, thequot; ^he = Tangent of
B 4

'Co amp; 60 from the (inall Tangents,aftd turn
that Extent 4 times from 12 both wayes, on
both the Horizontal-Iiues, and thofe Iball
be the Points for Sin the forenoon, and 4
afternoon.

Anilafllj^
The = Tangent of 7 j-, taken and turned
4 times from 12 to 7 in the morning, and
y in the afternoon, will fit and fill a Pla»»^
of 4 foot in breadth, with a Se6tpr of one
foot, fhut.

The next Dial, in the third place, is the
'DirtSl Eaji or Wtfi-Bial, which is reprefeA'
ted in the general Scheam by the Lin^
N Z S, whofe Poles are in the Line E ^
whofe Plain alfo is = to the Pole, amp;
in the fame manner as the Polar Dial was;
yet with this difference, the EquinocSialquot;
line, whereon to prick the hours, is not the
Horizontal-line, but is thus found.

^n, as AEFD, which diftance of the
CottipaffcSj make a =:Sine of 30 degrees 5
3nd take out the Sine of half the latitude,
3nd lay it from the South-corner of the femi^
Circle at A, to E; and draw the Lme C E
^or 6 a Clock hour-line: Then lay the =
Wof half the complement of the latitude
jrom the North-end of the Scmi-circle at
0 to F, and draw the Line C P for the E-
^nino(ftial-line, and another == to it. ( Or,
you might draw thefe Lines by theTriap-
Z^ler-^aadraxtf by applying the flat-fide to
the Wall, and making the Thred to play on
38-jc, and fi-jo, counting from dojo on
^he Loofe-piece and Moveablc-leg, as by ap^
Paying it, you will fee better than many
^vords can didtate). Then having the Equi-
'^ojSial^-Iines, and the Hour-line of
P°i*gt;talfo the place where i or ii fhall be,
^hentnake that diftance, -viz.. C i, in the
WcMial, or C 11, in the Eaft-dia!, a =
iangent of 7 j degrees; and the Sector fo
lay off the = Tangents of 75, 60, Ay;
^d if you will not niove the Scitor. take
^ Tangent out of the great Liiie that is
f^ght againft 30amp; ij in tte little Line of
^figents, or fee, by t!;er= Lineof Lines,
^quot;nat the great Tangent of 39 amp; i J is, and
one ^th part,

You may take the Tangents under
when the Sedtor is fet to the fmall Tangent,
and by turning 4 times, you have the re-
mainder of the great Tangent above 5
whm^he SedW is fet to the great Line o»
At, as in the Polar Dial.
^^nbsp;Oretfe-,

. Alter the Sector to the Radius of 45quot; quot;J
the great Tangent that goes but to 4f, and
take out the = Tangents of 30, andlayj^
from 6 both wayes, for 4 amp; 8 ; and the =f
Tangent of r j, and lay if from 6, botlgt;
xVayes, for /amp; 5»

-Anilafllx,
•By all thofe Points, draw Lines = to 0,
for the Hour-lines required; arid hufflbe^
the Eaft-dial with the morning-hours, and
the Weft with the afcernoon-hours,the Sn'^
is'tobe a Plate, or anupright Point; the top
of whofe edge, or point, is to be equal to the
Tangent of 45:, as the Sedtor ftood to pn'''
down the Hour-lines.

^ The fourth Plain next, in a natural order
of eafinefs to apprehend as I judge, is tl^
amp;m^mtalDlal, that lies withitamp;plaw-

the Horizon ; and the Z'cnith is the Pole
'hereof, reprefented by the primitive Circle
^•M.E.W. in the general Scheam, wherein
^^^y the Hour, Àrks, and.SdIe is requi-
red.

The Stiles Elevation is ahuayes equal to
jj^e Latitude, and therefore given ; the Sub-
file is alwayes in the Hour 12, being the
Meridian-line.

As the Sine of 90, the right Angle P N i, See the
to the Sine of P N, a fide alwayes e-
qual to the Latitude or Stiles eleva-
tionyi-jo;

So is the Tangent of the Angle N P i, .
If; or N P 2 30, amp;c. the Angles
at the Pole, to the Tangent of N i a
fide, or N 2 a fécond fide, the feveral
Hour-arks on the Plain required ;
found by the Artificial Sines and Tan-
gents, as faft as one can write them
down.

The Extent of the CompalTes, from the
Sine of 90, the Sine of the latitude 51-36,
^eing laid the fame way from the Tangent

15 gt; fhall reach to the Tangent of i i-'foj
^nd if you turn the Compafles the other way

aTablt^ and laid off by Chords or Sines in »
Semi-circle, fhall be the true Hour-points
to draw the Lines by.

But I Ihall not infifl further thereon,
fliew how to draw it morereadijy, and a'
truly by the Sedlior, th$ft.

Firft,draw a ftreight Line (in the M^''
dian, if the Plain be fixed ) for 12, as the
Line A B 5 then defign a Point in that Lin^
to ferve for a Center, as at C j then on the
Center C, erert a Perpendiculer-line to A P'
and draw it through the Center C, for the
two (J'a clock Hour-lines, as the Line D E»
then draw two Lines equally diftant fron^^
and to the firft Line A B, on either fide»
as large as the Plain will give leave, asV/
and E G i ( which may commonly ferve fo^'
margents to put the figures in).

Take th? diftance C D, and make it a
Sccant cf 00, and take out the = Secant oi ,
the complement of the latitude, and lay it 1
fromD to F, and from E to G, on the tW
Parallel-Lines, and draw the Line F G.

Then laftly^
Fit D F, or E G, as a = Tangent of 4f gt; ^
pid take out the = Tangents of every 3 deg-

4y mmuts, counting from 45 ^ and
JJofingtheCompaffing, and lay them both
^J^yes, from both the fixes, on both the
^ Lines, for all the morning hours from j
P; and for all the afternoon hours and
Starters from 3 to 9 ; then take out B E,
B G, and make this alfo, a = Tangent of
^Jj and take out the former = Tangents,
lay them both wayes from B, on the
■'-'fie F G, for the mid-day-hours and quar-
from 9 to 3 afternoon ; and by thofe
°ints draw for the hours and quarters re-
quired.nbsp;^

Por pricking down the Stile, Note^ That
'heTangent of 38-30, the complement
^ the latitude, as the Seftor {lands for thé
•^oon houES, laid from D to H, gives a Point
draw it truly by ; or the Sine of 51-30
latitude, laid from B at neareft diftance
?°out H, as the Seaor flood for the mom-
JJJg-Hours will do as well. The Stile is to
^ ® Plate,or a bended Wyre, cut or bended
according to the Angle H C B, and ereéted
•P erpendicularly on the Line 12, fo long, as
Sun being 62. degrees high may caufe the
^adow thereof to reach the hour of 12 ;
'quot;d then fet duly North amp; South,and Hori-
^ntal, the fhadow will fhew the true hour
the day. Nftetht Figurf.

Nott 4/gt;,Thata Horizontal Did ixwf^
for any one latitude, may fetve for any o^
tber latitude North or South, elevating or
deprefling the Stile, till it look to the Pole'
pome; that is, by making it to recline
Northward, or Southward, as much as the
difference of the latitudes, viz.. that the Dial
Was made for, and that wherein ic is to be
ufed, fhall be.

The next Plain to this, and moft like ih
is the Direa North and South Dial; whofe
Plain lies — to the prime Virtical, or

of Eaft and Weft, and its Poles in the South
and North part of the Horizon, and repre-
fented by the Line E Z W, in the general
Scheam, whofe Stile is alwayes equal to the
complement of the Latitude, as the Hori-
zontals was equal to the Latitude, and coti-
lequently given.

The Hour Arks on the Plai»^ art fon«'^ ^
the former Cano»^ viz.
. As the Sine of po, viz.. the Angle P Z E,
is to the Sine.of the Side P Z, the Co-
latitude or Stiles Elevation;

, The Arks on the Plain, found as before
j y Artifidal Sines and Taagents ; and be-
drawn into a Tnhle, to be laid oftquot; by
^«ords, or Sines, or by the Seéfor,
Thw ;

-.Draw-a Perpendiculer-line for the Sub-
tle, or 12 a dock Line j and in that Line
. fign a Point for a Center, as the Point A
^theLinë A B through which Point A,
Jaw another Line, crofTmg the former at
^ght Angles, for a Horizontal-line, and
two fixes, as you did in the Horizontal 5
on each fide,and equi-diflant from 12,
two Lines r:: to A B, as marginal-
J|i}es, as C F and D E i The d iftance A D,
the Parallel make a =; Secant of 00, and
out the = Secant of y i- 3 a, the latitude
the place, and lay it from C to F, and
îfom D to E, and^draw the Line F E ; then
jJ^akeDFaTangSntof 45, and lay off the
ours and quarters as you did in theHori-
^Ofïtal in all refpeas.

Make BFa=:Tangeatof 45', and lay
^ff the = Tangents of every hour and
quarters (if you picafe) from B, both wayes,
JowardEandF • and by thofe Points draw
^mes for the hours required.

The Angle of tL Stile may be laid off
by Sir.es, Tangents, or Chords, as before is
flie.ved, to ihe quantity of the complement
of the Latitude, and may be a Plate or
VVyre, as you pleafe, as the Angle G A B. _

The North Dial is the fame with the
Settthf for manner of making, only the
Noon-hours are neglected, and tl»e Mor»'
ing and Evening-hours, both before and af-
ter 6 on each lide only inferted; and the
Center of the Dial for that caufe appointed
in the middle of the Plain, and not on t'he
upper.part, as in the South, and the Stile-
points upwards ( as in the South it J^oia'^^
downwards), ifote tht Figures.

The next Plain to be confidered, being
alfoDireft, but not Eredf, or Upright, b»'^
Leaning from you; and may be either 3
North or South Recliner 5
Thatuthmi
As the Poles, of a Direft South Plain, ar«
in the meeting-point of the Meridian
and Horizon, viz.. the Point S. in the
general Scheam; and the Point N, j«
the fame Scheam, is the Pole of the
North Plain ^ and the Point Z, is the
Pole of thQ Horizontal Plain.

So the Pole of thefe'PlVins a Point in
the Meridian, elevated as many de-
grees above the Horizon, as the Plain .
lhall recline from the Zenich, oriip--
right toward the Horizon.

Suppofe the Hour-circle of lt;? in the ge-
quot;^tal Scheam, to reprefent a Reclining Pl^in,
^quot;e Point M, in the Meridian, is the Pole of
''•gt; being as many degrees above the Hori-
^ons, as Pis below the Zenith Z.

So alfo is P the Pole to the North Recli-
•^'ng Plain W E E,; for the Point P, is as
•^lUch above the Horizon N, as M is below
^«e Zenich Z.

Now therefore North Dire^ Redlnen
quot;ave their Poles any where between Z amp; N,
South DireEl Recliners have their Poles
^quot;y where between Z amp; S, according to the
•degrees and minuts of their Reclination.

This being premifed, for drawing the
•^^^ur-lincs, obferve, That for Souch Recli-
DirtU, the difference between the Re-
•^'inatiou and the complement of yourXati-
^'ide, is alwayes the Stiles height for that
declining Plain.

Butnotty That when thcReclination 'S
more than the complement of the Latitude,
thac then the contrary Pole is elevated,»»^-

But for North Recliners, the fumol tnc
Co-latitude, and the Plains reclination is al-
wayes the Stiles elevation; bat note; whe ^
the Sun is above 90, then the complewen
to 180 is the Stiles elevation ; but it mt»
be turned the other way, viz.^ contrary ^^
the nature of a North Recliner, for jn
Stiie will look downwards in the North
chner, and upwards in his oppofite Soquot;quot;^quot;
Incliner.nbsp;u

Not! alfo bj the »47,That when the SoutfJ
Recliner reclines equal to the complement °
the Latitude, it is called a Direa Pol^V
Dial, or rather an Equinoaial in refpe^Y
his Poles, ( but I mrnd not to be fingular gt;

And when the Reclination. of a Notj^
Rettiner, is equal to the Latitude, then t'l
Stiles height is juft 90 degrees; and tquot;
Plain, called an Equinoaial-plain, or rath«
Polar, in refpea of his Poles, ( being fir«
order treated on).

Thus underllanding, and right concei-
ving-Avhat ths Plair.s are, the drawing of
any cf them is the fame with the North an»
South J for the Sulcs height is alwayes to

^lt;5Untcd the complement of the Latitude,
^'^d by confequence you have the Latitude.

Suppofe a plain recline from the Zenith
foward the North part of the Horizon lo
^quot;^gvees, his Pole is lo degrees above the
/^o'izon ; and then lo taken from 38-30,
^«ere remains 28-30, for the Stiles EJevati-
5 or the Latitude to draw it asaHori-,
^oncal Dial; and di.30 for the Latitude
which place you are to draw a Dired E,
'flt;aSoudiDial.

^uppofe a South Recliner, recline yo de-
pees, being more than 3 8-30, the Co-lati-
' J^de ; then take the Co-latitude from
quot;ence, and there remains 11-30 for the
latitude

zontal Dial by; and 78-?o for a Latitude
[a draw a South Dial by ; but the Cock muft
Jook up to the North, therefore muft be
^^rned the other way,

For a North Recliner, reclining 60 de-
^•■ecs; 60 amp; 38-30, added, makes 98-30,
complement to 180,'is 81-30, the
C 2nbsp;Stiles

Stiles height; but the contrary way, as
you may-well perceive by the Horizontal.

Suppofe a North Plain recline 8f de-
crees; that, and 38-30, added, makes
ï2?-30 , whofe complement to i80j
5'd-30, the Stiles height : but put the con-
vary way, as a South Incliner, being almoi
i Horizontal Dial; fo that to drawth'
Dial, let ^6-30 be the Stiles height, or Cc^
latitude ; then, 33-30, is the Latitude t
draw a Direét South Dial by.

You may count the Stiles height the '
titude, and then draw it -as a Horizo^^^ '
Dial,. by taking out the Secant of the
latitude, and the work will be the iann«^
As,ui the Figure North Reclining û
grees.

This Plain being a Direé^ Plain alfo, ao^
taking no cognizance of Declination, C
therwife-fhould have come after); is on J
an Eafl or a Weft Plain, reclining or faJ^
ling fi-om you ; or incluung,

you 5 and the Poles of ,thefe ï^lains are ih
quot;^he prime Vertical, or Circle of Eaft and
^elf, as the other were iu the Meridian ;

this Plain is reprefented by the pricked
Circle NFS, reclining4j degrees, whofe
^oleis at CL 4 y degrees above W; for the
drawing of which, it will not be amifs, but
^ery convenient to draw a particular Scheam
h the laft Rule in the 17 Chapter*

With 45' degrees of the fmall Tangents,
^taw thé Circle'N.E.S,W. crolhng it in the
Center with the Lines W E, S N; then lay
the;half Tangent of 38-30, from Z to
and the half Tartgent of the Reclma-
quot;^'on from Z toE, and from Z to d.,
!quot;iddraw the Circles NES, and F P CL;
in which Scheam, P -F reprefents the Stjle,
F G the diftance of the Subftile from the
Meridian, and G P F the Angle between
the two Meridians , viz,. Z P N of the
place, - F of the Plain. All which.re^
quifites are thus found out by the Artificial,
Natural Sines and Tangents^

So IS the Sine of N P the Latitude yi-j®»
to the Sine of P F, the Stiles height;
3 3quot;? 7.

So is = fine of ZE, the Reclination 45quot;'
to — fine ofPF 33-17, the Stile^
height.

a. Tor the DifiAnce of the Suhflile frlt;gt;^
la, thfU) bj t/4rtifieial Sines
TnKgents.

So is the Tangent of the Stiles elevation»
P F 3 3-37, to the fine of F G 41-4°»
the diftance of the Subftile from 12.

clination4y; p , ^ , - .
So is = Tangent of the Stiles height
to_fine of the Subftile from

ftom the Movéable-leg, and meafute it on
the Sines from the Center, K fliall reach to

So is the fine of the Subftile from 12,
G F 41-40, to the fine of G P F y 8-7,
the Angle between the two Meridians.

If this Rule fails, ufe a Icfs Radius, or aquot;
= Anfwer, as is largely Ihewed before»

Firft confider the Scheam, where yo»*
fliall find the Reclining Plain to be repre-
fented by the Line.S E N, as the upper-edge

Meridiaii-line alfo; N or G is the place
where theMeridian cuts the Plain, being
the Horizon; Therefore here the Hour-line
of 12 is a Horizontal-line, and the.Sun.be-
ing in the South part of the Meridian, dotfgt;
caft his fhadow Northwards; and being iquot;
the Eaft, cafts his- lhadow Weftwards s
Therefore laying the Scheam before you, aS
the Plain reclines from you, youflialUee
that the Meridian miifl lie to the right-hand
from Z, toward N; quot;and the Subftile up-
wards from N, towards E, at F, And the
the Stile muft look-upwards, as the Angle
G P F doth plainljc fhew j and the morn-
ing-hours are chieflyquot; lit for the Plain, be-
caufe'theSun rifin^Eaftward, is oppofite to
thfe Pliiin. Thiis the Affe6tions and Scitua-
tioiis fH^the Cardin^l-lfnes are naturally and
demonttrativcly ihewed, ths Pelineation
(ollowes.nbsp;Firfl,

Flrfl, ■
Draw the Horizoftcal-hne.S N; and on Z,
® Center, defcribe a Semi-citle as S E N,
N toward E lay offquot; 41*40, the
T|quot;ance of the Swbftile from 13, and draw
Line Z- F for the Subftile alfo beyond
from FtoE, prick off 33-17, and
that Line for the Stile-Iine.
Then for drawing the Hour-lines, you
firft make the Table of Equinoaial-
^'ftances, or Angles at the Pole, thin J

Firft, in all Direft Plains, it is orderly
t ^^ s 3-4fgt; for the firft quarter of an hour
otn 12- 7.30,for half an hour J ii-iy,
1°'' three quartets; and ly degrees for an
i and fo fucceffively to 90 : So alfo
» \ - quot; he in all Plams, whole inclination of
^eridians is juft ly, 30,45, 60, 75, or
yu degrees, being even whole hours; and
lear as well, when it falls on an even quar-
er of an hour alfo. But when it doth not
Js here, then the beft Rule or Method I
'^«ow is thw;

Then right againft 12, fet down theln-
HiHation of Meridians; then fubftraft i y

degrees for every hour; and 3.45: tor eve;
quarter, as often as you can, fetting do^
the remainder; then draw a Linea-crol»

of iy,.or 3-4f ( for hours, or hours a»^
quarters) fet down on the other fide beio ^
the Line, as you fee in the TtibU followug
and fo proced, adding of 3-4^ to
fum,,for every quarter; or, ij degreesj
every whole hour, till you come to
wayes; fo is the Table of Hour-Arks at
Pole, compleated for all Hours that
come on this Dial, or on any other.

Thus you fee that the Subftile falls on
near a quartcr.paft 8,- or 3 hours 3 quartet^
and better' from 12; then if you will, fy
the former Canon, you may find all th^
Hour-Arks on the Plain.

So IS the Tangent of the Hours,
Table laft made, called Arks, at tIJ
Pole 31-^3 for tf, to the Tangen^ ^^
the refpeéfive Hour-Arks at the Pl^'f
from the Subftile p 0-0 for 6. !

So is the Tangent of the Hour from
to the Tangent of the Hour from tP^
Subftile.nbsp;_

Which being brought into a Tabie,
be pricked down in a^Seraucircle by SinC^»
or a Line of Chords, from the Subftile 0»
the Plain,

a Line at any convenient diftance
parallel to 12, as D 15 then at any conve-
lent diftance from the Center Z, draw a
Jjine Perpendtculer to the Subftile qwite
■^fough the Plain, as the Line K L.

. take the neareft diftance from that meet-
Point at F, to the Stile-line, and make
= Tangent of 4 j ; then the = Tangent
every hour and quarter, as in the TMt
^aken from the Se6\or, and laid fromF the
way, as the hours go, fhall be the true
oints whereby to draw the Hour-lines re-
quired.

. Buf in regard that this way will fome-
time be troubled wichExcurfions in fome of
J^c hours, you may help it thus; Having
^'awn lome hours, as fuppofe 6 amp; 3 ; or
^ amp; p ; or indeed any 2 hours, 3 hours di-
stance affundep, as here 6 amp; 9 ; take the di-
ftance between 6 and p, and lay it from the
Center to N on the Meridian, and draw the •
LinepN --tod at length, beyond 12;
Then, as before, make a = Tangent of
l-y, and lay off every hour and quarter, as
the South Ere6t Dial, both wayes from
Alfo^ make N p a —Tangent of 4^,
and do likewile laying the hours both wayes
from N, and you fhall have Points enough
to draw the Dial by.nbsp;Other-

Cpunt rhe complement of the Latitude
where the Dial is to ftand, for the Latitude
And the complement of the Reclination for
a new D/clitjatimj and then draw theiquot; as
Upright Decliners by the following Rules*
and you fhall do as well and ipecdily as an/

North-eaft Decliners; and Weft Recline»'^»
are North-wefl; Decliners; And Eaft and
Weft Incliners ( being the under faces) are
South-eaft, and South-weft Decliners.

true, and large, you may from thence Geo-
metrically find the Sabftile, Stile, Inclina-
tion of Meridians, and every hours diftance
on the Plain, by Scale and Compafs, thm '■gt;
As Captain Lankfori hath fliewed.

tirfi. For the Stiles Elevation fet off a
. Quadrant or quarter, as W.N. from B to I;
then a Rule laid from Z to I, cuts the Plain
at A; thai a Rule laid from A, to P amp; Fi
cuts the Circle at O and L; The Ark O U
is the Stiles Elevation, and mcafured by fit
Chords, gives 33-17.

Secondlyy For the diftance of the Subftile
from a Rule laid from Qto F, cuts the

J-ifUb or Circle at M, the Ark MN mea-
^.^edon fit Chords, gives 41-4°» theSub-
from 12.

'^hirily^ For the Inclination of Meridi-
a Rule laid from P to A, in the Limb
^'^es C ; the Ark W C J 8-7, is the Angle
ftween the two Meridians.

To find the Hour-Arks on the
: i a Ruie laid from CLto the interfeft-
of every Hours Ark ( in the Scheam )
the Plain as here, a Rule laid fromQjo
gt; lt;:uts the Limb at R, the Ark M R ip-o is
jje diftance of 6 from the Subftile on the
yam J and fo for all others, as 12 amp; 6 is
j R tfo-40, both hours and quarters if you
^ave them truly drawn on a large general
^^quot;eatn, as Mr.nbsp;hath done.

Thus much for Dlrdt PlaitiSy

FOr finding the Declination of a Pbiquot;'
themoti eafie way is by a Magnetics -
Needle, fitted according to Mr. Falles
in the WfAT of z Declinatory (ashec^'v
being 180 degrees of a Semi.circle, divide
on an Oblong-Board, or Quadrantj ?
longer Needle in a fquareBox, (or) fitted
wuh Hinges and a Cover ; after all wh^
wayes, you may have them made at the figquot;
of the SunMal in the Minories-, by Jquot;quot;^.
Brown ; or of any other manner you iha
think fit.

Box and Needle alfo to be fitted of another
form, in fome things more convenient.

5 ^ nole made near 4 inches Ions, i inch
* ^''oad, and 4 tenths deep for a Needle to
^itl 1quot; 'nbsp;degrees at each end 5

tot ^'^^fs-hinges, -and a cover, and a brace
auH ^ the Iid upnghrjamp; an Axis of Thred^
u a Plummet playmg in the lid, and a
g^fiZontal and a South-dial, drawn on the
Cover; alfo a hafp and glafs to keep
3 ^Needle clofe covered, and on the bottom
tenth of an inch deep, made
^^ as broad as one leg of the Seftor is.

is decliner ; and the contrary when iti
JPjy^d to a North declitier, as the play^
o ot the Needle will tell you better than
any words; then open or clofe the Rule,
' the Needle play right over theLine in
bottooi of the Box, (unlefs there be va.
ation, then you muft allow for it Eaft-
j-^^ms or VVeftwards what it is). Then, I
y? the quantity of the Angle in degrees and
^^'luts the Se6for ftauds at, above or under^
J IS the degrees and miiiuts of Dechnat:-
Dnbsp;on J

on; being counted from oo in the little
Semi-circle, as complements to theAng e
of opening; as in the 4th^/lt; ofthe5tW
Chapter is largely and plainly fhewed.

Thus you have the quantity of degre^^
and minuts of Declination : but to dete^'
mine which way, confider ;nbsp;,

If tlgt;e Needle will ftand ftill in
middle, when the North-end is toward tn
Wall, then the firft denomination is So«^quot;'
if not North.

When you know where North aquot;
South is, you may refolve which way ^^.r
Eaft and Weft is; For, obferve alwayeSj i
the North be before you, then the Eaft
the right-hand, and the Weft on the »
and contrarily. If the South be before y®quot;'
the Weft is on the right-hand, and the Ban
on the left.

If the Sun, being in the Eaft-point of tnc
Horizon, can look on the Plain, it ®
South-eaft Plain ; but if it beholds it whc'^
in the Weft-point, it is a South-weft Pla»quot;'
Likfwlfe,nbsp;,

IftheCrofs-endof the Needle will not
ftand toward the Wall (the Needle play'^
well; and the Sun being due Eaft, behold»

i but if it caniiöt look on the Plain,
due Eaft, then it is a North-Weft
ain, declining fo many deg. as the Seaof
at, under or above po, being alwayes
j^^^omplementof the Angle the legs of the
ftand at, and found by taking the
. quot;gle the legs ftand at, from 00, when the
^'^leislefsthahpo.

jj t aking po out of the Angle, when it
Jq at an Angle above po degrees, as a
n 7 at the little Semi-circlc on the Head

t}, quot;ppofelcome to a Wall j and putting
^^^ox and Needle on the Leg of the Se-
VV nnbsp;the other Leg to thé

hi: 7 (or on a ftreight piece of Wood, ap-
fto the Wall, becaufe of the Walls un-

Vav ^ ot fhe Sector ffands at, being al-
^^yes what it v/ants of, or is above po
^he r^^' is the degrees of Declination 5 and
b J which way, the Needle and Suns
quot;^Eaft and Well, tells you.

If the North or Crofs-end of the Needl^
be toward the Wall, it is a South PlaJ«'
and if the Sun, being in the Eaft, can be-
hold it, then it is South-eaft; if not,
South^-weft Plain.

Take the = diftance between Center an
Center, in the middle of the innern^quot;^^
lines, and lay it latterally from the J
and CO int two degrees more than ^^^ r ^
fheweth, after the manner of Chords quot; ^^
j)0 (at the fine of 45) toward the Coinp^ j f
point, and that fhall be the degrees
minuts required.

doftand at an Angle of jo degrees, ^ ^^
the Legs at 48, two degrees lefs^ the co^^^^ I
plement whereof is 42; as if yoquot;
thus from 45-, you will find, 4°

you like not the abating of twode-
then the = diftance taken juft be-
^^^en the two legs ^igbt agaiuft the Cen-
fhall be juft the —^ fine of 24 degrees,
^ counting after the manner of Chords,
every j degrees on the Sines, for 10 011
Chords backwards from 45 of the Sines,
^hich is 5)0 in Chords.

quot;you ufe the firft Rule, of the 4th ^/tf
the yth Chapter, viz,, by taking the —■
quot;J? of JO, and put one Point of the Com-
. in the middle Center in theTangent-
itj'.^Sj and apply the other to the Line of
you fhall find it reach to rhe fine com-^
^5'^ent of the Angle the Lines ftand at,
40 degrees and 2 degrees more, viz,.
''s the Angle or thing defired ; as pra-
With confideration will make eafie.
, fhias, by the Needle, you may find rhe
^^^'inacion'of a Wall, which in cloudy
^^ ather may ftand you in good ftcad; or.
^^ prove a declination taken by the Sun, to
.^^^siit miftakes. And if nothing draw the
.^^dlefVom its right pofidon, but that it
P^^ywell, 2nd you find the Angle tridy,
^^^ may come to Icfs than half a degree:

And this convenience it hath, that it
the Needle a comp^nt diftance from ^^
Wall, to prevent that attraition ; ^^ quot;
happen to be fo near a Meridian,or Eaft aquot;
Wefl-plain, that the ^ngle, by the Seaor.
cannot well be taken; then youmayoivy
apply the fide of the Box and Needle to tje
Wall, and the Needle it felf will Ihew th«
Declination, on the degrees on the botto
of the Box.

Yet forexaancrs, the way by the Sun »s
alwayegt; the beft, where you may corne^

ir.aice a good Oblervation, and then ^
Needle only is not to be trufted to ; a
ter way with opportunity offering it felf?

For this pm-poie you (muft or) ougW
have another Thred and Plummet, w^''.
Thred may be a fine even fmall Pack-thre j
and K is convenient to have it ready hang^
up near the Wall, fo far off, as theTn-J«!quot;!
Ur-Quadrant may pafs along between j
and the Wall, that you may not betroubl^^,
tohDldicup, and lay it down, and f
annoyed with the inconveniencies of yoquot;^

You muft needs take notice of the two
^^«ridians, viz,, one of the place which is
tUe Meridian, or 12 a clock; to which
place, when the Sun or a Star comes ^it is faid

And the other is the Meridian of the
^'ain, in which Line the Pole-point of e-
^ery Plain is, being 90 degrees diftant from
^l^e Plain every way, and inallupright-
^ials their Pole is in the Horizon ; and that
?egree of Azimuth in which the Pole-point
'les, counted from South or North toward
Eaft or Weft, is alwayes the declination
^Weof; fo that by finding the SunsAzi-
'^nth at any time, and the diftance of the
at the fame time from the Meridian of
the Plain, is gotten the declination.

The Azimuth of the Sun from the Meri-
dian of the place, is found by the 26, 27,
30, 32, 34, 39 tifes of the i Jth Chap^

But the Azimuth of the Sun from the
Meridian of the Plain, is found by apply-
^»g the Head-leg againft die Plain H6nzon-
'ally, flipping it to and fro, till the fiiadow
of the Thred, hung (or held) up, play nghc
over the Center of the Tnangtiler-Qjtdrant

on the Head-leg; then what deg. focvcr tW
thred cuts, counted from lt;5c]o on the Loolequot;
piece (being thePerpendiculerorPoIe-poif'^
of the Plain ) fhall be the Azimuth of the
Sun from the Meridian of the Plain.

This is the Operation; the Applicatioi»
or Ufe is worded feveral wayes by fevera»
men; I hope I fhall do it as fullyj and
briefly as fonie others.

The Sun, to our appearance, paffeth fro'^
Eaft, by the Meridian, to the Weft everf
day; therefore in the morning it wants
coming to the Meridian ; at noon it is for ^
moment juft in the Meridian, and in the
afternoon it is paft the Meridian of th^
place.

Even fo it begins to fhine on, and is di'
reilly againft, and leaveth to fhme upo'^
moft Plains, when it begins to fhine upof'
or is not diredly againft ; I fay, it wants 0'
coming to the Pole or Meridian of th^
Plain.When it is diredfly againft the Plai;i?
then it is in the Meridian or Pole of rh^
Plain; when it is paft, it is paft, orbcgi»^®
to leave the Plain.

Paft, or P being in the afternoon.quot;
, The other Three Varieties, let be Sha-
dow Want, Shadow Direa, Shadow Paft ;
which may be in feverai Plains at feveral
; that is to fay, at morning, noon, and

r. If the Azimuth and Shadow are both
^anting, or both paft; fubftradl: the lefler
of the greater, and the reftdm is the
Reclination, But if one want, and the o-
be paft, then the [um of them is the
quot;^clination,

^ If the Sun come to the Meridian of the
^ Iain, before it come to the Meridian of the
P'ace, it is an Eaft Plain. But if it come to
quot;^he Meridian of the place, before it come to
the Meridian of the Plain, it is a Weft
i-'lain,

3. i^^fum ox remainder, after Addi-
tion or Subftraaion, be under j»o, it is a
South-eaft, or South-weft Plain, declining
^ many degrees, as the fum or rejidtte is.
if the fam or remainder be above po, it
a North-eaft or a North-weft Plain, and
the complement of the fpim or remainder to

4. If the/«OTor remaiflier he 00, it
juft South; If 90, juft Eaft or Weft, But)
if it be 180, it is a djrea North Plain.

Suppofe thatonthefirft of
forenoon, I come and apply the Head-Is? j
the Trianguler-Quadrant to the Wall, aquot;quot;
•holding of it level, the fhadow of thj
Thred, held up fteady, cuts the Center aO^
60 degrees on the Moving-leg ; that is,
deg, want; which I prefently fet dowu
a Paper ready prepared, thtu-^

Then, as foon as poflible, or rather by
fomebody elfe, at the fame moment, fin«

the Suns Altitude, which hppoktobe^o
degrees; ( but if you are alone, and have J

Thred ready hanged up ; then take the Al-
titude firft, and the ftiadow will be had
prefently after, the Thred hanging fteadily)
fee that down alfo, as here you fee.

Then by theztfthar/* of the ifthC%-
you fhall find the Suns Azimuth at that
time and Altitude to be 94 degrees, and
after Subftraftion rcmams 34-0, for the
WaDs dechnation Eaftward, becaufe the re-
'iiainder is under 90, and the Sun comes to
the Meridian of the Plain, before it comes
to the Meridian of the place, or South.

. In a morning, June 13,1 obferve the Al-
titude, and find it if degrees, and inftant-
ly thefhadow, and find it to be 10 degrees
paft the Plain, t»»«.. on the Loofe-piece, to-
^vard the Head-leg, I fet both Altitude and
Shadow, with the day and time down thin;

June 13, Forenoon,
Altitude— ly—o
Shadow — 10—o Paft.
Azimuth—109—o Want,

And then find the Azimuth at that tiitic
and Altitude to be iop degrees; here the
terms being unlike, I add them together»
and the/«/» being above 90,1 know it muquot;
be a North Plain ; and becaufe the Sun
eomes to the Plain before it comes to the
Meridian of the place, it is North-eaft; and.
the complement ot 119 to 180, is 61-°
North-eaft.

Suppofe the fame day, in the Afternoon,
I find the Suns Altitude i y-o, and the iha-
dow ao degrees want; the Azimuth at the
fame Altitude, and the fame day, will be
near the fame number of degrees; but in
the Afternoon it is paft the South, or Meri-
dian of the place ; here alfo it is a North-

North-weft, becaufe it is againft the Sou^,
quot;efore it comes to be right againft the Plain.
. But if you happen to come when th« Sun
in the Meridian of the Plain, then the
^uns Azimuth is the Declination,Eaft or
^eft, as the Azimuth is.

If you take the fhadow, when the Siiri
is juft in the South, or Meridian of the place,
^he fhadow is the Declination ; if it is paft
t^e Plain, it is Eaftward ; if it wants, it is
Weftwards.

Thus I have (I hope) fhewed the true
quot;quot;•anner of finding the Declination of a Wall
by the Sun fhining on the Plain, as plainly
as briefly as the matter will bear, fpeak-
^quot;g to young Tiroes therein.

It may be done alfo, by obferving when
the Sun juft begins to (hine on a Chimny,
or Wall, or high place you cannot for the
prefenc come near, conceiving the Sun to be
then juft po degrees from the Meridian of
the place wanting, or juft when it leaves it
, b^ng then po degrees' paft the Plain, then
fake the Altitude and Azimuth, and work
accordingly to the former Rules.

F Or better illuftrations fake, Iwilld^a^v
a particular Scheam for this Dial alfo,
as I did for the Eaft Recliner; whofe De-
clination let it be 20 degrees declining from
the South toward the Weft, in the latitude
of 51-32 for

The Scheam is drawn by the former di-
rc(5lions; the Pole of the Plain being at D,
declining 20 degrees from S toward W,and
the Plain it felf is reprefented by the Line
A B ; the Circuler pricked Line D H P C is
a certain Meridian drawn through the
three given Points DPC, whofe Center
will be in the interfeftion of the Plain A Bj
and the Tangent Line for the hours, which
being drawn, whatfoever Z H is in the half
Tangents, Z 0 is the complement thereof,
in the fame half Tangents.

^ The Scheam thus drawn, Z H is the Sub-
PH is the Stile, QZ the diftance
^ from 12, H P Z the inclination of
■^Jridians, or Angle between the two Me-
Jfiians, viz,. of-Sie Place P Z, and of the
Jain P H, found by the following Canons.

Declination N C 20 degrees
So is the Co-tangent of the Latitude
P Z 3 8.28, to the Tangent of Z H
15-12.

the Plains Declination N A 70-0 ;
So is the Co-fine of the Latitude ZP
3 8-28, to the fine of P H 3 5 -4lt;Jj the
Stiles Elevation.

Plains Declination W A 20-0;
So is the Tangent of the Latitude NP
5:1-32, to the Tangent of A Q^, the
Co-tangent of 6 from 12,23-18.

So is the Tangent of the declination S ^
ao-oo,to the Tangent of ./EK 24-fquot;'
the Inclination of Meridians.

y. Then having made a Table of Arks ®quot;^
the Pole, by this Canon you may find the
Hour-Arks on the Plain.

As the fine of 90 P K, tothe fine of the
Stiles height P H 3 5 - 46 ;
I So is the Tangent of the Hour from

thefe rfirnißtes may befomi by tbe ce-
ntral Scale and, Seilor^tbe Canons where'
of in brief are thtu ;

As — Co-tang. Lac, 38-28 Z P,
To = fine po po-ooZN;
^ = fine Declination 20-00 N C,
io_ISubftiler iy.i2ZH.

^iflatice betmen 6 amp;
As — fine Déclin. 20- o WA,
To = fine of 90 po- o Z W 5
SoisrrCT. ofLat. ji-32 N P,
To _ C-T. 6 amp; 12 23-18 A (i
Tangent (îlt;5-42.

Inclinations of tMfridtant.
M — I declyiation 20- o S D,
To = fine latitude j 1-3 2 Z ^ i
So = fine ponbsp;po-oo Z S,

Thefe Requifices are alfo found by th^
particular Quadrant, very really and truly»
for thac Latitude the Rule is made for,
this manner,

Lay the Thred to the complement of t^®
Plains declination, counted on the Azimu^ ^
Line, and on the degrees it giveth the Su^'
flile .fromi2, counting from 6o|o ontP
Moveabk-Ieg.

The thred laid to 70, the complementquot;
20 on the degrees, gives 15-12 for tn
Subftik.

„ Take the Subflile from the particular
of Altitudes, and meafure on the
^^»muth-hne from po, and it fliall give the
^°iiplement of the Inclination of Meridians^
the Angle counting from 90, hej-g

4. To find the Angle hetmen izc^r 6 i
Take the Plains Declination from the
P'«icuiar Scale of Altitudes (lefs by the
j^'^^of the Declination, to a Radius equal

^ On the Azimuth Scale, and to the Com-
Hais-point lay the Thred, then oil the Line
^j^^^grees, the Thred gives the complemenE
th ^ Jnbsp;counting from 60 toward

5. A Rule laid from P to 0, cuts the
Limb ac T; the Arfc T E is the Inclination
of Meridians.

J. A Rule laid from D, to the interfei»'
oft^o^' any other Houff line, with thePia'.^
A jB on the Limb, gives Points, whofe
^ances; from C, are their Angles from
or ,their diitances.from F, or their Ao§
fro;?? the Subftile.

Thm hj any of thffe wajfi, having i^j^fj
the %tquifntSffroced to draw the ^^^
thw,

Firft, Draw a Perpendiculer-line on ^^
Plain C B, by a Thred ajid Plummet; j
if ii^be aSouthDecUner, at the
make a Center, as C^ and on that C^it^
defcribe the Arch of a Quadrant,as the At
D E on the Center C j dien jn tbat Arch ^

^ines or Chords lay offquot; from D die Subftile
y upon the SubftiieV the Stiles height.
I'ndtheHour-Iine of 6 by the Angle be-
ween 12 and 6; and draw thofe-Lines as
gt;0ü fee in the Figure, contrary to the Coaft
Dechnation; then draw two Lines paral-
to 12, as 6 G, and 12 H; then fit the
*i'itance of the Parallels from 12, iti tlie Se-
^aftt of the Plains déclinition 20;' -dnd
out the = Secant of the Latitude
J*quot;?«,and lay it from (Ï to G, and from C
the Line of 12 to F) amp; draw the Line G ^
P H, which Line is a to the Hour-line
y «; then make F G Radius, or the =
angeiHof 45 ; and before you prick oftquot;
Hours, take out the = Tangent of thé
quot;chnation of Meridians; and if it reach
F to the Subftile,on tlie Line F G, your
done is true, elfe not.
The»,

Take out the = Tangent of jo, amp; ly.
and the refpeé^ive quarrers, and S
before; then n,ake ÓÜ a = Tangent of

4y, and do hkewife as before, in the Hori-
zontal and South Dials, and to thofe Points
quot;raw the Hour-lines required.

The Requifitcs, as Subftile, and Stile, In-
clination ot'Meridians, 6 amp; 12, are found
the fame way, and by the fame Rules, as tnc
South Decliners are done. But when y«
come to dehneate the Dial, there is iojquot;®
alteration ; which I conceive is beft feen M
an Example, as Northeaft declining 3f deg*
l,at. f 1-3 2, at London.

Firft, as before, draw a Perpendicule«quot;'
line for 12 a Clock, as A B ; then about
the middle, or toward the lower-part of tnS
Line, as at C, make a Point for a Cent^)
as C ; then on the Center C defcribe the
Arch of a Circle, that way, from the Pquot;®
A B, as is contrary to the Coaft of declin»'
tion, as if the Plain declmes Eaftward, ^
here, draw the Arch Weftward from A
as B D; and the contrary way in North''
wtft plains y and on that A) k lay downth?
Subftile from 12, and the Stiles height above
jhe Subftile, and the Hour of d, by the
Angle of 6 amp; 12; and;then, by thofe PpJi^'^
and the Center, draw thefe Lines.

To Draw the Hour-Lines on ^
' 'Dial falling near the Meridi-
an^whoje Stile hath hat a fnt^^
Elevation^ and therefore ^^
Center.

/-plt;Hc former Examplts may be fuflicki^^
to the confiderate, to draw any Er^^
Declining-Dial having a Center ; but
the Stile happens to be lefs than 15 deg. °
Elevation ; then, if it be not augmented
catling away the Center, the ufefulnefs, aquot;»
handfomnefsof the Dial is loft; now
you draw the Dial by the former Rules.o» ^
Table, and cutoff fo much, and as manV
Hours as you care for, the work is '
fjrrned.

■cliners, ( but near to
the Souch.edge of the
Plain, in far North,
' decliners) draw a per-
pendicular. Line, re,
)refenting the Hour,
ineofi2, as the Line
A B in our Bxamfle,
being a Southweft de-
clining 80 deg.25 min,
then,in the upper-part
of that Line, in Souths
decliners; or about the
middle, or lower-part
in Norch-declincrsjap-
point a Center, as here
at A ; then upon A,
aj a Center, as large as
you may, draw an

only to be feen; and upon that, theStile-
line, as before: Then at any convenient
places, as far frOm the Center as you can,
' draw two Lines Perpendiculer to the Sub-
ftile, as the Lines C E, F G, for two contin-
gent Lines, (antiently and properly fo
led) ; then by the Inclination of Meridians,
by th»dire£lions in the Eaft and Weft Rc-
eliner, being the 7th Dial in the zd C^aftf*
make the r^^/* of Hour-Arks at thePole^
by fetting down againft 12, 82-30; an»
taking out 7-30 for every half hour, ^n
you come to 06 at the Subftile ; and thequot;
by adding 7-30 for every half hour, and *7
for every hour, to 8 4, as long as the S^n
fhines; which in regard it falls on an eyequot;^
half hour, is th? moft eafie, and fits the Pon'^^
in the Tangent ready made for houis an'»
quarters»

The next work, is to refolve what hou^^
fhall come on the Plain, as will be beft
tcrmined by the difcreet Orderer, or Si^^^
veyor, or experimental Dialift, as here
and I; and for thofe two hours, mark the
npper contingent Line in two places whe'^^
you would have them to be, as at E and C »
then take the — Tangent of 37-50 for
fi-om the fmall Tangents, and add it to tne
.— Tangent of 67-30, the Tangei^t tor i)

^Take the whole fpace C E, and make it a
[^Tangent of 72-? 3 ; then take out the
g- Ungentof 67-30, and lay it from C to
» aid take alfo in the fame common-line
S^t againfi the fmall Tangent of 37-30,
^quot;ichis in the large Tangent lo-jo, the
jr Tangent of lo-yo, taken for 37-30, be-
2^idfromE, the place for 8, will meet
fn fH; which Point H, is the true place
Jhe Subftile, to fit and fill the Plain,
'th the hours determined.

Seaor fo fet, Takeout all the =
them L^^ above 4 J, as in the Table, and lay
E .quot;J the right way from H, toward C, and
and f quot; ^^^^ = Tangent of fmall 45 ;

'etting one Point in H, ftrike the touch
Ta^ as at 15 then make HI a =
^^ngentm great 4^, and take out the =

hart ^^^ Subftile; then make H K
aaiiw, or the Tangent of 4^, and take
the Parallel Tangent of j-jd, the Stiles

H I, the firft Radius, and fetting one Po'quot;
in L, draw the touch of an Ark as by M »
then draw a Line by the Convexity ot tn
Arches by T andM, for the true Snle-hne. ;

Take the neareft diftance from the WUV
K,to the Line IM, and make it a Ta''
gent of 45, the greater Radius, and tas
out the =: Tangents, as m thtTatle^ ai^
lay them from K both wayes; and tB
laftly, by thofe Points draw Lines from
hours recuired.-nbsp;^

Notey'Thzz if in ftriving to put too ma''
hours, the lum of the two extream hoquot;^
come to above it will make the hours ^^^
«lofe together, and put you to much m .

^Ifo me, If your Rule prove too feJ^J
then take the half of the fum of both
Tangents, and turn the Gompaffes twice-^^,^
Aif», If yoii be curious, you may quot;-'e ^^
Natural Logarithm Tangents, inftead or t
Line of Tangents, but this will fa^e ve )

This is a gfiieral way 6f augmenting ,
manner of Dials, when the Stiles heigj^
low, as uader ij degreesand «s rea
way,as you fticet with m any
foeveti.

p O R the conipleat and tr^e drawing of
* thefe Pyls, that youtpay pjMiily feis
Aflfeaions'aad Properties, k \yill be ne-
- ^ary tohaj,e aScheam fox every variety j
11 whereof, I ftiaU follow the Method
Mr. mUt, ofnbsp;his ^rt.

/ ^b^ews i which will C9nQprehend any
«?f Reclining and Piedmi^ Oial, under
varieue? ^^ 3 South R^liiiers, and 5
ANortb Rcclinei-s ( the IncUnyig beins their
gfi^s, «0 other, as »fterwardsis

Whereij^J/haUb« very brief, vet fuffioiil
^tly plain^59a M?itb«agt;WKalnbsp;and

Jjral Sjt^s and .Tangents,draw the
by the Sedlpr, the fifteft Inftrumenc
' t^at ufe J Wiavoth«.occHrrait Obfer,

li And firft for a South-Declining,
dining Dial, declining from the South to-
ward the Weft degrees, and reclining
from the Zenith 20 degrees, bsing lefs than
to the Pole, vix.. falling from you,betwee«
the Zenith and the Pole: As theCirlt;:ie
A E B, reprefenting the Reclining Pl^'quot;'
plainly iheweth P being the Pole, and Z ^he
Zenith.

The manner of drawing this Scheam,
plainly Ihewed before, (Chaf.t.) both ge-
nerally and particularly for the drawing
Dials, and thenbsp;there, is the very

Scheam for this Dial; wherein you ma/
forther confider, That the Perpendiculer-
line Esright before you, and when youlfok
tight on this Plain that declines Southweft,
the North is befc^e you on the left hand,
the South behind you on the right hand SS
degrees, the Eaft on the right hand,the
on the left; the Line C D the perpendicttquot;*
Jer-line right before you, reprefenting the
Perpendicuier-line on the Plain, A B ^h®
HonzontaHine, Z E the quaimty of
nation, P F the- Stiles Elevdtionabove the
Plam, having the Souch Pole elevated above
the lower-part- of the Plain, becaufe the
^nbsp;North-

North-Pole Is behind the Plain, E G the diJ
^^'quot;ce on the Plain, between the Plains pcr-
fndiculer, and the Meridian, ( being to be
JdEaftwatds, as the Dial-draught fhew-
3 befides tlut general Rule before hinted,
f^twhenfoever a Plain declines Eaftward,
Subftile Line muft ftand Weftward, and
jj^ contrary; for the Arch whereon to prick
J^f Subftile,and Stile is alwayes to be drawn
that fide of the Plain, which is contrary
^ the coaft of declination) E F the diftance
the Subftile and Perpendiculer, to be
jj.'Q.the fame way ; G F the diftance on the
u ^Jn, from the Subftile to the Meridian, to

are found by thefe Canons Arith-
j.^tically, or by the Artificial and Natural
^es and Tangents.

To find the Diflaftce of i2j from the
Perpendiculer E G, or Heriz.on A G, hy
^ke fecend Axiome of (JUr. Gellibrand,
Viz. that the Sines of the B^fe and Tan-
gent of the Perpendiculer are proportio-
nal.

As fine 90 Radius, ZD
ToTang.ofIleclin.Plain ND
So fine of Reclin. Plain Z E 20-0O
To Tang. of Perp. Sci2 E G 1J
Wbofe complemencnbsp;AC

As — Tangent of N D Jf- o
To = Sine of Z D 9 o- o
So = Sine of Z E 20— o
To — Tangent of EG 13-28, tj^
diftance ciquot; i a,from the Perpendictile'^'

2. Tç the Vißance o» the MtrW^^quot;'
from the Pole to the Plaïn, P G, bj ^J'
3 Propofitions of Mr. Gellibrand, tquot;
Sines of the Sties are profortional tot
Sines of their ofpoßte Angles.

pendiculer from 12 ^ G B
^To the fine of dedinacion GZE ^
quot; Msthé fineof90 G EZ po-0
'he fine of che diftance7nbsp;.

« Which being taken from 38-28, the di-
«nce on the Meridian from the Pole to the

Ijenith, leaveth the diftance on the Meri-
2'an of the place, from the Pole to the
-'ain, viz,. 14-3 3, as a help to get the next,

hovt the Plain P F.
In the two Triangles Z G E, and P G F,
^hich are vertical^ by the fécond Conféra-
fy of Mr. geUtbrand j If two Perpendicu-
er Arks fubtend equal Angles, on each fide
the meeting, then the Sines of their Ihiy-
Pothenufaes, and Perpendiculcrs are pro-
portional, ( and the contrary ) j for the
Angles Z G E, and P G F are eaual Ar.gled
andZE,andP F, are both two per-
Pendiculer Arks on the Plain A B. ;
Therefore,

the fine of the Perpendiculer Z E j
So is the fine of the Hypothenufa P G, toi
the fine of the Perpendiculcr PF, and
the contrary.

To fine of the Reclination Zt 20 oo
So is the fine of the ArchO
on the Meridian, fromgt; PG
the Pole to the Plain j
To fine of the Stiles height PF si'^S

In the fame Vertical Triangle, having thf
fame acute Angle atthe Bafe, theTange«'«
of the Perpendiculers, are proportional
the Sines of the Bafe, by the fécond Ax'toi^
of

As.— fine of GÈ 13-28
To =:Tang. ofZE 20- o
So=TangiofPF 12-13!
To—fine of FG 7-58

j. T»find the Angle between tbi tm Me^
rtdiänsy of the Place and Plain j\a. thé
PF G.

, .Êy the third Propofition oïUt.GtÜlbrani,
quot; « proved j That the Sines of the Sides are
J'ofgt;ortionaI to the Sines of their oppofice,
^quot;gles, and the eontrary^

the fine of the Dift. on the7
- Merid. from the Pole to PlainjPG 14-3 Î
çO the S. of 90, the opp. Angle PFG po-00
^ is the S. of the Subft. fr. 12, F G 07-58
the S. of the Inclirt.Merid. F P G 5 3-2 8
Fanbsp;M

To •— fine of the Angle 33-28 F P quot;
The Angle between the 2 Meridia''«'

By Angle of Inclinations of Meridia^^»
make th^Tahlt of she Hour-Angles at t^^
Pole, by theDirea:ions,a«p.2. which be^
ing made as in the Ta^e, draw the Dial f
this manner j

'Upon A B, theHoriz-O'^
tal-line of your Plain, d^
fcribe the femi-circle AE^'
and from the Perpendicquot;'
ler-line CEof thePlaquot;^?
lay off 13-28 Eaftwar'l
for the 12 a clock Line,
the Plain, or the comply'
ment thereof 7^-32, frquot;^
the Eafl.end at Bto , ^
draw the Line C • ^^
ASet further Eal|'
ward from 12,
diftance of the Subftquot;
from 12, to F, anddra^^^
Che Line CF for the Suf'
ttftc; and beyond tba^

et off from F 12-15,the Stiles height above
Sublfile to G, and draw C G alfo.
The», .

. Draw a contingent Line perpendiculer to
Subfl-ile C F, as far from the Center as
you can, as the Line Fi I; then take the
^eareit diftance from the point F, to the Line
^G, and make it a = Tangent of ; then
^he Selt;S:or being fo fet, take out the = Tan-
B^nts of all the Hour-Arks in the Tahle, and

^y them both wayes from F toward H and
'as they proceed ^ then Lines drawn from
y Center C, and thofe Points fliallbethe
'^ouis required.

Bm ftoU, That this Dial is better to be
jiugmented by the loling the Hours ot ®
and 9 in the morning, which makes the
Hours more apparent, as you fee.

yilfoy theRequifites formerly found, may
Geometrically be found by the Scheam, be-
ing large and truly drawn , as before
Ihewed inthe other Dials, Thta,nbsp;,

X. A Rule laid from Q, the Pole-point o«
the Plain, to G the Point of li on the Plan'»
gives in the I.imb the point 12 ; D
J3-28, is the diftance of 12 a clock-line dj
the Plain from the Plains perpendiculequot;^^
line Z. D, ( and to be laid from the perpe^^
diculer-line on the Plain Eaftwards inth®
Dial); and the diftance on the Limb frquot;^
•A to 12, is the Meridians diftance froni th^
Eaft-end of the Horifontal-line on th?
Plain, namely 76-32.nbsp;.

gives the Point Sub, for the S^bftlle ;
the Ark Sub. 12,7-y S, is the diftance fro®
or the Ark Sub. D 21-26, the diftanlt;^^
fropi the Perpendiculer.

3.nbsp;A Rule laid fromQjo 6, the p'^^®
where the 6 a clock hour-line on the Sche^,
cuts the Plain, gives on the Limb the Po'quot;
l^,theArk6 13,25-38, or6D,38-Jo»'
Jhe diftance of the Pgur^Jine of 6 o^n JJJJ

^lain, fromthcHour-linc 12, ofthePer-
pendiculer.
4. A Rule laid from V, the Pole-pomt
the Circle QF P, toP 8c F, on the limb,
Sives two points IK, and the Ark IK is the
Stiles Elevation 12-13,

f. A Rule laid from P to Y on the limb,
gibes the Point M ; E M is the Inclination
of Meridians : or, a Rule laid from P,to the
interfedion of the Circle PFQ , and the
EquinoaiaHine, gives a Pomt ui the Limb
«ear C, which Ark C S, is more naturally
tbe An^le between the two Meridians,

If you like the way of referring this
Plain to a new Latitude, and to a new De-
clination in that new Latitude,

A Rule laid fiomE,toPand G, in
tne Limb gives L and O ; the Ark L O is
the complement of the new Latitude, being
Ark P G, the fécond requifite, in the
former Calculation being 14-33» the di-
ftance on the Meridian from the Pole to the
Plain.

7. A Rule laid from G to Qj5n the limb)
gives R, the ArkS R is the new decUnation

So is the fine of the old Declination,
the fine of the new, in this Examflh
being 32.37, and generally the fa®®
way as the old Declination is.

That when the North-pole is Elevate'^
on South Recliners, you muft draw theiP
as North-decliners; and North-weft an^^
North-eaft incliners, that have the South-
pole Elevated, you muft draw them fj
South-eaft and Weft-decliners, which will
direft as to the right way of placing tb«

In this place I fhall alfo infert the general
way, by Calculation, to find the new Laci-
t|jdc, as well as new Declination;

of the Plains old Declination ;
So is the Co-tangent of the Reclination,
orlnciin. to the Tang, of a 4th Ark.
Thf»,

In South Recliners, and in North Ineli-
get the difference between this 4th
and the Latitude of your place, and
copiplement of that difference is the
Latitude : if the 4th Ark be lefs then
old Latitude, then the contrary Pole is
~'evated; but if it be equal to thé old Lati-
^de, it is a Polar-plain.

But in South Incliners, and inMorth
Recliners, the difference between the 4th
and the complement of the Latitude
the place ( or old Latitude ) fhall be the
few Latitude, when the 4th Ark and old
®titude is equal, it is an Equinoéhal-pl^in.

Ï4-31, the complement of the new Lati-
tude, which will be found to be 7^-2?, tquot;®
new Latitude.

By which new Latitude, andnew Dech'
nation, if you work as for an Ereft Dial,yoquot;
ihall find the fame Requifices, as by the iot'
mer Operations you have done ; and the
diilance of the Perpendiculer and Meridia^^
wiU fet all right.

degrees Weftwardj and reclines 33-3 »
Now, to try whether fuch aPlainbejuft®
Polar-plain or no, ufe this Proportion :

As the fine of 90 . DA po- o
To Co-fine of Declin. NA yf- cgt;
So Co-tang, of Reclin. DE y 6-5 7
To Tang, of Latitude N.ï* fi-ja

Which4thArk, if it hittoberightthc
ï-atitude, then it is 9 deglining PoJar-plainj
°relfenoï.

If you have a Declination given, t6
^vhich you would find a Reclinaüon
^ake itPoIar, then reafpnthus.-

By tht StElor ;
As tbe Co.fine of theOeclin. A N 5 y- ö
To the Radius or Sine of AD 90- o
So is the Tang. of the Lat. P N f 1-32
To the Co-tang,of the Reclin. D E i

^ut hjthe Scheam, thefe three Operatiquot;quot;^
, arefonaihjäxavcingthe Seheam.

1.nbsp;For if the Line or Circle, reprefeiit-
4tig the Plain, cut the Pole P, it is a Fol»f'
Vial.

2.nbsp;If AB, rhe Co.declination, be given?
then draw tli^ Circle A P B, and it gives E j
thenZE is çhe Reclination, meafured b/
half Ta.ngents ; or a Rule laid from A to ^
on the Limb, gives an Ark from B ; which
Tnealyred on fit Chords, is the Reclmati-
on. .

. 3. If P, the pole-point, and ZE rhs .
^«chnation, be given ; then,, with the di- •
^ance Z E, on Z as a Center, draw an Arfc -
^'aCircle.in that Quadrant which iscqn-
t^ary to the Coaft of Declination, obferving.

letters in the Scheam; then by the
^onvexity of that Ark, and the Pole-point
fj draw the Circle P E, cutting the Limb
Into two equal parts, which are the points
St B, the declination required.

This being premlfed, there are tHo thingi
tequlfue to befoMnd, before jou cm drayv
the 'Dial, viz. the Sttlflile from tht
Perpendlctiler or Horlz.mj and the In-
clination of (Meridians.

fo the Co-fine of the Lat. P Z 3 8-28
the fine of the Declination P Z E 3 j-00
10 the fine of Subftile from Perp.P E 20-54

the diftance of the Subftile from thé
Perpendiculer, whofe complement
is dïe Elevation above the Horizon.

SothefineoftheReclin* ZE
To the Co-fineof Incl. Mef. ZP E
^Whofe complementnbsp;Z P Qjii-^i

As — fine of Reclinatiort Z E 3 |
To = Co-fine of Latitude PZ
Sossfuieofponbsp;PEZ 90-0°

The Scheam being true drawn, which be-
ing one degree and 15 minuts lefs than 30,
the quantity of two hours, I fet
sainil2; and by continual addidon quot;t

jj ^Cg. to I-If, and che increafe tfiertof,
JJJ^^eup the one half of my TatU, as fol-

If I take ly, the quantity
in degrees of one Hour, out
of 28-1 y, the Inclination of
Meridians 5 there remains
15-45, fornbsp;on

ceflively to it, amp; its increafe,
till it come to 90 ^ Tbtn, to
13-45, the refiduc of 15,
taken from 28-45 5 ^S
as often as you can to 90,and ^ ■P'^
thus is the Tahlt made.

Firfli Draw a perpendicu-
lerLine on your Plain, as
C B, by crofling the Horizon-
tal-line at Right Angles;

[ ] . rf
tlieii ffotti the perpendidJer^L-iiie by ^
from the upper-end, toward the left-ha«
(as the Scheam direfts, ZD being tbele'J
pendiculer, and Z N the Meridian, a»o
E P on the Plain, the diftance between, bej
ing toward the left hand ) 20-y4, .
Subftile-line, as CD; then on that
(any where) draw two perpendiculer
quite through the Plain, crolTing the Su '
ftile at right Angles, for two ]^uinolt;3i2'
lines, as EF,amp;GH.nbsp;.

Then confider what hours fhall be put
your Plain, as here is convenient, from ^^
in the morning, to 6 afternoon; ( th^'^A
the Sun may ftiine on it from 8 to 7, °
then the Lines Will be too clofe togecH^^
and the Radius too fmall). And alfo wbe^
you would have thofe two utmoft hours
be, as at E^nd F on the upper Equinocfti^ -
line; or, at.Gamp;H on the lowtr contnV
gsnt-line.

TMtiox 10, and add it to the ■—- Ta jj
gent of di-if for lt;J, on the Line of fquot;^^^
Tangents, and you fliall fee the Compa''
point to reach to the Tangent of 73-f ^ m
fum of both the two extreann hours, .w^
happening under will be fonvcn^^

Take the whole diilance E F, or G H, and
^ake it a = Tangent of . then the
lettor fo fet, take out the = Tangent of
, and lay it from the point E to 1, oa
J^e Equino6lial-linc ; Alfo, take out the
■^.Tangent 6i-if, and lay it from the
F ; and if your work be true, it muft
^^eds meet in the point I; then draw the
Jr'ne ik for the true Subftile, and from
hence lay the = Tangent of 45, to draw a
near y, for the Stiles Elevation, paral-
the Subftile; for being aPolar-
it hath no Elevation, bu' what you
P safe to augment it to, as herefrom I to L.
The»,

As the SeéVor ftands, prick on all the
hole hours, halfs, and quartcis, according
0 the Numbers m your Tthle, at leaft thofe
be above 45 i and for thofe under 45,
^ alee r= Tangent of 45 m fmall Tangents,
^ Tangent of 45 in the great Tauge ts,
then the Seéfor ftiall be fet to that Ra-
lüs, which is moft convenient for your ufe.

The next and laft kind oi South Reclli^^'
«re fnch as recline, or fall from you bclo^
the Pole, vlt. have their Plains lying betwe«quot;
the Pole and the Horizon, as by the ScW^
is mcrre apparent.

In which wórk, the drawing the Schcanj»
and thé things required, are the fame as'
the firft Example) as the Figure, and
lowing words, do make make manifeft.

The Example'^ers,- is of a Plain that
dines from the Sooth toward the Weft
dt^rees, and reclines upon its proper A^'
muth^ E, 60 de2;rees from the Z-enith. ^
- I. • Having drawn the Scheam, thent»f
fbr'the diftance of the Meridian from tP
Pcrpcndiculer, or Horizon.

To die Tangent of Declinadon N D 35'quot;
So the fine of Reclniation Z E 60-0^
To the Tang, of Perp.amp; Mcrid. E G 31-»^

Whofe complement is 58.48 AG, the
''Utance between the Weft-end of the Hori
^°atal-line, and the Meridian.

A Rule laid from Qjo G, cuts the limb
^} then D L, and A L, are the Arks re-
H^'red i D L from Perpendiculcr, and A L
Horizon.

A Rule laid fromE, to P and G, gJVJ
on the limb 2 Points, whofe diftance
twecn,is 2d-i3, the Ark required.

As — fine of GP itf-if'
To-fine of GA 58-4»
So =: fine of A N .55-00
To — fine of PF 25-02

Being found by the Scheam,
Rule from Y, to P ^ndF, on the
gives the diftance between being 2 j-02,
Stiles Elevation,

Bj the SeElar.
Co-tang, of the Declin. AN yj-oo
fo S.dift.onMer.fr.Pl.toHor.NG
Tang, of the Stiles height quot; ~

— Co-tang.ofDeclin.PlainAN fy-00
=:S.dift.on Mer.fr.Pl. to Hor.NG 25.19
Y® ^ Tang.of the Stiles heigh: P F 25-02
^ 0 = S. of the Subftile from 12 F G 08-03

elfe, the A.k M D,'is the diftance of the
quot;Oltile from the Perpendiculer 23-19.

^ the fine of the diftance. on p
y Mer. from Pole to Plain T
^ 0 the fine of the Angle G F P p 0-00
l°thefineofdift.of$ub.Fi-.iz GF 08-03
the fincof the Incl.of Mer. GPP 18-27

the Inclination of Meridians, by help ^
which, to make the Tahlt »f Hem-Arlu ^
the Pole, as before is fhewed, and as in ^^
Table following.

Pirftjfor the Affection s,confult tlie Scheam,
^'fherein, laying the Perpendiculer-linc C D
'■'ght before you, you fee that the Subftile,
^nd the Meridian, are to be laidfromthe
Perpendiculer toward the left- hand, the
^ibftile lying between the Perpendiculer
the Meridian, and the Stile or Cock of
^'^e Dial muft look upwards, the North-
Pole being Elevated above this Plain, which
^ill guide all the reft.

Firft, draw the Horizontal-line A B, and
C as a Center raife a Perpendiculer, and
K °ff by Chords, Sines, or Tangents, the
J^eridian or 12 a clock Line, the Sub-
^'le, and Stile, asexa6fly as you may; and
J'^aw the Lines 12 C, Subftile C,. and
Stile C.nbsp;■ . ,.

. As far from the Center C,. as youccmve-
quot;lently may, draw a long Line perpendicu-
to the Subftile, as the Line E H F; tlien
lifting one Point of a pair of Compa/Tes in
open the other till it tough the Stile-line
^^ 'he neareft diftance.

Make this diftance a — Tangent of 4^1
and take out the ^ Tangents ofevery who'^
Hour, as in the Table, aUar as the Tanged
of 76 wiU give leave; and then from t»®
Center C, to thofe Points draw Lines forthf^
even whole Hours j then to any one whquot;'^
Hour, as fuppofe the Hour-line of
draw two = Lines equally diftant on bot»^
fides the Line of 3, as IK, L M,
Then,

Count any way 3 hours, and 6 hoUi
from 3, as here 12, and p, lo as the =1quot;^
■may crofs the 3 remoteft hours, as here yf^
fee p and 12 a clock Hour-hnes do crofs tn
= line at I and K ; then take the diftan^^
IK, and lay on theHour-lmeof
to N, and draw IN L = to p C ; Wln'^J
' Work doth conftitute the Parallellograquot;^
KILM.

Then Uftly,
Make KI, andNI,= Tangents of
and ptickoff every hour, half, and quarts,
(aifd minut if you pleafe) on the two Lir»^'^
I K, and IL, from K and N both wayes, ^
before is already fhewed w the Erecl V '

- That to fupply the defedl on the other
pjte, when the point M falls out of the
J, «in, the diftance from I,to the Hour-point
'•■om 1virill reach from L to and from I
lo, from L to 8. This is general in all
^lals.

» -dlfo note. If you like not to lay off the
^tft Hours by the Tangents, having made
Jh^ Tui/e, as before, you may foon find the
|^°tir-Arks on the Plain for 3 Hours, as
quot;^quot;■e 12, and 9 ; Or, 4, i, and 8, which
J^'ould have made the Parallellogram more
and confequently more better, and
to draw the reft by the Sedor. Thus
may fee how your Work accords; The
^ay by the Table and Contingent-line, and
Way by the Seftor on the Parallellogram,
bv Calculation, amp; at laft ufe the Myftery
Dialling made plain and ready, to an

.The other kind, viz.. North Declining
^eclinersjhave alfo three Varieties; as thofe,
Tliat fall back or rtdine between the Ze-
nith

nith and Equino6^ial: ad. Thofe that ^
dine to the Equinoaial : And 3d. Thoi^
that recline below the Equinoftial. A''
firft of the firft Variety, reclining lefs th«quot;
to theEquinoftial.

The drawing theScheam, is the fame a
in the former, except in the placing of tf^
Points and Letters i For firft, thefe Plaquot;^
behold the North-partof the Horizon, aquot; J
then when you look on the Plain, the So^t
is before you, and the Weft on your rig'j''
hand,and the Eaft on the left; then
South and North are alwayes oppofite, ^
the point P, reprefenting the Elevated P^'
of theplacc, which with us being Norfquot;^^
muft be placed towards N downwards,
before in South Recliners it was upwards-

It is neceffary in the Scheam, to draw
Equinodfial-lme, by laying the half Tan-
gent of J 1.32 from Z to^5 thentKe'^'
cant of 3 8-28, the complement of Z E, la'
from iE on the Line S N, ftiall be the Cequot;'
ter to draw EiEW for the Equinoitia''

fine 90 Radius
Tangent Declination Plain
^ fine Reclination Plain
^ 0 Tangent Merid. amp; Perpend.

.Whofe complement AG, «Sj-yS, is the
^^fidians Elevation above the Eaft-end of
Horizon.

Ky the Scheam, A Rule laid from Q^to G,
the Limb gives L ; then D L and A L arc
Arks required.

By tht StÜer.
As fine dcclin. of the Plain G Z E
To fine dift. of Mer. amp; Perp. G E
So fine of the Radiusnbsp;GEZ 9°-°°

Which added to 38-28 ZP, makes »^P
G P co be 70-31. Or, By the Scheam, Ç
Rule laid from E, to P and G, gives ofl tquot;®
limbrf^j the Ark is 70-31.

from Zenith to the Plain \
To fine of the plains Reclin. Z E 20-0^
So fine cf dift. onMer. fromquot;^ GP

fothe = fine GZ 32-0?
^fhe=fine Z E 20-00
w -Tothe— fins FF 37.01.
the Scheam.
A Rulelaidfrom Y, toPandF, on fhc
^itnb gives c and dj the Stiles height.

4. To find iht difiante oftbt Suhfltlefrem
tht Meridian G F ; when it is abovt
90 deg. takj the comp. to 108 deg»

Tangent of the Reclin. Z E 20.00
Tofine of dift.of 12 from Perp, G E 26-02
So Tang. of the, Stiles Elevât. FF 37-01
To fine of the Subftile from 12 G F 65.24

Bj the Scheam.
A Rulelaid from Qjro G and F, gives on
^^e limb L F, the Ark required,

To fine Radius oppofite Angle GFP 90-
So fine dift. on Plain from 12 7 G F tff-^-f

A Rule laid from P to Y, on the liifif'
gives g, the Ark E g is 74-3 8, the Inchnati'
on of Meridians.

A Rule laid from P to K, gives h, S b J'
the Inclination of Meridians, by which to
make the Table as before- isfliewed, and a«
tblloweth.

For drawing the Dial, COit-
fult with the Scheam, laying
the Plain AEB, an d his
Perpendiculer C D right be-
fore you ; then note, S N ij
the Meridian-line, ZE the
Plains perpendiculer, with
the Meridian G on the left-
hand, and the Subftile F on
the right-hand.

o -^eams, and confequently the fliadow of
Q® Stile into the North ; So that though
oe the true Meridian found, yet it is the
°rth.part that is drawn as an Hour-line ;
H the Subftile, and other Hours, are coun-
a from the South-end thereof, as the Table
^»^d the Fi^re of the Dial, do plainly make

^nifeft J being drawn in this manner,
^rft, draw the Horizontal-line A Bjthen
^^ as a Center, draw a (emi-circle equal
of the Chords, and lay offquot; the Meri-
pn, Subftile, and Stile, in their right Sci-
^ations, as laft was dedarcd ; then draw
^quot;ofe lines, and to the Subftile ereft a Per-
Pendicuier^ as D E j then take the Extent,

Perpendiculer or Contingent-Lme '
drawn, cuts 12 and the Stile-line, and nws
it a = Tangent of 45 ; Then is the Sector
fet, to lay off all the Hours by the = A ^
gents of the Arks in thcTahlt, except
and 10, which do excuf.

2, 3 ; and draw them through the C^quot;'
tcr, on the other fide, they ftiall be tje
Hours of 12,1, 2, 3,4, amp;c. on the Mortn-
part of the Plain, where they are only ufeo'
As for the Hours of 10 and 11, do thui i.

Draw a Line = to any one Hour, wb»*^
= line may conveniently cut thofe Ho^ '
lines.

from p to 12, or from d to p, in that Ljquot;
laft drawn, a = Tangent of 45, and W
off hours and quarters, or elfe the who
Hours, by the diftances from 9 to 7, and^
for 10 and II, turning the Compaflcs tn
other way from 9 ; then to all thofe Pom
Lines drawn, fhall be the Hour-Unes r«^
quired.

the bare drawing of theScheam, yo«
^ that the Circle A E B, reprefenting the
in Plain, doth cut the Meridian juft
nic Equmoaial 5 Now to try by Arith-
quot;ick, whether it be a juft Equino^ial-
^'quot;ijornojfay:

As theCo-fine of tquot;he Dedin. A S
To the Co-tang. of the Lat. S G
So is the fine of ponbsp;AD, ^

But en the contrary, ifibt %tclindtiofi
in given, and a Declination rr^mredf to
make an £lt;^»inoElial PUin ; Then con*
^trarilj fa} thus.

Bythe SeQor.
Co-tang, of the Reclin.
To fine of 90
Co-tang, of the Latitude
Co-fine of theDeclin.

By the Point G, artd the touch of an Arch
JooutE, draw the Circle GE, to cut the
lt;tnb luto two equal parts, and you have the
^ Onus A B.

4. The Plainthtu made^ or proved to bi
EtfUvtoUial . to find 'he ^.Meridians
Elevation above the tioriz^on, A G •
Or, hit Difianct from the Perpendicu-
ler EG.

To fine of the Reclination
So fine of dift.Mer.Pole to Plain G P 9°' ,
To fine of the Stiles Elevation P F

Bj the Scheam.
. A Rule laid from Y to F on the limb
gives C, N C is 48-24, the Stiles height. '

. The diftance of the Subftile from 12,'
thefe Equinodlial Dials, is alwayes jgt; ^ de-
grees ; for a Rule laid from Q., the Pjle of
Plain, to G, on the limb gives (,; a Rule
J/fo laid from Q^ to F, the Subftile, on the
iixih gives i; the Ark ^ lt;i, is 90 degrees,
for the diftance of the Subftile from
and alfo for the Inclination of Meridi-
for the Subftile ftands on the hour of
S being part of the Circle E P W, which is
f^^ehourof 90 degrees diftant from the
«our of 12.

A Rule laid, as before, from Y to P, on
hmb, gives N ; the Ark E N, or W N,
PO, for the Inclination of Meridians.
Which being juft 90, the r«^/« is eafily
^^de, vii. ij, 30; 4J, do; 75,90;
^^ice repeated, from 12 to (J both way s.

On the Horizontd-line AB, draw an ob-
flt;;ure Semi.circle, and fet off the Meridiat^
as the Scheam fheweth, viz.. 5 o degrees
mm. above theEafV-end of the Honzontai^
line; but makevifible only theNorth-e»
thereof, as the line C 12 ; Then» po degt^
from thence, toward the right-hand, as ^
Schcam fheweth, when the Perpendicuie ^
line is right before you, draw a Line ti^
ferves both for 6 and the Subftile, as C '
Alfo, lay off the Chord of 3(J-47 from 6 ^
5), and draw the Line Cp alfo, which
found by Calculation, as before is flieweO'

Draw a Line = to 12, or Perpendicu^
to being in this Dial all one, asthe Ui
PEG; then fetting one Point in E the b«^
ftile, take the neareft diftance to the Sen '
line, and it ftiall reach from E to G, '
Point for p.nbsp;«

The fame diftance E G lay alfo on the
12, from C to H,and draw thehneG H^j
then make E G a =: Tangent of 4r,

Make the diftance of H G a Tangent
45, and lay the fame = Tangents both
^ayes from H, and to thofe Points draw the
quot;OUr-hnes required.

This third and laft fort T^orth-Recite
are thofe that recline beyond the E-
^quot;iinoftial, that is, he between the Equir
and the Horizon; and it differs
'omewhat from tlie other hve before, in the
^'^heam and Operation alfo.

For firft, the Ark of the Plain is extend-
below the Horizon, till it meet with the
quot;^orth-part of the Meridian below the Ho-
'quot;iion at H ; and the Center of the Ark
AQB, is in the Line ZD, as much di-
ftance from Q^, as the Secant of deg.

the Radius of the Scheam, being the
complement of Z Q_2j-o ; Here alfo the
»aine requifites are to be found as in the

A Rule I^id from Q^to G, gives ontbe
limb ai ; thenD a is the diftance fromtf^^
Perpendiculer f 2-18 ; and A a the diftaquot;quot;:®
ffom the Horizon 37-42.

Co-tang, of Reclin. Plain
Co-fine of Declin. Plain
Co. tang, dift.on Merid.quot;)
'rom Plain to Zenith J

^ Whofe complement 75-02 G Z, added to
^ »the complement of the Latitude, makes.

for the diftance of theNorth-pole
^ on theMeridian of the place, from (the
;Vorth-pole P to) the Plain hdow the Equa-
r at G ; which being more than po, find
complement thereof to 180, viz,. 66-30,
^ «»ng the diftance on the Meridian from P
jj^ePole, to the Plain on the North-partof
j^? Meridian, viz,. P H, found on the
quot;leam, by laying a Rule from E or W, to

Zenith and Plain ƒ
To the fine of the Reclin. Plain Z E df-^o
So the fine dift. Mer. from ? P H 66-f

A Rule laid from Y, to P and F on th«
limb, gives «i and r; the Ark de is
the Stiles Elevation,

4. Tofind the Stibftile from I2y viz. F G
framtht South part, or H Ffromtbe
North part.

Bjf the Seüor.
Tang. Reclin, of the Plain ZE (îy-co
Jo Co-fine dift.Mcr. amp; Horiz. EG yz-iS
Tang. of the Stiles Elevât. P F 5-9 se
fo fine of the Subftile fromquot;) F H 38-30
North part Merid. ƒ

y A Rule laid from Q, to H and F, on tha
hmb, gives ƒ and e ; the Ark ƒ and is the
Subftiles diftance on the Plain from I2.

By the SeBor^
fine dift.Mer.fr.Pole to Plain P H lt;?f-oo
fo fine of 90 Radius PFH90-00
Jofiae of dift.from Subft.amp; J2 FN 38-30
^0 fine of Inclin. M«rid. F P N 42-4 j

A Rule laid from P to Y, on the Ii«»quot;
gives then VJ g is the Angle of the Ijj'
clination of Meridians, vix., 42-45' ; '
which make the Table, as is feveral tii»^
before (hewed, and as followeth.

On the Horizontal'
line A B, defcribe J
Semi.circle, and
oft' the Meridians W
vation, in its prop^
place, as the Sche»^
directs; and then
Subftile, and Stile,
yond the PerpendiC^^
ler

12	42-4j
i
2	72-4f
3	87-45
4	77-1 y
5	62-1^
6	47-iy42-yy
?	32-iy
8	i7-if
9	02-15- i.ylt;5
10
ii	2-iy
12	4^-45

as by laying
Perpendiculer-Li»equot;

J®«; the Line Z F, reprefents the Subftile
the Horizon: which things being ob-
.^ved, and the method thereof underftood,
^ ^iii prove better than dry Rules, or Pre-
'^epts, without fuch reprefentations, and far
ï^ore eafie to apprehend and conceive in the
''^agination.

, Then prick down the Hours of 9 amp; (f,
p the Table on thé Contingent-line, as be-
or on the Serai-circle, having Calcu-
'^ted only thofe two Hour-lines, by the ge-
neral Canon.

Draw a Line = to 12, at any convenient
^fftance from it, as G H; Then, take the
^'iftance between 6 and 9 in that = Line,
lay it from the Center to I, on the 12 a
JJock Hour-line, and draw theLineHl;
then make the diftancesGH, and J H, fe-
^crally one after another, = Tangents of
45quot; j and take out the =± Tangents of 45,
30, If; and lay them both wayes from 12
and 6, on thofe two Lines, as hath been of-
Jn fhewed, m the former Dials; then lines
drawn to thofe Points, fliall be the Hours
Required.

Ncllning flaws are but the under Faces of
i quot;k^c/jwj, beholding the Nadir, attn^
lame Angle that the Recliners behold
Zenith.

And the raafcing of them differs nothi''^
from the Recliners already mentioned;
all the Requilites, as Meridians Elevation»
Subftiles diftance from 12, and the Stiles
levation, amp; the Hour diftances, are the faflquot;^
both in the Incliner, as they were in the
cliner, and have the fame Numbers allb fet
the Hour-lines j So that in drawing of thd^
Dials that have Centers, if you draw aU

as thus, in the Example of the declining
quinodial, being a North declining Eaft 5'
degrees, and recTining jf-fo. If you dr^
the 24 hours, being done at the fame tini

flrokç, by drawing the Hour-hnes
^Jroiigh the Center on the other-fide, and
.quot;^Subftik and Stile alfo, as here you fee
Dial annexed j being the Equino6\ial-
i^^al, belonging to the fécond Variety of
^orth Recliners declining Eaftward 5 y,and
^^clining Jf-yo^ the Lines drawn through
Center, and complemented to 12, is a
°Uth-€afl: declining, and Inclining 3 y-jo,
Alh

, -ir you turn the Paper, and look againft
and then the North-caif becomes
North.weft Decliner yy, and reclining
^^yo ; and the South^eall becomes a South-

Declining and Inclining as much,
v^ Thus you fee, that every Draught of a
will ferve for 4 Plains, that is for tht
I ace you draw it, and his oppofite ; and
. another Plain, declining as many degrees
2 contrary way, and reclining as much
to, and for the oppofite thereunto, as by
Draughts of the two fides, may
Kïnbsp;^PP^^r. And the like holds

As a North-eaft and a North-weft, a
^°Uth.eaft and a South-weft, dechning 30 5
Dial drawn romid about, ferves all 4
. ; But note, that no South Ereéf or Li-
'ningDial,. can have the Sun to fhine on

any Hour^line that falls above the Honzott-
tal-line 5 and thofe hours on the North^re-
cliners, that fall below the Horizontal-hn«»
belong alfo to the South Dials.

But for a plain general Rule, to know
what hours belbng to any Plain what»quot;'
ever in any Latitude, do thus.

Firft, draw a genewl Scheam to y®^
Latitude, as this is done for fi-jaj^^gj
mark the 4 Cardinal-points with E-W-N-^
S. as is ufual for fetring the Scheam right D
fore you.

For all Declinirig Upright Dials, draW.oJ^
ly a ftreight Line for the Plain, Perpendi'^jj
ler to the Line that doth reprefent theP^^^
of the Plain, counting lo many degre^^
the Declination of the Plain fhaU haprj,
to be from S. or N. toward E. or W.
all the Hour-lines, of the Scheam that t^
Line of the Plain lhall interfea, are tP
Hour-lines proper to that Plain.

the Line E and W, being PerpendicU^
to S and N, the Poles of a South and No«!^^^

P|aw; doth therefore reprefent a South^'
plain on one fide, and a North-plain on the
^ther.

If you conceive the Sun to be in C^w^f^
®quot;d gomg of his Diurnal Motion, at his
filing about a quarter before 4, beholds the
^^oxth-fideof the Line EW, and continu-
^fn fo to do till 25 minuts after and
hen it fhines on the South-plain till 3 y
J^inuts after 4, and iheil begins again to
Ijne oii the North-plain, and fo continues
quot;quot;Sun fetting.

i when the Sun is in the Equinoaial,
.^^oeholds the South-plain at the Rifing, be-
at 6 a clock in the morning ; and fhines
jt all day, till Sun fet,being at 6 at night j
then the North Dial is ufelefs;

^°uth-eaft Plain; then if you count 30 de-
pj^es from S toward E, forthe Pole of the
'^'ri; and 30 degrees from W toward S,
, jj^ from E toward N, and draw that Line
jj fliall jeprefent the Plain; then yoit fliall
[ tliat the Sun being in Gatscer will be--
i

gîii to (hine on thisPiain, juft a quarter be-
fore 5 in the morning, and continue till near
half an hour after 2,nbsp;.

ftiine on it till a quarter aher 4, viz,'
Sun fet ; and all the hours after a, belong tquot;
the North-weft Plain that declines 30 de-
grees, and one hour in the morning a»'quot;'
viz,, from a quarter before, till three quarter
after 4.nbsp;..

Draw a long Line, as A B, and crofs
with a Perpendiculer in the Center C, an
lay off from C, toward A and B, theTa^^
gentof 45 ; or the Semi-tangentnbsp;'

qual to the largenefs of your Scheam ; tn«^
lay off tbe Seniii-tangent of the Reclinati^,
from C to D, up and down, botbwayeS|
then take out the Secant of the compleifl^'^j
of the Reclination, which will be a Rad»
to draw the Arks A D B, which Paper yo^
muft cut out, and apply the two Points ,
the Paper ADBD, to the two Points °
Dcclination of the Plain , noted in t'
Schcam with A and B j that is, put A to ^^

^odBtoB; then the round or convex-edge
the Paper, reprefents the reclining Plain j
^d the fame edge, on the other part next
^jie Horizon Southwards, reprefents the
5»outh-weft Incliner.

Suppofe 1 make the Paper A D B, to re-
jy-yo, the Reclination of theEqui-
•^P^tial-plaini theii, firft fet the Schcam
^'ght before you in its right fcituation, and
putting the Points A, in the Paper, on A on
he Scheam; and B in the Paper, to B on
he Scheam ; I fhall find it to be even with
reclining Circle A E B ; then following
«e Tropick of Canctr, I find thac it fhines
the North Recliner from the Rifingtill
. 2, at which time it leaves the North-
Inbsp;declining Eaftward, and begins to

quot;»ne upon the oppofite Plain, viz,, the
^ouch-weft Incliner, declinmg yy.o, and
j^echning ay-jo, and fo continues till Sun-

4L ^^^ That if the Line that reprefents
I quot;fie Plain, cuts-the Tropick twice, as the
thquot;c E ^ North-plain; then, though
V n ^^^^^nbsp;morning, it

'ill fhine on it again in the afternoon.
^ ^otcalfo. That a Norch-eall Recliner, is
^prefented by the other Convex-edge of c'ae
I 2 quot; Paper,-

Paper, as here a North-eaft Dechner pi
*ina Inclining 3 y-yo, the Sun will ftiint bquot;;
till 3 quarters alter 8 in Canctr ; but 1
Cafrkorn it fliines till half an hour after 9'
andccmesno more on it that day : A«
note alwajts. That when it leaves any Plaiquot;^
that then it begins to ftiinc on his oppoW^^
as here the oppofite to this North-eaft A'»'
eliner, is the South-weft Recliner, being
prefentedby the fame Line or Circle A D
that the North Recliner was : Only,
muft count that fide of the Line next to tquot;
Horizon, the Inclining-plain; and that nO
next the Zenith, the Reclining-plain 5 poj'
the Line that reprefents it, having no bredt
can bc no othcrwife diftinguifhed,
you will malie a material, Armilary Sph^^ ' '
of Paftboard or Brafs, as the follo^^'J
Difcourfe doth plainly demonftrate, in tn
feveral Operations, for the better conceiV»»^
of thefe Mathematical Excercitations.

Thus you have the way of maki«^
manner of Sun Vials, upon any plain
perficies, the Axis of the World being ^^^ ;
fuppofed Stile to all thefe Plains; As ^
thofe curiofities of Upright Stiles, and
tical Dials, and drawing of Dials by .
Horizontal, or Equinoaial Dials, you U
them in the Works of Ui-Samutltf j

^nd others ; and m Kirk.ers Ars magna,
But I intended not i Volumn of Sha-
Jf ws, but only a further improvraent o£
Je Tr,angaler. Quadrant, as you will fee in
nenext Chaper, of drawing the Furniture
Ornament^ of Dials; which being but
^Idom ufed, I fliall here crave an ApoJo^^y
the brevity therein, fearing,' left chat to
young Praclitioner it may feem fome-
^«at hard to conceive, though to the exer-
quot;^'^ed in thefe matters it may be plain e-

^ Then fora Conclufion, you fhall h?ve an
p^'ie Mechaniek way, to draw a Dial on the
eilingof a Room, that heth Flat or Hori-

To fur nip? any pw/, mh the ufnal
Mathematical Ornaments hythe
Trianguler-^adrant, (ts Faral-
lels of the Suns Declination J of
the Suns place, or length of the
Vay^ to find the Hm:^ontal and
Virtkal Lines ^and Points p draP
the J;^muths^ and yilmicanters;
the fewifh^Italian ^abylonijh
Hours, and U Houfes on anji
Tlain before mentioned.

J, To draw theTrofickji or ParaBels of
the Suns Declination, or the length quot;f
the Day Artificial, on any Dial.

THe fpeedieft and cafieft way for this, is
■ by the Trjgon, or by the TrianguUr'
Quadrant^ (which is a proper TrigonJ the

Axis or Stile being firll fet into the Dial;
For, if you apply the Moveable-Ieg to the
, Axis, and extend a Thred of a competent
length, fafiened in the Center of the Tri-
atjguler-Quadrant, and caufe it to cut the
Suns place in the Line of the Suns place, or
the Line of the Suns Rifing, at an even,
whole, or half hour, in the Hour-line for
the length of the Day ; or the degree of
Decimation in the degrees; and at the fame
time the Notch, or place, where you intend
the Nodus or Apex fhall be : Then, I fay,
if the Thred fliall be extended to touch th§
Plain, and the Superficies of the Trianguler^
Quadrant, and any one of the fore-mentio-
ned Points on the Rule, and the place of
^he Nodus, you may find Points as many as
you pleafe on both fidrs of the Subftile, by
which Points, by the help of a thin Rule
that will bend to them, you may draw
thefe Parallels of the izSi^s, length of
the Day , or Suns Declination required j
one of which Lines, viz.. the Equinoaialr
line, or T and rii^ on the Signs; or the Line
for 12 hours, the length of the Day, will
he a flreight Line.

But nete. That if it be a Perpendiculer
Stile, whofe upper Point, or Apex, is to be
the Nodus to'^give the Shadow j then you

muft ftiain a Thred very hard, or apply a
Rule for the prefent whereon to reft the Mo-
ving-leg on, inftead of the Axis; or, elle
you may do it thus, as Mt.punter Iheweth-

Firft, ttmake theTrjgon, if the Rule or
Quadrant frtve too Urge for jour ftn*^
Vial.

On ailieet of Paftboard, or Slate, draw'
a long ftreight Line, as A B ; to which
Line ereâ: two Perpendiculcrs, one at the
upper, and the other at the lower end,

C D, and E F ; then make A B a Tangent
cf 4 y degrees, (then having firft made theie
httle tables that follow, by the Trian'
guler- Quadrant, which is oidy the Suns De-
chnation, at his entrance into the whole
Signs, or at an even half-hour of Rifing ) j
lay of both wayes from B, the Tangents o»
the Suns declination at t bquot; i ffi, as in the
Table following ; and draw Lines to thefe
Points from tl^e Center A, as in the Figure
annexed ; and then fet the marks to theii'j
andthisis thenbsp;Figure I,

Then from the Center A, any way on
the Line C D, at fuch a convenient diftance
as you think may fit the Plain, fee olFthe
Point G; then making G A Radius of
4y Tang.let off on A B from A,the Tang.of
the Stiles Elevation to F, and draw the
Line F G, as an obfcure Line.

Then come to the Dial Plain, and mea-
fnre from the Center to the place on the
Subfiile-line, where you would have your
remoteft Line of the lign 5o or vjquot; to pals i
and take this dillance between your Com-
pafl'es, and carry it'in, above, or below the
Line F G, firft drawn and produced to
or y?, till you find one Point to ftay in A
and the other in A G, fo as to draw a Liquot;»^
— to F G firft drawn ; if that doth not fif»
then dele F G, and draw this = to it in rti
ftead, to fit and fill the Plain with the Tro-
picks to your mind, to make them large and
yet convenient.

Then note, The point G reprefents the
Center of the Dial; AG is the length ot
the Stile from the Center to the Nodus; a.
Perpendiculer let fall from Ato F G, fhewS
the point ff ; G H is the meafure on the
Subltile-line on the Plain from the Center to
the Horizontal-line, H A is the Perpendicu-
ler height of the Stile; A the Apex or top
of the Stile or Nodus to give the fliadow.

Draw a Line from 0,/=: to AB, a'
IK; and anywhere between A B, draw
L M ■= to A G ; and whcrefoever F G cuts
L M, make a mark as at M; then make L M
a = fine of o a degrees, and th: SeStor

Take the meafure from G to F, and lay it
On the Dial from the Center on the Subftile,
and draw that Line precifely Perpendiculer
to the Subftile, for the tiue Equinoctial-line
on the Plain.

The meafure from the Center of the Dial,
^ the crofling of every Hour-line, and the
«Quinoftial-iine, taken and laid from G, to
Jhe LmeABjgi ves Points to draw the Hour-
hnes on the T'jgon ; As in the Figure,
fVhertin joH may note.

That if the Subftile happens to fall on an
even, whole, or half hour, then one Line
will ferve on both fides of the Subftde ; but
if not, you muft draw as many more, and
fet Figures to them, to avoid confufion.

Then, I fay, that the feveral diftances
from G, to the crolfings of thofe Hour-lines
^aft drawn on the Trygon j and the Signs
being laid on their correfpondent Hour-lmes
the Center of the Dial, lhall give

Points in thofe Hour-lines, to draw the figns
oftheZodiack, wich a thin Rule that will
bend to thofe Hyperbolick Sections.

The famé way feivcs to draw the Paral-
lels of the length of the Day, if you lay
the diflunce from G, the croffings of the
pricked Lines and Hours on the Try go», aquot;quot;
IS as true as any other way by Calculation?
which muft afterward be performed by
tràélion in .this manner.-
' Thus you liave the way to proportion the
Height of the^ Stile, to fit the PJaiu, and the
place of the Honzontal-line in all Ert^'
Dials, which is alwayes Perpendiculer
12, and drawn through that point arct^l^

Firft, The Horizontal-plain can have
none, nor many other both Reclining ^^^
Inclining, whofe Reclination or Inclination
is above the complement of the Suns Meiquot;-'
dian Altitude in-v)», if the Stile have any
confiderable Altitude.nbsp;^^

way is thm;
The Dial being fee in (or,as in) his place,
apply the Moveing-Ieg to the top of the!
Stile amp; one corner of it to the Plain; and at
the fame time let the Thred play evenly on
6o|o, and the corner at the Plam will make
as many marks as you pleafe to draw it by.

Otherrvlfe KoUf
That wherefoever the Hour-line of 6,
and the Equino6tiaI-line domeer, there is
one Point: Then find at what hour and
minute the Sun doth Rife or Set at, in the
beginning of any other whole Sign, moft
remote from the firft Point, and that ftall be
another ; and fo as many as you pleafe to
draw that Line by : This is general for all
Plains :nbsp;'

Qjtadrant,
Lay the Thred to the Sign given, and in
the Hour-lme is the hour and minut re-
quired ; Thus the Sun being in r, rifeth
and fetteth at d, or i quarter of a minut
before, or after ; and in bquot; at juft y, and fets
7; ini at 9 minuts after 4, or fets 9
minuts before 8: The like for Winter figns.

The unequal Jewifh, or Planetary
divide the Day, be it long or fhort, into ^
equal Hours; for the drawing of wbi*^quot;»
intheEquinodlial the common hours gi^^^
Points. For the Tro picks do i ,
Divide the number of minuts in the longe«'^
and fhorteft dayes by 125 viz,, divide
the minuts in one day in s at London, by
12, the Quotient is 82 4 • and divide 4J4r
the number of minuts in one day at LoH'
don iiivy, and thé Quotient fliall be 3/IJ
then if you fallen an Index, or lay a Rule to
the Center,and to every i hour and 22's
© from 12 j and to every 37'! in -vy, ''
ftiall give Points to draw the Jewilh or Pla-
netary hours required, according to thi«
7Mhlt) thus midt (or London^ by the Line pf
Numbers; againft 12 fet 6, and the reft
order as the day proceeds, for our 12 is th«
according to the Jewes.

Extend the Com-
pafles from 16-26^
the length of the
longcft day in hours
and minuts to i, the
fame Extent fhall
reach the contrary
way from 60, to
S)8lt;J, the Number of
minuts in one day.

Which properly is one hour 22 inmu«,
the length of one hour in Cancir ; pen tnc
fécond hour,IS 2 hours 44 ' i the th»rd hour
is 4 hours and 6' A from 12 i and fo for t^
reft, as in the foregoing for Lonion.

But if you drw the Parallels, of
length of the day in the Dial, you ihall
find thefe hours to crofs the even Hour-hnes
and quarters in the Parallels for 15 and?
hours, as well as in the Equinodtial.

Firft, draw the common Hours, and the
Parallels of the Signs, or rather the lengch oi
the day ; Then note, that thefe Hour-lines
meet with the common hours in the EqU'^
noftial i only the Italians who account froiij
Sun-fetting, call our i a in the Equino6tia'
18; And the Babylonians, who reckoquot;
from the Sun-rifing, call our 12 m theE'
quino6lial 6 hours.

The Sun being in either of the trop'icks,
doth rife ins, two hours 13 minuts before
6 -, and in -vy, fo many hours arid mmuts

rom the Suns
Rifing, or the 18th from
his,lafl Setting, and foi
fucceffively 13 minutsafter
every hourr \

But, if you dfaw^the'
Parallels, of the length ; ofj
the Day, then yo\t -fhall
find the i8th hour,.ifter.
Sun Sdtting, to tcut the
Hour-line of 10 in the
P^allel of the Day, be-
ing 8 hours long, and 12
in the Parallel of i2.h'ours
long ; and the common
Houj-iine of 2 in the Pa-
rallel of 1Ö hours long,
and fo fucceffivcly for the
rctf, for fo many hours
from the laft Sun-fetting i.
For, from 6 the lafi jiight
in the Equinoéfial, to l a
this noon, is 18 hours •
Knbsp;but

but inyf, from 47' after 3 at Sun-fer, tothe
next noon, is 2ahours and 13', as mtn-
Fifurt foregoing. ,nbsp;,

for the Baigt;)M*fh.homSy\vho reckon
by. equai hours irom the Sun Rifing, ^^
forey count 2 hours and i? minuts alter
in -v^ V and 2 hours and 13' before é in © i
and;uft (5inT, and that fhall draw the
Linajof^tlic Suns rifing ; then count 3 hours
and after ,6 jn and 7 in V,
hour 13' before ^«in and that fliall be
the firft Hour after SUîtrifing, and fo
ceirivfely till night.- 'ç

Bvtt if you ufe thffPaà-al'kl of the length
the-dayj' tbc work kcafier;. for then f»
and(5j)-yiiVEheParâllielsiof ilt;î, 12, and ®
hou'rSj fhali be Points-ferthe firft frOinSquot;quot;'
rifing ., aiid6, 8, dndf«6^ ifliall ftiew rhe
fécond hour from ^un-rifihg,. and .io for-
wards,-as in the following.

lhall be the Azi-
muth or Virtical
Circles requiredj
counting theMe-
ridian, or la a
Clock Line, as
South, and the
reit in order.

T)ids, bothDi-
rc£l or. T);cli-
ners, deal with
the Declination
of the Plain, as
you did with
che Liclination
of Meridians j
and at the Meri-
dian, or 12, fee
the Plains decli-

^ I deg. 15' as often as you can; and what
fi^E laft number wants of n-i y, fee on the
°ther fideof the Subftije, and to chat add
*quot; -1 y till you have enough, as in the Table
^'inexed for a Dial, whofe declination was
Is degrees Weftwards.

Hour	Paral	Paral	Paral
from	of 16	of 12	of 8
rifing	hours	hours	hours
—1.
I	S	7	9
2	6	8	10
3	7	9	11 quot;
4	8	10	12
S	9	II	I
6	10	12	. 2
■ 7	- II	I	3
8	12	2	4
9	I	5	y
lo	.a	4	6 ^
	3	5	7
I2	4	6	8
	S	7	9
H	6	8	10
	7	P	II
16	8	10	12

the Stile Radius, or Tangent of 4J, and on
the Horizontal-line lay off the-3 Tangents
of the Rumbs laft made , in the

tical Circles , .
Rumbs, do meet n
a Point ( called
Vertical Point /
found in the l^f'^
dian,ori3 aclo^
' Line, right over 0
Incliners) or und«
( in Rechners) tn;
Apex or top of |

Stile, that is to give the Ihadow, when fet in
ifs right place, right over the Subftile-iine ;
And as far off'the foot of the Stile ( being a
Point in the Subllile, Square, or Perpendi-
lt;^u!er to the Apex or top of the Stile) in a
'^ercical Line drawn through the foot of che
Stile, = to the Perpendiculer Line of the
Plain), as the Co-tangent of the Reclinati-
on, making the Perpendiculer lieight of the
Scileto be Radius or Tangent of 45 degrees.
Alfo, The Co-tangent of the Reclinatiou
the Plain, to the fame Radius, laid from
^befooc of che Stile, in the fame Virdcal-
j-ifie, fhall give the Point in rhe Vertical-
hne, to draw rhe Hori2.ontal-iine by ; for
^Rule laid to this Point, and the cro0iug the
^quinodial-lme and hour of 6, fliall draw
the true Horizontal-! inir

Then make the .tnce between this
1 oinc, and che meeci ^ oi the Equinodlial
aadd, a - Tangeu. of the Well or Eaft
Azimuch m che TahU, and then the Seftor
IS fee, to lay off all the reft, by taking the
= Tangents of the Numbers in the fablt,
and laying them from the VerdcalTpoint in
the Horizontal-line, both wayes on thcHo-
rizontal-hne,nbsp;s

For, from hence you mnj note, That the
Sun, being in che Equinodlial, doth nle and
K 3nbsp;fee

fet near 6. and alfo doth rife near the Ea i-
point, and fet near the Weft; therefbre the
fame Point in the Dial, muft be for the hour
6 in the morning ; and the Eaft Asi®quot;^quot;'
or the hour 6 at night, and the Weft
muth, according as the Plain declines t»quot;'
wards or Weftwards.

Then Right Lines drawn from the Verti-
cal-point in the Meridian, and to all thele
Points in the Horizontal-line, fhall be the
Azimuth-lines required.

A thus for Example in the Figure annex-
ed, being the Third fort of South-Recliners
before-going ; Declining 3 5 degrees Soutn-
weft, and Reclining 60 degrees,^ C H is the
Subftile, CG the Stile, H the Foot of th^
Stile, I K the Vertical-line drawn through
the foot of the Stile I, the Vertical-point Jquot;
the crofting of 12 ( and the Vertical-line )

and ytt right under G the Apex ( con fide r-
ing the Reclinaticn) and the raifing of G the
Apex,Square, or Perpcudiculer toH the loot
of the Stiie; Then, I fay, a Plumb-line
fall from G, will reft in I,che Vcitical-point^
The Diahbeing fet in its due placc.

made a = Tangent of 45 ; HI is the Co-
tangent of the Rtclinarion, c/t. 30; aquot;
K tlie Tangent ol the Reclinaticn (So, be-

ing the Vertical-point in the Horizontal-
Imc, from whence to lay the = Tangents,
of the Rumbs in thsTable laft made, into
the Horizontal-line. -

Then lines drawn from the Vertical-
point I to thofe Points in the Horizontal-
ine fhall be the Rumbs or Points of the
Compafs, Vertical CirdeS, or Az-imuths iq-

Wnen you have made the Tables of the
Angles at the Zenith, as before, you may by
this Canon make Tables of Angles at the
Vertical-point, between the Verncal-lme
and the Rumb, to be drawn on the Plam.

This Table being made,, you may fet one
Point in the Vertical-point, and defcribe a
Circle to any Radius, and therein prick off
from tiie Vertical-line, the feveral Chords
of the Rumbs, as in the Table you ftiallmakc
by the laft Canon.

T»hit, Ihemng at what Hour a»d M^ff^f
Sm is»«, in an even Axjimutby or jPW' quot;J
the Comfafiin ffi,'for y 1-32.

Laftly, by help of this Table, being generaf
- far all Dials in the Latitude j 1-32, it ^
donethtii', . ,

note. The Table doth fliew tlic hovif
and minut, together with the Suns Altitude
in y^} Ti ^quot;d s, when the Sun is juil ofi

even Rumb, or Point of the Compafs -
fo that having drawn the hours, and quar-
•^^rs of hours ( by theSeftor eafie to do)
through the Plain in light Lines, make pricks
in yp, T, and ffi, as the Table fheweth.

Then you fhall fee that a Rule laid to the
Vertical-point, and any one of thofe three
Points fhall cut the other two, if tlie former
Lines be true, and you eftimate the minuts
^vell.

That this laft Table in the EquiV
lodfial, is thus readily made, by Sines and
Tangents.

As the fine of p o, to the fine of che Latitude;
So IS the Tangents of the Azimuths from the
Meridian, being the fiift column in the
Table,

So that you fee the Azimuth of 45-, or
Rumb of S.E. will nofs theEquinodtjalat
mmucs paft 9, as in the Table ; which
Table is ealily made by tbe Trianguler-Qua-
diant, by the Rules in Chap. XV.

Fiift, on the Equinoaial, thefe
fhewing the Suns Altitude, cannot
fH-efTed.nbsp;. ,

On the HorUontal Dia/ they are Orde »
making the Perpendiculer height of theStn
Radius, or Tangent of 45 ; prick off on the
Hour-iine 12, from the foot of the Stue,
the r= Tangent of 10,20, 30, 40, yo,
amp;c. Then one Point of a pair of Compa|i^^
fet in the foot of the Stiie, and the other
opened to 10, 20, 30, amp;c, draw thole
Circles for the Parallels of the Altitude re-
quired.

diftance from the Nodus, tothecroTing ot
the Horizontal-line, and Azimuth-line, on
which you would prick down the Altitudes
fhall be the Tangent of 45 ; then the
Seaor fo fet, the Tangents of 10,20,30,
Sec. laid from the Horizontal-lme on the
rcfpcftive Azimuths, ftiall be Points to draw
the Parallels óf Altitude by (or by apply»«g

tiie Rule to the Nodus and Plam, and the
Tihred to the Almicantcr ) as afterward is
plainly fhewed.

But if the Stile is not in, then the Secants
to the fame Numbers and Radius, that
pricked down the Azimuth Lines, fhall be
the feveral Radiuffes to ufe as before; where
you may note, Thar the Suns Meridian Alti-
tude in the whole even Signs, will help to
prove the truth of your work.

The Eaft and Weft ErcU Dials, are fitted
with Parallels of Altitude in the fame man-
ner; for the Perpendiculer height of the
Stile, is a Tangent of 45, and the = TaS-
Sents of ii-if, i-'SO, 33-4J, Stc. laid
from the foot of the Stile in the Horizontal-
line, draws down-right Lines for the Azi-
muths; and the Secant of ii-ij, 22-30,
33-45, amp;c. fliall be the feveral Radiuffes
to prick off the = Tangents of 10,20, 30,
40, yo, 60, ( or what you will ) on thofe
Perpendiculer Azimuth Lines, for the Al-
micanters,or Parallels of Altitude required.

But for Dfc/,quot;«;»^ �lim!?g Plains, you
muft firft draw the Azimuth Lines,as bdore
is fhewed, and then find alfo the leiigth of
the Axis of the Horizon, as Mr. Cnmer calls
ifj which is thus done;

Make the length of the Perpendiculer-
Stile a Tangent of 45, viz.. G H Fig,ll.
then H I is the Co-tangent of the Reclina-
tion, and HK the Tangent-of theRechna-
tion j and then, as the Se£for ftands, the
Secant of the complement of the Rechnatioquot;»
fliall he the length of the Axis of the Hoquot;-
zon required, viz,. G I, or by the Sines aw'l
Tangents Artificial.

So is the length of the Stile on the Lin^
* of Numbers, being taken in inches and
ICO parts, to the length of the. Axis in
the fame parts.

Which is an imaginary Diagonal Line,
reaching from the Apex to the Vertical'
point.

This being found, you muft find the
Angles between this Axis and the Horizon-
tal-line, on every particular Azimuth ; and
laftly, the diftance between the Vertical-
point, and the Parallels of Altitude, one-
very particulcr Azimuth laft drawn.

For the doing whereof, you muft work
as you did before, to lay off the Signs, or the
Parallels of che length of the Day, for thefc

Alfljicanters, bear tbe fame refpect or pio-
portion to the Horizon, as the Parallels of
the length of the day have to the Equator,
andare defcribed in the fame manner, as
followeth. See Figure III.

Firft, draw the Lme A B, and make A B
a Chord of 60, and fweep the Arch of a
Circle, and lay off 10, 20, 30, amp;c. and
draw the Lines from A the Center, and
mark them with 10, 20, 30,40, yo, 60,
the even loth degrees; or, 45 for equal,
2(5-34 for double, 11-19 for y times, the
length of the flxadow and objcft, orwfrat
you pleafe.nbsp;''

Draw AC Perpendiculer to AB, and
lay oflf the length of the Axis of the Hori-
zon from A to C ; then make A C the Co-
fine of the Rcclination, and as the Sedlor
ftands, take out the fine of the Reclination,
and lay it from A toD, and this will be the
diftance from the Apex to the Horizon;
Alfo, the fine of 90 fhall reach from C to
D, the diftance between the Vertical-point
and the Horizon ; alfo the neareft diftance
from A to C D, is the Perpendiculer height
of the Stile A H.

point on the Plain, to the Horizon 0»
particular Azimuth Line, and Jay them in
the Trjgon, or III Figure, from C to the
Horizontal-line A D, produced if need be ;
and draw thofe obfcure Lines, as in the Fi-
gtire, and mark them with the Names of
the Rumbs. to avoid confufion then is
your Trjgon made ready for ufe.

Take the diftance from G in the Tngonj
to every croffing of the Azimuth-line and
Almicanter, and lay it on the Plain from
tlie Vertical Point I, on its proper Azimuth,
finifhing one Alnlicanter before you meddle
with another, and the work with patience
and diligence will be performed; the lines
are to be drawn from Point to Pdint, wi^
a fteady hand, or a bending thin Ruler, be-
ing Conical Se^ions,

That when the Vertical-line of the Plam
falls on an even Azimuth, then half the
number of Rumbs will ferve, beina; laid
each way on both fides at ones.

Having a TMe of the Angles at the Ze-
nith, the fame as you made to draw the A-
zimuth-lines, draw a Line at any conveni-
ent diftance, Parallel to A C ; the further

the laft Line EF, as at D; make DE a
I^arallel fine of 90, and lay oftquot; the fine
complements of the Angles at the 2enith in
the Table, from E towards D, and draw and
'iiark the Lines, as in the Figurt.

Othtrwife,
The Stile being fixed,, and tlie Dial fet in
its place where it muft be, or at leaft fet ta
the fame Reclination, and Declination that it
niuftbe; then if you apply the fide of the
Trianguler Quadrant to the Nodus, and the
corner at the end of the fame edge that
toucheth the middle of the-Nodus to thfe
^'Wn; and at the fame time, the Thred and
Plummet p^ay^ng neatly on the Almicanter
you would draw, you may find as many
Joints, and mark them as you pleafe. With-
out sll the former trouble, and 1: may be
'ivery whit as true; if the under-fide be in-
convenient, you may ufe the upper; only
be fure, that rhe f.de you app y, and the
Tlired and Pkimmf day at the Angle of
the Altracanter required.

The Civclcs of Poficion, or 12 Houfes,
meet and crofs one another in the croifing of
the Meridian and Horizon ; therefore the
Horizon is the begining of the ifl and
Houfes, beginning at the Eaft, and reckon-
ing under the Earth, by Imam Cali, to the
Defendant, or 7th Houfe, at the Weft-pafC
of the Horizon ; and fo to Medium Coelf*
the beginning of the loth Houfe, to the
Afcendant, or Horofcope, the beginning o»
the xft Houfe.

To draw thefe on the Horiz.ontal'Dial,where
they are Parallel Lines to the Hour
do thus ;

Take the diflance from the Apex to the
Equinoftial-Iine, and make it a = TangcO'^
of 47; then the = Tangent of 30 degrees
laid both wayes on the Equinoctial, ihall
give Points to draw Lines by, rrtoiJ, for
the Houfes required.

For all Vprlght Dlt;W/and Reclinersm^^
Centers, flreight Lines drawn from the
meeting of the Meridian-line and Horizon,
to every two hours, counting from 12 quot;quot;

^ot Eaß and fVtft Dialsy take the Radius
3s before, viz,, from the Apex to thefiqui-
öoftial-line on the Plain, which here is the
Meridian (and but the length of the Stile)
® Tangent of 4y ; then the = Tangents of
30,60, and laid from 6 on the Equino6tial-
line, gives Points to draw Lines Parallel to
the Horizon, for the Houfes required.

For Saß and fVeß Keclinersi the Perpen-
diculer height of the Stile made a Secant of
0; quot;then the Secant of the Stiles Elevation,
ftiall be Radius to prick off the = Tangents
of 30, do, on the Equino6tial-line from the
foot of the Stile, whereby to draw Lines
parallel to the Horizon for the Circles of
Volition required.

All thefe Lines may moft elegantly and
eafily be drawn and exprefled, on a large
Ceihng, with competent exaftnefs in this
manner following.

Firft provide a Quadrant of Brafs, or
thin Wood, of about a foot Radius, or 14,
If, or 16 inches; alfo, a Semi-circle of
J^afs, of about half an inch broad, and a-
bout an inch lefsRadius-than the Quadrant:
tbe Semi-circle muft have at each end, fome-
What more than to make up 180 degrees,
Lnbsp;to

i .. ;
to nail to the Tranfam, or ftroke of the Win-
dow, where your Glafs is to lie.

Alfo, to one Ray of the Quadrant muft |
befartened two ftrong Wyres, to faften the
Quadrant to play after the manner of a |
Cafemenr, one Point in the Rav of the Qua- L

Tranfum of the Window, by putting two
little bits of Wood under the ends, that the
iluadrant may play evenly and fmoothly on

the Scmi-clrcic- to almoft the half-round, fbif
quite the half-round will not be neceffary, or

'^ni tht DecUnatteHi ef ratbtit
•ut {JMttidian-Une.
Mtirant till the edge be juA
gt; and at the lame inftant get
ith i then if you count fd
ns Azimuth is, on the Brafs
ora the place the Quairant
ght way, a Line drawn frottl
le Serai-circle, or QaairaHti
is the true Meridian Line;
)u muft carefully find by twd
and then mark it with Ink
Dn the Brafe Semi-circle, tO
ace, in fstting die Qjiadrant
:imuthj at every hour and
Points you intend to draw
; which a crooked Rule fet
iemi-circle, to pafs to and fro
'ra»t, will make eafie*
ng a Tahlt of the Suns Alti-^
tude and Azimuth, at every hour in that
Latitude you draw the Dial fot;

t ] .
to nail to the Tran[itm, or ftrokc of the Win-
dow, where your Glais is to lie.

Jlfo, to one Ray of the Quadrant mult
befailened two ftrong Wytes, to faften the
Quadrant to play after the manner of a
Cafemenr, one Point in the Ray of the Qiquot;*'
dram next the Center, flicking in the hole
where you intend the Glafs fhall lie ; a^
the other end fattened to a piece of Wood
nailed on the two upright Pofts of the Win-
dow, fo that howfoever you turn the Q^i'
drant, fixed on thofe two Points, it may be
precifely Perpendiculer, the Semi-circlc play-
ing all the while through a hole in theo-
ther Ray of the Quadrant, thai lies Horizon-
tally ; having a Skrew to ttay the Quadranf
at any Azimuth, as in Flgurt IV, is plainly
cxprefTed to your view.

Then having degrees on the Semi-circle,
and alfo on the Q«lt;i(/rlt;i»f, and having fitted
the Quadrant on his Points to play precife-
ly Perpendiculer, which the Plummet in
the Quadrant will fhew,by turning it round
about, and put in the Semi-circle through
the hole in the HorizoYital Ray of the Qua'
drant, and nailed it fo to the Stoole or
Tranfum of the Window, by putting two
little bits of Wood under the ends, that the
Quadrant may play evenly and fmoothly on

the Semi-circle- to almoft the lialf-roünd, fbt
quite the half-round will not be necefläry,or
ufeful.

Turn the Qifadrant till the edge be juft
againfttheSun, and at the Came inftant get
the Suns Azimuth j then if you count fd
much as the Suns Azimuth is, on the Braß
Semi-circle, from the place the Qnairant
ftands at, the right way, a Line drawn froOl
the Center of the Semi-circle, or Qaairanti
'b that place, is the true Meridian Line;
which p ace you muft carefully find by twd
Or three tryals; and then mark it wich Ink
or other wife, on theBrals Semi-circle, tO
count from thence, in fetting the Quadrant
to the Suns Azimuthj at every hour and
quarter in thofe Points you intend to draw-
on the Ceiling; which a crooked Rule fet
to 00, on the Semi-circle, to pafs to and fro
with the Quadrant, will make eafie»

Then, having a Table of the Suns Alti-^
tude, and Azimuth, at every hour in thai:
Latitude you draw the Dial for;

marked Meridian-line on the Semi-circle,
and there skrew it faft; Then extend the
Thred fattened in the Center of the
draiit, till it cut the Altitude of the Sun at
the fame hour and Azimuth, on the degrees
of the Quadrttnty and extending the Thred
to the Ceiling, make a mark for that Hour
and Altitude 5 that Point at that time, gives
the true place where the refledki fpot wiU
fall, at that Hour, Azimuth, and Altitude
on the Ceiling of the Room.

This work repeated as many times as there
be hours and quarters in the Summer, and
Winter Tropicks, for about y hours, (and
in the Equinoftial, and any where between)
if you pleafe) fhall give Points enough to
draw the Dial, and alfj the Tropicks, and
Azimuths, and Altitudes alfo, if it were
convenient to mark it ; Or, to any other
Altitude you mind to have at that Azimuth,
all at once, or atmofl with two flips of the
Thred ; the Ital'iM, Bakjllt;mifh, or Jervijh'
Hours, as eafily drawn by Points found in
the other Lines.

Alfo, On tlie Meridian-line, you may add
the day of the month, or any thing that
depends on the Suns Meridian Altitude ;
which work being well done, and drawn
,nbsp;with

witb fraoth Lines, and well ornamented.
Would be a comely, Sr. pleafant Ornament to
a Ceiling, and far cheaper then fome free
Ceilings are done, and more ufeful.

Luftly, When all is done, to pur,the Glafs
in right, the Foile being firft rubbed oft, to
to caufe it to give but one fpot, let tfee Super-
ficies of the Glafs lie jufl fo high as the Cen^
ter of the QuadraKt was, in the drawing
the Lines, and put foipe Putty under it,
andtheSunflvining, mgke it toplayrighc
on the true Hour, Altitude, and Azimuth j
or, if it be juft at noon, then bend it on the
Putty with your finger, till it fall juft ou
the Meridian, and day of the Month alfo in
the Meridian-line.

Alfo note. That look what Altitude the
Sun hath at any time, the fame will the re-
flected Altitude be, at the fame time, if the
Glafs lie true, which two Obferv^tors at the
fame time may carfully prove.

The making of the Tablet of the Suns
Altitude and Azimuth,is very largfly fliew-
edin the i jth Chapter, Vfe the 37tb and
38ch, where you have wayes both general
and particular, for anyone or niore^ti-'
tudes.

A, the place on the Tranfum for the Glais
to lie on, and in the middeft thereof one
Point, in the Ray of the QuAdrant, is to
play : 1H, a piece of Wood to be nailed
faftat H andl, for the other Point to play
in at G: L, the hole for the Plummet to
play in, being cut through the thin OHa'
drant y BandC, the ends of the Stmi-
circle, nailed on the Tranfum or Stoole :
K, the hole in the Quadrant for the Semi-
circular-Ring to pafs through: FE, the
Pofts cf the Window: D, the beginning of
the degrees on the Semi-circle: A M, a
Thred extended from the Center of the
^ndrant to the Ceiling.

Thus you have the ufual wayes of Dialing
in a competent meafure, plainly, and
praftically handled, which may lgt;e ufe-
ful to many a Learner j and 1 hope
will be as well accepted, as with free-
wUl (though with little ability, and lefs
leifure) readily imparted.

The Defcription andfome Ufes of tk
Sphear for Diallings anji for thi
better underjlanding of the
ral and particular Schems.

NExt the Foot and Scmi-circle Frame for
fupporring of it, you may coiifider j
1. Tiic fixed Horizon, to which thePffj
is fallencd with 4 skrewsj numbred aquot;''
divided into ; 60 degrees, or four yo df^r
whofe count belt;;;ins at the Dividccs fide quot;
the Meridian-Circle.

a. The Meridian Circle, whofe fore-W^. i
at the Nadir-;point Hands in the Center'»
the Foot; this is alfo divided into 4 po'»
sgt;nd begins to be numbered at che Sout 1 a»^
North part of the Horizon,upwards tow at'*
the Zenith, and downwards toward tb^
Nadir ; w bich Circle is alw ayes fixed as th^
Horizon is.

3. The Equinoaial Circle, made faft
the Eaft and Wert Points of the Horizon»
moving up and down upon the Meridiaii'
Circle, according to the Elevation of tb«

SUinoaial in any Latitude ; this is divided
Jlt;e\vifc into four 90', amp; numbred from the
^eridian each wayes to the Eaft and Weft
öints of the Horizon.

Poles, to be elevated or depreffed fit to
Latitude of any place ; on the Fiducial-
r'igeof^hich, isfaftened the Thred, repre-
Sing the Axis of the World, at any Eleva-
^'on of the Pole.

y. On the 2 Pole Points, is fattened the
^our Circle, which delineates or reprefents
^^e motion of the Sun, or any fixed Star,
quot;Moving in its fuppofed Diurnal motion a-
?out the Poles of the World, and may not
'Improperly be called the moveable Meridian
Circle, or Hour Circle, divided as before.

lt;5. The Moveable Horizon, that moveth
about to any Azimuth, and ftidech or moveth
the fixed Horizon.

7. The Plain, fixed in 2 oppofite Points
to the moving Horizon; being fet, either
Horizontal, when it lies Parallel to the fixed
Horizon i or Ereft, when Perpendiculer
thereunto j or fet to any Reclination or In-
clination, by help of the Semi-circle of Re-
klination, faöened to the backfide of the
Plam in the 2 Poles thereofnbsp;„ ,,

8. You have the upper moving Semi' ^
circle, in turning about of which, whatever j
degree the fore-fide of the Semi-circle cuts,
the Perpendiculer-point cuts the complc' \
mfint thereof, and to be called the uppequot;^'
Semi-circle, or Circle alwayes Perpendicu-
ler to the Plain.nbsp;, i

The Horizon; The Meridian;
Equinoctial Circles; The 2 Pole
and Axis; The Hour Circle, or Moveab''^
Meridian; The Moveable Horizon; T''®
Plain; The Semi-circle of Reclinatio'^'
The upper Semi-circle, and, The Thred.

Note alfoy Every Circle is divided
4 times po, and numbred the mofl ufef'*'
way.

Suns place, his mean true place, fuffici-
^ly true for any illuftration in Maihema'.
Mpraftice.

t. To reftifie the Sphear to, anyLati-
Me, cotmt the Elevacion of the Pole or»
^^c Meridian Circle, from the Horizon up-
wards, and downwards from the North and
^outh parts of the Horizon ; and there
■^ke faft, with the help of the fmall skrew,
^he Fiducial^edge of the Poles Points, car-
quot;^ying the Hour Circle fixed upon them, theo
'ue Pole is rightly elevated.

a. Count the complement of the Poles E-
^evation on the Meridian, from the South
part of the Horizon, and to it fet the divi-
ded fide ofthe Equinodlial Circle, then is
that reaified alfo (in the Northern Hemi-
fphere, or in the Southern, if you call the
North Pole the South Pole).

3. Extend the Thred or Axis pafling
through the Center to the South Pole, and
there make it faft, and then the Sphear is
»eitified for many Ufes in that Latitude.

Lay the Thred in the Center of the Pla»«
on the day of the Month, and in the Lhic
of the Suns place, you have his place.

On tbe fth of Ntvembef, it is 23 dcgr^
in J- J or if the Suns place be given, look ioi
that, and juft againft it, in the Months,
the day required.

The Suns place being i y d^rees 1 look
for it in the Line of his placc,and iuft againft
it I find 28 day.nbsp;^

Look for the day given, and right againft
it in the Line of Declination, is his due D^
dination required.

Augtiji the yth; The Declination is
degrees y minuts from the next Equinodfial'
point, viz,. Ki.

1 the Sun hath North tJechnatjon ; or
' Southern Sines, or Winter-months, the
' ^iin hath South declination. Or if you have
'he Suns declination, find that in the Line
^clination, and right againft it in the
Months is tk day required.

21 degrees South declination, beginning
from the Equinoaial towards the Winter
Solfticc, I find Novemh.i'i. The like work
j had been, if the Suns place had been given,
' find his declination.

i This is ufually reckoned from r tor,
round, in 24 hours; but twice 12 is as
Ufeful, and then k is thtu ;

Find the day amongft the Months and
Dayes, and jufl againft it, iigt; the time of
Hours, is the Suns Right Afcention; (but
Kotty it is not right ^ured for this ufe)
counting onwards from t, or the lOth of
March, to the 13th of Septtmb. and frotn
thence to Aries again ; Likewife the degrees
are to be reckoned from T onwards, as the
Months proceed.

On the 12 of LMaj, what is the Suns
Right Afcention ? Lay the Thred on the
jithof May, and in the Line of Hours it
cuts 9-57' counting (tomArltt onwards}
or in degrees jj-ijquot;, counting as before»
Thus, if any one of thefe 4 general things
be given, the other may be found,

Tht Suns Diclinatien and Latitude hii^i
given, to find the Suns Meridian Jl'
titude.

The Sphear being redVified, count the
dechnation on the Meridian, from the Equi-
nodtial, that way the dechnation is, either
North or South; and where the count ends,
there is the Meridian Ahitude required fo'
that day, or Declination.

y«w* II. Bechnation 23-30'; Count
23-30, from 3^-30, the place where the E-
quinodial ftands^ for y 1-30 Latitude, and
the account will end at 62 degrees, the Suns
Meridian Altitude at that Declination
Northwards : Bur, if it had been 23-30
South Declination; then count as much
from the Equino6fial downwards, and the

count will end at ly degrees, for the Suns
Meridian Altitude, at 23-30 SouthDecIi-
quot;ation.

The Suns VecUnation and Latitude being
given, to find the Suns fifing or Set^
tlngy and Amplitude, Eafl or mfi.

Count the Suns declination oa the Hour-
Circle towards his proper PoIe,that is South-
dedination toward the South-Pole, and
North-declination towards the North-Pole;
and thereunto lay the Thred that is faftned
in the Center ; then bring the Hour-circle
and Thred both together, till the Thred
touch the Horizon ; then the Thred on the
Horizon fhews the Amplitude, and the di-
vided-fide of the Hour-Circle, fliews the
Suns Rifing and Setting on the Equinoaial,
counting the Meridian alwayes 12, and the
2Eafl aiKl Weft-points lt;5,and if degr. for
an Hour, and every deg. 4 min.

June II- Declination 25-30, the Sun
Rifeth at 13' before 4, and the Amplitude
is near 40 deg.

April 10. Declination 11-30, the Am-
plitudeis 18.30 from the Eaft to the North,
j^jnbsp;and

The Declination amp; Latitude, amp; Suns Alti-
tude given,tofind both Hour amp; Jsjimntb-

Reilifie the Sphear, and fet the Plain ho-
rizontal ; that is. Level or Parallel to the
Horizon ; then apply the Thred to the D^'
clinadon, counted che right way on the ^
Hour-circle ; then turn the Hour-circle and ,
upper Semi-circle about, till the Thred cuts
the degrees of Altitude in the upper Semi-
circle, and the Hour-circle, fhews the ho«t
in theEquinodtial, and the Semi-circle cuts
the Suns Azimuth in the deg. on the Hori-
zon or Plain.

Declination lo. Latitude y 1-30, and the
Suns Altitude 30 ; the hour will be 8-27gt;
and the Azimuth 66, from South Eaft-
wards if in the morning, or the contrary ^
in the afternoon. '

Hour being given, fee the Hour-Circle to
the hour on the Equinoaial j then bring
the Thred to the Declination, counted on
the Hour-circle ; .then bring the upper Semi-
circle, till the fore-fide do juft touch the
Thred, and the Thred on the Semi-circle,
fliall Ihew the Altitude required ; and on
the Horizon, the Azimuth at that Hour,and
Altitude.

But if the Azimuth be firft given, then
fet the upper Semi-cii cle thereunto, counted
on the.fixed Horizon; then the Thred laid
to the dechnation, on the Hour-circle, and
turned.about till it touch the upper Semi-
circle, there it fhews the Altitude ; and the
Hour-circle on the Equinodtial, fhews the
hour.

Set the upper Semi-circle to the Eaft or
Weft-Point, or po degrees of Azimuth;
then lay the Thred to the declination on the
Hour-circle, and then bring it and the
Thred together, till it juft touch the upper
Semi-circle, and it fhall there ihew the Al-
tuude at Eaft or Weft required.

Set the Hour-circle to 6 on the Equino£ti'
al, and the Thred to the Decimation ; thei^
bring the Semi-circle to the Thred, andic |
fhewesthe Altitude at 6 required. .
Example.

At 23-30 Declination, the Altitude
18-15 above ch: Horizon in North-declina-
tion ; and as much under in South-declina-
tion ; for, you muft obferve that the lurelt

working is from the upper or divided-fideS
of the Rings, cn every occafion to ufe it.

Reaifie the Sphear, and fet the Plain
Parallel-to the Equino6tial-circle ; then fee
the Meridian-circle due North and South,
and the fliadow of the Axis ftiall on the
Plain, flievv the true hour.nbsp;^^^

At the true place of the Suns Declination,
on the Hour- circle make a msrk, or ftick the
point of a Pin , then turn the Sphear a,-
oout, till the lhadow of that mark, fall on
the Center; ( the Sphear flanding Horizon-
tal, as near as may be ) then the Hour-
circle lhall, on the Eqainoftial, £he\v the
hour of the day required.

Note, A fmall Bead, or knot on the
Thred, will do the bufinsfs as well as may
be.

Thus any tlie like Queftions may be
Wrought for the Stars; or the manner of
raifing the Canon for any Spherical Triangle
whatever, to work the fame exaCily by the
Logarithms.

Suppofe I.would make the Canon, or
Propofitton, to find the Suns height in the
Vertical Circle at any dechnation. Firft,
The Sphear being reilified, and the Plate
fet Horizontal, bring the upper Semi-circlc
to che Eaft-poinc, and laying che Thred to
the dechnation on the Hour-circle, bring it
and the Thred together, till it juft touch the
upper Semi-circle.

wherein you have,
ZiE, 5:1-3 0 the Latitude, the Angle at
the Equinoctial; and ZE po, the upper
Semi-circle; and A B 23-30, the Declina-
tion, part of the Hour-circle ; to find B E,
)art of the upper Semi-circle : Now this
jeing a right-angled Spherical-Triangle,
and the parts which are given, being one
right Angle, wz,. The Angle at A, and the
Side A B,'the Suns Declination; and the
Angle at E 5'i-30, to find the Side BE i
Now the Sines of the fides of Spherical-
Triangles are proportional to the Sines of
their oppofite Angles, and the contrary.

And the like for any other, as by com-
paring the Rules, in Mr. Norwood's Trigono-
raetry, and the Circlcs of tlie Sphear toge-
ther, the ufe and convenience thereof will

Only note thû fltln Obfervatien:
That the fide of a rigk-Angled Triangle,
xvhich fubtends the Right Angle, is molt
properly called the Hypothenufa ^ the other
which you make or fuppofe Radius, the
Bale quot;The other, the Perpendiculer. Or
^ort (hort. The Hypothenufa and Leggs:
Therdtfre if the Hypothenufa and one Leg
lie given, the proportion is wrought by
■ines alone; but if the two Legs be given,
■nd firft and fécond in the Q^eftion, then
he Proportion is wrought by Sines and
Tangents together.

As the fine of T ©
To the fine of S ^ 23.31
So is the fine of T to- lo-oo
Tothefineofnbsp;R 11-31

So is the fine of t R 27 j4
To the Tangent of Rbquot; 11-3^
The Declination as before.
But if the one acme Angle, and his oppoJ

fite Leg or Side be given, then tbe Proporti-
on is made by Sines only, as m the bre-
going Example,

In Vertical Triangles that have the famC
acute Angle at the Bafe, as the Triangle
P bquot; S, and r bquot; R, being equal Angled
at b- ; the fines of the Bafes are proportio-
nal to the Tangents of the Perpendiculeti
and the contrary.

The Sines of the Perpendiculers, as pro-
portional to the Sines of the Hypothenufaes,
and the contrary.

As the fine of the Bafe bquot; S 60-00
To the Tangent of Perpend. © P 66-29
So the fine of the Bafenbsp;b-R 11-31

Alft for the Second, viz. Hypothemfaes
and Perpendiculers.
A.S the fine of Hypothen. Pbquot; 78-29
lo the fine of Perpend. P s 66.29
So the fine of Hypothen. vbquot; 30-00
To the fine of Perpend. T R 27-y4

This being premifed, when to ufe Sines
alone, and when to ufe Sines and Tangents
together, you may reilifie the Scheam to
your prefent purpofe, and fee there how the
Triangle lies m its Natural parts, very plain
and demonftratively to be apprehended.

To this purpofe, you muft take notice,
That the Sphear is very excellent to
demonftrate that Art; efpecially all thofe
Dials whofe Stiles have any competent Ele-
vation.nbsp;,

The Sphear being reftified to the Lati-
tude ; Then, firft the Plain, or Broad-plate,
is to reprefent any Plain howfoever fcituate,
either Horizontal, or Ereit Dire61:, or Dire(it.
Reclining or Liclining, or Eaft and Wefl
Ered, or Reclining, or Inclining, or EreCt
and Declining, or South Declining, or Re-
clining or Inclming, lefs or more than to the
Pole or North Declining; or Re-inclining
lefs, to, or beyond the Equino61;iaI.

^ain is fet even with tbe fixed Horizon, and .
'quot;e Notch which the Semi-circle of Rech-
l^tion pafieth in juft againft the Meridian ;
l^en if you ftretch the Axis ftreight, and
quot;^Hg the upper Semi-circle juft to touch the
^^s J then the Axis, on' the Scmi-circle,
^eweth the Stiles height j and the Edge of
Semi-circle on the Plain, fhews the Sub^
'^le to be in the Meridian.

Set the Hour-circle to every hour and
5^arter on the Equino6lial; and then if yon
quot;ting the loofe Thred, fattened in the Cen-
ter of the Plain, along the Plain till it juft
^tich the Hour-circle, then on the Plain it
'hall fhew the Angle from 12, for that re-
|Pe£five hour and quarter the Hour-circle
ftands at on the Equino61:ial, accounting
for a quarter, and 7-30 for half an
hour, and 15 deg. for every whole hour,
as was hinted before.

2. Far AH EreU DlreEl South or
quot;^iorth.Dlal,
Juft as the Plain ftood before, that is to
f^y, the Notches of the Moving Horizon a-
gainft the Meridian; turn the fixed Semi,

circle, till the divided fide of the Horizon
cuts no deg. on the fixed Semi-arcle, then
the upper-edge of the Plain refpc6ts the Ze-
nith, and the lower the Nadir; and the
two Notches in the MoveingHorizon ( be-
ing alwayes the Poles of every Plain) arc
juft in the Meridian; therefore it is a DireC'
Plain, and Erecl, becaufe Upright without
any Rcclination, as the fixed Semi-circje
fiieweth. Then being fo fixed, and made fai*
there, pull the Axis ftreight, and bring the
upper Moving Semi-circlc juft to touch the
Thred or Axis; then on the upper and
lower Semi-circles, the Axis fheweth th®
Stiles Elevation 5 and on the Plain the Sen^''
circle cuts the Subftiles diftance from i^j
viz.. 00, becaufe a DireCf Plain.

And for all the Hour-Arks on the Plain^
fet the Hour-circle to every hour, quarter,
and half hour on the Equinoftia j a»»
bring the Thred eafily along the Plain, til'
it juft touch the Hour-circle ; then on the
Plain ic ftieweththe Ark from'12 required-

Alfo note. The feveral TriangleS made on
the Meridian, Equinoftial, and Hour-circle,
at every hour ic is fet unto.

Suppofe at the Pole, I fet P ; at the cut-
ting of the Equinodfial, and Meridian, :

the upper-end, or Zenith, fet Z ; on the
Meridian, and where the Hour-circle cuts
i ^he Equinoaial, at i amp; 11, fet i j ; at 2
8c 10,30; at3 amp;9, 4fi at 4 amp; 8, 605
J amp; 7, 75 ; and at lt;S amp; 6, 90.

But note, Becaufe the Hour-circle cannot
pafs by 12, you muft turn the other-fide,
' or half, for the afternoon hours.
I Alfo note. That if the back-fide of the
Pkin do not well reprefent the South-fide,
being the more ufeful Dial j then if you
hold the Sphear with the foot upward, the
Zenith becomes the Nadir, and the North
Plain a South Plain, to appear more Plain to
the apprehenfion.

before, and let the upper part of the Hori-
zon cut the Semi-circle of Rechnation, ac-
cording to the Plains Reclination, and there
make it faft j then the Axis drawn ftreigW»
and the upper Semi-circle brought to ir,
iheweth the Stile and Subftile; and the
Thred and Hour-circle,laid as before, givetquot;
the Hour-Arks on the Plain, and ftiewecquot;
alfo honv the Proportion runs.

To find any Requifite alfo you may ob-
ferve for all North-Recliners and South-If'
cliners, that the complement of Latitude a«d
Reclination put together, doth give the
Poles Elevation, or Stiles height, for a'*
thofe Plains, which fometime will be above
90from the South part of the Meridian;
and then the complement to 180, is to be
fet from the North part of the Meridian J
But if it be a South-Recliner, then fubftract
the Reclination out of the Comp. Lat. and
the remainder is the Stiles Elevation : Bquot;'^
if the Reclination be more than the comple-
ment Latitude, then fubftradt the comple'-
ment Latitude out of the Reclination or In-
clination, and the remainder is the Stile'
Elevation.

Note alfo. That the upper-f^ce of the
Plain, that beholds the Zenith, is the. Re-
cliner; and the under-face that beholds the
'nbsp;Nad».»:-

l^adir, is the Inchuing-plaliir Ani mte;
That both Plains, viz,, both Iiicliners
Recliners have tbe fame Requifices in
^achof them. But, the hours proper to the
décliner, are not to be put on the Incliner ;
for tvhen the Sun (hines on the one, it can't
flline on the other. Therefore to know what
hours are fit for thefe or any Plains what-
ever, do thus ;

The Sphear reaified, and the Plain fet to
his true fcituation, lay the Thred on the
Suns declination, on the Hour-circle ( ac_
lt;^ording to what time of year you would
^know when the Sun begins and ceafes to
fhine on any Plain ) and turn the Hour-
circle, with the Thred fo laid, till the Thred
do but juft touch the Plain, and the Hour-
circle doth on the Equinoèlial, cut the Hour
and Minuit required ; when the Sun comes
on the Eaft.fide, and when it goes off from
the Weft-fide of the Plain.

Suppofe you have a Direél North-plain
that Reclines from the Zenith towards the
Equinoaial 2 y degrees, you fhall find the
Stiles Elevation to be 63-30, the Subftile
from 12. The North-Pole to be elevated on
the Recliner, and the South-Pole on the
Incliner ; and that the Sun ftiines on the

[ ] , . ^
North-reclincr in the longell; dayes,
23-31, declination, from the Rifing 13
before 4, till 10 j and then it begins tlt;j
fhine on the South-incliner, and fhines tiH
a afternoon ; then ic comes on the North-
recliner again, and continues till it fets.

But in the fhorteft dayes, when theDquot;'
chnation is 23-30 towards South, then
the North-recliner it fhines not at all, but
only on the South-incliner,- from Rifing
Setting ; and fo doth ic all the time the Suf^
hath South-declination.

Note, Thac the Circles of the Spheaf
fiiews the Canon to work this Queftion
exaidy, whereof you have a large Difcourl^
in WeUs his Art of Sbadom, from pag. 39
to 408, in 3J

4. For A DlreB Eafi or Wtß ErtÜ-DUl'
The Sphear being reäified to the Laquot;'

ön, but muft be fet Parallel to the Plain, at
quantity you pleafe, which is to be the
Radius of a Tangent-line, whereby to prick
uowu the Hours; and that the Subftile or
.place where the Cock or Stile muft ftand is
6, being the Hour-circle, dll it be juft a-
gainft the upper Semi-circl'e, touching the
Thred, and in the Equinoaial it cuts 6, th,e
true place where the Stile muft fbnd.

Alfoy By the fore-going Rule you fhail
quot;nd the Sun ftiine all the year from the Ri-
ling, till 13 on theEaft-fide j and on the
^^clt-fidc from 12, till his Setting, ^ ^ ^^^

. Turn the moving Horizon to 9Ö degrees
the fixed, as before ; then fet the Plain tb
quot;'s due Reclination, and make it faft therej
and pull the Axis ffreight, and bring the
Jjpper Semi-circle juft ro touch it, and
«traitway you have the Stile, aad Subftule^-
^d 12, the Inclination, Meridian, and
nour-Arks on .the Plain.

As far Example.
. AnEaft-plain reclining from the Zenith
towards the Horizon 4J degrees, hath his
Meridian, or la aclockLineintheHotizon;
tor ifyou extend the Thred from the Ceiitc^
' Nnbsp;td

to the fore-fide of^the^ Meridian, ^^
the 12 a clock Line muft alwayes be, ^hicft
in this Plain lies in the Horizon. The Sub-
ftile doth lie 41-40 from thence upward, as
the upper Semi-circle doth fliew ; tl^e In-
clination Meridian is thus found • Bring
Hour-ciicle, till it ftand even and parallej
to the upper Semi-circle j then on the t'
.quinoitial it cuts 58-7', the Inclination 01
rlie Meridian, with which you muft make »
Tfhlt of Hours, or Arks at the Pole, to cal-
culate the Arks on the Plain, if you worK
Arithmetically.quot; But by the S^ear, Set
the Hour-circle to the hours on the Eqquot;''
noftial, and the Thred being brought aloquot;^
the Plain till i: toudi the Hour-circle,
ihew on the Plain the Angle from the H^'
rizon or Perpendiculer; or with fome mot®
trouble, from Subftile or 12.

It fliews, that the North-Pole
Elevated on the Wefl-reclining j and th^
South, on the Eaft-inclining oppofite there
unto i and that the Recliner in S5, ihe^^j
from p in the forenoon, till 8 at night; and
the Eaft Incliner from the Rifing, till 9 for^'
noonin Summer; and in Winter, tiUai*
in the forenoon.

Now to make thefe Plains, asEre£l:De-
ciiifcrs, let the complement Latitude become

a new Latitude; and the complement De-
clination a new Dccliuation 5 then they may
fccomeEi-ca Dccliners, as in the next fo/c

By Decliiiatwn^ I mean the quantity of
^he Angle that the Meridian or Pole of place
Jiakes betweel! the Meridian, or Pole of the
Plain J therefore to fee the Sphear to any
decimation, do thtu;

^ The Sphear being fet to the Latitude,
l^u the Sphear as. well as you can guefs, to
fcituarion of the place; that is, put the
jjorth pare of the Meridian towards tbe
and tbe South ,part towards the
°uth ; then turn the Notch of the movable
«Qrizon, alwayes to the degrees of the
' iains Declination, from North or South,

alfo as m amp;ed Duls : Then is rhe Plain fe
g his Declination, t,;^. the diftance of the
^orizon between the Meridian, orPole-
PJacc, which »s alwayes 12 a clock, and the
jviendian, and Pole of the Plain, being al-
J^Ves juft where the Notch is iu the Mo-
Horizon.

Now aceoriifjg to thefe. HhUs, A Maijl
tliat declines 3 o degrees firotaSoudi to Welt,
The Stiles Elevationnbsp;TheSubftUe

from %2, 21-40. The Inchnation oi tW
Meddian 3^-24. The South Pole « eleva-
ted on the South-fide, and the North lolc
on the North-fide : And the Sun fhines on

the South-fide, ^from.grto 7 at night; ano
on the North-agam, till Shn-fetting,
working-as in the former Directions iseX'
prefled.nbsp;,0

Note^ In thofe Ereft Decliners, whoj
declinations is above 60 degrees, you ib^
find the Stiles Elevation to be very fmall gt;
therefore to make it exad, you muft ufe A^
tithmetical Calcularion j for the doing quot;
which, the Sphear, with due confideratioH»
gives the beft direitionSj with thefe Frafquot;^'

As fine 90 ,
To fine Declinatioh
So Co-tangent L^utMe
To Tang. Sübft. ïforn 12.

Wofr, If you fet P, at the Pole. Z, at
the Zenith. N, at the North-end of tlj^e
Horizon, at the Declination, Or Pole-plain'.
H, on the Plain, juft againft the moving
Semi, circle, or Subftile. A, at the Plain on
the Horizon. jE, on the Equinoctial. I, at
the Hour-circle, cutting the Equinoilial, fet
iuft againftquot; the upper Semi-circle. Note, Q^
is to be fet on the Plain, right againft the
Hour-circle, being fet to the Hour.

Having, I fay gt; by thefe Ru/es, and the
like, made and foynd the Requifites, then
proceed to draw the Dial thus j by help of a
^ftor with Sines and Tangents, to 7-5 ;
fuch as are ufually made.

The other 6 forts, yet behind, I Aall de-
monftrate only in two of them, which do
properly enough comprehend them all i and
the work of one, is as eafie as the work of
the other,, efpecially by the help of the
Sphear, where the hardeft is as plain as the
Horizontal. Therefore,

1. For Sotub Reel'tHers, they may recli«®
Ihort of, to, or beyond the Pole, at any Dc
dination, as the putting up and down the
Plain, doth plainly demonftrate. There-
fore, firft.

Set the Notch, or Pole of the Plain to the
Declination, and the Reclining Circle to it»
Reclination, and there make it faft; then
extend che Axis fl:reight, and bring the up-
per Semi-circle juft to touch it, and the
Hour-circle exa^ly even with the moving
Seini-circle.

The Thred brp.ught along the Plaiji
■while it touches the: Meridian, and that
fliews the Meridians Elevarion above the
Horizon , on the North Recliner to be
76-32 ; or its DcprelTion below the Hori-
zon in South-Reclincrs, and that from the
£aft-end, as the Sphear ftieweth.

3, The Subffite from the Perpendiculer
1-ine of the Plain,js 2i-tf, as the upper Semi-
circle flieweth; but from the hour 12, or
J^leridian 7-y 8, and ftands on tbe Eaft-fide
of the Meridian.

The Inclination of the Meridian is 3 3-25»,
as the degrees on the Equinoctial, between
the Meridian and Hour-circle, fhew.

All the Hour-Arks are eafily found from
the Plains Perpeudiculer Eaftwards and
Weftwards, by applying the Thred to-jhe
Hour-circle and Plain, being fet to the
Hours on the Equinoctial.

The South Po e is elevated in the South-
Reclmer, and the North, on the North In-

IT yt)u let Letters to the Sides and Angles,
uccording to the former difcourfe, yon will.
fee how all the Canons in the Arithmencal
Calculation lie, as I fhewed you before in
the Declinin^g Dials. And as again thus;

On the Pole fet P. On the Zenith Z. At
the Weft-end of the Plain, fet A. At the
Eaft-end B. ArtheSouth Pole of the Plain
C. At thé North Pole D. At the Eaft-end
of the Horizon E. At the Weft.end W-
At the North-end of the Meridian, fetl^L
At the South-end S. Where the Hour-circle
cuts die Plain F. Where tl?e Meridian cuts
thePlain.G. Where the fixed Semi-circle
cuts the Plain, fetE. A^ in thenbsp;be-

Tor tht Hours %» aU Dials, fay thus;
As fine of po.
To fine of Stiles height;
So Tangent of the Angle at the Pole,
To Tangent of the Angle on the Plain.

For thefe Plains alfo,you muft redifie the
^phear to the Latitude, and fet the Plain to
his Declination, and Inclination, which is

given, and for which you are to malcc a
North DecUning Rechning Dial. As yoa
did in the South-Rcdiner,- fo work m all
refpeéls, as you fliall bring forth the Qae-
ftta's, cither by the Sphear or Arithmetical-
Calculation, as is largely fhewn. Anlt;i
a Plain that declines yy dc^liCC t
Morth towards the Eait, and relines 30
from the Zenith, you fhall find the Requi'
fites to be as followech.

1.nbsp;The Meridians Elevation above the
Horizon, is found to be 63 deg. 5 8 min-
But yet obferve. You muft make ufe of that
part oP it which is below the Horizon, be-
caufe the Sun being Elevated high on the
South-part of the Meridian, muft needs eaft
a fhadow on the North-part thereof; there-
fore in drawing the Dial-part, part is onl/
to be made ufe of for the Sun to fhine on.

4.nbsp;The diftance of the Meridian of the
place from the Pole of the Plain, is 70-33'
The Angle between the 2 Meridians 74'3
The Hour-Arks are found by applying of
a Thred to the Hour-circle and Plain ; the

The Norch Pole is Elevated ; and in re-
gard the Plain dedines to the Eaft, the Stile
Q^uft be fet towards the Weft, and it ihines
On the Plain in Summer-time, from the Ri-
fing unto 12: But in the Winter-time, but
a few hours.

Notealft, That thefe Declining Reclin-
ing-plains, may be referred to a new Lati-
tude and Declination, wherein they fhall
become Upright Decliners, as before is hin-
ged.

quot;^he Poor^Mant Dial-Sphear gt; Or a-
notber way to demonjlrate theMyfiery
of Dyalling, both for Declining and
Inclining Vlains, in a very plain,
eafie way, for one 6tb part of tbe cojl
of tbe other Brafs-Spbear.

^ A S to the Vefcrlption, the figure an-
nexed, and a few words ftiail fuffice ;
^■herein confider;,
Firft, The plain flat-Board, reprefenting

Secondly, the two upright piecfc, as
Eaft and Weft-points, as A,F, andbt, to
fupport the moving Plain.

Thirdly, The Moving-plain, moving to
any Inchnation, on the two Points E and F,
with 180 degrees upon the Plain, and noted
hyABEF.

Fourthly, Alfo a Brafs-cirde as G,
ftenedto the Plain, to fet it to any degree
of Inclination; and a skrew, as at H, that
may ftay it fteady, when fet to any Reclina-
tion.

Fiftly. On the middle of the Horizon-
«al-board, is fattened at the Point M, a true
Horizontal-Dial, drawn fit for your Lati-
tude, and to turn round on the Point Mj as
IMKL.

Sixdy, A Thred fattened in L, the Cen-
ter of the Horizontal-Dial; and in N, the
Center of the Plain ; to be both a Stile for
the Horizontal-Dial,, and to reprefent the
Axis of the World ; alfo a fmall Wood-
den-Quadrant will be ufeful, fuch a one as
half the Plain is, to draw Perpendiculcrs,
and meafure Angles, as afterwards in the
Ufes. The Ufes follow.

Apply the fide A B to the Wall, and
hold the Inflrument level, ^s by help of a
Point Plwmmet, fattened at N, and the
Point playing right onM, it is eafie to do;
then by the Triangfiler-Quadrant, having
firft obferved the true hour, turn the Hori-
Xontal-Dial about on the Point M, till the
fliadow of the Thred (or Axis) fhew the
lame Hour ; then the Point onquot;the North-
end of the Horizontal-Dial, fhall fhew the
true Declination of the Plain. For any
South Decliner, the ufe is obvious.

But for North-Decliners, you muft turn
the Plain out of the way of the Thred, ftill
keeping the fame fide, AB, to the Wall;
and if the Horizontal-Dial hinder, put a
Parallel-piece between, as y our Rule, or any
other thing, and you ftiall have die Point
give the Declination on the Southern Semi-
circle on the fixed Horizon.

Tht Dtd'tnauontfanj ErtB DtcH^'tr gi-
ven, to find the SukftiUj and Stile, In'
clination of Meridians, and every hour
and quarters diftance from I2,
the Perpendiculer Line on the Plain.

Firfl, Set the Point at 12 on the Hori-
zontal-Dial, to the Declination of the Plai»,
toward the Eaft or Weft, and fet the Plain
Upright.

Apply the fide of the Quadrant to the
Plain, and caufe the fhadow of the Thred
to play Parallel to the perpendicular Ray of
the Quadrant, and at the fame time itfhall
ihew on the degrees on the Plain, the true
Subftiles diftance from 12.

Suppofe the Plain decline 20 degrees
South-weft, you Hrall find the Subftile to be
T J deg. and 11' from 12, and to ftand on
theEaft-fide of 12, in a South declining
Weft 20 degrees. Latitude y 1-30.

j4gain,for the Stiles Elevation.
Apply the Quadrant to the flat of the Plain,
on the Subftile Line, fo as the Thred may
cut the Center of the Quadrant j and then

The lhadow of the Thred when it cuts
the Subftile ly deg, 12' on the Plain, fl^all
on the Horizontal. Dial cut i hour ^(îmin.
Which reduced to degrees, is 24deg. 50 min.
the Inclination of Meridians.

Turn the whole Liftrument about, (as it
is then firft fee) till the fhadow of the Thred
ftall fall on every hour and quarter, and
then the fhadow ftiall cut on the degrees on
the Plain, the diftance of every hour and
quarter from 12, for that declination, in de-
grees and minuts ; which you may draw in*
to a Tabley for your , ufe and purpofe ; or
hereby examine your more exadl Calculation,
and prevent all grofs miflakes in your for-
mer work.

North-Dials, the Stile ihould look upwards,
which will appear fo to do, if you turn the
Inftrument the bottom upwards, for the fur-
ther help to your fancy ; but obferve that the
Hour. Arks, and Angles, are the fame for
the North, as for the South, only the dif-
ference is in the Scituation, as to the contra-
ry-fide, and looking upward inflead of the
South Décliner, looking downward, as by
turning the Inftrument appears -, fo that if
you draw the Dial as a South-weft, when
you would make a North-weft j and let
right figures, and the right way, and then
your work is effected to your mind, to the
right intent and purpofe.

Set the Point at 12, to 30 degrees Wefl:-
ward, and apply the Square to the Plain*
till it juft touch the Thred j and on the de-
grees on the Plain, it cuts 21-40 for the
Subftile 5 and at the fame time almoft half
an hour paft 2 for Inclination of ' the Meri-
dians ; and applying the Quadrant to the
Subftile-Line on the Plain,, and to the
TThred ; it cuts 32-35' for the Stiles height,
being the fame, and the fame way found
as for the ?owth Decliner Eaft,

out ehfcrve^ That for the Hour- Arks^you
»nuft note,Thac the North-Dial cannot ftew
at Noon, nor any Hours yery near
Noon, which will be feen on the South E)e-
tliner Eaft ; Therefore 4 in the morning, is
here called 8 ; and y is called 7'i and 6 is
6; 7 in the morning, is called f ; and 8 is
to be named 4: And if you turn the Inftrur
i«ent, that the fhadow of the Thred may fall
On thofe hours, it will alfo cut on tbi de-
grees on the Plain, the true Hour-Arks re-
Siuired.

For 8, it fheweth it not 5 at 7-,it ftiCweth
77-00 J at 6, it Iheweth 58 deg.f x minuts 5
®t 5, it cuts on the degrew on ,thc Plain
45'-3 8; at 4 in the morning, it cuts on the
Plain 3 5'-27; but. the ftiadow falls then on
Hour of 8, on the. Horizontal-Dial.

Alfo note. That tliefe numbers are not laid
from the Subftile, but from the Plains Per-
)endiculer, which in all Upright Plains is a
'Perpendiculer Line j and in all other Plains,
a Perpendiculer ^ to, the Horizqntal-Linp,
drawn on the Plain. And thus proceed
^ith any other; the affedfions are beft feen
^'hen you turn the Inftrument the upper-
part downwards.

Set the Pointto 8o;.as before S. W. then
theThred and the Qtradrant lhall (he^^^
38.2 for the Subftile V and 82-8 ontheHo-
Titontal/ for the Inclinatibit of Meridians;
and 6-'i2^-for theStiiesBevation ; ■ and the
lhadow of the Thred on the Horizontal'
Dial, will ftiew ydfl ho# clofe and inconvc
nicnt the Hours will be,'if not helped by the
formet^ dire6l3on$; and in like manner wiU
the Nbfth-Eaft ofWa be, and Iikewiic
helpfid,

To find tht Rtquijites, and Hottr-Arkh
from tht Ptrptndicfiltr of a Dtclinitii
Inclining Plain, with its AfeBiens.

Set the Point at 12 to theDeclinatIon,antl
move the Plain by help of the Arch, o^
Circle of Brafs, to the Inclination, and witP
the skrew make it faft: and fteady in tha^
place.

Then for the Subftile, Apply the Qua'
Vdiant tQ the Plain, and alfo Perpendicul«^

to the Axis, as the edge of the Quadrant
• being Aicjc, will neady fhew ; then the
Thred wiD fliew on the degrees on the Plain,
die diftance of the Subftile from the Per-
pendiculer, or the complement thereof fronj
the Horizon j which Point note with a fpot
of Ink; for, when the fhadow of thé Thred
falls on that fpot, on thé Horizontal Dial,
it fhe\mh the Inclination of Nieridians ;
that is to lay^ on What hour and minut, the
Cock of the Dial flxould ftand right over.
. i^lfo. The Quadrant, applied to the
Plain and Thred^ on the Subftile-Line,
^eweth the true Stiles Elevation above the
Plain.

hftijy making the fliadow of the
Thred to fall on every Hour on the Hori-
^ntal-Diaij it ftialf at the fanie time ftiew
now many degrées and minuts on the Plain,
■that Hour-Une ought to be from the Perj-
pendiculer, or from the Horizon; and alfp
quot;which way, either to thé Right or Left,Eaft
-orWeft; or from the Subftile, or 12; if
' you will trouble your felf to count it, from
tne|»Iace found out for the Subftjle, or

The Subflile, by applying the Square,you
ihall find to be 30 degrees on the leftjiamJ
of the Perpendiculer Weftward, and the
Inclination of Meridians 48-20, theStiks
height y 1-3 lt;5, and the Meridian on the
right-hand of the Perpendicuier-line 11-3°
Eaftward j and the lhadow of the Thred
playing on every hour and quarter, on the
Horizontal-Dial, will fhew on the Plain the
quantity in degrees from the Perpendiculer-
Line,nbsp;quot;

Set the Inftrument as before, and find the
Subftile, Stile, and Inclination of Meridi'
ans as before j But note,as to the Aftelt;£lio»sgt;
which way do thtu •

Turn the Inftrument the bottom upward)
and as near as you can guefs, turn the Plain
to its fcituation j then you ftiall firft fee the
Stile to look upward in the North-eaft Re-
cliner, which before was downward in the

The Subftile ftands on the right-hand
of the Perpendiculer, 30 degrees Weftward
( for obferve this alwayes, If a Plain de-
clines Eaftward, the Subftile will ftand
Weftward, and the contrary).

That the Meridian-Line is to be drawn
quite through the Center on the other-fide ;
becaufe, when the Sun is in the Meridian
above, it muft needs eaft thefhadow of the
Axis,or Stile, the contrary way downwards.

To find what are the mofi Honrs, that
the Sun can fliine on any Plainj whatm
foevir.

Firft, on all South Dire^, or Declining
IncUning-Dials, the mid-day-Meridian is
proper to it, unlels it incUne above 75 de-
grees, and then it becomes ufelefs in London
Latitude; then what hour foever you can
make the Sun to fliine on the Plain, and
.Horizontal-Dial both together, (the Sun be,
'ng at that hour above the Horizon) by ben-
ding or turning the Inftrument any way,

( when the Point at 12 is firft fet to the De^
clination) that, and all thofe Hours are pro-
per to that Plain, at one time of the year or
other,

That feveral Hours that ferve for the
South-plain, do, at fome time of the year,
belong to the North-plain alfo ; as by turn-
ing the Inftrument about, you may plainly
fee, cither by the Sun-fhine, or tv the
Thred, and your Eye cutting the Hour-
Lines and the Plain.

That if you would delineate a South Re-r
dining Plain, you may bring the Plain
toward the Thred, till it becomes a Polar-;
Plain.

But if it Redines below the Pole, then
conceive it to becopie a North Redining-
Dial, and work ?s is before directed, and
you fhall obtain your defire; for the Dials
will be the fame, the pne as the other, as
before was hinted at, in the Inchning-
Pl9ins.

Jit Dtcllnation of «ny Plain gmv, tofni
wb^ %tclin(itt9n willma^ it a.Ttlarquot;:
Dial, ^ttithf coHtrarj.

Set the North-point to the Declination,'
and bring the plain to touch the Thred;
then on the Brafs Circle is cut the Reclma-
tion required.

Or contrary ;
Set the Plain to the Rcclination _given,
and then bring the Thred to the Plain by

In like manner you may difcover a de-
clining Equinoaial, but not fo eafily when
the Subftile and Meridian are po degrees
aflunder; the Subftile bemg.then alwajes
in the hourpfd, as by movmg the Plain,
if the Declination be given; or by moving
tbe Jhted, if the Indination be fiiven, till
the ^uare, touching the Thred, it ihall
fhadow or bourn, juft upon 6 on the HQs
riz9ntal-Di^l.

That Eaft and Weft Recliners, and In-
cliners are difcovered after the fame man.
quot; quot; 0 4

In Eaft and Weft-Dials, that the Stile
hath no Elevation, but is parallel to the
Plain ; and how the Meridian lieth in the
Horizon, in Eaft and Weft Recliners, and
Incliners.

Many more Ufes might be infifted on,
which I fhall leave to the fcruteny of the
induftrifius Praaitioner» in the /irt of
Shadtws,nbsp;.

HoTigt; to remedy federal InconyeZ
niencies in the uftng of the
Jrtlficial Lines of Numbers^
Sines and Tangents^ as they
are ufmlly made*

1 der one degree of Sines and Tangents,
then the Line of Numbers will fupply it,
^ving due refpedf to the increafe of the

So is the fine of ideg. ic', the Suns di-
ftance from the Equinodfial, to 0-28,
the Declination which falls beyond
the end of the Rule.

is70minuts. therefore by the Numbers
/4r, So is 70 minuts, the Suns diftance from
{he Equinoaial, to 28 the SunsDechnation
on the Line of Numbers,, obfervjiig to ex,
tend the fanie way, as from the firft to the
fécond tertjj.

2.nbsp;When you have occafion to ufe a luie
above 90 degrees, then you muft count the
fine of 80, for the fine of too ; and 70»
few no ; andlt;So, for 120.

So alfo, the diftance from 90 to do in the
Sines, ss the Secant of 30 degrees ^ and the
diftance from 5Kgt; to jo, is the Secant of 40;
or the Point beyond 90, that reprefents th?
Secant of 40,

3.nbsp;If the Extent bç too large for youf
Compafles, as from 4 y or 90, to 3 or 4 de-
grees; theninfteadof 90 or 45, make ufe
of a Point in the Sines or Tangents right a-
gainft the middle ï in the Lme of Numbers,
where you may have two Brafs Centcr-pins,
viz,, in the Tangent o£ j-43, and the fine
of y-45 ; and jhe extent from thence back-
ward or forward, ftiall reach in the Num-
bers, to the 4th proportional Number re-
quired.

Inftead of which, you may faf. As the
fang, of 5-43, toi-5i on the Numbers j
^is the Tang, pf ? f, po 6.43 on the Num-
bers dimini(hing a Radius; for as Tangle
to 1.1 y, a greater than that; fo is the Tang..
Pf 1 f, to a greater than ij alfo, vtx.. 0-43.

Secondly, in Sines amp; Tangents, or Sines
only^ where there is another Caution to be
obferved, As fine po, to fine 10 j fo is fine
io, to fine of 3,241.

To work this wjth fmall Compafles on a;
large Line, dethta; Note, ;hat at 10 on the:
^ine of Numbers, or Sine of po, or Tang.

45-, is one compleat Radius; but at the
'diddle I, on the Line of Numbers, is a
place, or Radius, Ms; wherein the Loga-
'ithti, Sines, the Chara6teriftick is 8. A-
Rain, at the fine of 0-341, the Charaaeri-
^ickis7, ( and at 3 minuts it is d,) which
do note the feveral decreafings of the Ra-
Jmlfes; Therefore fet the difianc? from one
Number given, to the next neareft place a-
gainft I, or next Radius, as far from a
^eater or a lefs Radius, as your occafion
lerves, and note the place.

In this Operation, the extent from the
lomtatj-45 on the Sines, to the fine of
degrees, I fet the fame way from the

thus; As the place againft the middle
inftead of 90, tothe place laft found for
10 ; fo is the fine of 20, to fine ol 3 oeg.

But in thofe Lines of Numbers, Sw«,
and Tangents, where the Number is double,
this is performed by working a-crofs only.

4. When the laft term in Tangents hap-
pens to be above 47, then the remedy
two wayes,

Apply the end of the Rule, next pO,
clofe and even with any thing on which the
Point of the CompaCfes may ftay, till yoö
take from thence to 4J, for that diftance
laid from 47, fiiaU reach to 49-07, reading
the Tangents as numbred beyond 4J.

The CotrvpafTes being fet fröm the fine of
ao, to the fine of 90 ; fet one Point m the
Tangent of 45, and turn the other on the
Tangents, and keep it there fixed ; then re-

iiovethe other from 4y, and clofe it to the
ïhird term, being here the Tangent of 3,0 5
tiien this laft Extent laid from 45:, fliall
'each to 49-07, the Tangent required.

Take the excels of the'firft Number a-
hove and fet it the fame way from the
Second Number ; then the Extent from the
lecond Number to 4f, fliall be the true di-
ftance between the firft and fecond terms.

' For the Extent from 45-, to fi-jo on'
tne Tangents, fet the fame way from 30,
does reach to about 24-30 ; then the Ex-
tent from thence to 4y, fliall reach from
40 to 21-04 on the Tangents, the 4th
Number required.

If it had been from a Tangent above 4f j
to a fine, the fame way would have re-
medied the defcél.

The Compafles fet from the firft tern*
fine poj tofine of 3 o thefecond,alels ; then
fet one Point in the Tangent of 45, and
extend thé other backwards in the Tai^ents,
and note the ^ce^ keeping one Point there
elofe, the:other to 50 théthird term (being
abave by coiintingbackwards) Then, I
fay, that Extent laid froth Tangent 45-, (hall
reach to Tangertt 30-43^ the 4th proportio-
nal Tangent required^

Jf the Proportion had heen increafiijgj
then there had been no trouble at all.
tAt[ö Mtèf
That working a-crofs, or changing the
terms, is a good remedy alfo.

As fine pó, to Tang. 50; which is proper-
ly incrcäßng, for the Tang, of yo beings
more than the fine of po, yet taken on the
Rule from pd to 40 j the complement there-'
of^ as if it were decreafing ;
amp; is fine 30, to Tang. 30-48, the contrary

7. Laftl/j When one or two Radiuflci
(ôïAlteraticais of the Charafteriftick) feUs
oetweCB the firft and fécond term.

Firft, By thé Line of Nuniibers only •
As 8000 is to 10, So is yooo to lt;î i, or

To work this properly, and naturally, thtf
tiiiite on the Numbers fhotild be four times
tepeated, which is (eldom more done than
twice, as here : But this, and any other, by
the Line of Numbers is not interrupted, ha-
ving a duerefpe£t to the Number of Places.
For to work this, the beft way, is changing
of terms thus; As 8000^ to yooo in the
feme Radius ; fo is 10, to 6-2^ in the fame
Radius alfo. Or, without changing ; As
8000, to the next I ; fo is jooo turning
the Compafles the fame Way, to Ö-2 f.

Example. As fine po, to t degree (or
Uflder); fo is fine 30 degrees, to fine 30
rninuts, This being too wide an Extent for
the CompafTes, the third Rule is a remedy
for

repeated, as in Mr. Ougbtred's Cirelts of
Prof onion, is as eafie as may be -, being fure
to remember the number of RadiuUes be-
tween the firft and fecond term, that you
may have fo many between the third and
fourth term alfo.

Much more might have been faid as to
this ; but this Obfervation being alwayes
kept, That as the Extent from the firft term
to the fecond, is either increafing or de-
creafing ; So alwayes muft the Extent be
from the third to the fourth, increafing or
decreafing, in like manner, when you ulc
Sines and Tangents; And Numbers alio,
except, as before, in a few particuler Rules i
then you will be truly refolved.

where it performs the Ufes of the
Davit-^adt ant, the Crojs-Stajf^
Bojv, Sinicalnbsp;SeHor,

with is much cafe and exadnefs
as any, or all of them, will do in
Obfervation of Operation, Natu-
rally or Artificially.

Being firildius Contrived,and made by John
Brown, dwelhngat the fign of tlie Sfhiar
and S««-DM/ in the CMinorics, near to
Aldgatt, London.

^■ondnn. Printed by John Darby, for John Wipg-
are ro be l.ild ac his houfe in Cmtcbed
fryers ; and by John Brown at the Sphctti anc
Sni'Dial 111 the Minories j and by Jebn Selie
.^Uj^frmitage-flairs m fVarpinS'

THe Defcripcion of the tnflrumenti is
largely and plainly fet down in the
firft Pan, and Fir(l Chapter.

But, in regard that is the general Defcrip-
tion of all the tines that can conveniently
be put on, and thofe neceflary for this ufe
being far lefs, I lhall repeat the Defcription
again, as far as concerns the ufe thereof for
Sea-Obfervations.

I. Firft for length, it ought to be twci
foot long at leaft, when ftiut together, and
not above 3 foot at any time forSea-ufesj
(but for Land-ufes it may be 6,S, lo, or
12 foot in length, td find Altitudes or di-
ftances to Seconds of a degree certainly).

i. the Form of it is the fame, as before,
vU. an opening Joynt of about an iiKh and
quarter, or half quarter broad eaeh Leg j
and 6 tenth parts of an inch in thicknel?,
with a Loofe. piece of the fame length,
breadth, and thicknefs, to make it an Eq^'-
lateral-Triangle. Aitbe Figure jheweth.

3. The Lines neceflary for Sea-ufes are,
firtl, the 180 degrees upon the moving-Leg
and Loofe.picce,numbred as before is ihew-
ed. Alfo, 60 degrees on the innermoft-
edgeof the Loofe-piece. The Kalendarot
Months and Dayes, and degrees of the Sun«
Place, and Right Afcention, on the move-
able-Leg.

For the fpeedy and ready finding the Sun?
place, and declination, which you may dquot;
to a minut at all times, by help of the Re(9:i-
fying Table, and Allronomical Gautions ol
Time and Longitude.nbsp;.

Alfo, on the Head-leg, is the general
Scale of Sines and Lines, to the great and
kfler Radius, m in tht Figure. And thus
much will ferve both for Obfervation and
Operation, as in the following Difcourie
will fully appear.'

4. To this Inftrument doth chiefly be-
long the Sights for the Obfervations at Sea,
^nbsp;where

[ 229 ] ^ . ^
where the Horizon is made ufe of in the tai
king the Sun or Stars Altitude.

And to this Inftrument belongs the In-
dex and Square, thac makes it a moft com-
pleat Sinical-Quadrant, for the plain and
eafie refolving of all plain Triangles.

Alfo, a weighty Plummet and Thred,
and a pair of large Wood or Brafs Com-
paflcs for Operation.

Thus much for Defcription, being all put
on one fide only, unlefs you ftiall be pleafed
to add the Artificial Numbers, Sines, and
Tangents on the outer-edge, and a Meri-
lt;^ian-line, and his Scale on thfi inner-edge ;
3nd Natural Sines, and Natural Verfed-
Sines on the Sedor-fide : But thefe as you
pleafe.

THat the Difcourfe may be plain,
brief,and general; there are lo teriu^
to be named and defcribed, before I conie
to the Vfes and Examples, which are as fol'
loweth.

1.nbsp;Firft, the Head-leg of theInftrumeflt
in which the Brafs-Rivit is fixed, and abo«'
which the other Leg türns, as A B, in th^
Figure; on which Leg, the general Scale 0»
Sines and Lines are yfually fet.

2.nbsp;The moveable-Leg, on which tbe
Months and Dayes be, as in the Figure, nO'
ted by B D ; which Leg turns about tb^

7'he Loofe-picce that is joyned to tbc
Head, and moving-Leg, by two Tennons at
each end thereof, noted by D A in the Fh

the round-part of the Head-leg, being Cen-
ter to the 60 degrees on the in-fideof the
Loofe-piece i which Point is kno\vnbyB,
in the Figure.

J. The Leg-Center, being near the end
of the Head-leg, which is the Center to the
degrees on the moving-Leg, and out-fide of
the Loofe-piece, being in all j8o degrees 5
and noted in the F/f«rf by the Letter C.

6.nbsp;The great Radius, or greater Line of
Sines, ifluing from the Leg-Center toward
the Head, having the Tangents on the raove-
able-Leg to the fame Radius j and tbe mea-
fure from the Leg-Center to the Tangent
on the moving-Leg, a Secant to the fame
Radius i asC E in the Figure.

7.nbsp;The little Radius that ifTues from the
Leg-Center toward the end, having the
Tangents, on the out-fide of the Loofe-piece
to the fame Radius, and the meafure from
the Center to thofe Tangents for Secants, to
the fame Radius i asCF.

8.nbsp;The Turning Sight alwayes to be
skrewed to the Head, or Leg-Center, known
by his fhape and skrew-hole, as

J) The flidina Honzon-fight to Hide on
the raaving-Leg and Loofe-piece,noted with
its bi^uefs and hole to look through, as
quot;nbsp;p ^nbsp;10. Tlif

io. The /Ladow Sight, and 2 others, to
pin the Inftrument together, which you may
call the Objcft^Sights, alwayes fixed in the
two holes at the ends of the moving-Leg,
and the Head-leg ; and the Ihadow-Sigh:
is to fet to and fro to any place required ;
noted in the Figure with and the other
two with And

Thus you have their Name and Defcrip-
tion at large, which in brief take thus for
eafie remembring,

1. The Head-Leg. 2. The Moveable-
Leg. 3. The Loofe-Piece. 4. The Head-
Center. y. The Leg-Center. 6. The great
Radius. 7. Tlie lefs Radms. 8, The turn-
ing-Sight. p. The Horizon fliding-Sight.
10. The fhadow-Sight, and thetWoOb-
jeft-Sights; the open-part in one is next to,
and the other remoter from the Rule, to an-
fiver to the upper or lower-hole in the turn-
ing-Sight, according as you pleafe to ufe
them in Obfervation.'^

To find the Suns, or a St«rs Altitude, iji
a forward Ohfervatitm^Hi hy a Fore-fiaff.

Center, and put the objcdl-Sight into the
hole at the end of the Head-leg, and put the
Aiding Horjzon-fight on the in-fide of the
Loofe-piece ; Then fetting the turning-fight
to your eye, and holding the Loof^piece
in your right-hand, and the mov£able-Leg
toward your body, then with your Thumb
on the right-hand, thruft upwards, or pull
downwards the Horizon-fight, till you fee
the Sun through the Objedf-fight, and the
Horizon through the Horizon-fight; then
the degrees cut by the Line on the middle
of the Horizon-fight, lhall fliew the true
Altitude required.

Alfoohfnve, That if you Hke to ufe the
upper or lower-edge of the Horizon-fight,
inftead of the fmall bar a-crofs the open-
hole, after the manner of the ends of a
Fore-ftafF, that then the degrees and minuts
cut by the edge of the Brafs, is the Alti-
tude required, to be counted as it is figured
from the Ohjedf-fight, toward the Horizon-
fight ; the degrees between them being the
Angle required.

Note alfo. That if the Altitude of the
Sun, or Star, be above 3 c degrees, you will
find it a hard matter, to behold the Horizon
and Sun with a bare rolir^ the ball of the
eye toly, and -« ftirring of the head, will

cafily Caufe a ftiriing erf the hand, which
will fpoil the exâétnefs of Obfervation, un-
Icfs the Inftrument fhall be fixed to a Ball-
focket and Three-legged-ftaflf, which is not
ufual at Sea.

Therefore to remedy this, you may ob-
ferve with the open oval-hole in the turning-
fight fet to the eye, or taking the turning-
fight quite away j Obferve juft as you do
with a Fore-fbffe, fetting the round part ot
the head, to the hollow-part befide your eye?
fo as the Head-Center-pin may be as near
the very fight of your eye as poiftbly as you
can ; which Center is the Center to the de-
grees now ufed in a forward way of Obfer-^
vation. Or, rather ufe this way when the
Weather will fuffer, by a Thred and Plum-
met, which I fliall add as a fécond Ufe-

To obferve the Sun or a Stars Altitude^
by a forward Obfervation, ufirg tht
Thred and Plummet,

Skrew the turning-fighc to the Head-Cen-
ter, as before, and put the two Objeit-
fights into the two holes at the two ends of
the Rule; and on the Leg-Center-pin hang
the Thred with a weighty Plummet of two

Then hal^M^thf^Trlanguler-Quadrant^
fetting the fma 1-hole on the turning-fighc
clofe to your eye; and if the Sun,or Star,be
Under 2y degrees high, then look to the
Sun or Star through the turning-fight, and
thatobjedf-fight, ^vhich ftands in the end of
the moveable-Leg, letting the Thred and
Plummet play between your Thumb and
Fore-finger, as a Brick-layers Plummet in
his Plum-Rule doth in a bendid hole, that
you may keep it in order whilft you look at
the Sun or Star, and the weighty Plummet
will pull the Thred ftreight, and let you
know by feeling which way it is playirtg,
till it playeth evenly and truly, whilft you
have the Objed precifely in themidft there-
of, whether it be Sun, Moon, or any Star,
or other Obje6V, \vhofe Altitude you would
obferve; Thefn, I fay, when the Plummet
playes well, and you behold the Objcdl
right, bend back the Quadrantf and fee what
the Thred cuts on the degrees on the move-
able-Leg, which fliall be the true Altitude
required ; And in my opinion, muft needs
be more exaft than any other way of a for-
^vard Obfervation, becaufe you are not trou-
bled to mind the Horizon and Sun both at
at once.

An Objeftion may be, Tht holflerom
windsy and the roullng of the Ship, will hin-
der fuch an Obfervation.

Again, I anfwer. One Objedl is better and
more certainly feen, than two at any time
together; and though the Wind blow
hard, if you can (land to obferve at all,the
heavy Plummet will be fure to draw the
Thred Perpendiculer; and for ought I
know, you may come as near this way as
any other; however this, at moft times, may
confirm and prove the other, and may be
ufcfful in Rivers, and Harbours, and mifty-
Dayes, when you may fee the Sun well e-^
noiigh, but not the Horizon at all.

To find tht Suns Altitude by a backward
Obfervation, a* vtith aBackrJIaf^ or
Uivis-Quadrant,

Skrew the turning-fight to theLeg-Cen.
ter (orCenter to the degrees on themove-
able-Leg ) j and fee the objeif-fight to the
long ftroke by oo-lt;Jo on the out-fide of the
Loofe-piece, and put the Aiding Horizon-
fight on the out-fide of the moveable-Leg ;

men hold the Objeft-fight upwatd$,and the
fmall-hole in the piece turning on the «dge
(or to the fmall-hole in the middle) of the
Horizon-fight ( which you pleafe) clofe to
your eye i and looking through that hole,
and the middle-hole of the turning-fight,
to the true Horizon, turning your felf about,
and lifting upj or preffing down the Hori.^
ion-fight, c ofe to the moVeable-Leg, till
the fhadow of the upper-edge of the £ha--
dow-^fightj being next to the Sun, fall at the
fame time juft on the middle of the turn-
ing.fight ) Then, I fay, the edge or middle
of the Horizon-fight, that you looked
through, fliall cut the true Altitude of the
Sun required.

Skrew the turning-fight to the Leg-Cen-
ter, and put the Objed-fight, whofe oval-
hole is remoteft from the j^adrant, in the
hole in the end of the HeajXeg, or rather
»n a hole on the general Scale, between the
turning-fight, and the Sun j and put the

Horizon-fight on the out-fide of the move-
able-Leg i then hold the turning-fight to-
waratheSun, and the fniall-hole inthe
edge of the Horizon-fight to your eye j then
look through that hole and the turning-fight,
tilt you fee the fliadow, the ObjeóVfight, to
fall juft on the turning-fight, or the fha-
dow of the turning.fight to fall juft on the
objeft-fight, which is all one, though the
firll be more eafie, becaufe you ihall fee the
Horizon through the turriing-fighr,and that,
both at once; Then, I fay, the degrees cut
by the Horizon-fight, fhall be thj Suns di-
ftance from the Zenith required; Being the
very fame work, and done in the fame man-
ner, and producing the fame Anfwer, viz,'
the Suns diftance from the Zenith, that the
Davii-Quadrant doth.

Note, That this way you may obferve
very conveniently, till the Sun be 20 de-
grees diftance from the Zenith j and by the
adding of a tfoArch, as in Davis Qua'
drant, of to 45 will be enough, it will do
as well as any Davii Quadrant, being then
the fame thingi

But I conceive, the complement of thé
Altitude being the fame, will do as well •,
which Altitude is better found by this In-
iuument, than the diffance from the Zenith

To find tht Sum Altitude when near thé
Zenithy or above po degrtis abovtfowt
part of the Horiz^on.

in fmall Latitudes, or in places near the
Equinoéfial, or under it; the Sun will be
found to be in, or near the Zenith: and if
you count from fome part of the Horizon^
above 90 degrees diftant from it; then in-
ftead of fetting the Aiding Objeft-fight, to
the long ftroke at 00 on the Loofe. piece,
you muft fet it 3Ó degrees mere towards the
Head-leg ; then obferve, as you did before,
and whatfoever the Horizon-fight cuts, you
muft add 3 o degrees more to it, and the
j'um fhall be the true Altitude required.

Suppofe that in the Latitude of 10 deg.
North, on the loth of Jane, when the
Suns Dechnation is 23 degrees and 31 min^
Northward ; Suppofe that at noon, I obferve
the Suns Meridian Altitude, skrewing the
Turning-fight to the Leg-Center,and fetting
the Objeft-fight to the 30 degrees on the
Loofe-piece, near the end of the Head-leg,
and the Horizon-fight on the movable-Leg ;

then hold up the Quadrant, with the (ha^
dovv-fight toward the Sun, and the fmall-'
hole,in the Horizon-fight toward your eye,
and look to the Horizon through that, and
the turning-fight, 'the lhadow. of the right-
edge of the fhadow-fight, that cuts the de-
gree of 30, at the fame time falling on the
middle of the turning-fight, you lhall find
the Horizon-fight to cut on 46-29 minuts 5
to which if you add 30, the degrees, the
fhadow-fight is fet forwards, it makes up
the Suns true Altitude on that day
in that Latitude; 76-29 the Meridian Al-
titude, and 33-31 the Declination, added
together, make 100 deg. 00 ; from which
taking po, there remains to, the Latitude
of the place,

1.nbsp;/«fA« Ohftrvatlon^ firjl you may nott
this, That if you had flood with your back
toward the South, you would have had 103
degrees and 31 minuts, for the ilidmg Ho-
tie.on-fight would have flayed at 73 degrees
30; to which if you add 30, it makes
103-31; which a Davis Quadrant will
not do.

2.nbsp;In the holding it, you may lean the
head of the Rule to your breail, quot;and com-
mand it the berter, as fo fteady holding.

. 5' You may turn the Turning-fight
bout, to any convenient Angle, to make it
fit to look through to the Horizon, and al-
fo to receive the fliadow of the Ihadow-
fighr.

If the brightnefs of the Sun o^'end thé
éyes, ybu may eafily apply a red pr a blue
Glafs, to darken the Sun beams, and the
Sights may be painted white, to make a iha-
dow be feen better.

I'o fni the Latitude at Sea^ hy a forward
Meridian Obfervation if the Alcttuds^
according to Mr Gunter'/ Bow.

Sk'tew the Turning-fight to the Leg-Cen-
ter,\and fet the'lhadow-fight to the SunS7
Declination, and the. Horizon-fight ,ro the
moving Leg(orLoofe-piecc), and theTuin-
ing-fighc CO your eye,; then let the Ihadow-
%ht cut.the Horizi)n, andthe Horizon-fighc
the Suii, moving It higher or lower till ic
fits ; then whatfocver the fight Iheweth,
adding 30 degrees to it, is the Latitude of
thjt place required.

Suppof: ortthe,ioch of tJAfarch^ when
^he Declination is oalyf io' to the North-
er 'nbsp;ward,

ivarcl, as in the firft after Leap-year ins;
then fe: the edge, or ftroke on the middle of
the fhadow.fightto lo' of Dechnation to-
ward the Head, and che Horizon-fight, on
the fame Leg toward the end, and Aide on-
ly the Horizon-fight till it cuts the Sun, and
the other the Horizon 5 then fuppofe t lhall
ftay at 21-30 : then if you count the de-
grees between the two Sights, ic will a-
mount to the Suns Meridian Altitude; but
if you add 30 degrees to what the Sight cuts,
itfhall give the Latitude of the place where
the Obfervation was made for 21 and 30,
to which if you add 30, it makes 51-30,
the Latitude of London, the place where the
Obfervation was made.

Note hire. That in fmall Latitudes the
, Sun will be very high, in Summer time efpe-
cially, and then the ftiding-fight, muft be
fet on the loofe peice.

/ij thfu for ExampUi
Suppofe on the icth of May ilt;J70, when
the Dechnation is 20-7 in the Latitude of
30 Degrees, I obferve at a Meridian Alti-
tude, i fliall find the Hiding-fight to ftay at
00. on the loofe peice ; then it is apparent
that 30 Sdded to o, makes but 30 degrees
for the Latitude required.

pais beyond oo on the loofe pelce,then wbacquot;
'foever it is you muft take it put of 30, and gt;
'lie remainder is the Latitude required.

Suppofe on. the i ith of June 1670 you
j^ere in the Latitude of 10 degres to the
^^orthward, and ftanding wich your back
to the North, as you muft needs do in all
forward Obfervations in more Northern
Latitudes,, you fhall find the fight to pafs
jufl: 20 deg. beyond 00 on the Loofe peice ;
therefore 20 taken from 3 o the refidue is 10j
the Latitude required.

Suppofe that in t he fame place you had
obfeieved on the nth of December, \vhca
the Sun is moft Southwards, if you fet the
One fight to 23-31 Southwards againft the ,
1 Ith of December j then if you obferve for-
^vards with your face toward the South, as
before, you fhall find the moving-Sight to
ftay at 20 degrees beyond cb on the Loofe-
riece^ then, I fay, 20 taken from 30, refts
^o,the Latitude required, becaufe the fight
Pafled beyond 30 on the Loofe-piece.

Laflly-, if the Moving-fight Ihall happen
to pafs above 30 degrees beyond 00 - on the
^oufe-piece, when the other Sight is fet to
the

quot;your face toward the South (part of the Me^
ridian) ; then, I fay, the Latitude is South-
watds as many degrees as the Moving-fight
ftands beyond ? o, on the Loofe-piecf to-
ward the Head-leg.

So that the general Rule is dtwayes, ill
North Latitudes (obferving the Suns Mcri'
dian Altitude, to find the Latitude,by a for'
ward Obfervation, according to
Borv J, your face muft be toward the South}
Although that thereby in fome Latitudes»
the Altitude may feem to be (as indeed it '
above the South-part of the Horizon) abov«
degrees.

If the fliding-fight ftay any where on th«
Moving.leg, or Loofe-piece, fhort of oOf
add it alwayes fo 30, and the fum fhall b^
the true Latitude North ; if it pafs beyon'^
00, then fo much as it doth^take out of 30»
and the remainder is the LatitudeNorthi
but if It fhall ftay juft at 30 on the Loof^'
iiece, then the Laticude is 00 ; but if it pa^''
)eyond 30, then (o much as ic is, the Lati'
tude is Southward.

The fame Rule ferves, if you were i^
South Latitude, then you muft in forward
Obfervations, to find the Latitude,as with'
Cuntirs Bov, ftand with your face to ^^

Korth, and in fetting the fixed-figbt to the
Dechnation, you muft count South Decli-
nation toward the Head j becaufe thofe that
have Southern-Latitude, have their longeft
dayes, when thofe that live in Northern-
Latitude have their fhorteft dayes.

The fame Rule ferves for the Stars alfo,
for being inNorth-Latitudesj and obferving
a Latitude forwards, have , your face al-
Wayes toward the South, and fet one Sight
to the Declination, counting the Scars Nor-
thern or Southern Dechnation, the fame
Way as the Suns,(and the contrary in South-
latitude) ; then holding your eye clofe to
the great-hole of the Turning-fi^ght, flide
the Moving-f^ht till its middle-bar or edge
(as is moft convenient) cuts the Star, and
the other the Horizon ; then whatfoever the
edge of the Mpving-fight curs fliort of o,^
added to 30, or beyond 00, taken out of
30, ftiall be the Latitude required.

Suppofe the middle-Star oWrhns-Glrdle,
Whofe Declination is 1-28 South, beingin
the Meridian, I fet one Sight to 1-28 of
South-declination; and flipping the other-
fight till it cuts the Star, the fixed-fight be-
ing fet to the Horizon, you fliall find it Itay
in Che Latitude ofnbsp;at 2^-30 on the

Note, That if the corner of the Inftru-
ment be inconvenient for the fight to Aide i
on, as for about y degrees it will, then you
may remedy it by flipping the Sight fet to
the Declination 10 degrees more, any way '
that is convenient, increahn^ or decreafing J
Bett the» note. That inftead of adding even
30, to what rhe Moving-fight ftayeth at,
you muft add 10 degrees more, vis.. 40,
when you flip it towards the end of the ,
Moving-leg ; or 10 degrees lefs, viz.. 20
degrees, when you flip it 10 degrees more 1
toward the Head, as is eafie to conceive of.

. To find the Latitude at Sea, by a bac^ari
C^eridian Obfervation, according te
the way of Mr. Gunter'i Bow.

Skrew t^e Turning-fight to the Leg-Cen-
ter, and fet the fliding-Horizon-fight to the
Suns Decimation ( the middle or edge of ir,
as you can béfthke of) and the Shadow-
fight on the Loofe-piece, or Moving-legj
with your face alwayes to die North, in
North Latitudes ; or fuppofing your felf to
be fo, though it may be you are noi.

Then looking through the hole in the Ho,
rizon-fight ( ftanding at the Declination )
and the Turning-fight to the Horizon, with
your hand gently Hide the fhadow-fight till
the fhadow fafl juft on the middle of the
Turning-fight, as you do in obferving the
Altitude with a Davis QuairaHt; then, I
fay, whatfoever the lliadow-fight fliall ftay
at under o, add to 30 ; or over o, take out
of 30, and the fum or remainder, fliall be
the Latitude North : but if ic happen to
ftay at juft 30, the Latitude is 00; if be-
yond, it is fo much to the Southwards j
This is only the converfe of the former, and
needs no Sxample, but a few words to de-
monftrate it; which may be thus.

Suppofe that: on the nth of December, in
fayling. toward the Eafi-Indies, about the
Jfleof St' Mattkevps, luppofing our Ship
to be in North-Latitude, t fee the Horijon-
fight to 23-31, South Declination; and the
Shadow-fight on the Loofe-piece, then ftan-
ding with'ojy face to the North, as another
then would do, as ac other times, and looic-
ing through the Horizon, and Turning-fight
to the North-part of the Horizon, I find
thefliadow-fight when it playes well over

Confider that the diflance between the
wo Sights, is the AIntude of the Sun a-
bove the South-part of the Horizon ; which
if you do count on the TrtanguUr-Qua.
droKt, you will find to be 36-29, and 33,
which put together, make lt;^9-29, for the
Suns Meridian Altitude ; to which if you
add 23-31,his declination, it makes 93-00
the diltance of the North-Pole and Zenith,
or 3 degrees of South Latitude; for had
you bsen juft under the Equinoaial, the
Altitude would benbsp;or had you

been more Northward, it would have been
lefs; therefore by confidering, you may
foon fee the reafon of the Qperation.

If the Shadow-fight be too near the cor-
ner, or too far from the Turning-fight to
eaft a clear fhadow; then, fet the Horizon-
fight that ftands at the Declination 10, 20,
or 30 degrees more toward the end of the
Moving-leg, and you lhall fee the inconve-
nience removed ; but then you muft take
10, 20, or 30 degrees lefs than thefhadow-
fight fheweth, for the reafon abovefaid.
The reafon why even 30 is added, is be-
caufe

faufe that o degrees of Dechnation, ftands
at 60 on the Moving, leg, inftead qf ^to,
or 00.

Note-, If you had rather move the lower-
fight than the upper, then count like Lati-
tudes and Declinations from 00 on the
Loofe-piece toward the Head-leg, and un-
like the contrary, and then fet the fhadow-
fighc fixed there ; then obferving, as in a
back-Obfervation, the Horizon-fight fhall
fliew the complement of the Latitude re-
quired, without any adding of 30.

Thus you fee, That the TrUtigttler-Qua.
dranty containing 180 degrees in a Triangle^
brings the lhadow-fight near the Center,
and with one manner of figuring, gives the
Suns Ahitude above the Horizon, backwards
or forwards, and his diftance form the Ze-
nith, and the Latitude of the place South
or Northgt; or North or South, backwards
or forwards, by the Sun or Stars, by one fide
only, as conveniently and with fewer Cau-
tions, and as exadfly, if well ufed, as any o-
ther Inftrument whatfoever ; So that by this
time you fee it is a Fore-ftaff, Quadrantgt;
and Bow. The other Ufes follow.

X» fi»i the Latitude by a Meridian Obfcrquot;
vatian, by the Thred and Plummet, by
tbe Sun or Stars.

This way of Obferving without a Hori-
zon, mufl be done by an AjlroUbe, which
is a Plummet it felf, or elfe with a Plummet
fitted to another Inftrument, and at fome
times may do better fervice than the Horizon,
and for an Altitude barely, is ihewed al-
ready.

Count the Declination, which is the fame
with the Latitude, from oo on the Loo'e-
piece toward the Moveable-leg ; and con-
trary Declinations, both of Sun or Stars,
count the other-way toward the Head-leg,
and thereunto fet the edge of the Horizon-
fight, that hath the fmall-hole on it.

Then let the Sun-beams fhine through
the fmall-hole on the Turning-fight, to the
fmall-hole on the Horizon-fight, the Thred
and Plummet duly playing, fhall fhew the
Latitude of the place required.

But if you look at a Star, having the
fame Declination, then fee your eye to the
Horizon-fight,, and behold the Star through
the Turning-fight, and the Thred fhall fall

on the Latitude required, when,you look
toward the South, being in Northern Lati-
tudes.

So alfo. When you tuyn your face toward
^he North, in obferving thofe Stars, it is befl
done when they come to the Meridian below
the Pole \ but for their coming to the Me-
ridian above the Pole, then their Declinati-
on is increafed by the quantity of their di-
ftance fronr the Pole, or the cornplement of
then Dechnation.

The Declination of the Pole-Star, when
in the Meridian below the Pole, is 87-20
from the Equinodfial; but when the fame
Star is on the Meridian above, the Pole,
then it is p2 deg. 40' diftant from the fame
Northern part of the Equinodfial.

So that if you make 60 on the moveable-
leg, to reprefent the North-pole ; then you
may count or obferve any Star that is
degrees diftant from the Pole, both above or
beiow the Pole; then adding 3 o degrees to
what the Thred falls on, fliall be the true
Latitudes complement required ; becaufe
you have removed the Pole from po to 60,
30 degrees backward.

The declination of tie uppermoft Star in
the great Bears hack^, is 65-45'; that is,
2(5-If below the Pole; or, 25-degrees 17
minuts above the Horizon, when on the
Meridian below the Pole ; but the fame
Star, when on the South-part of the Meri-
dian, is 77-47 above the Horizon,or 2.6-1^
above the Pole.

The Star being below the Pole, you may
fet the hole in the middle of the Horizon-
fight, to the Declination, counting 90 the
Pole, and looking up to the Star, as ufually,
the Plummet will fall on 38-28, the Lati-
tudes complement required.

The fame Star being on the South-part of
the Meridian, above the Pole, 1 count 60 on
the Moving-leg for the Pole, and26-i y be.
yond that Poje further, vix,. to 8511 c.
which is as far as you can well go, counting
lt;Jo the Pole; then obferving, as you did
before, you fhall find the Thred to play on
08-28, the Latitudes complement required,
for the diftance between 08.28, and 86-1 y
is 77-47, adding 30 degrees, becaufe of 60
inftead of po, for the Pole-point.

[ i^? ] .
Notty That the Thred playing near the
corner, may prove fomewhat troublefome to
bijferve, without help of another perfon ;
but if you will be exact in this or any other
Obfervation, a Staff and a Ball-focket,
fhould be applied to this, as well as too-
ther Inftruments, to fiand fteady and fure
in fhe time of Obfervation.

thefe wayes are ready and eafie, with-
out taking notice of thofe Regulatfons and
Cautions, which are to be obferved in find-
ing the Altitude, barely, as in the Seamans
Kalender, and Mt; Wrights Errors in Navi-
gation, is plainly feen.

But if you know them all, and had ra-
ther ufe thofe Rules in thofe Books; then, I
fay, a Thred and Plummet by this Inftru-
ment, will do as conveniently as any other^
or the three Sights and Horizon, as before is
fliewed, to find the Altitude.

THus much as for the way of Ohftrva-
tion 9 now, thac your Operation may
be true alfo, ic is neceffary that you have i
7 able of the Suns Declination, for the firft,
fecond, and third year , after the Leap-
year.

But in confideration^ that the/lt;rfi7«(f after
the Leap-year, is mean between the other
three-, I have riiade a Table for thzt, and
the Months on the Triangaler-Q»airant
are agreeable thereunto • and for the firft,
third, and Leap-year, have added a ReBt-
fjing Table to bring it to a m'inut at leafl to
the real truth, wherein I have followed che
Suns place, according to Mr. Streets Table
of the Suns place, for itf^tf.

In which Table, you have degrees and
minuts ; and a prick after, notes a quarter of
a minut ; and two pricks, half a minut ; and
three pricks, three quarters of a rainuc
more.

Now, by the Rule, you may count to a
minut, and the �ifyiKg Table tells you
how many minurs mare you muft add to,
or fubttraét from the degrees and minuts
the Table or %ule fhall fhew it is, in the
fecond year.

'A Table of the Suns 'Declination every Uy 4f Noon
for London in the year 1666,the fecondyear after
the Leap-year, according to Mr. Street'/ lafgt;les of
Longitude. Calculated by John BrovPn, 1668.

s 0 s	Janu.
Q-

w
n	D.M.
I	It 4?---
1	11 3Ó
3	21
4	11 14-
5	21 O^-'
6	20 51
7	20 40
8	20 17--
9	20 If
10	20 01•.•
11	1948-
12	19 34* •
13	19 20.
14	19 otf
il	18 SI
itf	1835...
17	18 20.
18	18 04
»9	17 48
20	'7 3''
21	17 I4--
21	16 57.
	16 40
24	16 22.
iS	16 Oj...
26	4Ï
17	15 27
28	15 08
	14 49
	14 19 .J
	(4 10-

I
1

6
7

If
i?
ts
17

II
21
»3

laj

A TMt of the LMaimtHdtSy Right afcention In
Hours and Minuts, and Degrees and Minuts, and
the Declination North or South of^'} fixed Stars.

Pole.Star.or lad in irtcle Bur.
Andromed»s Girdle»
Meduraei bead
Pctftns right (ids
Middle ot the Pleidej_'

BuUi eye
Hircu or Goat
Otionilcft foot
Mid.fta in Orions Girdle
Orioni right (hoaldtt

To find the Suns Declination for the year
itf/o, on the 12th day of May : Firft, if
you divide 70 (being the tens only of the
year of our Lord by 4, rejefting the 100')
you fhall find 2, as a remainder, which
notes it to be the fécond after Leap-year i
and if o remain, then it is Leap-year.

Look in the Table of Declination for
i666, the fçcond after Leap-year, as the
year 1670 is, and find the Month in the
head of the Table, and the day on one fide,
and in the meeting-point youihall fùid 20
deg/3 J min. for the Dechnation on that
day at noon required.

If you ufe the Trittuguler ^Madrttfit,
extend the Thred from the Center over the
%2tho£May, and you lhall find itTto cut in
the degrees jufl 20 deg. 31 min. the true
Declination for that year and day.

Nate, That if you have occafion to ufe
the Declinatiori before noon, then obferve
that the difference between ftroke and
ftroke, is the difference of Declination for
one day ; and by confequcnce, one half of

Suppofe I would have the Suns Declina-
tion the, i8ih ofnbsp;1666, at 6 in
the mïrning ; he^e, you mufl: note, that the
18th Rrokefrom the beginning of Augt^»
reprefefits the i 8 th d^ at noon juft.;, ^

Now the tinrie required being 6 hojifl be-
fore iioori, Lay thé THred one JouHh païC
óf the^ diftance for ône;d'ay-y toward
17thquot; diay, and then, in,the' depréâ, quot;tlic
Th^ed ftiall cut onnbsp;whereas at iibon

juft^ it AvÜl be but 9-3 8 ;' and the next, pr
ipthdày at noon, it is p degrees 16 mih.
and 3 quarters of a min. as the three Wlckj
thus .m the Tablé dqtK, plainly/jhew.
but by the Rule, a miriiit is as müfl^'^S«^
be feen, and fo nea? with care rùa^ .you
come.nbsp;, ;.nbsp;. . ...

Nott aljJb farther, . ,
That if you ftiall iife it^iii places tliirfre
4 hours, lt;î or 8, 10 or 12 hours more Eaft-
ward, or Weftward in Longitude, the fame
Rule will tell you, the minuts to be added
in Weftern-Longitudes, or to be fubftraJî-
ed in Eaftern-Longitudes, as Reafonand
.Experience will diàatequot; unto you with due
.confidcratiori,

.As far ExAmfW,
To find the Suns Declination for the year
i(Î70} on the i2th day of May : Firft, if
you divide 70 (being the tens only of the
year of our Lord by 4, rejefting the lOO')
you fhall find 2, as a remainder, which
notes It to be the fecond after Leap-year ;
and if o remain, then it is Leap-year.
then,

Look in the Table of Declination for
i666, the fecond after Leap-year, as the
year 1670 is, and find the Month in the
head of the Table, and the day on one fide,
and in the mectiiig-point you fhall find 20
deg. 31 min. for the Dechnation on that
day at noon required.

If you ufe the TrUnguler i^airimt,
extend the Thred from the Center over the
I2tb of Maj, and you fhall find iîTto cut in
the degrees juft 20 deg. 31 min. the true
Declination for that year and day.

Note., That if you have occafion to ufe
the Declination before noon, then obferve
that the difFerencc between ftroke and
ftroke, is the difference of Declination for
enedayj and by confequence, one half of

. ^['Asthmfor È^cample. , , ^
Suppofe I would hayé'the Siins DeclinaJ
tion the ; iSch ofnbsp;i666, at 6 in

the m'^ning ; he^e, you hiufl: note, that the
xiSth Rrbkefrom the beginning of
reprefehts the i 8 th d^ at noon juft.;', ^ ; ;

Now the tinie required being 6 hoU« be-
fore iioon, Lay thé T%'ed phe\fourih pkrc
of the' diftance for ôfie ; day, toward thb
17th day; ' and thai in .ifhe^ de^rééà, the
Th^ed ftiall cut onnbsp;whereas at lioon

jijft^it wUlbebut 9-38! ; and the next,
ipthday at noon, it is 9 degrees 16 min.
and 3 quarters of a min. af die three ^ickj
thus m the Tablejqth; plainly jheNv 5
but by the Rule, a miriuti is'as mUçli^'^S?^
be fecn, and fo near with care may ^you
come. , -,nbsp;■ . ;. ■■ -

That if you ftiall life iVin places tltàrfre
4 hours, lt;î or 8, 10 or 12 hours morcEaft*-
ward, or Weftward in Longitude, the fame
RulewilUell you, the minuts to be added
in Weftern-Longitudes, or to be fubftra6t-
ed in Eaftern-Longitudes, as Reafonand
Experience will diàate; unto you with due

For if being Eaftwards, the Sun comcs
to the Meridian of that place before it comes
to the Meridian of London ; then hy the
Thred as in morning hours : But if the
place be to the Weftwards where it comes
later, then lay the Thred fo many hours
beyond the Noon-ftroke fornbsp;as the

place hath hours of Weftern-longifude more
than London, counting ly degrees for an
hour, and 4 minuts for every degree j and
then fhall you have the Declination to one
minut of the very truth.

But if it happens to be the Leap-year, or
the firft or third year after the Leap-year,
then ;

Stippofe for tbc yth of OBabtr 1671, be-
ing the third after Leap-year, I would have
the Declination.

Firft, if you lay the Thred over the yth
of OÜober, in the degrees, it gives 08 deg.
42n»nuts,. , for the Declination inthefe-
cpnd year after Leap-year ; then, becaufe
this is the third year, look in the Retlifying'
Ttible for the yth of Oüohtr, and there you
find J. 4 .., lor fubftradt 4 minuts and a
half from 8-42 ,. rcfts 8.38, thetriiede-
clination required for the yth of Oüohtr
^671, The ike work ferves for any other
day or year; but for every yth andjoth

day, you have the Declination fet down iti
a Tai/f for all 4 years, to prove and try the
truth of your Operations; and by that,
and the Line of Numbers, or the Rule of
Three, you may continue it to every day
by this proportion.

As f dayes, or 120 hours, to the diffe-
rence of Declination in the Tab/e, be-
tween one 5 th day and another ;

So is any part of y dayes, or 120 hours,quot;
to the difference in Declination to be
4uUfd or [nhfiraBid to the $ dayes De-
clination immediately fore-going the
day required.

Suppofe for the i8th of Tthruar) itfdp,'
the firft after Leap-year, I would know the
Declination by the Table made to every jth
day only; On the 20tb of February, I
find li ' On the lyth day, 8-47;
the difference between them is i-yj ; then
the Extent oF the Compaffes from j, the
Number of dayes, to 1-J3 .. the minuts
difference (counted properly every lothfor
lt;S minuts) fhall reach from 3, the dayes
from ly toward 18, to i degree 7 minuts
and a l^lf^ which taken from 8-47', the
Declination for the lyth day, leaves 7 de-
grees 38 minuts and a half, the true Decli-
nation

Or.bf the hint of Numbers thw ;
The Extent from y.the difterçiiçe in dayes,
b 113 , the difference in min. for y dayes,
ftall reach from 3, the difference in dayes,
to 68, the difference in minufs for 3 dayes,
to be added or fubftraôted, accordmg to the
increafing or decreafing of ch.e. Dechnation
.^t that time of the year.

Proved thm
' If youfubftraft 5'i from /deg. 44«quot;
the declination in the fecond year, there re-
mains 7 deg. 38'i, the Decrmation for the

tion about ; theBquinoaia), where the dif-
ference is tnoft i and in jw' aod December
,pot at all to. ^e perceived. , V ^ _ .
. Thus you inay by the Hm ReBlfyf
^na Table, find the Suns Decliiiation to^
ipimut at any. time, withoiic the trouble of

Ufe l.' \
To find hovf m*ny Ltagutf, »r MiUst Oftfvfer
to one quot;Dtgrcf of Longitude, in any Lati-
tude between the SquinoBii^aniFolt.

many Leagues or Mite «re in one De-
gree in the Meridian or Ec^uiHoftial, which
Mr. JVowOTÜ'and Mr. hith ftated ju
bout 24 leagues, or 72 miles.

If yóU'lcletf the old nHtftbtrJ making the
miles greatér, MiE.. 60 miks; èr.20 leagues'^
then the proportion, by ttó Numbers, SineS
and Tangents, -runs thm j 'quot;nbsp;-

So is Co-fine of the Latitude, to the
leagues, on the Numbers, contamed in
one degree of Longitude in that Lati-
tude.

Take the latteral 20 for leagues (or 60
for miles) from the Line of Lines from the
Center downwards; and makeit a parallel
in the fine of po, laying the Thred to the
neareft diftance.

The neareft diftance from the Co-fine ol
the Latitude, to the Thred, meafured lat-
terally from the Center, ftiall fliew the true
number of Leagues required.

So is = fine of Co.lat. to — if, the
number of Leagues, after the exp«!-
ment made by Mr. Nervoed, of which
true meafure you may read more in the
Second Part of the Plain Scalet by Mr.
Coamp;ins.

If you multiply the Natural Sine of the
Co-lat. by 2, it gives the Leagues 5 or by
it gives the Miles in one degree, cutting off
che Radius from the Produ^.

Note alfo. That if you take the Natural-
Number of the Secant of the Couife or
Rumb, and multiply it by 2, cutting off
the Radius from the^Produft, it fliaUg'«
the Leagues requited, to raife one degree, at
the rate of 20 Leagues to one Degree of a
SrmCudc.

To fini how many Ltagues, or Mlleh
' fxter to Raifi, or toDifreft tht Pel*
degree on any %t}mb from tbe Mtri^
dian,

f 'A.S the Goufin.e cf - the Rumh from the
- t - Mcritiian, to . 20 leagues ( or 24
' Leagues)oj^th€ Numbers,!
--jSo is the fine of po, to the number of
Leagues required.
Which, when you have fayled on that
Rumb, you Avail railc or deprefs the Pole
ciK degree. '

.. ^ by thtTria^guler-Quadrant, thtu ;
^s — 20, taken from the Line of Lines,
-Ltu oir any equsd parts, to the Co-line of
3gt;tbeRumbyJaying the. Thred to the
quot;Ho-.neareftdiftance.
u-the fine of 90, or neareft diftanc?
quot; irom fina $K}, to tlie Tfared,r to —-
i ic Nuolber.pf .Leagues required, lt;0 fey^
on that Rumb, and to rai^e:;the Pole
•U 6Se degree.nbsp;^^^^^^^

To find how many Milts or Ltamtanfm^
to any ntmbtr of dtgrtts i» a»y faraUtl
of Latitude.

Suppofe yoii fayling in the latitude of
48 degrees, have alter^ your Longitude 30
degrees, and would then thereby know
how many leagues you had failed.

Firjft, bring (of reduce) the 30 degrees to
Leagues, by mifltiplying them by 20, or 24,
(the leagues refolved to be in one degree)
yvhich makes lt;Joo, (or 720),

The Extent from the fine of po, to 4i
the Co-fine of the Latitude, fliall teach the
fame way, from 600 on the Numbers, to
400 the leagues required ; or from 720, to
480, according to Mr. iVorwMii,

Bjthefrianguler-Qjtairantl
, Take (Too ^rom the Line of tines, or
^t^y equal ^arts, and make it a =ir in the fine
of po, laying^t^ Thred tpxhe nwreft di^.

(the Co-fine of the Latitude to the Ihtt^)
and itfhall give 400 on the Lines, or equal
parts, the leagues required.

As — tfoo, to = fine 90 ;
So is = fine 4a, to — 400, as frequent-
ly before,

Notcy That in this »//rt of Navigation^
or Plain Sayling, that the Angle that any
degree of the Quadrant, or Point of th«
Compafs makes with the Meridian, ^
North and South-line, that is called tht
Rumb or Courfe.

But the Ancle that it maketh with the
Eaft and Weft-iine, or parallel, is called the
complement of the Rumb or Courfe.

Nott, That in plain Triangles, the Sines
and Tangents give Angles, and the Num-
ters give Sides.

t/flfff »ott, That the difference of Lati,'
tude is counted on the Meridian, and the
difference of Longitude or Departure from
the Meridian,is counted on the Equinodhal,
Or on a Parallel of Latitudes.

One of which Lines, in plain Triangles,
is called the Bafe ; and the other, the Per-
pendiculer. The Bafe being a fine, and the
Perpendiculer a fine complement.

Notcalfoy That in North Latitude,Sailing
Southerly, the Latitude doth decreafe ;
therefore you muft fubftradl: the difterence
in Latitude, from the Latitude you parted
from J but if you fayl Northerly, then you
^uft add it to the Latitude you parted
quot;■om : The hke in South Latitudes.

But when one Latitude is South, and the
other North, then you mul^add them both
together.

Note alfo, That the difference in Lati-
tude and Longitude, (and Departure) when
given in degrees, are to be reduced to
Leagues,by multiplying by 20, and counted
alwayes on the Line of Numbers, or equal-
parts, when you ufe t\\t Trianguler. Qua-
drant.

So then in ufing the/»(/fx and in
Plain Saihng, the diftance fayled, is alwayes
counted on the Index from the Center. *
S - quot;nbsp;The

The complement of the Courfe or Rumb
is-counted on the degrees, beginning at oo
on the Loofe-piece.

When your number of Leagues exceed
100, you mufl: double the Numbers on the
Index, the Square, and Head-leg, or count
lo for a lOO, amp;c.

Problem I.
Tbe Courfe, ani Diftance ran on that Courfe,
being given, to fini the iiference in Lati-
tude, ani Defanure^ or difference in Lon-
gitude, *

As fine of po, to the diftance run ( or
Leagues fayled) on the Line of Num-
bers ;

So is Co-fine of the Courfe or Rumb, to
difference in Latitude on the Num-
bers.

?ois the fine of the Courfe, to the De.»
■parture, or 4^fference in Longitude.

As — Leagues failed, to = fine po, lay-
ing the Thred to the neareft diftance;

Sois=: fine of theRomb orCourfej to
Departure, or difference in Loi^i- '''
tude.

Set the Index (being put over the Leg-
Center-pin ) to the Courfe counted on tlie
degrees from the Head, toward the Loofe-
piece.

Then Hide the Square perpendiculer to thg
Head-leg, till the dividea edge thereof cuts
the diftance run on the Index; then Jhall
the Index, on the Square, give the Depar-
ture or Difference in Longitude; and'the
Square on the Head-leg, fhall fhew the Dif-
feixn'ce in Latitude.

Problem II.
The Cmrfe and Diference of Latitud* gi-
ve», to find the Diftance run, a»d De-
parture,

= Co-fine of the Courfe ;
So = fine 90, to — diftance run.
So is fine of the Courfe, to the De-
parture.

Set the Index to the Courfe, and the
Square to the difference in Latitude ; then
on the Index, is cut the Diftance ; and on
the Square, the Departure.

The Courfe and Departure given, to fnd the
difiance run, and different of Latitude.

Set the Index to the Courfe, and Aide the
Square perpendiculer to the head-leg, till the
Index cuts the departure on the Squarejthen
the Index fheweth the Diftance, and the
Square the Latitude on the Head-leg, coun-
ting from the Center,-

The Diftance ruK, and difference in Lan.
tude given, to find the Conrfe and Ve-
parture, .

As ■— Radius, or a fmall fine of po, to
= diffance run on the Line of Lines;

So is =: difference in Latitude, toCo-fine
of the Courfe, meafured on the ftnall
Sine.

Set the Square to the difference in Lati-
tude, and move the Index till fhe Square
cuts the diilance run on the Index; then
fhall the Index fhew on the Square, the De-
parture ; and on the Degrees, the Courfe
required.

Problem V.
Tht Viftance ran, ani Veparture givenquot;;
to fini the Courfe ani Difference in La-
titude.

As tbe Diftance run, to fine po ;
So is the Departure, to fine of the Courfe.
Then,

As fine po, to the Diftance run ;
So is Co-fine Courfe, to the Difference in
Latitude.

Bji tht TrIanguler-Quadrant.
As.— diftance run, to.=: fine pb ■
So .-T- departure, to = fine of the Courfe.
So is = Co-fine Courfe, to — difference
in Latitude.

By the Squart and Index.
Slide the Square and Index, till the Index
tuts the Departure, op the Square, and the
Square cuts the Diflance run on the Index ;
Then,

On the Degrees, the Index fhall Ihew the
Courfe ; and on the Head-leg, the Square
lhall fliew the difference in Latitude.

TheVifersnceof Latittide, and tbe De-
parture given, to find tbe Courfe ^nd
Diflance run.

So is = Departure, to -— Tangent of
the Courfe on the Loofe-piece from
whence you took 4^. ~
Then,

Set the .^Square to the Difterence in Lati-
tude, on the Head-leg, counted from the
Center j and bring the Index to cut the

Departure on the Square • then the' Square
on the Index fhews the diflance; and the
Index, on the degrees, gives the Courfe re-'
quired.

In all thefe 6 Problems, which Mr. Gutt'
ter makes 12 Problems, of Plat» Sajllng,
I have fet no Example,nor drawn no figure,
becaufe the way by the Index and Square is
fo plain; and of it felf makes a figure of
the work : For the Index is alwayes the
Diftance run, the Hypothenufa, or Sccant:
The Square fheweth the Departure; and
the Line of Lines on the Head-leg, the dif-
ference of Latitude: And you may net on-
ly perform the work, but alfo fee the reafon
thereof, being a help to the fancy of young
Learners in thefe Nautical Operations: And
if your Square playes true, you may be
more exa6t than you can by Scale and Com-
pafs, and much more quick and ready ; not
only in this, b,ut any thing elfe in right-
Angled plain Triangles, as in Heights and
Diftancesj and the like.

(hort VlflattciSy may come very near the mat-
ter j as in making a Traverfeof the Ships
way from place to place Coajbng, as in the
Screighcs, and the Channel, and the hke:
I^utfor grtAt Dijlancts, itii not fo certain
as the Sayhng by iMttcnters Chart i there^
fore to that purwfe the Meridian-line was
invented, to reduce degrees on the Globe, to
degrees in FUn», as Uv.fVrigbt hath largely

On the innermoft-edge of die Rule, of
Trianguler-Qyadrant, you may have a
Meridian-line fo large, as to. have half an
uich for one degree of the Equino^iialj
and the inches for meafure, to go along by
it • or rather you may have it he near to the
Line of Lines on the Head-leg, as you fliall
thmk .moft convenient, for then it will ba
the fame as Mr. (jttnters is, and perform his
very Operations,as near as may be, after hij
way, by the Thred.and Compafles, or Index
and CompafTes, .

Tvfo places being propounded, one under
the Equipo^f^, the other in any Latl-
tude, to find their Meridional difference
in degrees and minutSy or loo parts.

Look for the Latitude of the place, fci-
tuate out of the Equinoftial in the Meridi-
an-line, and right againft in the equal-parts
1 is the Meridional difference of thofe two
places.

Let the River of Amaz^ones, under the
Equinodfial, be one place ; and the Lix,ard^
m the Latitude of Jo degrees North, another
place ; look for yo on the Meridian-line,
3nd right againft it, on the equal-parts, is
y7-J4, for degrees 5^4 minuts j or in
Decimal parts of a degree

Any two places having both Southerly er
quot;Northerly Latitude, to find the Meri-
dional difference between them.

Extend the Compaffes on the Meridian-
^ine, from one of the Latitudes to the other ;
die fame !^tent laid from the beginning of

the Scale of equal-parts, by theMendian-
line, fliall reach to the Meridional difference
required. 1

The meafure from the leaft Latitude, to
the beginning on the Meridional-line, lhall
reach the fame way from the greater, to the
difference on rhe equal-parts.

If the Latitude of one place be 3 o degrees,
and the other jo degrees; Extend the Com'
paflcs from 30 to fO on the Meridian-line,
and that Extent fhall reach on the equal-
parts, from the beginning of the Line, to
3.6 degrees 26 minuts.nbsp;'

when o»e place hath SaHth Latitude, and
the other North Latitude, to find the |
Meridional difference.

Extend the CompafTes from the begin-
ning of the Line of Meridians, to the lefTer
Latitude; then that Extent applied the
fame way on the Meridian-line from the
greater Latitude, lhall fhew on the Scale of
equal-parts the Meridional difference requi-
red.

Suppofe one Latitude be lo deg. South,
and the other 30 deg. North 5 The Extent
from O to 10 degrees, fhall reach from 30,
to 41° 31', the Meridional degrees requi-
red.

The Latitudes of tm places, together with
their difference in Longitude being gi-
ven, to find the %umb direiling from
one to the other.

Let one place be in the Latitude of yo
North, the other in ij deg. and 30 min.
North, as the Lix.ard-point, and St. Chri-
fiofhers j and let the difference in Longitude
be 68 degrees, 3c minuts; and let the
Rumb, leading from the Lizard to St.Chri-
flophers, be required*

Firft, by the Meridian-line, and the Scale
of Equal-parts, by Problem IL find the
Meridional difference in Latitudes, which
in o\xx Example will be 42 degrees, and 12

The Extent on the Line of Nunribers^-
.from 42 degrees and 12 minuts, the Meri-
dional difference in Latitude, to 58 digrees
and yo minuts, the difference in Longitude
lhall reach the fame way from the Tangent
of 4y, to the Tangent of 5:8 degrees and
2(5 minuts, the Rumb from the Meridian of
the Lizard Weitwards, being two degrees?
and better, beyond the yth Rumb from cb«
Meridian.

the —r Tangent of 4y, taken frort^
the Loofe-piece, is to the = Meri'
dional-difference in Latitudrs on th£
Line of Lines;

So is the — difference in Longitudes, to
the — Tangent of the Courfe yS dC'
grees 2y minuts.

But by ihe Index and Square, thü is
wrought very eafily and dtmonfira-
tivtiy thm-.

Count the Meridional difference of Lati'
tudes on the Head-leg down-wards fro®..
the Center, as 42 and 12 on the Lineo^
Lines, and fet tlie Square to it»

Count the difterence of Longitudes on
the Square, Wis., 68-jo, and to that Point
lay the Index ; and then the Index on the
degrees lhall cut the complement of the
Courfe, w'x,. SI-SS3OX 58.2j,if you count
from the Head.

As the Co-fine of the Rumb, to the true,
difference of the Latitudes, (on the quot;
Numbers) ;

So is the fine of 90, to the diftance on
the Rumb required, ( on the Num-
bers).

Being given in degrees and Decimal parts,
and brought to Leagues by multiplying by
zoy or 24, according to Mr. Norwood, as
before.

Note alfoy That the true difference of La-
titudes, is found by Subftraaion, of the lefs
Latitude out of the greater.

As the diftance fayled, in the degrees and
ISO parts,counted on the Lins of Num-
' bers, is to the true difference of Lati-
tudes , found as before, by Sub-
ttraétion j

the Rumb;
So is the Meridional difference of Latitudes,
to the difference of Longitude required.

As — Tangent of 4^, to == Tangent of the
Rumb, ( firft laid on the Lines from the
Loofe-piece) }

is the — Meridional difference of Lati-
tudes,to the difference of Longitudes.

Problem VW. , ;
Bj one Latititdc, Dlfiance and RumÉj
to find the other hatitude. ■

. As fine po, to the Co-fine of the Ruihb ;
So is the diftance, to the true difference of

As — Co-fine df the Cdtirfe, to = fine po ;
So is diftance, in degrees and parts, oni
the Lines, to the —■ true difference in La-
titudes, to be added or fubftraófed from
the Latitucic you are in, according as yovi
Tnbsp;haye

By the Index and Square, work as m the
Jth Problem laft paft,or ad of SUin Ssjl^S'

When two places are fcituated under die
fame Meridian ( or Longitude) and on the
fame fide of the Equinoftial j then fubftradt
the lefler Latitude out of the greater, and
the remainder fliall be the diftance in de-
grees required, counting 20 (or 24) Leagues
to a degree on every great Circle of the
Sphear.

When one place is on one fide of theE-
quinodial, and the other on the other fide j
and yet both on one Meridian, as was the
former; then the two Latitudes (wV the 1
North-latitude, and the South-toude ) '
added together, fliall give the diilance m
degrees required.

Problem III.
When the two places differ enly in Im:
gitude, and are both under the Equiiiôàialj
then fubfiraa the lefler Longitude frçm the
greater, and the refidue is the diftànee in
degrees.

Problem ÎV.
When the two places have both orieLatI»
tude, or near it, North or South, and differ
only in Longitude*

Sois the fine of half the difference in Lon«
gitude, to the fine of half the diftance.

B; tht Tr'iMgHltr-QMAirMt, or StBor.'
As — Co-fine of the mean Latitude, to
the = fine of 90, laying the Thred to
the neareft diftance j
So is = fine of h-alf the difference in Longi-
tude, to — fine of half the diftance.

Problem V.
When both places have different Longi-
tudes and Latitudes, as thefe Three Wayes
following,

to the fine of 90; . , ,
So the = Co-fine of the Latitude, to the
pi Co-fine of the diftance.

when both tbe places' have either North or
Somh Latltttde, that u, both tottari
one Pole-, thtrtthtu.
As thefine of 90, to the Co-fine of the difc.

So the Co-tangent of the Fefler Latitude,
to Tang, of a 4th Ark.
Which 4th Ark, muft be taken out of the

the difference of Longitudes is lefs than a
^adrant, or added to it when mere, the»»

of the fth Ark ; '
So is the fine of the lefTer Latitude, to the
fine of the diftance.

to = fine of 90;
So is the = Co-tangent of the leffer Lati-
tude, taken from the' Loofe-piece, and
laid froni the Center, and froir^ thence
taken paralleliy to the ^— Tangertt e^quot; a
4th Ark, which do with, as b?fore-is
fhewed, to find a yth Ark,
And then.

= Co-fine of the y th Ark ;
So is the fine of the lefler Latitude, to
= Co-fine of the diftance,

III Way.
But rvhtn ent Latitude is on me fide thi
EquineBialj and the other on the othtr-
jide, viz. one having^ North- latitude^
and the other South. Then,
As the fine of po, to the Co-fine of the dif-
ference in Longitude ;

Which taken out of tht other Latitude,and
po deg. added,when the difference of Lon-
gitude is lefs than a Quadrant, but added
to it if more than a Quadrant,and that fufiJ
or difference ihall be the j'th Ark,

fine of the fth Ark ;
So is the fine of the Latitude, firft taken, to
the Co-fine of the diftance in degrees.

As the —• Co-fine of the difference of Lon*
, gitudes, to — fine of 90 j
So the = Co-tangent of one Latitude (be-
ing firft taken from the Loofe-piece, or
Moveable-leg, and laid from the Center
downwards, and from thence taken
parallelly) to the —Tangent of a4th
Ark.

fine of the yth Ark J
§0 — fine of the Latitude, firft caken, to
s=Corfine of the diftance,nbsp;Th^t

That IS, when the 4th Ark is fubftraacd -
OP, to the Co-fine of the comp. diftanéc
when added.

Suppofe I'would know how far it is from
the LizMrd, to the Capt of Good Heft j the
Li»jard having yo degrees of North-Iati.
tude, and the Capt of good Hope 3 5 degrees
of South-Iaritude, and die difference in
Longitude 30 degrees.

As the fine of 90, to the Co-fine of the
difference in Longitude 3 0, being beft
counted from po backwards;

$0 is the fine of CO, the Latitude firfl
taken, to the Co-fine of the diftance
8p deg. 3 min. the neareft diftance in
^e Arch of a great Circle.

Note, That here you will have occafion
to make ufe of that help mentioned f.218.
As thus for inftance.

The Proportion being as the Co-fme of
to the Co-fine of 88-J9V which «
all one, as the fine of 5-4 and fp, to the fine
of I degree and i minut, which is too large
for ordinary Compafles, on ordinary G»»-
Urs Rules j therefore firft lay the diftance
from the fine of po, to the fine o£ y4 and
f P, the fame way from the fine of j degrees
and 4y minuts, aiid note the place.

jilfo, Lay the diftance from the fine of
po,to the fine of 50, the fame way fronj
the fine of y degrees and 4y minuts ; and
note that place alfo.

As the Extent firft noted for y4.yP) is
to I degree and I minut, the Co-fine
of88-5p;

So is the fécond mark noted for yo, to
Sp- 3 J the diftance in degrees requiretl.

Which multipiyed by 72, gives the di-
ftance in the Arch of a great Circle, ' f/«-.
15412 fïiilps Statute-meafure i Or, y34°
miles, whereof 60 make one degree, on a
great Circle on the fuperficies of the Sea.

To fi»d tht dlflanct of flacts hj the Na^
titral F'trftd Sints in the way of a StBor
en tht Trianjrultr-Quadrant, being
much more fafie than the two form^
wajet.

Firft, by the Pes, find the fum and dif-
ference of the complements of the two La-
titudes, and count that fum and difference
on the verfed Sines latterally, and talje the
diftance between your Compafles, and make
it a parallel verfed Sine of i8o degrees.

If you have not the Line fet on from the
Leg-cenper, then the fpall Line of Sines be-
yond the Leg-center, being doubled, will
do the work, by taking the diftance between
the fum and difference, and fetting one Point
in the Center.prick at two times the Radius
of the Sines from the Leg^center; ^nd then
laying the Thred to the neareft diftance, or
the Line of Right Afceneions under the
Months, is a fit Line.

Lonitn and JerttfaUm, two places in
North Latitudei Lonilon S^-i^ferufalem
32-0, whofe two complements 38-28 and
y8 added, make9lt;J.28 forafum, -and one
taken from the other, leave 19.3 2 for a dif-
ference.

Now the — diftance between the verfed
Sines of 96-28, and 19-32, make a ver-
fed Sine in 180, keeping the Sector fo, of
laying the Thred to the neareft diftance,
(and noting where it cuts in degrees).

The = diftance between 47, the diffequot;
rence of Longitude between the two places,
lhall reach on the verfed Sines from 19-3^2,
the difference to 39-14, the diftance re-
quired; Hvhich, at 72 miles to a degree,
makes 280 f miles.

Nvtt, This one Rule comprehends -all
the Three laft Wayes, and is not troubled
with half fo many Cautions aS the foitner.

JiAVvng the Latitudes and Diftance of t»»
places, to find their Difference in Low
gitude*

Find the fum and difference of the two
Co-latitudes, as before, by Additioji and
Subftradlion; count them on the verled
Sines, and take the ■— diflance between,
and make it a = verfed Sine of i8o.

and dlfiance on verfed Sines, fhall flay at
the == difference in Longitudes required.

Let one place be Burmudof Ifle, and the
Lantude thereof3y ; let theotherplace
be the LUard^polnt, and the Latitude thereof
fO degrees; the Co-latitudes are ^7-3 y and
40-0; the fum of them is p 7-3 y ; the dif.
ference between them is 17-3 y. The di-
ftance in the great Circle, according to Mr.
Norwood, is 44-30, or 88 tf Leagues, coun-
ting 20 leagues to one degree.

The — diftance between the verfed Sines
of 17-3 y, and 97-35 made a verfed
Sine of »80, theSeitor isfet,

The — diftance taken between 44-30,
iand 17-37 on the verfed Sines, and carried
parallelly, fhall ftay at j'y,the difference in
Longitude required between thole two
places.

IN the Book called, The Geometrical Sea^
man, by Mr. Phillips, is a very ready
Figure to ftiew in a Quadrant, or more, by
what Longitudes and Latitudes a Ship is to
pafs in any long-run, which is contained
under po degrees, or I20 difference of Lon-
gitude, and the two places having both
North Latitude.

Which Figure, or Quadrant,is neatly and
readily performed by the Trianguler-Qua-

Upon the back-fide of the Index; beforg
fpoken of, may be graduated from the Cen-
ter, two Tangent-Lines, one equal to the
Radius on the Loofe-piece, the other to the
Radius on the Moving-leg ; then in the
ufe, count the fiducial Line in which the
Leg-Center-pin ftands, alwayes for the Me-
ridian of one place; and fome wherein
that Line, according to the latitude thereof,
counting the Leg-center the Pole of the
World ; and the Index being hung thereon^
by the Tangents prick down the Latitude^
there, I fay, knock in a Pin to ftay a Thred
for one place; then, on the degrees, count
the difference of Longitude from the Head-
leg, and lay the Index to it, and bring the
Thred fattened, as before, till on the Index
it cuts the degree and pare of the other la-
titude, and there make the Thred faft with
another Pin in the Loofe-piece.

If you move the Index to any degree of
Longitude between thofe places, the Thred
lhall cut on the Index the degree of Lati-
tude that anfwers unto it ; or if you make
theThred cut any degree of Latitude, the
Index gives the Longitude required for that
Latitude,

Note, If the Latitude be f2nall,asbctwecrt
to and 30, the fmall Tangents are moft^
convenient ; but if it be between 40 and
80, the greater Tangent Line is bcft.

Note-, That two' Tlweds and a pair of
Compaifes may (erve; but the Index is much
better and quicker in Operation.

Let the two. places be the Summtr-lflanii
and the Ltx,lt;trLfoint; the fame Examflt
that you find in Mr. Norvood, pag. 126 )
and in Mr. PhiUip s Geometrical-Seama»f
fag, yjquot;. that you may the more readdy
compare the truth thereof by^ dieir Opera-
tions.

Hanging or putting the Center-hole of
the Index over the Leg-center-pin, and
counting the fiducial-line on the Head-leg
for the Meridian of one place, count on the
Tangent Line on the Indes the Co-tangent
of one Latitude, as fuppofe the Latitude of
ihe Ll^ard-mnt (the Center alwayes coun-
ted

Then count 70 degrees, the difference
in Longitude,on the degrees from the Head-
^ leg, and there fiay it; then draw the Thred
put over the firft Pin, till it cut the com-
plement of the other Latitude, and by help
of another Pin ftay it there, which; you may
conveniently do by one of the fliding. fights;
then the Thred being fo laid, flide the In-
dex to every fingle djegiee, ot fifth degree of
Longitude, and th?n the Tlired lhall fliew
on thelhdei,; the Clo-tangent of the Lati-
tude anlwerabk to that degree of Longi-
tude, as in theï^é/* annexed.

Al[oy If you would have equal degree?
of Latitude, and would find the Longitude
according to it f then Aid« the Index to
and firo till theThted cuts on the Index an
cvendegree of-Latitude J then on*gt;j:he de-
grees you have the difference of Longitude
from either place.

jUftnttt, Th^ the drawing of one Line
Kgt;nly onthcTxianguIar hfirument in tbe be-
ginning, accoiding to the direlt;aions of lay-
ing of the Thred; with the Thre^^and
Compafles, will perform this work alfo^

If this worjc fit not any cafe that «ay
happen, there is another way mentioned in
fagt 7% of the Geometrical Seaman, by the
Steriographie^ProjeBion ; and that Seheami
is drawn the fame way, as the H^igt;contal-
ProjeBion for Dialling was, and fomewhat
cafierj and any two Points given, in a

C-irjle, you may draw a great Circle to cu^
them, and the firft Circk into two equal-
-parts, by the directions inP^^^iy; And
the Application thereof you have very
.plainly in Mr. Thillips his Book, to which
I refer you, haying faid more than at firfl I
intended, which was chiefly the ufe thereof
inOhftrvationonly.nbsp;,

So for the prefent I contlude this Dif-
i courfe, and ftiall endeavour a further Ad-
tvantage in the next Impreifion, according as
Time and Qfiportunityihall offer. Furttvtl,

Ttniraw an Ereil Eafiar mfi.Dtaly
To irAw A Horix^ontal'Dial.,
The iemonfiration of the Canon for Honrs, ib,
Tu irar» a DirtR EreSt South or North»
Dial,nbsp;30

To draw « DixeU EaSi or mfl Recliner, 3 7
To m^e fhe Tahle of Ar^s ae the Pole, 42
.Ta refir tkife Dieiis (e.a new Latitude, and 4
new Decl\natlon^rtherein (he*j»at become
E^a Deiffne^y Vs vV^ivnbsp;^^

To find the Requijites by the Scheam, ibid'
To quot;find: the Declination of a Plain by tht

Precepts to find the Declination by the Sun,
.-aymd Examples alfo of the fa:me.,, 58
Te draw Oft EneSb melirnng^Dialy ? 6i
The Jropertiont fn^ the Rtijuifites of irS
DecUntrtx Vnbsp;64

.^Ttf draw the Lines on a North Decliner^ -yO
To draw the Heur-Lineson uPlain, tiiar^'
clintj above ^odeWeesif -nbsp;73

To find the ufeful Hours in att Plains, 130
To drarv the Mathematical Ornaments en aU

To draw the Planetary or fewlfh H»ms, 141
To draw the Italian Hours,
To draw the Babylonifh Hours,
To draw the Azimuth Lines,nbsp;ij^

Tbe Defcription and, Ufe of the Arrnilarj-
Sphear for VjaUing, feveral xvajes, 172
The Difcription and Ufe of the P^oor-man's
Dial-Sphear for DjiaSing, and feveral
Ufes thereof,nbsp;^ 20 j

For afore'Obfervation with Sights, 333;
For a fore-Obfervation vflth Thred, ani
Plummet,

To Jjnd the Latituie by a MeriiUn Obferva-
tion, rf itb Thred and Plummet^
Tofindtbe SmsDeelinationynbsp;254

A 7Me of the Mlt;ignitHdes Declinations and.
Right Afctntions «ƒ 33 fixtd Stars, in
Degrees, and Hours, and Minuts, 26 i
The ufe of the Trianguler-Quadrant) is the
Operative part of T^vigtaion, 267
Of Sayling by the Arch of a great Circle,

)ri,x,theendsoj the Arch Q^P. p.iz.i.i4.f.-
heginSy t.heing. pajJ.i?-'- Latitude, r.Co-laii-
ttidenbsp;Sun^ T.fum. p.-^^.l.i^.f. inclrntr,

r, Uietinatro* igt;fAtendia»s. p. . I.ij. f. pliace, r.
fLdn. p.6i.l.io,f. Or.Q. p.ó^.U.f.I,r.T. p.éy.
f. 7.f. ii,r.7. f -pz gives a marl^ near ^^vthofè
meafure on tbe'Limtfrornfi. p.87.1.8. f. gries r.
£iveJ-UJ-add at R near C. p 93.1.15 f.F-E.r.fe
p.iao.l ai.add c d BSM^iw/, p. rov.lnbsp;r,.

p.i'ij.l t^ vfe^ r. kave. p.ioamp;.l.c f. r.
ZMiith. p.iri. Lj. f. «f/ii, i.cutttbefubftiJe.
p-iij.I.i^f. DF,.r.DE. I igf.T.r.cT. Alfa
inLi8.r.CT.vp.ti2.t.(5.f;E,r.F. p,111.1.15,14,
r5,i(5, addS'tJie. p ii8 r.i.é.f.I,r.L. p-ijp/Ufi,
add,lt;»i' in thii Examfje. p.i4Q.l.(?.adi4 ta. p.170,
LioJ. divides, x. divided, p.igi.l.io.f. popficitn,
t.propoytion. 'p.ii9.l^7-i- from, r. on. p. 193.1.6.
f. beirg., r. b^ing. p.197 I-4.- f- elevation, r. incli-
nttiion. p.iool.34. f. C,r. G. p.io4.l. 3.f. F, r E.
p 109- add in tlre laft line, or by the upper part of
the Plain. p.i24.1.i8- add, or remcve the Thred to
turn it further when it reclines beyond the Pole.
p.238.1.7. add of. I.io.addi«/. p.^47.1,i3. r.
and much better in fntaS Latitudes, p.148 l.^.i.
»pi.r. fw9. p.i5i.l.i^.f. io,r. A|R»,I,i7. f.
40, r. z7'

The ufe of thefe, or any other Inflrutlicnts
concerning the Mathematical Praaice,or further
Inflruftions in any part thereof, is taught by
Jihn Colfon near the Hcrmitage-ftairs.

Alfo by Ewlide SpeidweU dwelling near to
Wbite-ChappU Church in Capt. Conti Rents, or
ac'the Cuflom-houfe.

Alfo, by William Korthhall Mariner, dwelling
at the Crooked-Billet is Mceting-houfe Alley
on Green-Bank near Wapping.

Alfo the Infliuments may be had at thé houfc
of John Brown, John Seller, or John Wirgfield,
as in the Title-page is cxprefled.

There is no\v extant a large Treatife of Na-
vigation in Folio, defcribing the Sea-Coafis,
Capes, Head-Iaiids, the Bayes, Roads, Rivers,

IHaibours and Sea-marks in the whole Xorthern-
Kavigation; ftcwing tlie Courfes and Diftanccs
from one Place to another, the ebbing and
flowing of the Seaj with many other things be-
longing to the praftick part of Navigation.

A Book Cno*^ heretofore printed in
Colleäed from the ptaftice and experience of
divers able and experienced Navigators of our
Englilh Nation. Publifhed by John Seller,
Hydrographcr to the Kint^s moft excellent Ma-
jefly, and to be fold by him at the fign of the
Mariners Compafs at the Hermitage-ftairs in
flapping, and by John Wingfield right againft
St. elavet Church in Crouched-Fryers.

Prailual Navigation; or, an Introduaion to
that Whole Art. Sold by John Stlltr and John
Wingßtld aforefaid,

alrtoil 3 '1 -K j;; 'i
^fehife^-.» nX.'l .'.■■Vv-tti X a-r
.bsTS••gt; J: T ■ , '

Î yjasilivi ba; îiiiiiuD jiiii/.o
b:!quot;; ^iir' i-» -Xiinbsp;0'. t ' •gt; .

1UO-io j:ütbü(»£l4 Jjjdjii »■].•.J lt;-•■ i.ji. Z'lj.'ilj
/•■sW:^.nbsp;I;'H'IiM:;-! .rquot; ' ' jiinjiT Î

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	61 jZ	6	3M5
12	5807		35 38
	54 22		39
	JO 37		43 08
	4lt;5 f2	5	f?
JI 43			5038
	39 22		54 23
	3? 37		y8 08
	31	4	Si
I.C	1 28 07		6s 38
	24 22		lt;5? 23
	20 37		7S 08
	Ilt;s J2	3	7lt;J 55
9	13 07		80 38
	09 az		84 23
	oy 37		88 c8
	ox f2	2	pi SI

	/ / ^v
	/ / \
	i^A \
/ /	\
	7^ \ ]
/ /
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/ gt; \ / ' \		-'fl \
/ ^
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	gt;P \
	N \ / /

F^ 12,s	quot; V / / /
	/ ^ I r^V \ ^
		8
	.■.lt;.,tKt, 1 'ft
	r /	y
	£ \ \ ^^^ --J-\-lL_-i»_	10 11

^uns declinations for the Parallels of the length of the Day.
Hours	Declin
1(5-26	23-3*
16- 0	21-41
ly- 0	i6.f5
H	11-37
I?	y-y3
12	0.00
n	W3
10	?i-37
p	i6-yj
8	21-41
7-34	23-31

A Table to		divide
the Planetary hours
in 05 and in yy, for
London ,		51-32
Latitude.
H M	hou	HsxM
843	i	5 10
P 28	2	631
lo id	3	7 y2
10 44	4	9 »5
22	5	10 37
la 00	6	12 00
12 37	7	I 22
i ly	8	244
X 53	9	4 06
2 31	xp	S 29
5 8	it	tf yi
3 47	12	8 13

H.	8	12	IÓ
—			__
7	I*	I	3
8	12	2	4
S	I	3	y
lO	2	4	6
U	3	y	7
la	4	6	8
'3	y	7	P
H	6	8	10
ïy	7	9	11
16	8	10	12
Ï7	9	IÏ	I
I8	IC	12	2
	XI	J	3
2C	11	2	4
	I	J	5
22	2	4	6
	5	y	7
-4	4	6	8

Febr.	March	April.	May.	funt.
p.M.	D.M.	p.M,	D.M.	D,M.
13 50-- 13 30.. 13 10- 12 49..- 12 29	03 29. 03 05.. 01 42 01 18. 01 Ï4...	Ö8 31. 0853- 09 IJ. 09 36... 09 58	18 02 18 17. 18nbsp;32 19nbsp;46... 19 01	25 n 2J 14-. 23 18- 23 21 25 23...
12 08. II 47. n 25.;. II 04. 10 43	01 31 31 07 00 43- S. 19. • ti. 04-■	10 19' 10nbsp;40..- 11nbsp;01 II 22 ■11 41-quot;	19 14-•. 19 18 j 19 41. 19nbsp;H- 20nbsp;07	23 2Ó 23 27-.. 13 29. 23 30. ii 30-•.
10 2l 09 59- 09 37 01 15 08 S2	00 27 •. • 00nbsp;51.. 01nbsp;15 01nbsp;38.. 02nbsp;02-•	12 05 12nbsp;23 n 43- 13nbsp;03 13 22.	20 19 20 31 20 42-. 20nbsp;54 21nbsp;04.	13 31 25 30.. i? 3cf 13 29 23 27..
05nbsp;29-.• 08 07-• 07 -(-j... 07 2'2 06nbsp;59	02 2Ö 04 49. 03 13 03 36 03 S9- ■	13nbsp;4i 14nbsp;or 14 20 14 38- 14 57	21 IJ. 21 25. 35 21 44. 21 53.	23 15.. 13 23 23 20-. 2317. 2314
06 56 06 13 05 50 05 iOquot; 05 03.	04 22.. 04nbsp;4y.., 05nbsp;o,i. •• 05 32 05: 34..	is 'S 'S 33 15nbsp;50.-. 16nbsp;08 «6 2;-	22 Oli-. 22 10- 22 17... 22 2,'. 22 32..	i3 10 23 05.• 22 01 22 s/... 21 so-
04 39-•■ 04 1Ó- 03 jï-..	06 17. 06nbsp;40 07nbsp;02- • 07 25 07 47	16 42 16nbsp;58,. 17nbsp;14 17 30... 17 47	12 39 22 45quot; 22 22nbsp;57 23nbsp;02	li 44 22 37,.. 1231 22 13... 22 itf-
	08 09-		23 06. ■

Long	D.L.	Latiiude
300	op	32-25
305	OS	3f-f2
310	10	38-51
	15	41-^24
320	20	43-34
32r		4f-H
330	30	4(î_f4
33Î	3S	48-07
340	40	49-04
34f	4S	49-47
350	SO	fO-If
3S5	fy	50-31
3^0	60	fo-33
05	6s	f 0-23
10	70	fO—00



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	1

H.	8	12	IÓ
—			__
7	I*	I	3
8	12	2	4
S	I	3	y
lO	2	4	6
U	3	y	7
la	4	6	8
'3	y	7	P
H	6	8	10
ïy	7	9	11
16	8	10	12
Ï7	9	IÏ	I
I8	IC	12	2
	XI	J	3
2C	11	2	4
	I	J	5
22	2	4	6
	5	y	7
-4	4	6	8

H.	8	12	IÓ
—			__
7	I*	I	3
8	12	2	4
S	I	3	y
lO	2	4	6
U	3	y	7
la	4	6	8
'3	y	7	P
H	6	8	10
ïy	7	9	11
16	8	10	12
Ï7	9	IÏ	I
I8	IC	12	2
	XI	J	3
2C	11	2	4
	I	J	5
22	2	4	6
	5	y	7
-4	4	6	8