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_TREATI&E. 

TRIGONOMETRY, 

Plane and Spherical, ^ 

THEOlCETlCALandPRAC^lCAX, 

In which Ac ft vcral Cafes oiPioH^ znd Sfher 
rical Ttiin^ts ^c (qWcd, Injitumentally 
znA Arithmemcdfyr' ^ '' 

AslikewifeaTREAtlSE - : 

•v OF 

r 

Stemoghaphicic and' Orthocraphick 
Projedion. of the S p k e R e. . 

In which the Principles afd Theonms on which they 
depend, are clearly Demonftrated, and the Praaice 
naturally deduced from thofe Demonftrations. 

Illuftrated in the StereographickPioie^bnof the feve^ 
ral Cafes in Right and OhUqiio. Angled^ Spherical 
Triangles4^ So.tiiar tMft Reqijufites may be f4?un4 
without Calcularibn, by Scale and Cbmpafi. 



Tbe Secmd EJithHi cdrefiillf correSed h the Author 
SAMrHEYNES, Ttacber (f the 
MATHEMATICKS.. 



LONDON^ 
teihtedfbr RandW MmntyZcAT.PageiinPtfi^ 
Rmyon Timer-HiU. MDCCXVI-/f/^ 




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4 



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mt 



^h 



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R 
S 

f 

T 

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V 

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this Treatith, 

'YAdcUtl<»i. 

I SvA^xamosi^tbi Jtfireme iftk tmt^jDtm^ 
I ttttef betvfeen Vfbicb 'tispk^d, 

I Multiplication. . 

I Majority, or greater than. 

Radicality, or Sifuare Root. 
2 Reftangl^ or. ProduB. 

Co-finct 

Secint* n ' 

Qvfecant. 
Tangent; 
Co^tangenei 
Yorfedofine. 

Verfed^flne cf tie SUffiemnU 
Uight-angle. 
Sum of tbeJLMs. 
Diffietcsce of m JLAgu 
Sum of i&e Angles* 
.IHiKronce of, the Ar^ku 






mmmmmmwmfmm'^m'^^ 









C^ipt. Edm. Halky, 

Fellow of tb& Rjoyal Socisty. Ue], 
SIR 

nr HI S Treatife of Tr^o. 
^ mmetry and Stereo^ 
p'apUck ProjeBion h hnm- 
bly Dedicated to You; I 
being: fflad of this Oppac- 
tanity dF more PuMickly 
Teftifying the Reipe<ft t; 
bear to Yo«, and my Obli- 
gations for the Kindneflk\ 
cc^err'd bv You utK^ ^^^ ' 




•» > v> 



Obliged Servant 

Sam, Heynes^ 



TO,T HE 



R E A D E R. 



TKs Treatife is accommoJa^d bctb U4beTbewttich 
and Fra&ick Redder ; a PraBkalTreatife mljy 
tMild never pleafe tbelformery mr a Tbeorick the latper. 

In order to the former End I have made ufe-oftbe mcfi 
Cmcife and Elei^fnt Demmfiraticns J could fnd^andpar'^ 
ticularlyfrom that imtrnparabk Treatife ofTfigonometrf^ 
Written by the Uamed Mr. Caiwell, A. M. 0/ Oxford. 

And that I might anfwer the latter End the better^ J 
h^me given ffys aEtual Soktfions'of the fiveralCafes bdb 
cf Tlane andSpfierical Triangles in Numbers^ -Natural 
and Artificfaly'^ and likevnfe ^ tbeGunttifs Scale ; and 
have taken great care thai the feverdl DireBions t9 pro* 
Bice jhcoilahe Jo plain andeafy^ that they might be made 
fife cf effeBMfybyl^rfons that are not fo ipoell grounded 
as yet tn Geometry^ dec. as to underfiand the Demons 
Jhations. 

As to the other Treatife of Sterographick Proje&ion. 
of the Sphere^ vfiidb is a Branch of PerfpeSive ; I have 
JJlenionfirated the Principles and Theorems on which' 
it depends^ and from thofe ^Theorems have Jhevf*d the 
riafon of the feveral Methods made ufc of in PraSice^ 
lowing thereby the Projection of the Sphere upon the PlOfm, 
^ any great Circle to become eajjf , andlikewife tbe^Men^ 
fi$raiim of any are fo ProfeQed* 

"•Andlaftfyy Ihavejbevfdyou how to Profeft the feveral 
Cafes of SphericalTrimglet both Right andUhli^e^ the 
Reafonsofall which Precepts naturally flowing from Pre-^-^ 
oeeaino Propofittcn^ : There bath been nothing wrote in 
Engliih giving, foJullAccofsnt of this Mattery which is 
tumblySubmiitea to the Judicious Reader. By 



t4- 



<•) 



r- ^' 



SOME FEW 

Geometrical Prot)lems, " 

Necejfary to be knonpn in Order tafind the 
Requifnsof Right andObliqueTriajogleSy 
by a Sczlt of Equal Parts, and a Line 
^Chords. 



*■** 



I 




PROBLEM I. 

Frtnn a Pcifttin a Lint given^ to ereB a ' 

_ Perpendicular. > 

. , . . ■ ■ 

E T the line given be «^, and the Poijit gi- 
ven e'y take two Points r, 4» equally 
diftant from <r: Then putting the Plate h 
JFoot of your Compalfes in <^ and 4 ^'J^* !• 
topen'd to any diftance. greater than ctf, and 
ftrike two Arches cutting one another in /• Then 
draw the Line /e,, it Ihall be the Perpendicular re- 
quired 

PROB. II. 

To divide a given Right-line into tv:o 

equal Parts. 

* 

Let the Line given bctf ^; upon the Pointt^ iviti 
your CompaiTes opcn'd to any Diuanoe grea* • • 
tpr than half the Line a by difcrbe the Arch Plate !• 
c 4 then putting the Foot of your CompaiTes Fig. II»I 

in the Point b^ and with the iame opening de« 

fcrih^ 



< « > 

fcribe an Arch cutting the other in c and i^ the Line c ^ 

^1 bUTeft the line elTen inc. . 

?rob: m. 

From a Point giim without to let fad aferfen^ 

dknlar to a Line^ven. ' 

Let the Mnt gim be ^« and the Line given ah\ 
putooe Fqi>I w* your Compares in g^ and 
TToHT. with any DiSance, gteater tnan the neareft 
Jig* III. Diilaaee of that Fbint from the Line given, 
crofs the Line given in the Poiots. r and X then 
patting the Foot oFyour CompafTeson the Points r and 
^ with the MiiK (i|ieiiiiliig uFyvnzrCoRipafl^ defbribe two 
Arches cuttina on« anoth^ i« ^-a 6lul|ir being laid to^ 
and^ draw Oe Line^ &> it ihan be the Perpendicular 
re^jujrtU 

PR OB. IV. 

* Pr&atthe endiffdLhiegrven^to ereSta 

PerpinMcular. 

• ~« 

' I. Upon the Fointi wixhyoor CompailestpQcfd 1» 

any OtSanoe ^ r, ddKndbe the Arch f 4 (?s then 

'jlat^ It your Compaiifs at this Qpenii^ will lejKk 

li^.nr* mnxi c to ^ and from^toV^ uponthqP«iit$ 

i and ^ -and wi^ the Ame opening of your 

CompafTes defcribe two' Arched cutting one another in 

/;//jMlUethftPeipett«crfMw -. 

2. 1 our Compaiies. Beipg open'd at any Di&nce, 

put one Foot in 2, aik( the other anj where 

/Mf L iA 1^. ihflo turi^ aho«t ite Booc'that w^bi A, 

fig. V. tiHi(kcrd& the Uoe« h ghrsnin 4. di^aw tht 

. June rf4JifeAar«r/ss«^ Aefti»/*'aP«fr* 

jwdkiilar# » 

PROR 






'JT-^ 



A I 

B 

Ig:V. 
e 

-^ 

A "^ 



\ f»JI- 






Fi,A!nr.X 




^^x 



df.- 




probI V. 

Frm a Poim given to dratj a Parrot! ti^ . 

Lnii given. 

Letttie Point girenber^ and the L!ne gives «it 
Promthe Pointr as a Center,ftrike an Arch 
cutting the given Linetf 5 in 4, thps take fldtii. 
a rin your Compaffet, and puttb^ onft Point Fig. VI«, 
any where in the line a bt teppQle an ^, 
iviththe other deftrihe* an Ar^ Aentake 4^ in your 
Compoflef, and puttu^m Foot ia<^ with Ae other 
cut die former Arch. in Oy aLiatdrawn thio* ^ and a 
is Parralkl to* the line givea * 

PRO a VI. 

with a Line givenzh^ to nuAemAigk qfanf. 
Numier pf Degrees giMBy Snffqfe 45 Degrees. 

Open yoor Comptfis to 60 d^. ot yburLtoe of 
Chords (60 deg* of the Chonbbci^eqaal 
to the Radius of that Circle^ the Chttds MiirL 
ivere deiigif d to maafttie) and pntd^g on» fig. VtX^ 
Foot in 1^ with the other ddbibetii0;Arch 
e iBl;^^thea take 45 d^. from your Sedeof Chotds, tad 
putkfixNti «u^eiadieAxcbyditwtfaBLIiiD4r,aad'dft 
done. * ^ 

Henoe an Angle being already inade^ you may m^- 



Ai){(k* 



VROB, 



PA OB. VU. 

Th^i Side4 giwn to piake a Tria9igki ^ 

Lee tlieSi^ given be <^^ and^> with which it is re- 
quired to make aTriaAgK >n^e i ^zzthe Side a^ then 
. taking the Sid^ ^. in your CoaipafleSy .with 
fhi'eL one Footiha defcribe anArch, and witfathor 
JRgWUL DiAance ^ putting anyFoot in ^,cut the former , 
Arch in r, then drawing ac and Air and 'tis^ 
done. 

pROB.^viiL : 

Xol^if^atgf three Ptnnts mt fituate in a Ri^ht--' 
Ihtey into the Circumference of a Ci^cU * 

Let the three points giveti ht abc^ take above half 
the Diftan^e oetweeili a and b in the Com« 
TJateX, .paffes, then placing one Foot of your Com-;. 
J%.IX« paiTes in «^'' with the (aid Diffance defcribe^ 

. . N the Arch e '4' witfa the fame I>Uhuioe'and 
one Foot in b defcribe" the Arch <r d, cutting the former 
i^>e and%^ and draw the Line ed: Then taking aboVe 
half the Diftance betweien by Cy and one Foot being"^ 
placed in c defcribe the Arch /,^, and mth the fame ' 
l>ift|tnce> one Foot bdng placed in the Point by crois' 
the former Arch in tl:^ Points f and g^ and draw the^ 
^'^/i* ^w where the two Right-lines 3 f sxAfgy'* 
I|iter&4 each other j(vi^. in the Point 6) is the Center ; 
of the Circle^ wUch was required. i 

DEFINITIONS. 

Itf A Circle is fuppos*d to be divded into 360 equal 
XjL ^^^ ^^^^'^ Degrees, and each Degree inta 
60 Parts cairdMinuites, and eachMinuite iSito 
Plate I. 60 Parts caird SecohdsjCi?. any Porti6n of 
Fh. X. which Circumference is call'd an Arch, and is 
Jlcafur'd by the Number o£ Degrees, it contains. 

2. A Chord is a Right-line joyning the Extremities 
©fan Arch, at AC is the Chord of the Arches ABC> 
ADC 3* A 



(1) . 

i* A Right-^iine is a E^ght'line dmvn from one end 
of an Arch; Peroendicalar to that Diameter naflSi^ 
thro' the other did ; or it h half the Chord of^twice 
the Arch, A £ is the Right^ of the Arches AB, AD; 
and. hete 'tis evident, that the Sine of 90 deg^ (which 
iseqdal to the Radius or Semidiameter of the Circle) 
is the greateil of all Sines, the Sine 6f an Ardi greater 
tU^a QsiAdrant, being kg than the Radiiii^ 

• ■ . . . ^ 

4. A Vcrfed-fine is the Segment of the Diameter in- 
tercepted between the Arch and its Right-itne, £B is 
the Verfed-finc of the Arch A B, aM ED of the Arch 
AD. 

' ' 5; A Tartgent of an Arch is k Right-line dra^vn Per-* . 
^i)dicular tp the end of a Diameter, pafUng thro' onef 
ehd bf aft Arch, atid its Length is limited by a.Rigfit- 
line drawn from the CcQter thro' th^ other end of the. 
Arch, and this Line is call'd the Secant ; thus BM is the 
Tangent and F M the SeciOit of the Arches A £^ AD^ 

• 6l The d[l!ereac6 of an Arch from a Quadrant, whe- 
ther it be g^^ter pfleis is call'd its Complement,^ GA^ 
is the Complement of the Arches A B, AD ; HA is the 
Sine of that Complement orCo-fine, GI thie Tangent 
of that Complement, or Co-tangent, F I the. Secant oi 
that Coniplement or Cc-iecant. 

. 7^ Tiie difference of an Arcli from a Sermdircle is 
caird its Supplimenttf 

t 

8. That Part of the Radius which is betwixt the Cen-i! 
terand Right-fine hzz to the Co-fine, FEitHA, 

■ - . ' . . • •' • 

9* If an Arch be greater or leifer than a Quadrant- 

. jthe^ rJJJJ! > of the Radius and Co-fine ii equal to the 

Verfed-fine. Thefe things will be farther Illnftrated by 
the following Figures 

B U 



i 



■»SA; 



In a Trian^^le are fix Parts, v/^. three fides and three 
A49gks» any three of wBcb kdi^ ghren ^cmpt the 
thtee Angles of a Plane Triani^') tbp other thite tnay 
be {oitnd, eidier JM^dua^tftfjr by «hc Mp «f a Scale of 
equal P&ftsand Line of Chords, 6r by mn ^itbmetick 
Cakttlatio9 (if fuppofing the Radius dmded into any 
Nuaifeettif equal Parts <ife kn»\v hoW manyof thofe 
equal Paitsare intiie5in^ Tar^cnt, or' Secsmt of any 
Arch proposWl) /the Art of iniemng which is calM 
Xrigonomctrjty and is either Plane or Sphxrical. 

"fhitahle'oflStaturaiSinesy'fangems.&cc. 
may be ihm made, 

r; • \. AE die Sine ofran Arch Wng given to findlis 
^^i^-'tVfine ttA, 

sF E=H A, or TTC^pT 




-2. A E the Sine of an Arch bring giv^cn^ tofiiidBN 
.the Sine of I the Arch. ^^ 

FE the Co-&* is knowl) by (ths feft) aftd confc- 

quently E B ; then ^ AE f-\- fe U jr== A B> at)d|AB* 
.r=BNi.e.|/5'i+Vir=SiArch. 

' o, BN the Sineof an Arch being given, to find! A E 
the'Sineof tvnce that Arch; , , ^ . 

FN -the Co-fine is known (by the firft) and the Tri- 
angles F B N, A B E are Similar ; therefore FB, FN ; : 
AB. AE. i.e. R.;S : : 2 S. S 2 Arch. 

4^ S c and x P, the Sines of two Arches SD, x (5 being 
given^ to find x^ the Sine of the Sum of the Arches. 



^y f 







/ 




■is 


/■ 7 


f. . 








1*4 


7 / 




1%, 


/ / 




li 


/ 




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i 


\ 


? 


/ / 7 






^ 


,..,.j 7 y 






p 


7 j~---/ ''-^ 


p' 












;ph 








^ 


^T 




^^ -■■- A". 










/ 


/ 


/ 


^ 


f 


V, 


k 


1 




^ 

y 


/ 


^ 


1 


V 

- : 




r^£ 


-vs 



■wive 
^ I 

A"/ 

eye i«,*t«?j 





(7) 

The Qhfidei FOu FP are knowii (perHr&y -Aeh Fj?. 
FP : r^tf.Pr.and the Triangles j' P x (P ux, ^aF> 
o^B are Similar, therefore F^. ¥o : : xP.xjU then 

F>JcxP-j^FPx ^c^'- 
x/'-j-Prrix^ir •*-*--'-'-r---^---*---**-'i--*ti|at' is the 

FP 
Siiwofthe ona AjrchintirA^ QttetsGQ-fi?aTt Sine of 
the iDther Ai'ali hlttt ihi' Ui'-firie of the firft XrcB, an* 
this Sum- i^HWeSby^ilhtf Radluari^fiFtlfe* StiA of t^vo 
Arches ; and if the Sines of the fame t^vo Arches were gi- 
vitiy and it wtife xte9nr*d t!c» fihd tbeSine of their Dif- 

ference,it would b e * •a: & oftfarf 

difference of two Arches... . . \ ^: . 

5. Suppofe the Archi(iwf.=: 30^,?fiij^ ^ =: « ^, 
and i& 0^ the Sine of A L, and ^6 K, its Co-fine, alfo b d 
the Sine of b L^ and ^rits Cb-fioe.; drawF^x cutting 
bb inx, then 4^ h ^'^(4 ehy^)zzkby qzzbCy q-^e hy q •, 
therefore 3 ^ A, f — & e, ^, and^ bz^ 3s* e, that is 
g ^x 1.732050807— ^e ; andii>]-|-^ i&x 1.732050807 
^ b dy i, «• tlie SiUe of an Arcfi leS than 30 dsg. ad- 
ding V^ 3 X S of the Defe<a^ makes the Sine of an Arch as 
much exceeduig 30 <%• therefore if the Sines of all 
Arches lefs than 30 degi be known, the reft as far as 
60 ds^ may be had by one' Additioxi, »id a Multiplica- 
tion into the Rdot of 3^ 

I * 

6. The Triangles^ b f tnd b ebsxt $imi1ars but tijc 
AnglcA/^r^FLzr yjdtg^ ^crtfbrethe Ab^ hhc 
zz 30 dig. and fuppofing a Circle on the CetUr ^^ 
defcribedWough be by & e theChbrdof ^will beV 
qualto^^ the Radius, therefore, r b "[- b ^^K, b, i.e. 
the Sine of my Ar6h i^ D Icfe than 60 dig. iidftng the 
Sine of the Oefe^ b a^ make K h. the Sine of an Arch as 
if{v^h. exceeding 60 ikg^ : - 

, Confcquentlj, tbe^nesof nil Aithe$leftthan<So d-^g^ 

JB 2 being 



(8) 

h^T% kiwwji, ijie reft w 90 <fe{. ■Qarbefraodby om 
Addition. For In&aDce. 

S, 50 ibg'^-S, 10 <t^= S, 70 A/. 

.ifn AffUcation of what has ^emfaidt to tb^ 
aButil ntakh^ the afertfaid Tah,k. 

The Radius is eqaaito the Cborf of 60 <fc/. and f 
R,= S» 30. then by the Secoiid arc known the Sines of ■ 
jljflArcbcs, ■ 

Ag. * " '" iv y 
15 00 00 00 00 CO 
7 30 00 bo 00 00 
■ ^ 45 00 fco 00 00 ' 

1 52 30 00 CO OQ 

■ ' 56 1$ 00 00 00 

28 07 30 00 CO 

-'^ " 07 01 52 30 00 
03 30 5« 15 00 

©1 45 So 07 30 
52 44 03 45 

Sf> that by 1 2 Divtilons \fe come to Sines, which have 
tlip fame fenfiblc Proportion as thdr Arches, for the 
laft Sine lave one, is double of the Ia(t Sine to all Senfc, 
as one Arch is doubk of the other. ' 
~No»v the Number ^ Alinutes in 30 dag. is iSco, 



times, tbere[bre i.f 1803=12048x^2. 4.4. 03.4^* 
therefore 1800. 2048 ; : Arch 51, 44, O^. 4?'. Arch 






(9) . 

And thus by a cpnttnued. BifTe^ion of Arches, the 
Sine of one Minute being had (by the 3) is (owM {he 
Sine of two Minutes, then (by the 4) the Sineof thifec 
Minutes^ and fo to 30 deg. then (by the 5) to 60 ^g» 
and (by the 6) the reft to ga^g. . 

Having found thi^ Sioes^ the Tangents, and Secants 
zmy be found by the following Pr<qportions. 

FE. FB : : EA. BM? CS-R: S. T. 

FE. FB : : FA. FM>thatis^2.R: • R,/, 
Sfi,BP : : Fa ^ \ >. . /T. R; : R,t. 

LpM. I. 

f V X R= S f I ArcHs: i S Arch * T. f Arch. 

1, EB.BN : :AB.FB : :(f AB.)BN.| FB. 
ue. V. %\ Arch : :S.|-Arcb f R* therefb|» 

fVR=SffArch. 

2. fAE.BN: :AE. AB: :BN.B;k.theterore4AE 
y B A=S J f Arch= I S Arch x T fArch. 

L E M. II, 

I R vrs S f I Arch= f S Arch xr f Arch. 

1. f ED, FN: : (ED. AD: : AE. AB: :)FN.PB, 
Jherefore f ED x FB = F N f , that is |Rv=:S ff 
Arch. 

2. FG. Gz : : AE. ED, therefore f AE x Gz— | 
EDxFG~FN^.=|SArc^XT|Arch, 

L E M, m. 

Jbe Taftgenis oftm Archts A ani^ arereclprtUfy fro^ 

fortiohal to their Co-tangents. -^ 

for T, A. R : : R.t,A. 
and T, B. R : : R,t,B 



■T < II ) 



jher^fereT, A.xt A=:T,BxtB 
^herefoteX; A. t,^.j: tBtA. 



(lo) 

LEM-PT. 

Tbe O'finet ef two ArcbcsAmfdBr ^u^. recifrveajy pr9^ 

fffriional ta tbar SiC4uUs* : 

A.R: :IU/:A 



t^mmmfmm 



therefore g A x/A = S B x/yB 
tfierefbce Jg A J^B : i-Jj B / A; 

iPtane anj SphcricalTri^ocmnetry are ii4v«4 lyy th? 
help of four Fundbunentat PropofitioDs calf d Axioms. 

AXIOM I. 

Fir JPlam Triffrnfitetry. 

In a Wgbfr-arglcd Triangle, if one Leg* of the 
,BJght-angl(^ be made the Radius of a Circle, then 
ihall the orfjcr Leg, he the Tangent of the Angfe bppp- 
iite to it^' and ehe^Hypothraule (or fide oppootetothe 
;Ri^ht-angle) its Secant. 

• But if the Hypotbenuie bemade the Radius of a Circle, 
then will tlie Legs (or Sides including the Rjght-angk) 
ibe the Sines of the Angles opfo&te^ which \v^^^ ^ 
'mdent by the Figure. 

Upon this Axiom'dq)endb tkft Solution of the 7 Calb ^ 
of Right-angled Triangles. 

Andiin the fbtlmTing Proportions, I fuppofc that tiyo 

Lines eftiimtedinpftrts of any Meafure^ 

■« ■ . 

flatt IJ, For ^Toamt/ey in parts of the Ta^>le, arc 
JTgl. Proportional to themfelves reckoned accorr" 
ing to any> other Meafure; ib A' Bieckon'd as 
Radius, of parts 100000, is to BC the Tangent of 
the Angle A 3oWcf^.of Tabular parts 57735; f© the&me 
•AB of 213 Feet, ta BC 123 Feet ahnoft. j 

* 0' 

Note, That the three Angles of a Triangle make 2 L 
or i8orf:g. and therefore in a. Right-angled Triangle^ 

\ : ' one 



one of tte acute Angles ^s the Complement of riicf ' 
other to 90 ieg. confequentl^^enebei^ gxven> the other 
is kno\yn; and inan Obliqueyngled Triangle two An- 
gles being givei^ the third is alfi> kncnvn asbeii^the 
Supplii^nt or Refidue of th|p Sum of the other two. 

For tl^ more eaiie making the Proportions for the 
Solutions of Right-angled Triiu^ks. 

* - ^ • • ■ ^L 

Ob&rve, That asdifierent Sides are made Radius 
fo the other Sides acquire different Names: Whicti 
Names are either Sines, Tansents or Secants, And are 
to be taken out of your Talxe* 

To find a Side, any Side may be made Radius* 

Thentay, 

As the Name of the Side giTen, 

Is to the Name of the Side requk'd. 
Soisi^e Side given, ^ 

To the Side requir'd. ''\'^ 

But to iind an Angles one of theg^ven Sides mud 

be made Radius. 

Then, . 

As the Side made Radius, 

Is to the other Side: 
So is the napi3 of the firft Side, 

(Which is Rad.) to the Name of the Second Sixicir 

Which fourth Proportional moft be fimnd smapgfJt thi 
Sin^ or Tangents, tSc. to beDirtnimin'd by the Side 
made Radius and againfl: it is the Angle requir'd* 



??^ 



F^ 



(") 



%e Propihrtiont for the Solution, of the 7 Cajf^ 
of Plane Right'-oiagled TriuHgleKy , 



f^ 




jiver 



AB 

A& 

C 

AB 

A& 

C 

AB 

BC 

AB 
BC 

AB 
AC 

AB 

AC 



iieq. 



PropoJIS 



ons. 



jKadZ 



:5 A. S, A: : AB. BcIaC 

BC I R. T, A : : A B. BC AR 

T A. R : : A B. BCBC 



AC 

A& 
C 



2 A. R ; 

R./, A : 

r" A< 0" A : 



AC 



A& 
C 



BC 



AC 
A 
C 






B 



AB. AC AC 
AB. ACAB 
AB. ACBC 



A B. BC? : R. T,AAB 

Cotriplemenc is C. 
B C. AB: : R. T,C 
Complement is A. 



Cafe 



BC 



R.T,AAB 



\ B. BC : : 

then, 
g A. R: : A B. A C(AC 
or -/: A B^-|-B Cq—k C 



ACAB 
\ B.AC 









R. S A 
R. /. A 



\ CAB : : R/g A 

then, 
a. T, A : : AB. BQAB 
or v^: ACf— AB^=BC 



^. g,A : : AC. A 5 
r, A. R; : AC A I 



AC 

AB 

AC 



AC 
\B I 



tf- A. tA : : AC. AUBC 



immmm 



5 



■7 






in) 

to 

Cafe I. given AB. 'i6parts'J 

t* GccmctricaBy mtb State aadComj^a^. 
From a Scale of equal Parts lay dowo tlwXine AB z 



the Point A make the Angle BAC=:^o Ag,^ -Flatc II. 
produoe the linp AC tilTit meet tbeFtX" Fig. IL 
pendieaiar BC in C^ and the THangle isde- 
termin'd, and the Line BC heii^ Reafur'd by the latoe 
Scale hy which youPfeOeid, AB tvifl give its length. 

%£jftbeTMccfNaiwr4l^9i€i^T4Ui^ts^t¥u . 

R. T, A ? 5 AB. BCr 

ii^n) I00CXXXX3* 57739O) I : 5& . 

34641018 
28867515 



32.3316168 ihafit 5^nt«,&fr 

rrBa 
nvhich is (bund by Multiplying^^ and , 

one another, and dividing the PtodiiarbythefirQt 
To m^ Pnttiilcy] ihallgivenetno^Sdtttidnsby 

Ae Table of Natural Siir% £&• . \^ 

I 

3« Bjf the Canm of Art^cial Stnes^S^ and Logaritbm 

Jfutabcn 

RULE* '\^' - ^ 

Add the Log$f itbi» rf the d and 3d f«in togfl^ 
from that Sum SubtraS the Logwithm of the fcift 
Term, tho remainder AaU te^c ifC^^thm of the 
^ Term required. . . u 

c *y 



(14) 

By LOGARITHMS. 



T. loJkg. 9.7614394 
56. 1. 7481880 



Sumi 1. 5096274 
Subt. Rad. 10. cooDooo 



Rem. 1. 5096274 theLogariAm of 32. 33=SC 

Cafe 2.givenAB=:56/wf 1*7 

X,A. 30^^. >TofindAC. 
/^^ <^-B; 6odsg9 3 

I. Geometrically* 

Tlate II. The Triangle being Conftruaed as be- 
Bg^ IIL fore, Meafure AC by the Scale you Plotted 
AB by> and you hare its Length. 

2. By Logarithms* 

SA.R::AB.AC 

R. 10.000000O 
AB. 1. 748 1880 



Sum. 11.7481880 
SubtgA. 9.9353204 



, Rem. 1.8128676 the Logarithm of 64*992: AC 
Cafe 3.gtwn ^=^ 7p^^ g^ ^^^ ^g^C^ 

X. GeometrkaUy. 

Draw the Line AB, an4 put 60 parts from A to B 

upon the Point 6, let there be the Per- ' 

PlateU* pendicularErefted BC>lay off 3 5 parts from B 

f^.lV. to C, then draw the Lioe AC, and ^fca- 

fure the Angles A and C by the Line of 

Chords 2' By 



2. Bf the ZogarUbtm. 
AB.BC;:R.T,A. 

BC. !• 5440680 ' 
R. 10.0000000 



, Sum. 11.5440680 

i Sttbt.AB. 1. 778 1 5 12 • 

^ Rem. 9.7659168 which is the Logarkhm Tan- 

gent of 30 deg. 1 5 ffiifu the Angle; the Complement of 
which 59*45 is the Angle C 

The Triangle being Conftmfted, as in the 
former Cafe, Meafure AC by your Scale of ^late \l. 
equal parts and 'tis done. Hg» V« 

a* Bjf Lcgariibm u 

Find by the I^ Cafe the Angle A. 
Then by the fecond Cafe find AC. 

Ofe 5. giircn AB=8o^ | parts, to find A and C. 

u GetmetrtcdOj. 

« 

Draw the Line AB putting thereon 80 parts from' A 
toB^ upon the Point B to the Line A Bere^ 
the Perpendicular BC, open your Com- , Plate Ih 
paf{e% to the diftanoeofioo equal parts, and. Ffg.Vh 
puttinjg one Foot in A, with the other crofe 
the Line BC in C, then draw AC, and Meafore the 
Angles A and C by your Line of Chords. 

C 2 2. Ilf 



'X 



AC.AB;:ILSA 
AB. (^9090900 
R. laooocxioo 



RMM«>«««I 



Sum X 1*9090999 
Sobt. AC* 9^3000000 



•mt 



Rem. 0^30900 tbcLiiarithm 

Cafe 6, given A B=8o 7 _./.,-. ^ 

/ AC=5a»JP^Tofii4BC. 

Plate II. The ooiifc«ftiMiMi« the lame as in tbe 
J%.Vn. Jaft, cbere&ne^ M»fiw«C tjrfror Scale 
oxjP^uaI parts. 

. ^ tcgmtimi. 

By the laft Cafe find the Axgle A: 
Then by thei. Cafe having die AMk A and AB, 
Given, findBC "^ -» * 

Cafe 7. given A C— 50 parts 7 

X^A~3$4fep, >T«&8d^k 
X,C:-554*^ \^ 

t« <f€mittricatf. 
A ««J *c Right-fine A C, malm^ the 



#^.ynr. to C,ftemCte4kir*e perpendicular CB 
^_^^ to the lAie AR cutting Ab In B, AB 
»*« By jwur Sale </e5ua! Pam anfivet? the 



2 «^ 




(17) 

AC uig^foo 
. Sum* 1 1. 6123345 

jute iR« f 0*0000000 



Rem. i«6i23345 theLogarithinof4a95 parts, 
=AB. 
Note^ all thefe Cafe may eifily W workt by the 
GuntersSctl^ «rliefconi« a Une of Logarithm Num- 
bers^ Logarithm Sine^ afd LDgaritbm*Tar)g,eiiti» 

Extend your Conm^flcs on this Rule from the firft; 
Term to the fecond Term fa t^Propcrttan, and the 
feme extent wiU mdi fr^tm th* 4iri T^fili «r the 
fourth Temi cpqwi^i" 

Only obfervetJwtifyoiir 6wwiT«iin v|^^^ 

tiiantk&^ tiefiwtfc T^x» muft te||J**^ ^ 

dufi Che thirl, and thrtefive vftm fouf Cotoptl^ 

are placed on the third Term> cxtpiathem lW way 
the Number muft be greater or kfi a;s tbi3 Rule diredv 

VTheve dbe l^iQpMC]0i) k R. T, A : t A 6. BC 
EtHend |iour OompUGst from the Rdfios of the Tan* 

gents (which is the Tangent of 4$^.) to the Tangent 
of the Angle /L %r, 30 *^- l6e |»me Exteht of th& 
Compares Mrm icach from the djird Term AB=5^ 
fanff0 .to J2, 33i^^^s;BCi«^«ii«l» 



1 



( i8 ) ^ 

Example in tbe2.Cak. | 

The Proportion is g( A.R :: AB. AC 

£ztend ycur Coitipaffes from the Co-fine of A, which 
is the Sine of 60 dev, to the Radius of the Sines (^ which 
is the Sine of 90 £g») the fame Extent wilUeach from 
the third Term AB '^6 parts, to the fourth Term AC 
Sz 64* ^9 parts* 

After the lame way you may work them all by your 
Gunter^s Scale. 

A X I O M II. , 

Of Oblique Plane Triangles* 

In any Tri^ngle^ the Sides arc Proportional to the 
-Sines of the Angles oppofitc 

, DEMONSTRATION. 

Produce the leffer Side A B to F, making A FziB C, 

let fall the Perpendiculars F E, B D upon 

Tlate IIR . the Side AC produced if need be, then 

JVg.I. andll. will F E be the Sine of the Angle A, and 

BD the Sine of the Angle C to the Ra- 
diusBCrrAF. Now the Triangles A B D, A F E are 
Similar ; therefore AF (BC; AB : : F E. BD that is 
:: S>A.S. C,Q.E.D. 

Othenvife thus, A B, R : : B D. S, A 
and B.C. R .-: BD. S, C 



'»m' 



therefore A3* S, A = BCxS, C 
therefore A B BC:iS,C. S,AQ:E.D* 

AX I O M III. 

The Sum of the Legs of an Angle is to their (Kf- 
ference^ as the Tangent of ^ the Sum of the Angle/op- 
pofiteito thofe Legs, to the Tangent of half their Dif* 
l&renoe. 



••> 



Tig: I. 



Fw.^. 





%V. 



%.vr. 




'•-^-- -^gfl' 



^» 





(»9) 

DEMO NSjf RATIO N. 

• Ift the Triangle ABC, niakeBl>=B A, and dnivjr 
the Line A D, let fell the Perpendicular B F 
uponthe Une A D, which will bilfeft AD P/«ff HI- 
in F, draw P E Parallel to A C, which will tig* lU. 
bilfeft DC, the ditferenoe of the Legs 
inE,theAngleBAFisftheSjmioftheAngl«BAC 

andC, and F AG f their differcnc^ "^"^ ^V% 
Radius, B F will be the Tangent off the Sum, & F Ks 
of the \ Difference. NcKv the Triangle B G Qiscut 
Proportionably by the line F E. 

therefore BE. EC::BF. FG. 
or2 BE.2EC::BF. FGe^, 

- That is. 

As the Sum of the 2 Sides to their difference, 
Sais the Tangent of \ the Sum of the Angles op- 

(pofitc to tho& Sides^ 

To the Tangent of | their difierencC) 
Q.E.D. 

Or thusy 

Upon B with the diftance. B C defcribe the Circle 
. BC£D,Froduce AB toEandD,andCAtoG,draw 
D C, and A F Parallel to D C, draw likcwife C E. 

Tis plain tjiat A E is the Sum of the 2 Sides AB, 
B C, and A D, their difference, the Angle 
B A F is I the Sum of the Angles oppofite PWff HI. 
to thofe Sides, and the Angle F A C | their Ilg. IV. 
diffeienTe. 

Now-the Angle D C E in a Semi-circle is Right =: 
A F C, and therefore if A F be Radius, F E willbe the 
Tangei^t of the^ Sum of the Angles, and FG the 
. Tangent of the ^ difference. 

Now the Triangle E D C is cut Proportionably by 
the Line A F. Therefore AE. AD : :EF. F C. Q. E. D. 

PROB. 



PROB. 

If tbiAmtoABC togim^ini At LofHrMkim 
•T the Legs ABy C B » aiMl it be Mtqirfred to £ndl tte 
t»D iHOibnq|Aii|^B A C^iad BCAjtvUckit^ 

B U L & 

^At ItokAtLegA&tothegiVatoBC;: felsRa^ 
IkiftotbeTang^ofitttArth-y ftomiMch taker stira 

Tten asRad.totheTanE«of tbertiRiiainifigAfdL 
: f ^if the T^ off the Som of dto Angles 
A and C, Co the Taqiiefit of I their diSbienoc. 

DEMONStjLATlON. 

^kU&UL DfMV B H perpendicttlaf to ABrrA B 
Jii^HnC ss Bland IK perpendicular to HL 

•O'hcnBH. BE : : R. T, Angle BJffi, whence ta^ 
king away the Angle BHIzr 45 deg. there willl remain 
^ ^ the Angle IHK, 
/iE;/5AndR.T,IHK:fHI*IK::AH f ^HtVlK ; a ABT.JS 
^ -? : (by tHi^dJmm) T,|Zm^ <>• T.|XLL<f«<5ftir 

AXIOM IV- 

Having 4i*wft a Perpendictthr B H from an Angle 
^ ilftopppoSfeBftie^theBaieACftatlbetaUieSimiof 
Cht -oriier dfo fi4ef A E, as their didl;rence CI ^ ia t^ 
the didbrence of the Segments of the Baib QA.' 

DEMONSTRATION. 

t6r by the j^^PtopofitioiiafchtthirdorJitfitfi tbe 
ReOAl^ under Ao, A £ ii equikl to lh» Ritftan{^le 
iindtf wr A and A C 

Therefore A C. A £: : DA.GA.Q.E.D. 

Tbi 



( g> ) 

7%e Prcfortionsfir the Sktuti^is ^the 6 C^ei\ 
in OHique Plane Trum^les* 




yiveniReq 



•MHftaAi 



AB 
BC 

■It II >(■< 



AB 

BC 

ft 

C 



A 

C 

BC 



«iB*i 



B 
AB 

BC 



B 

AB 
BC 

r»n, rn, 



AB 

BC 
I AC 



AC 



AB 



A& 
C 



AC 

A 

B 
C 



l^ropottiote. 



MMarita 






AB. BC ! : S,C. S,A 



f 'I'M WM-Uri I'll i 



A «• B C : : S, C. S> A 

Hetke by Subftf ^aion the 

Angle B will be known. 

S, A. »,«: :BC.A C 



•Mmm 



■MMMNM 



*«■ ■!■ I ti 



S, A. S,C:: B C.AB 
BC+AB. BC— AB,*: 

T|XXi,oprthett, 
i'lS-j-f Xstgreacef jt, A 
|Z-.|X=:leirer X, C 



fcin. I ii 



iriih^^ 



^ then 

S,C. 5,Bi : AB. AC 



■Mm 



ACBC-f-^BA: :B C- 

BA. OC 

thenj A C-|-|0 CzrD C 

fAC-|OC=:DAthen 

AB. AD::R. gA, &| 

CB,DC::R. g C| 

D 



Cale. 



I 

MHMMfaSiS, 



•Mai* 



iM>ii**Mak 



5 

* 



J./< 



i./^ 



f. 



/c 






^•/t 



( 20 

ExMfiei in the Cafes tfOhlifie Tru^ksl 

Cafe L giTcn AB=30 7 XofindtheAn- . 

J. GeotmtricaUj. 

Draw the Line C A produced as far as neoeilary 2 
At tbe Point C make the Angle A C 3== 
Plate IIL 25 deg. 30 mnh let otf 40 pai^^cs from C ta 
Fig.\. * By take 90 parts in your Compailbs from 
* the fame Scale, and putting one point in 
By crofs the line CA, with the other, whichit. will do 
in two points A and A. Draw the Lines B A, B A ; 
then the Angles Al^ CAB, lieing meafured by a 
Line of Chords, will give the Anfwer. And here 'tis 
xnanifeft, that the Angle A^ in the one Caiie is Obtufe^ 
in the other Acute. 

^ But here Note, That if the etiii|p/Angle be Ob- 
tufe, the Angle required is Acuie«^ But if Acute^ 

andopp^itetothe^^^^^fide the Angle re- 

qmred is^ DouSfiil \^^ whether Acute or Ob- 
tUie 'muft be determined before the Operations* 

^.Toio this by the Canon of Artificial or 
Zogaritbmich Sines. 



..1' 



Obfenre^ That it has been demonftrated in the /r- 
cmd Axiom, that the Sides have fuch Proportion one 
to another^ as the Sines of the Angles oppofite. 



There* 



c 2J ) • 

Therefbn^ Hrfiy 

As the Sine of any one AngH 

Is to the Sine of any other Angle ; 
So is the Side oppofite to the firft Angle ; 
, Tq the Side oppofite to the fecond Angle. 

' As ftny one Side» 
Istoanother'Side; 
So is the Sine of th^ Angle oppofite to the firft Side, 
Tothc Sine of the Angle oppo&t^to the fecond 
Side. 
Therefore, If you would find a Side, . h^in your 
Proportion Syith the Sine of an Angle jj and ii you 
lvott]d (nd an Angk^ begin \nth a Side. 

AB. BC :: S,C. S^. 

BC. 1.6020600 
S,C. 9.6339844 



Sum. 11.2360444 
AB.iub(t 1.4771212 

■ I 'll ■" " . i ) 

9.7589252 



Cafe J. gjiveB^ AB— 30 "7 

BC=40 >To fod AC. 

C=2?i3o'b , 

1, Geometriedltf. 

The ConftruAion is already fliew'd J Plaieia. 

ThoKfottt nwafuie A Q and you have fig. VI. 

- - — - ' p 2 ■what 



what is defir^d, that is the greater C A, if A be acute>tb& 
Icffcr if the contrary. 

2. Bj tbe C09m cfl^mibm Simt^ ^. 

Find the Angle A by the laft Cafe ; For In|lance A 
was found, 

35 i 2' 7 .£. C Aeute, 7 and confequently C 1 19, 28 
J44: 585 iObtufci, KBCzz \ 9.3a 

» 

S,A. S,B: :BC. AC. 
S,B. 9-939839^ 

I ■■■! HUM m 

Sum. 11.5418996 
SuMt $,A. M589519 

Rem. .1.7829477 ^ <^6a5A C if *e Angle A 

beAcvte. 
S>B. 9.2191164 

*B C i»6o2o6oo 



Sum. 1 1.82 1 1764 
Subfl:. S, A. 9.75895 19 






1.0622245 11* i4ssACif A be^Ofatufe^ 

. Cafe^§ivenBC;:;?6 7 

A;=33U5i;> ToftidAB. 
C?;:26 z^ 

f, Giometrietltf. 
The Angles A and C being known, the Angle B muft 

Sf VIT ?'■'"'' ^ i^"^ CB, andtheJeonput 76^*^. 

: «»«te*Ji« Angle 9$;A55a^3f;ind the Ai- 



gle C B A — i5to. ^^ the lines CA and BA will concur 
in A, and the Triangle if Ajtemtia'i, then ifyoumca- 
fuic A B by your Scale, 'tis done. 

S,A.S,Cr:BC AB. 

S,C. 9,6426182 • 

BC. 1.8808 1 36 



■•"•iw" 



Sum. 1 1.52343 iB 
[SubftfS,A# 9*7390129 



m» 



Rem. 1.7844189 tbeLogarlthmofARi^S;^!. 

Ca&4. given BCszjt "7^ f a^t. < . 

BA=<o CTofinathe Angtei 

Draw the Line B C and from your Scale put off 7$ 
from B to Cy then draw the Line B A mak- 
ing the Angle C B A = 88i 00^ and fet Plate IIL 
bff 50 Jtf J. from B to A, and draw the Line Jfgf. VIII. 
AC and the Triangle is determined. Then 
the Angles A and € may be meafiur'd by a IJne of 
Chords. 



%. Bj 



(^6) 

BOhAB.BC— AB:: Tf ZZXop.Tf XZX'sTOp. 

Thati% 

125. 25:: T46i TfXZZ'sOp^ 

25. 1.3979400 
T. 46£ iOjoi$id2S 

Sum. 11.413102s 
Subft.Lqg. 125. 2J0969100 

Rem. 9.3 16192$ Tvhich is th^ Log-Tang-of 
fthedifflbf AandC=: 11^^42' 
Nowf tiieC+f tiiediK7 C 57^ 42^the Ang. A 
Sum qf A & ^ ii. 42' ^^< 
B^6d. C^ \ / 34^ 18' the Ang. €• 

This hatfa beeA prov*d in the third Ajdom; 

Tlate HI. ^ Caie tfae$. Given, the fkme as befbx« : 
Figm EC. To find die fide AC. 

The Angks A and C being found as in the laft Caic. 

A=:$7i42^ 
C=34 18 

J|f Ztgaritbms. 

S,CSi,B::AB. AC 

S,B. 9*9997354 
AB. 1.6989700 

Smn. 11.6987054 
Subft. S^C9.7509i4o 

— ■■ " . 
i-94779i4&e Logarithm of 88. STtfi* 

2Z A v^» 

Cafe 



Cafe 6, ^ven AB= 687 

BC=i2o>TofiiiddieADgfcf A, Cya. 

, j« Geometrical^ 

With the 3 Lines AQ CB> BA make PAif^ m; 
the Triangle ABQ then by tbeWlpof a f^.X. 

Line of Cnords^nieafure tbe Aif ksxeqiiii^d. 

ACBC-l-B A:;BC— BA.OCtliedifl&reiice 
of the Segments of the Ba& 

Tfaati^ 

156. 18S :: $2. OC 
188 2.2741578 
52 1.7160033 

Sum 3.9901611 
Subft. 156. 2,1931246 

Rem. 1.797036s = ^52.67 =r O C the diflerencs 

of the S^inents. 

TheftfAC + |OC=DC=: 109.33 
— =:DA=: 46.67 

Now theObliquc Trian^ AB C is by thePferpen- 
dicular BD divided into two R^ht-angkd Triangles 
BDC, B D A, in each of which the Hypothentifc and 
Bafe isgiven : Therefore by ti« 5 Cafe. 

The Operation of which I ftall Icaye to the Learner 

to avoid prolixity. 

Tbif 



(28 ) 

Ibis UfiOifcmiff be Solved ijfour other tbeoremr* 

Vor the Analyfis and Demonftration whereof} Sup- 
pofe m^ n. tixe Legt of the Angle re- 
quired bits Bafe tf,i» (/> « the Seamentsof Plaee TV. 
the Bate made hyp^ a Perpeumeukr let Fig.h andll. 
fall from another Angle^ whether it fall 
within or wichoos the Trias^ 

a 

thcref. i^zz^^ -{-nn-^bb. 

-- ' 
2n 

therefore R.g iim.tn m^Ynn — b bi: 2wn.m m^\- 

— II I i „ ^ 

2 n 

*i» —.^ *tb«re&r© 2 i»ii,( 2«r» -1- «»«+»;» —^i) 

compotifiding and dividing: :R.R -J-^'=±— bj^dcf.pth 

. -|Rf- f R^l-.5Sf f ADg.?bythe 2d Lemma. 
|RV5-*-2 S^|Ang.ib/ the ift Lemma. 



^hereforeX 5^ x— * ^ ? .^ f J^ f | Angle. 
».R,::1 i^-Xtr^'illlAille. 




/ 



There* 



< »90 



/ • 



1. 4Mrii.z4-£j^1i^^::Rf.;gf.|A]]g1e; 
or dividingtbe ift«nd2dTtrmby 4. 



' ^ ■ • 



ill Wwdii Aitt, 
As the kea«tf 1e of die Sldei ifidudint tlie Angfe 

btb tiks ReAmgle^l tiie Sttrit aftW 3 Sides 
MHMplittil by f €h(ft Sum cfthej j^made 
Icfe by the Bafe! • ^ 

So is the Square of Ibe tLMus, 

To the Sqttarc of the O^fm ff li tin Angle 
esquired. ... 

3.4^^(-i^x«ir-^x:: RfvS^f A4|(t%. 

Or M before^ 

, -. - > 

InWpfdetMis^ 

Asthe Re^ifbof the Sid«» tedhidief tW Aiggb 
fejquirea. 

htdtbb Re&ai^ :or| the Stunofd^B Sides^ 
made kil by one of the Sides {ndMing the 

, Ai^MuktpM byf thdSiittortUi i MdA 
te^ leTsby ^ ott^Si^e tedttdfaftbe Aft* 

Sq is tbe,Souare of the Rai^ ) 
'e<^m suture ofthe Sine off tht Aig^iiquiied; 



^ !• *+* 



Uo) 



< •• 



3. z-|-ixz^t.&4-xxi«x::Rf.T}f Angles 



In Words thus, 

A$| the Sum of the SidesmulttpUedby f tfaeSumof 
the Sides made le& by the Bai^ 
Is to halftheSumofthea Sides^ made left by one 
of the including Sides^ multiplied by f the Sum 
jfthe 3 Sides made lefs by the other incUnins 
Side : -0 

So is the Square of the Radius. 

\ To the Square oftheTai^nt of I the AmIc re- 
quired* • . • ^ 

__ Or, 

In Words thus, 
;As fthe-Somof thej Sides made left by one of the 

gtthrttJ Sidis made Icfi by the oAer inclu^ 

fctoftheSumofthe 3 Sides MultipKed by J the 

.^^ ^Sum oftheg Sides made left By the Bak 
5o» the Square of the Radius, 

Jo the Square of the Co-tangent of I the Angle 
.«.:.. ieqi|iieicU -'^~ • 



N 
4 



Now 



, For |x +1 xs: m 

tliercfbre |2Z — fxx s: «« 
ani Confcq. iz— xx=:4«ivii. 

If therefore you have an Angle given, with itsBai^ 
and the Sum orDiflerenceof theLe«, you'll hive by 
thefe Theorems the Square of the Sum or tHflfeienccii 
land fo both Sum and Differences and consequently tha 
very Legs. 

For Inftance, in the Jirft Cafe if the Angle indudedi 
between the Sides i» and n were given together withz 
tbeSum of m' and ny then the 2d, 3d and 4th Terms 
are i;ib&lutely known, and confequently the 1 Term 
^mnrs known, if from z z I Sub(lra£k 4 ^ » the remaixi- 
der is XX the Square of the DilFerence^ then . 

|z-f-|xgreaterSidc^ 
f z — IxfelferSide. 

5. Jh the Theorem which firft comes out for the 
finding the verfed Sine of the Angle required. 
r». 211111. *^—xx: ; R.V 
Let it be changed into thefe two Proportions, 

ziob — XX 
Vtx* 2m. A-)-x : : ^ — ap.G— — — t»— 

: : R. V =: RX**-.x:P 



• bb^xx\ 

then* l^ G I =: J 

\ 2m J 



2I9MI 

Or into thefe Two. 
b^\-Xm 2n : ; m.Hzz 2m n 

bb^xx 



/ 2mn \ 
thenH = ( ■ ■ ).* — x.R.V=:Rx 

\bJ^x/ 2mn 

which Proportions may be workt by the SeStf or 
C«itf^i^s Line E2 Sghmc 



(ifO 
Spheric Trigmmetry. 

ASfhem Trumle, is fhaf; wUdi boontaanrd 
Mr«eo tb^ Axidics 9f 3 grrit Circles <>f the 
Spbre* 
A ^M^ ^Uglc i» tbt; bam with tJitloclmeiaDof 
tt» PliLpe$ of thole a Circkt whidi eooilki^ 

^ffeHms tfSfbirk Trumiks. 

%• When t Circle falls on another Circle iSbfi Sum of 
|}se two Aisles made thei^by is ::; % ^. 

:^ Wben a Circk erofl^a Circk^ $he Vertical An- 
gles made thereby arc mutuary Equal* 

3. The greater Angle is oppoiite to the greater Side : 

^ AQllofeetoTnapgle hath its two Angles at the 
Bale mutually Eqxwl ; $m on tbs contrary, if a Triangle 
bas 2 Angles equal, it has two Sides equal* 

(, T>vo Trianglf) mutually Equilateralj arc alfo 
BquianguUr one to anotbert 

6- If there be two Triangle and in each one Angle 
and the two Sifles including refpe^vely Equal \ or if one 
5id<^ an4the two Angles adjacent b0 feverallv Equal ; 
then the two Triai^les are equal; for if laid one upon 
another they will agree. ' 

7* TwsQ Sides of a Triangle are bigger than one ; for 
the Arch of a great Circle is the Ihorteft'piftance be- 
tween two Points on the Surface of a Sphere ^ a ftreight 
line is betwixt two Points in a Plane* 

Theie (even Properties are common fo plane Tri« 
aaiigles, and have a like Ocmenibacion* 

2. All great Circles mutually cut eadh other into two 
fOttal FartS) for the common Se^iqn pf ^their Planes is 
|Diameter of the Spb^pre, and cdifife^den^y the two 
$(;^lQns of their Peripheries^ aip ^t a Se/uesrcle JM- 
ftancef • -■ ^ - r *- 



^\ 



< 35 ) 

Hcooe it &Ibws that every Sic _ 

it left than a Semicircle. DB is left than cb 
ff0^eTV. ScmidrclcDC. 

Ik. ill. 9* The oppoiite Ai^teat tbv Seftiois of 
twoCirdes arc equal XD = XC for tte 
Ame Planes make both Aiig1e& 
)0. Ip any Sphme Triai^le if tdie Smnof the I^ 

of an Angle DB, BAbez: >than#S(Pn»4NleDBC 



tbe intern Angle, attbe Baft D, isaoeDr^fagQrs;^ thyn 



the oittw^ oppDfite AmlpB A C afidoonfinuetfly the 
Sum of the two intemad Angleiat theBafc D, D A B ikatl 

Jear >diantwo^J» 

D B MO NSTR AtIf>N. 



If pjj-LB A be ;;: >tbaaDC,tbcnBAJ8z: >than 

BC; therefore the Angle Q or which is equal to it D 

V^SthanBAQand theiI>i-XOAB- Vthan 

(L^AC+XDABrz) 2-f* 

lU Condi. In an Ifofedii TiiaMilfl^ if one of the 

equal Legs is c: Vthaa a Qaadiaat, the Angle at the 

Bi^ is ::; >thaa ^« 

I a. The Sum of tl^ ijuoe Sides of a Triangle is lefs 
than a Circle forBA<;BG+-ACi therefore DB -h 
pA-}-BA<pBC+PAC. 



(j4) 

T 3. If from the Point of an Angle as a Pole, you dc- 

fcrifee a great Clrck, or which is the fame^ if you dc- 

.fcribe a Circle at thediftaxice of 90 Deg, from that 

Point, the Arc of this Circle intercepted between the 

Legsofthe Angle, is the meafureof the Angle. 

1 4* The Poles of the fides of any Triangle G H D, con- 
Ihtute another Triangle n x /», which we may call Sup- 
plemental to.the Triangle G H D ; For the Supplements 

°^ ^^ iSi^^ C <>f *« ^ » « » aw equal to the 

ixS^sJ-f^ Trine GHD. 

D E MO NST" RATIO N. 

From the Points G H D as Poles, defcriheon the Globe 
3 great Circles, X A/, RTmii, xBn 2^ then Mr being 
the Pole of G H, mjr is=Qiiadrant:=A x, x or E being 
the Pole of G D) therefore adding mkan both fides 
«w x.*ri Aji^ StipfHement dfC A^ ^ meafute df^he L 
H G D« 

Z»=:Qiiadrant, C» being thePole of HD) PUteJV^ 
n B X aiKl adding B » on both fides, n x Fig. VII* 
r= BZ =: Supplement of B K the meafure of 
the Angle H D G. 

n T=:Quadrantz=#» R, and Subftrafting mr T on both 
fides n mzz T R the meafure of the Supplement of X G 
H D. And here obierre that the Triangle n^m eon- 

iftituted between the 3 next Ptole^ hath its 3 f a^? 

= f «dfl^} Of the Triangle GHD, five that the 

grcateft fide i» Mr is the Supplement of the Angle H 
and the Angle E of the fide G D. 

15. Any Angle of a Triangle mth the difference of 
the other two is <:^2-J for x n<^ x /» 4- mn>> i. e» 2 -^ 
— D<:C2^~G-|-2**— H, and taxing awsiy 2 *J 
on etch fide. — D <: 2 -J — G — H, and adifing G 
and H on both fides 0*4- H — D <;2 -J* 

— - • 16 2. 



( ?5 > 

i6. 2Triaiig1es A andB iiiaiuallyEquiaiwilar, art alio 
tnutuaHy Equilateral, for becaufe they are Equiangular^ 

thdLr Supplimental Triangles «» C ai^ 
Tlaie IV. Equilateral by the 14th, and tbere&xe 

f/^.IV*andVL Equiangular by the 5th; andtbere^ 

fine tbe proposed Triangles A and B 
tvhofe fides are the Supplements of thofe Angk% are E- 
quilateralby thei4tk 

17. Three Angksofcrery Triangleaie^ 2 V* and 
<C,^^ for n z-f^x jM^- » ^ ^4*^ (hy the lathj^tis 
2-J— D -{-2<^ — G'Y- 2 <-J— H 4C 4 -J that is, 6^ 
— D— G — H<l4*-'and taking away 4<^ on each fides 
2U— D—G^H<:oand adding D»G,H> 00 both 
fidcs2-J<:D+G-i-H. 

2ly. The Sum of the Intern Angles is le&than the 
Sum of the Intern andExten^ both which In. all make 
but 5J» 

Orthus^by the Figure to the uth, the Supplement 
o(n tn is the meafure of die Ati^le H ; £ «r tbe msafuner 
of the Angle G, and E it of D. 

But the Supplement oSnm added to him makes but 
2«>' therefore the Supplement n m added to E » -f^E 
is p> 2 -J> that is G4-0:>a*i. 

Orthus^ by the xoth, the Angle B AC 

Fig. V. is i: "> than tbeAngWD. In this firft cafc^ 



make? AC =; D and produce ABtoE. 

Then D F+A F=2-J, thprcfdre BF+ A F <:2-i» 
therefore Z F B E=:D B A:>^F A B > but F AD^-F D A 
= 2-J, i. e.PAB"|-BAD-hBDA = 2-J. thtar^fore 

I>BA4-BAD+BDAr>^rJ- * 

2.IfBAC=:n, then BAb+BDAr: 2-4, thew. 

fore B A D-j-BD A-l- ABb^:^-*. 

3. If B A C<CX>, then BA DH-B D A> 2^^, there- 
fore B A D-|^BbA-|-ABD>2-J. 

18. Of feveral Arcs of great Circles failing from the 

.' fame Fbint of the Spheres Surface on another 

tlate IV. Circle, the greateft is that which paifes thro* 

FfJ. VIU. the Pole of the Circle and the next tothis, 



J J •> 



U p€Mt ditufhat MUdbhhtAardSi Vtt bpMk P 
tIttFMeorthe CittH C ftf D atid * tben^le df D PC s 
thcDi* AD(Ap4-'PB)'>A B> AE> AC anidx: 
Ai«Bi» C:> BPt>6D. - 

i^ Aptatt Qircld }>affif)^ tftnf the Poles dfaDMiier 
|ite« Cirekeatt it «t Ri^ht Angfesj andonthe oon* 
tf «7, ifiteittt it it Right Aogbsit paffes thro if& Pofes* 
For the Angle of the Inclixuition oTthc tufo Plaiiet is then 
Ught xrBD-^^rsPGDtePDB-ftAC. 

flO» In aft OMi^tfe At^lti Triaiigle, if the Angles 
M ftdBttfe ai« MI»)thtf ts>hotb AcatBy or both OUul^ 
thtpMfaidfettlar let &11 from the o[^t Angle falb 
Mriiliithe Tii«ii|g^ tnd the Qpsidfantal Arc without : 
but if they be unlikx^ the Petpendieular ^Oi"^ without 
«i9il the Qjiaaribit tirlttdn, ibr the Triangle £ A F has 
X X fiy F^^eiilr and the Perpendicular AC fat1$ wltlu% 
and the Q^rant A m without^ alio the Triangle B A Gr 
1iatXX..B,G, Obtuft, atxittieFBipendicularAOviiil^ 
>ll^ and €fae Qaadrant A i^ ivithout. 

But the Triangle B A ^ has the X X B,E, of difleient 
ki&di, andthe Perpeodleular A C without^ atid the (^* 
4UanrA#wkhia 

By the fa^ie Figure may the Ahit^uilSes' of Right 
Ai^M l^da^fesbo Sotted; ^ 

SQLV TICKS. 

u The Legs of the Right Angle irct)f ti^ fame kind 
^ith the oppolice Angles, ib i|i the TrlangleB D A, be- 
iaiSfeDA>^QjtadrantDP,thc Aiffilel3»A>» -JDBl^ 
iwd in theTriangle B C A,becaufe A C<:C&adraot P <i 
J^CBA<:-JCBP. ,T 

2. IftheLe^s (and confcqiiewty tte^Angtes) are of 
-1^ Sinkybf ofdUleretitkini^ ; tht? IffWbenufi is accof^ 
dinjly ^ ;> Quadtatit. Sd in tUTisk^}»9 £ D A^ 
•E C V ^he l^f i^^ A E is <C Qsnli^ant, but in th^ 
T riangfe B D A, tbcrj^fpt^rtejtt/ir A B 
' ^rii!l^^r/<r,iftheJ^5««lfj^^^ 
SimilaiS' and the 2 ObSmic Angles aite Sii^ilar j but if 



CJ7) 

the HMbenuJe be ^ 90 6tg. the Legs are Diffimi. 
Jar^ and the Oblique Angles Diffimllar. 



4mlS Ht^HffHhcnufi be ^ 90 Deg. each Leg Hull be 

^E^Marl ^ ^^ adjacent Angle, this follows from 

^ the ift, and 2d. Sol. 

. 5-Andonthecontrary, ifaLegbeJgg^^J 

to its adjacent Angle, the Hjfpotbsnufe will be ^ 90 deg. 



Fcr viewing the Siiie$» Co-Sines, attd other Right lines of 
ArcSy which are not vipble in a common Sphere^ Hu con^ 
veniettt to ufe tbefUfiCs^tbtAtcs of three great Circles 
cfCardlifibjf^a^"'^^ \ ^^^^millar Sphere^ 




As Tuppofe in the preceding Fig. 2 A res, B P, B A, and 
thai BP H the Plane of the greater Arc \veit turn'd 
xoond B H> till that a Right Line fallingfromTP perpefn* 
dicular to the Plane B A H> may fall on foxne Point of the 



S ky fuppofe on D ; forin that Pofition P A B mil 
be a Sjrfieric Triangle, Right, Angled at A, B P the Hj* 
fotbenufe, B A the Ba% P A the i^cppendicular Arc. 

F And 



Afrit riMMfrPA.Nuaw2.s PA^tfcTil MH g ft n il 

Atted according.^ te Ldtaa ifairMiV MkOntm A U, 
P R Pcrpaidicular to B Ci jfe A.E^P F, P D win bethe 

aflesofrfcAfwBA, BP,FA, adi thefr Cb-fiftcs t»ai 
be £ QG QI>C vthefethifl|ftbdfl|dpmor WMskd^ 

afftf^ aad c^nftqoently the Demonftration of the i6 
es cf Might' Ji^kd Triarnks \ ta which End let the 
Aie m m Mim. ^ be atfo fitted in the Solid accotdii^ 
to its Uetcn. Then ia tikgi%J^b^;^A^k(iS|Airk 
T^tianglwt p B A> flr B A having the 6me Acute Angle 
BattkeSdib 

AXlOlit i 

The Sines of the Hypothemfes are proportional x6 the 
Sines of the Perpendiculars 

PF. PD: t^^^C-arcT. 

A X I 6 M IL 

The Sines of the Bafes are prc^oitional to the Tmgcntt 
of the Perpendiculars. 

A £• AG : : * C. ity* 

For the two 'Bi^bt Angle Triangles F P D, c v S^ are 
Simillar. Alfo, E A G, C Ay are Siirtilar. 

For the Solution of the following Cafes I fuppofe BP A 
a Rhbt Angkd Triangte^ziA its Sides producai to Quar 
drants, BN BM, AD; fuppofe alfoP E, PF, NG, 
and E G Quadrants. PU te V. Ftg. I. 

AKss ^;4B ManiFEsX FP&£ii^BPA,ui OD 

:;^NU::;;X.B^aad^Al«}e$flFA3<l*H>C»? '' 



Hoie fo&i^Ki the TiMt of «eB«fMim foft tl»&i» 

■~ $xamfes 



?1 


z. 


-n.J 


/ ^ ! ^\ 






#• /^'T\ 


> 


( ''^ ) 




'/A 


> 


U 




■ /-VX' 


t 


V^v 




B\.X_— ^ 



■I 

















Girci. 


^r^ 


Reqd. 


2 
1 

F 

2 

3 


f 


BA,PA. 


B 


... 


i 


f. 


BA,P». 


B. 


1 


Z 


f 


BF, P. 


B. 


4,X. 


Z 


9 


BP,P. 


PA 


i^i. 


Z,idj. 


4 


/ 


BP,P. 


BJ 


S, i. 


t.op. 


5 


/" 


PA, P. 


B] 


<.Z,;«Jj. 


i 


6 




Ji P 1 
B,P. 


BP 
PA 




, 




w 


iZ. 


i 15 




■ 


xz. 


f li 


i 






■^ 


fitween p. 


S« 38.35- 


• 



1 OF 



V 



• 



*- • 



r?^)' 



M^amti 



d,f 



CAfcl. Given B A =36.007 p^ -o-o 

Plate V, PRO PORT 10 K 

■ R.SBA:.^PA;;gBp. 

S*BA. fJ>J7530« 
g P A. 9'9'ir^^7 



iMfll 



Sun. i$J8$4So6) • ^ 
Sjabft. R* IO.OOOOOOO 

^mxk 9J8H«o*$theSof38.x8=:BP 



of Artificial Sines, to the Sine of 60 deg. fwhiAisfhe 
Complement of B A ) the feme Extent of the Compaf- 
<cs will reach from the Sine of 65 (which is the Com- 
plement of P A ) to 5 id 42 f the Complement of 38 deg. 



f^ Cafe.?. 



C 40 > 

TJateVi Cafe 2. Gircn A B— 30. oo7 o^i • 

PRD PORTION. 
S,BA^ R t :T,PA. T.B. 

R. 10.0000000 
T#P A* 9.5686725 

Sum. 19,6686725 
*^.5ubft. S, B A. 9.6989700 



Rem. 9.9697025 T, B — 43 cfeg. 

BjtbiiSSoh 

Bj tie Gwtten 

Extend . your Cotnpailes from the Sine of B A 30 ieg» 
to the Radius, the Siheof9odeg. the Iknie Extent oftlie 
Compffes will reach on the Line of Artificial Tangent^ 
fromthe Tangent of P A 25 dcg* to the Tangentof B» 

43 deg* 



^ -^ Cafe 1^ 



rUtteV* Cafe3. Given BP=6o,oo ?„„,„. 



&.in. P: 

PRO PORTIOJr. 

R.S BP: :T,P.T& 

S BP 9.6989700 
T, P, iao76i865 

S«m 19.7751565 

Subft.R.10. 

— - d 9 

Rem. 9-77515^5 tlieTB=:59. 1^. 

By tbe Qimter* 

. Ob&rve the Method in tbe former Caies. 

a r 

P=z 

p 
^^•iv. proportion: 

R. ;gP::T,BP. 

« . .. ' -^ 

?P 9.8080675 
,BP 10.2385606 
t « ^ 1 III u ■ 
Sum 20.0466281 
Sub^R. lo........ 

'■ d f 

Rem. 10.046628 iT, P A=r 48. 4* 

iSi^A tbe 4tb. 

Cafe 



Cafe 4. Given ^^^^V^jRe^uirelP A. 






Cafir 5. GifOi BP=:6o. cx5?Rcqi 

Pw$o. 005BA. 






proportion: 

S, BP. 9'93753otf ... 
S, P 9.8842540 

Sum. «p I93sti7846 4 ^ 

Suht.&a RjemioxL9#9ai7^6 J!^ BA =4)/ 33 

•■J f 

Pacgaooj^^'*^'^* fg^ VI, 
T Ji\) f RTIO N. 
:iSP,.R:tT^PA.T,BP, 

TjPijL 9tJ86f^|o 

Sum 19.8612610 
Subft.gp.9.93 75506 

^em, 9.9237304 T,lPi=39.5^ 

* 

•' Ca<e7,' 



• ■* 






PtdteV. . d r 

Jw.VII. Cafe 7. GiTttPAc:3i6»Q07B^^ - 

g PA 9.907957(5 
S,P. 9.6989700 

Sum 19.6069275 d- ^ * 

Sufaft. R. item. 9,6069276 ^ tTtiAca 

d. ' 

%.▼«. B =37. 3o5***»*'^*^ 

r RO P OUT 10 K. 

R, S»BA::T, B.T,PA. 

S, B A 9.9375306 
T, B. 9.8849805 

- ■ ■■ I 

Sum I9.S225III d ^ 

Subft, R. Rem. 9.8225111 T^VJkszUr 3^ 

il^iLi lib . 



C^t^-9i. 



i r 

jPUteY. Cafe 9. Given P A=i6. 17 jRe,d.B A 

pROPOMTiaK 
T, B.R.:sT,PA.S,BA. 

R. JOOOCXXSOO 

T,P A 94^55386 

Sum 19465538^ 
Sub. T, 59^20434 . , , - 

RiBnu 9.8o34952»S,BA=39-30.0ri4o.3O 

I*«t CWi #» Ambiguous, 

1 > 

JP/*<e V. Cafe .10. Given P A=: i6, 17 1 Rcqd.B P 

TROTORTION. 
S,B.R::S, PA. S, BP. 

R. laooooooo 
S,PA. 944775^^ 

Sum 19*4477586 
Subft. S, B. 9.6204884 

d ^ d r 

Rem. 9.82727029$, B P:; 42.1 2, or 1 37. 48 

,' ■ -y 

Cafe 



SC^41^ 



, . d ' 






I 

R. io.oooO(j«> . : H 

gB. 9.95844** ,•.1.2 






Sum i9*95S44i;c^ 
Subft.gPA.9.9822aQ* : i/ : '. 

^. Rem. ouiH5tSMfc$»F55 7i-AT*VPAfans 

between N M and fi, if not *ds io8« 47 jlmbigucus. 

. T,BF.|li::T.PA,|g». 

i [ R. lOxxDoqoQo 

S T,P A, 9.8771144 . V; 

Sum. I9.8771I44 . . , r.,,ry 

S»bft.T,BP.io.oi5i«tI 



' ■ / 



Mm. ' 9S6ffiUtT ?: 43.- tfc 

, SftbeidSelutuif, 



ftOeV, Cifcij. QirmPBir37.oo7p,^- 

fJ^.Xm. pB=4tf.ooJ***J** 

r MO f OUT J ox, 

R* 10*0000000 
§,PA- 9-779463a 

Sum i9.77945}D 
Subft. S, B P.9.9579}4* 

Rem* 9*92i$289S,B:;=56 35 * ^(?/. S« 

i'ia^f V. Cafe 14. Given P A— 37. bo ? p ^^ „ . 
F^.XW, . ■ PB=4«. ooj'****'** 

I1iOtX)ti'IIOK. 

« PA.R::SPB.SBA, 

Ri laooooooo 
g P B 9-84177x3 

Sum 19^8417713 
Sut g P A. 9*90^348* 

RCQlt 



9*9394a27gBA4=s29 34 

£jf tfe 2a S9Utth 



P* 



C47> 



■ a ■» 

. CafeiJ. Gi»aiB=46.oo3j.^,-p thuy, 

T»P.R::rB«S BPi 
R. iaoocxxx)0 

T 8.9*9848372 

Siint 19»9848372 
SttbftT^.10.2212263 

— «— 4 r 

JUxn. 947636109 ;§BPd( $«• 33 M ft* 

i J 

JPtiUeV. Care x6. GiVeDB=:4^ oq7 q^ « «, ^ , 
%.XVL P=59* 005 >«V*P^ 

\ FRO fOlkt lOK. 

S, RRttJ^B-SSPA, 

R. tOXHX>000O 

;§B. 9-841771? 

■I * r, 

Stim. 194817713 
Stthft. S, P. 9.93 30656 

^_— ^ d r 

Rem. 9.9087057^? A == 35 $2 

Any Oblique Quadfanul Triangle ZtiP 
Jtlaie V. whole fide H Z is a Quadrant^ may be le- 
%XVIL duced to the Rigbt-angled Triasgk H N P 
by producing the other fide 2 P to N, fo thaC 
ZN be Quadrant) (or taking auray a Quadrant ifZP 
Ind been bigger than a Quadrant ) then m the Right 

G 2 angl' 



<4S) 

angkd Triable H N P,H N=:X Z, and PN the Com- 
plement of 2; P. and ^ P^ N the Complement of 

TfteLoi^MMr tmn i jSfi^ Obfervadon af the 
*fr$fi$itmsfct Solvin^tte Cc^lMore-mention'd, found 
they xmAth^Vii^hfclkOeimMr^^tim. 

For the Underftanding of which, 'tisnecefitrytopre- 
mife that he dOfr theturib i^ { m^» the Sides inclu- 
ding the Right Angle j togetl^r with the Comylements 
6f the other 3 parts vi^. the Jtjf^bemujk and $ Ohlique 
Angks)/fv( urcular PartSy taki0| tto Notice of the 
Right Angle as being known; ani it the 3 parts wjbtoh 
enter the Queftion ( of which 2 are {jtven and: t nctfuirM ) 
have no interrumioo C ^ Right Angk coming between 
tiekv accounted totit) thatjpojt' imch tt between the 
'Dctiertwo, is c^4the fflAddfe patft, UfA^ otber two 
£ttremes Cmfufit*phixt if there be an Interruption, that 
part which is^ieparaked (mm the other two h eaird tihr 
MMiMi^thtdOKt 2 txtnrt^sOfp-iit or Pk^mS. 



F^r MiUftM^ trgk^M ttimi^c B F A. 



f/B A 



PA 



PCQnip*B 

p 






If 



Com.? he mid. 



I 



CoJti.BC 



ExtrJDisj. 
rComp. P 
Comp.BP 

Comp.BP 
B 



^ ^CamtSPiu 




i 



Contt^ B 



Comp.B 
BA 



1 




(4») 

JheTUri Neper'/ Qtneral Prifeftioft 

istim, . > 

■ . .. ■ » 

ArAKx.SymUk puts: O Cofim oftbeJEi* 
tremes Disjun^ 

Ttetk, Tfe Airi^ mubhjbdbffk Sim ^fh Mid- 
dU f^t $$ emtsl t$ tJk McS^k «r ProJkB cjj tb^ 
Talents cf the Mxiremes OmfaS^, Mi equal U tim 
ReBoiigk er FroduS of ibc Coynes of the Extremes 

JJjtace, To find ^ Middle patrt> when tbe ockr 
twoparts given arc ekber j |[SS^^ 

RULE I. 
li.T,onc Extreme: :T>otbef Extreaie^ S, Kid. part 
R. g one Extreme : : g other Extr. S, Middle part. 

Her^ jMft nmy obferre, tliat when the MUUle part 
IS to be found, you mat Wgln with the Radius: Bat 

RU LE IL 

T,^n»E^r.R : t S» Ma*. part, T, other Estt. 
g, one fixir* R ; : S> MiM. p»rt,5g otter Extr. 

S»tbtf wfaeoiNie£xtreit»ei» tobefound^ theotfaerEi-' 
|li«M wufl: te Ate firft Term. 



n 



Seme fcvi Examphs to ExereifitBi Jbfie. 
=|fjRcq«i«dBP. 



Cafe i.B A 

PA 



Here the Compfetn^nt of B P lequiicfd is the middle 
put, and the other two parts given are Extremes di£- 
jana^ therefore by the 2a part of Rule i& 

' R. ^ oneEjctreme: :J^ other Extreme S mid. pare 
Thatis, R-gBArrg P A.g;BP* 

the (kme Pr^pottion a^ befbr^ which being workt in 
Kumbers ftlreauly, tvotild be oeedkfi to repeats 

Cafe 2. Where tbereare the (ame things gireii to fbd 
the Angle B. 

Here B A is the middle part, and the Complement of 
B and P A aHe Extremes Conjunct* 

Therefore, by the ift part of the 2d Rule. 

T, one Extream R. : : S, middle part, T, other Extreme 
thAtisT,PA^R.i ;S,BA.tB* 

Cafe 3. GivcBBP^6oi7R^^^^B^ 

Here the Complement of B P is the middle part, and 
the Complements of B and P are Extremes Wjun^L 

Then by the ift part of the 2d Rule. 

T, one Extreme R. S, middle part, T, other Bxtreme. 
That is,T P.p. :: S BP.tB. 

Here though the a laft Canons are not exa<5bly the 
fame as thole produced in the forftier Solution of them, 
derived tVom the Frodudlion of the Sides of the Trian- 
gle 



(51) 

g]^ yet thdur Co-incidenot with them will appear eafi- 
1/ from the 3d Lemma, * 

I prefume^twill he ne^kft to .give any farther In- 
ftaoces of this Geoeral Rule «f the Lord Neper. 



m'm^ 



Of the Solution </ Oblique Spherical 

Triangles. 

OBlique Spherical Triangles may he reduced to two 
Right-angled- Spherical rrumgks, by letting fall 
a Perpendicular, which Perpendicular ri- 
Pldte V, ther divides the Oblique proposed into two 
Jj^.XVin. Right, or makes two Rjght^ by adcUng a 
Right-angle Triangjc to it. 
In Oblique Triangles there are 1 2 Ca&s» 10 of wHeh 
may be folved ( bring thus prepared ) by die 2 firft Aki- 
omsy or by Rules deduoedfroqi tbem, 

R U L E I, 

TheCorftiesofthe Angles at thcBafe^ aver propor- 
tional to the Sines of the Anglos at the Vertex. 

Note, Th^t B A, DA are the Ba&s of the Right-an- 
gled TriaiJgles B P A^ D P A. 
The Angles B, and D are the Angles at the Bale ; 
the AnglesB P A,D P A are called Angles at the 
Vertex, B P and D P are palled the Sides. 

For by thp 7th Caf^ 

FlateyL k.SPA: :S,BPA, S B 
Hg. I. R-SPA: :S,DPA. g D. 

Thcrcforc^SB.S,BPA: .;5D.S,DPA- 

For 



MO 



ForFiDi»itiouwhid) aie the ftme to mt tUrd^a^ 

tbe iajne one to another* 



R u L E n. 

fprtyftiP it Cafe, 
R.SPA: :gBA.gBP, PtotfVI. 

ThBrefoxe, 2 B A* )g BP: ; ;ig D A^;gDK 

R U L £ IIL 

The Sinc^ of tbe Baics are reci{U'ocaIl^ proportional 
toite Tanjptsof Uie Aisles at the Bafiu 

Fcr b9r cbt t4 Axi«m» 

S,BA.R: :T,P^T,B 
S,DA.R: :T,PA.T,D» 



»"»fc 



TlKidoie, S, B A «T, BsR « T, P AsS, 

DA»T,Oi 

TherefiMt, S, B A. S.0 A : : T, D. T, B. - 
for E^ul Rc^Ut^les haVB tbdr ipit» itdpracally Proy 
portiooaj* 

RULE IV. 

The Tangents of the %liiC% arc reciprocally Proj^rw 

^onal to tbe Confines vfthi Ar^lei fti the Yoftex. 



Fof 



/ 



I 

I 



r 1 



.1 
•f 






-4 



For by the 4th Oi^ 

Hole Vr. T,BP. R: :T,PA.^BPA. 

Ii£. in. T.DP.R: :T,PA.S DPA. 



Therefon^ T, B P * :5 B P A= Rx T, P A =T, 

DPxI^DPA. 

Therefbie,T,BP. T, DP : : gDP A.;^BP A. 



AXIOM III. 

In any Triangle ( whether Right-angled or Oblique) 
the Sines of the Sides are proportional to the Sines of 
the Oppoiite Angles* 

For by the ift Axiom, 

S,BP.R ::S, PA. S,B 
S,DP.R ::S,PA.S,D. 



Therefore, S,B Px S, B-Rx S,PA=S,DP xS,D. 
Therefor^ S,BP.S,DP : : S, D, S,B. 

Obferve this Rule for the letting fall your 

pendicular. 

Zet H he let fall from the End of a given SidCy and 

Oppofite to a given Jingle. 

For by fo doing you have enough given in one of 
the Right-angled Triangles to Determine any of its 
unknown parts. 

H The 



Tbe two firft Cafe are »Wei each by one Operati- 
on by the 2d Axiom foregoing, the next eight Cafe axo 
folved each by twoi Operatioin. 

And tkul jFoumiy k?)^v» nAuit part of the R^hfi* 
anglc4 Tmvgte fp find ¥i. yQ«« 91ft Operatloas^ Olk^ 
/frv^ |9 psA fiiA & i'tfri, Aff h whm fiund, the ether 
f0ri Qptbe tbkig rcamrei^ ( wiUb U tabt 0Akdt9y or 
fubfiraSed from ttk l9P^^ fiuud) miff come under 
meeftbc Pour Rules mentioned. 

This \yUl be o^y uadbNrftood from this General 
Vibwofthe Proportions for the Solution oftbefirft to 
Cafe of Oblique Triangles^ annexecL 



mmmmmifm 



PropoK'' 



Pr^fmioHs fir the Ahin^ theJStfi lo Cafes 4 

tihliifue Triof^s. 







3P. PD.B 



ileq 

D 

PD 



BD 



ProportionSfc 



s,pb.s, B : iSj B p. s> a«j*rf*;? 



an I » "■ 



S> D» $, HP : : S, B. S,PD. k/»%. 



lip fcl 



ii i *^*i 



BP.PD.B 



Flf fti find B A. by thfc 4tll Cafe. 
R.JEBtt TjBP.T,BA4 
Andthcn by the 2(1 Rule, 
25BP^BA::SDP.:SDA. 
ThciiBAjHDA = bD, 
according "sS the Perpendicular 
fells mthin or without tte Tri- 
angle^ Which is doubtful, unlcO 
the kind of the Angle D were 
knowt). 



BI'.B. D 



Firf^ findB P A* by the 3* Cafe 

?BP.R: :tB.T,BPA. 
henbyihe4tbRulek 
T, DP.T, BP t : 2 BPA.g DPA. 
Then, B P A^fjp P A = P B d. 

and here the falling of P A with- 
in or mthout is dottbrtul, un 
lefs the kind of the Angld D be 
knoWm 



Calt 



itt .»— 



I 



5 



Firftiind B P A by the 3* Cafe, 
2Bt>.R: :tB,T,B PA. 
Then by the ift Rule. 
S B.S,BPA: :gI).S,bPA. 
ThenBPA + PPAzzBFp, 

thatC B and D are alike. 
Is ifl B andD are ludikt* 

H 2 B P 



\ 



(;6) 

Prof onions JQT the Solving the fir ft lo Cafes of 

Oblique. Ti^angles. 



Giv( n. 



B P. C.D. 



B p. B. p. 



B.P.BP.px: 



I'ec 



;.D. 



Proportions. 



iril, fird B A. by the 4th Cafe. 
^B.R: .tBP.tBA. 
fh-n by the 3d Rule 
r,D.T,B:: S,BA.S,DA. 
(^xnB.A-hD ArrBD 
if JSandD? CHke 

i B and D 5 ^^^ \ unlike. 



O. 



I 



"^if^, $nd 3 P A, by the 3d Cafe, 
^Bp.R::^B. T, BPA. 
then by the i ft Rule 
S,BPA.S,DPA::gB.;:SD. 
lFBP a is greater than B.p D & 

Sut ifB P Ais <: B P D, and B 
5 Acute ? p. . C Acute 
LQ%fe.|'^''l0btufe 

?irfl, find B JfrA. by the 3d Cafe, 
5BP.R::tB. T, BPA. 
then by the 4th Rule 
5 D P A.S HP A: :T,BP.T,DF 

'^°PA^^funHke}sCorPA) 
■^P^fgreaterl^^^-Qpadrant, 



])aie 



B P. 



(57> 



*mm^ 



Prof mions for the Solving the firfi lo Cafes oj 

. Oblique Triangles. 

Given 



BP.BD.B. 



Req. 
OP 



Proportions. 



■^— ^ 



_ 



Firll, find B A. by the 4th Cafe, 
;5 B.R::tBP. T B A. 
Then by the 2d Rule 
;? BA.SBPt:gDA.gDP. 

I^I^Ais{5Sfy,,}(PA)XB 
then BDis '^ ^ ? Quadrant. 



ZIEf 

9 



BP. BD.B 



D 



mm .■ 



Firft, find B A. by the 4th Cafe, 
2B.R::tBP.tBA. 
Then bv the 3d Rule 
S,DArS,BA::T,B.T,D. 

ifBA.is 5^^BD.thenD 

. Clike 7o 
^^iunlikc^^' 



10 



Tl 



mate VI. Caie i. Given BP— 34dl5(/ 7 

•^ 6^:30 07 3 

S,D?.^B:: S,BP.S,a 

S^B. 97O04981 
S, BP. 9-7<^8i5 

Sum 19*4572796 
Saba.S»DP. 9^39844 



•«M**ii*H 



Rem. 5.«23295lAeSineofD5 '^J'^ ^^ 

4138 i^ 



.| 



Acute, 
Obfufe. 



If* V- Af . r. ii?* ^=80 00 >Rjeqd. P D* 

m I>=--A7 Id \ 



te47 10 

S,D-S,BP::S,B.S,DP. 

S,BP. 9.8080675 
S,B. 9.9933515 

Sum 19.8014190 
Subft.S,D. 9.8653021 

Rem. 9.9361169 thcSineofPD.=:59i.4i' 



Cafe 



(59> 

Ca(o3^(XvmB]in=:t9ii3o7 PbAiVL 

/, ^ PD=36. 20>Rcqd.BD I^.YL 

FirO, R. :g B: : T,,BP. T> B At 

g B. 9.9607302 
T,BP.9.9i6io45 

Sum 19.8768347 

Rem. 9.8768347 the Tangent of3 A. 36A ^f^ 

Then bj the 2d RiiK 

SBP.gBAsigDp.gDA^ 

S B A. 9.9024438 
gDP. 9.9061 107 

Sum 19.8085545 * 
Suhft. SB P. 9.8874061 

• Ren. 9^11484 the ^t^Asr^^d. 30^ 

accorfii^ aa the PerpcwMcidtr. fel V^itirii or without 



Cafe 



r 60 ) ^ 

Cafe 4. Given the fiine as before. RequiredF. ' 
''^FMl, S>BP-R • ' ^^-Tj BP A, 

IL io/xx)OOCX) 
tB. 10.35141^9 



»•■ 



Sam 20.3514169 
Subft.^PB. 9.8874061 

Rem. 10.4640108 the Tan. of B P^A Jid. 03' 

Then by the 4th Rule^ 

T,DP.T,BP::SBPA.;^DPA. 

T,BP. 9.916104$ 
*5,BPA. 9-5"5397 

Sum. 19^.276442 
Subft. T,DP. 9.8665644 

Rem. 9.5610798 the g DP A. 68i 40^ 

according as the Perpendicular falls withan or without 
the Triangle. 



Cafe 



Cafe 7. Glren B Psc 6od.io'? Phte VT 

Firft,SBP.R ::^,«.r, BPJU 

• - • . 

R« 10.0000000 
tB. 100761865 

$imt 9&076ifi(J.{ 
tul)ft.gBP»j.«j;i7745 

R«Mfc «0.37^t4oT3PA=df7il2i' 
Then BPD>.BPA=EDPAc3$d<2:4ff. 

Then hy the fttft Rule. 

S,BPA. S,DPAt{;gBL;^a 

S,DPA. 9.8889612 
15 B. 9.8842540 

t 



"BMtirilltaHiU* 



Sum 1 9*773 2^11 a 

Rem. 9.8080726 = Jg Ds 5<5ai OO'* 

Firi^gBP. Ri:»ar,BPA. 
: I 2 Then 



Tlvn by the 4tb Rulc^' 
g DPA- SBPA::T,BP. T,DP. 

gB PA. 9.58^574$ 
T,BP. 10.24U830 

Sum 19.827057$ 
Subft.SDPA.9*8oi20i$ 
■ ' 

Rem. 10.0258560 T, b P. =: 46A 42^ and 
muft be le& than a QuadranC^ becaufe D P A isof the 
fame kind with B, that is of the fame kind with P A, 
otherwife P D \yould have been more than a Qnadrant 
by the5th 5(;/«^i^ii. 

PW^VI. Cafep. Given B'P=:6oAio'7 

J/V.XI. ,1 r h B=4o 00 >Reqd.PD. 

•^ AC' C'V' BD=87 30 i 

« 

Fir(|k,2B.R::TBP.TBA. 

Which will be found as in the 6th Cafe from the Data 
53 A 1 1% and Confequently D A = 34^ ^^ 

Then by the 2d Ruk^ 
2BA.2PB::^DA,SDP. 

gBP. 9-^9<5774$ 
gOA. 9,9169455 

Sum 19.6137200 
5ubft.SB^ 9.7776128 

Rem. 9.8361072 gD P = 46i 43^ 
Becaufe D A is like B, or P A otherwife it would have 
been greater than a Quadrant. 

Cafe 



( ^t ) 



• T 



•a-.r -•.. v^j.-^ ,, ) 



Cafe 5« GivepBP=:6o. 10*7 PUUeW 

; , - Firi^ S B P. R : - tB. T, BPA.: 






R« io«cx)ooooo .. ' - . f . 

tR 10^761865 r; /. . 



Sbnt 20.0761865 
Subft.)^ B P. 9.6967745 






Rem. 10.3794120 the.Tan.otBP A. 67^21^ 

Then by the Firft Rule, 

SB.S, BPA:.;gD.S,DPA. • 

S,BPA. 9-9^51426 
S D. 9.8080675 

Sobft. g B. 9.884^540 

Rem. p.88895tf, S,i!>PA=5o^. 45^. 

Then the Sum of | «7^^^| z,/^. , , g. 06 = P. fince the 
Angles B Md D are of the fime kind ; but if they had 
beenof <yferentJdnds, the diflerenceifthefc two An 
gles would have been equal to P. 



I • Cafe 



Cafe 6* Given the luMas before. Required B I> 
Fiirfl^ 9? &B. t > T BRt B A. 

« 

R. lOAOooooo Plate VI. 

Sum i9.7$85i7o 
Sttbft.;^ B. 9.8842540 

Rem. 9.8742630 tfae^ B Asstjil 11/ 

Tl9»|i3»ti>e3iRa]eb 
T,D.T,B::$,B A.S, DA. 

ff 

T,B. 9-9238135 
S,B^ j.90}af»? 

■■ ■ 

Smn. 19.8272058 
.Sttbft.T,D. 10.0761865 

Rem* 9.75 10193 S, D iL $4^1 19^ 

Therefiirc tl» Sum oflg^ ^* j^| wj. 87* 30* 

= 6 D firice the Angles B andO aie of the fame 
otlicrwrifc.theii:dift; 5: RP* 



cue 



C <57 > 

Thcrcfo»AE,AC;:(DL,D«>::*L,p«r,therd: 

be evident from the Conftfttaiw* of the F!gtfri> the turo 
Ri^t^Dgled Triangles B F ft or Cj' are Sumlar,and 
Aweior^ ,.-•**-'' 

PF, Fp::4rC,tfjr 

6crtf.WF»«fF— F^i:AC,*C— <?i 

and confcquently «* tv«^ C : : ff », j^ «u 

Cafe II. The 3 SMer 4rf*«7Si*erk Triangle being ^^^ 
gjmii to fcii an Aj)g^ 



i««B*««a 



3,«xSARf;:SiB4-|diff: ^rxS^lBafe-^ 



diir.rr.Sf|AngIe,Deiii.A£xi»F>Rf ::(bytlto4A 

iM,$th.aniii}:)::Sf Baie-t-|di£ irfxS|Baie-«-| 
difi; tfr,Sf I Angle 

Inftanoe wftiUHDeffc 

- . . . - - 

DP. 46 42 > Rcqd. P. 
B n. 87 30 3 



IthediCof theSidesis 6 44 

^|Bafe+|diflr. rr.is 50 29 

|lttlr«i-.|diff.cr.is 37 01 



R f • 



Rf» 200000000 

Jf^^jsi^ X ^^^^zrpSrTr, 19.666932$ 

— —■ ^ I 

Sum 39.6669325, 
Subft. S^rnxSyth i9.3oo25 34 

Rem. 19.8666791 

S f, *. the Angle ft.* tl* | of which 9'9iZ^239'i 
is the Sine of 5 9^03^ '| ;^ Aif le required, which tie* 
11% doubled is 1 184 w£^, . 

COROLL. I. 

Sttfypofewr, 0y thcLegsoFthe Angle required 6 the 
Bafe^'ZzTM^^ X=:«f — ii."5 it | Sum of «i, jiI| ^. 

Then^S^rnxS^n^ Rf: : S, ) — mxS,$ — n«S f f 
Angles 



•^^i 



ThcrcCS f f AngkriRf xS,)— .j!ifxS,3-w/. 



>■■ ■* 



SjxtxSq 



^ UJiirf .prefcribes this following Rule^ by two Propor- 
tional which comes to the fame 9 

S) IV* 
t^ en ^ /S,3 — «rxS,3*ir^\ 

Rx&>< — wx S, ^ n. 



Then Qx Rs=S f | Anglc= R f xS,i|— i»x $> ^j^n. 

S>mx5) }s 



COR- 



y. Cafe^. Given as Ijeftrjj to find D» 

I. BA,asintbe9thandtftbCa£ss»from PhieWL 
Che fame DatOy will be found 53^! 11% and Fig* XIL 
Coniequently D Azz 94^ i /• 

Then hy the 3d Rule. 

S,DA. S,BA: : T,B.T,D. 



S, B A. 9.9033923 
T,B, 9>9238i35 

Sum 19.8272058 
Subft.S>DA. 9.7510991 



/ 



Rem. 10.0761067 T, D. 5oi o:f 
for bereB A ls<C than BD,^and confeiuently BandD 
9XC like. 

L E M M A V. 

The difference of the Verfed-Sincs of two Arcs x f R, 
=: S I Sum of thofe Arcs Multiplied by the Sine of the | 
diflP. of the Arcs. 

Suppofe the 2 Arcs A F, A E. the diff. 
of the Verfcd-Sines, A G— A K=zH G= Plate VL 
E BtheSineof the |Z of the Arcs is AD Fig. XUI. 
the Sine of the f X of the Arcs in F O, now 
the Triangles A C D, E F B are Simillar, therefore A C, 
Ad; :FE,FBoriAC,AD::iFBCF6;.FB,therf. 

|ACxFB=FOxAD. 



AXIOM 



(66} 




Sappofe B tlie Ancle Required, B A aad B P (sBM) 

* its LegS) and fimpofe PA the Bafe to be Obliautf to 

chePlane B A> tfaes wiQ P B A be an Oblique Tnangkk 

. Let fall P g perprntttciilar to fl m and ^y perpesal- 

ciilar toC Ay aoa P i perpenoicular to b A, aal mt L 

perpendicukrtoC A^tber^nre/tS^jis perpendicular to 

CA, and^LssiA-LArrVBafe— VdiiF. Legi. 

Now AE,ACi:BUCm2 ^^ ,^tiplyiiig the 

correfpondent Termsof both Proporthm* 
AExmFy AC XAC;:(^bljxCm.C f»xjlAy;:bU 

J A. 

Now thatAE, A C :: 3 L> f w will be evident from 

ck Similitikk of the Triangles A £ C> D L ^» D i ^ 

There- 



I 



:iL 



i 



!-A 









.♦' ' 



C<69) 



>L» 




H. 



Putting ju^Compliment of m. Cuisior sciblvcs the 
nth Cafe by this fingle PrqportiQn. ,1 



* ^ 



S, ^* or ».. J. :jg Bc/> S, /it-|-«. V« 
For by .fbc fourth Axiom. . 

S,mx S, ».r f ': : V B c/> V iw — ». V. 

And dividing thtf ift and 2d Term by S n, ^ 

• S,m.rr/:;:^^fi\i : V B cjt) V / » — 1>, V . 

Thier£5>>9«0-» .: :.;5B^S»ft*f-».,V» 

The three lafl third Terms being only diflferent JOe* 

l^nadpos of the {aine<)(i^BtLfy. ^ For that VB ^Wm-^i 

zz'S B</>JlS>^-T»'>^^vident on In'fpe^on of\the 

Figure; and thatji^BpoJ^^.iii^i^zr ;2SBooS,|lc — » 

vrUl bp evident, if we prove that )S w — ?* — S, /x ^[- ;S 
tvhich may be done thus r 

. .Suppofd f =: a Quadrant. . Then the Compktuent of 



i9»— nisf — m^;f, and in the room of w, putting 
^— -M, to whiph it is equal by Jfypotbcfisd S, ^ -f-n — 

S,^--«r -J- ;>, the Arcs be equal, the Sines are equal. 

» 

K COR. 



being a PropoJftio*! like tfee 4^ Ag*mv ^ ^^ ^ ^ 
Forrefuming the lift Figure, aduce the Pkw B P 
to the (ame ptome with B A. fo that they mjy make one 
Circle, andfctfallPxpqpeniieularto ACthen*X:i= 
xA— BA, that is V. ? c r— y.Ba^ 5 l?ut 

AE.AC: :*A. pPTand Multiplying the 
an4 ifiF.^ : :#!?•.?• yCorfcfaoi^ntTcnm. 

whavc AExiarp•Rf::3Px*^•3P«v.anddlVl- 
whavci^fixiwF.Rf ;;>A-V* Angle leij^urrf. 
orSj*V$^ii. Rf ;:*;^v.- 

Now .that AE. AC:<4a. ?P wiU tewident from 

she Similitude of the Triai^ks A E C, D * ft D P A. 

... • , • ' 

Therefore, 
AE.AC::DA;D-0t:D>:.DP::D*+Dx. 

AP +p ?''? ^r *^*^ r* 

Thcre&reAE, AC: :Ax.DP. 

Andt6atii» F^R: : J^Ja*. wftl be ciwda« ftcwvjthe 
Simmtude of the Rigbt-ai^ed Triapglcs ^f P, >C ^. 

PorFP.C^::Pp.C^i:F3PH-?g.Q'«r+Cj^ 

i. f . : : P p. j^ «•• 

anibeqfUtftftFttM F.and Cnrs; R^aodf ^^e-v.. 
Therefore im F. R : : ^ P. v. of the Angl^ 

TUfbdnf piov'dj hmce felkm*^ 

COR. 



^ 



•eORVLL. w. 

S,iwxS»-Rf::S,1|ac.ie44ffirer^xSi|Zrr-|Balc 
Forby thelaftCoroLS JS». xS i». K::h\.v: : >xx|R. 

* S, I Z f r— I Bafe. Jg f . f Angle. Hence, 






2 

C R b L. V. 
C O R O L. VI. 



» / 



S,5xS,? — B.S,$-.i»xS,f_«::Rj.Tj|Anglc. 
F&V the 5* kridiftt:«fon. 

S,$xS,5— B.2 f f Arch:^(SwxS».Rrf)::S,$— /» 



Therefore S^^x^^B. S^^^jnxS^i^u; : 

text bec^tife ^. S : :X'X. . 
Thejrefor^ • 

S, 5«S, I— B.Sj-^u-iwxSjf-tt : *Rif. i" f | Arch. 

C^ROt L. YIl 



*'?f7S^''r'^?''^5''-^-'^*-'^ff Angle, 
Tluf follow from the laft,^ becaufe T, R . : R. t. 

K 2 COR, 



C Q R.O L L. Vni. 

VZ-.VX..2R:: V^-JVX VX. Or 
:: VZ — VB. V. X. 

Which is the Prafticc of Fofier^ mth his Line of Vcp- 
fed Sinti : . 



J>EM0NSTRJT10K 



' . * - 



By the 'Fourth Axiom. 

• / ^ - f i / 

VB — VX.VX::S,«xS,n.R4r:: (byOr.3.) 
V Z— VB. V X : ,• the Sum of the firft and fifth Tcrmsl^ 
Sum pf the iecond and iixtb.. u.e. : ^ V Z -^ V X 
3 R. 



-* i 



C O R O L. IX. 

%Z^ S X.R...- .-gB v>SX.VZ.or 



-•-2 . 

This follows from the eighth CoroUaryy and may be ea- 
iily underftoodfrpm what is faid in the JBxplication of 
thcjecond ConUary. 

But here note, That if the Sum of iw and n be 

\ grater \ ^^*^ ^ Offadrapti ^ g of the Sun* 

muftbeunderflood as|^^^^^| andaddcd, or 
iub(lra<£^ed accordingly. 

CASE 



(73) 

- Gafe XIL k. C ft ■ 

• Ik • ^ 

^ty Three Angles being givifh tofindaSidi. 

The Anglet adjacent to the Side required, call Legs; 
and the Angle oppofite call the Baib ; then work as ill 
the nth Cafe. 

. For fuchistbe Opehition io the Supplemental Tri* 
at^le, whole Angles and Sides are equal to the Sup- 
plements of the Sides and Angles of the Triangle pro» 
pofed ; and Arcs and their Supplements have the (ame 
Sines and Tangents. 

Of the fevered M^hodsfor. the Mutton of the 

nth Cafi, mentioned in the Nine CoraOaries 

from the 4th A^tom ; the Reader need only 

commit fome. one. of them to Memory for his 

confiant Ufe : Such aa G>r. I. and Cor« 

* V. The refl being mentioned only that the 
Reader may know the Reafon of the fe^ueral 
Methods made ufe of by different Authors. 

■ • 

So Uhviife the^ Lemma'/ and Rules following 
are not to be thought of Abfolute Necefjtty^ 
fince the Solution of att the Cafes may befoU 
ved by what has been done before, they being 
( as l^ore ) inferted to give the Reafon of 
' other Rules for the fame Purpofe ; which 
a Theorift would not willingly be ignorant 
of - 

LEM- 



(74> 
L E 1(! M A VI. 

the Sum bf <he$h)^>rtttoAirH< b ttt tbi B^ 
fence of the Sines, as the Tangent of half the Sum 
of the Ardis to the Tai^ent of bldf the Oiftienoe 
oftbem. 

Z^Rrfes 2 Ahs. X. S!nes::T>f ZAk*. T,|XAtcs. 

Let fife 4 Arcs Be £ A, J* A ( in d* Fig. tt Xe<». V. ) 
tfien-iWilUje, 
'£G-f-FH.fiG-FH::EG-|-Frf. EG-Ftl 

. . , 2 , 2 

Tii5?r«fo ^feV/ r/.i»M^ nK^ V^'-'d. 

• • • 

LEMMA Vri. 

, 'i,(Vfu*s. X, Co-fuw : s T f Z Afcs. Ty f t Arcs. 
If the Arcs are lef»than oo Degrees* For in the fere- 
$tt!i«<A«l Fl«ire, the '2 'Arcs %emi AE, A F. G C 
aadCHm tHeir C6-«tfei ; it^tflll be 

CH-|-Cq, CH— CG..- •• CH-fCG. CH— CG. 






. 2 ' . . . • 2 . 

•i.e., . cui tntli : fty 0;^F f j-s^N. JIK. 
TheVr. i:<«.«H: :tK NK. 

]^ut iftlie Arcs are t>igjger than 96 Degiiees. 
it, Co-&net*X H^^"^* •• * , | i^ Aki. r | X ArCs. 
IffieTaiigHrtSof 2 Ar&sBeihg fedprocarfy propbrti- 
/ C^tangitrts, fey the *)Kri iiriS^iSwtf. 



\ 



>oU to iheifCti.{kn^rft! 



LEMMA 



2, Tangents* X^TaBgcnts : : S, Z Arcs. S. X Aricsi* 

Pi».Kau :EL-ELj .-EG. F H. 
Theref. jp m, S,«r -.t E Q, F H. Q.E.D. 

S,Arci:S,f Arcxg f Arc. 






For in Fj^io. P/4ff the ift, AE.Fif.* .• Bn. |FB:^ 



Theref. A E=:Bii-^F n-S, | Arc x^^krtu . 

f FB !«. 

LEMMA X. 

* 

The Sines of two Arcs A, E, arc |myportioml m 
the Kx;,^^aq|^, ^i^ct t(]Q Sines and Cd-fos qfOieir 

; S;, A. S, E .• ; ( by the muflf UftnM ) S^ f A *g f A. 

^R« 
S,|CxSfEf -SjIAxI^ |A.S,fE-xVtE' 

" -•■,—■. 

LEMMA 



(70 
LEMMA XI. 

Tie Tangents of two Arcs A, E," are pn^rtional 
Cd the Reftangks under the S^s and Skcants of thofe 
Arcs* 

ue. T,A. T,Ei:S,Ax/;A. S,Ex/,E^ 
PorT,A.T,Ej.-S,AxR.S,ExR.-.:S,A,xSExR. 

"sa"* gE 
S,ExgAxR;.-S,AxSE.S,ExgA::S,A*/,A, 

For by tbsfiuftb.ZemmOy the Co-fines of two Arcs are 
xecipiocally proportional to their Secants. 

RULE V. 

The Tangents of the Bales are proportional to the 
Tangents of the Angfc at the Vertex. 

|5jr.xyiiL Forijjrtheyewii Axiom, 

t!xBPA*T,BA(::R.8,PA)::T,DPA.T,DA* 
Therf. T,BPA.T,BA: ;T,DPA. T,D A. 

R U L E VI. 

• • • ■• • ' 

S,Z>Cr.S,XCr :: t | Z X X at the Vertex. 
T^iXXX, atthe Vertex.- .- r |X XX, attheVertcau 
T, I ZXX, atthe Vertex. 

J) K M O N S T R ^ T 10 jr. 

* • 

For, hy the fourth Rule, 
T,BP. T,DP.-;S,DPA.g,Bf A. 

There- 



I ' .-' r -^ r 



SDPA-gBPA. 

That fc, by Zemvia Vllf, and VII. 

AsS,Z(>. Sj3^Cr :: t |Zii, kttlie Vertex. T,|X 
ZLy at the Vertex. 

RULE VII. 

Ty^icr. T,|X«-:: T.AZXijatBafe/TP.IXix, 
atBafe. 

T|Z«r.y|X«^»TrfKi£,atBsfe' T, fZXX, 

i* Bafor 

For, by the third KAom, 
■ I, rili ^ D*> f : %D. S, A 

S^ B py-S, ©K' S, Bfi-*^ P » : ^ »^i #. ^ b 

— S, B. ^ 

Therefore by thefi^tb I^mma. 

2 2 2 2 



2 

ROLE ^IH. 



T I' Z/X, Vertex. T, | ZXX, daft : : f, fXiiLBtfc. 
T,iX-MyVeflafc 



For, by the jirB Ride. 
S,BP^S,DPA::;B;ft^IX 



I'hert- 



(78) 

^ BPA-4-DPA. S,BPA— S,DPA : : XM^X^ 

Therefore by temm. Vl, ViL 
T, BP A^-DPA. T, B P A (/» D P A : : t, B+D. 



^•m^^mmm^m^mmim 



2 

T, B v)D. 



thcref. T, B c/iD. T, BPA c/> DPA : : /r B+D. 



i^mmmkfm. 



i 



( 



2 2 \ 2 

Tj BPA-|-DPA\ :: tBPA-|-DPA. T,B-hD. 
i 



PROBLEM I. 

In awf Sfharic Triatigk B P D. Given the 
AngkP, and the Sides BP,D?. To fihd 
the tuo Angles at the Bqje. 

S, I Zrr. S,\lLcri : t f P. T,|XZZ, Bafe. 
AnaSlZrr.;^ fXrr: : t|P. T,|Z XX, Bafe. 

DtMON^TRATION I. 

Rule 6, S, Z f r. S, X f r : : r f P.T f X XX, Vertex* 
Rule7* T, iZrr.T,fXcr:;TiZXX,Bafe, 

TAX XX, Bafe, • 

Rule 8. T, f Z XX, Bafe, Tf X XX, Vertex ::t|P 

T,|XXX,Bafc. 

Therefore, Multiplying the Homologotts terms into 
each other, and Dividing the ill and sdProduas ; as 

likcwife 



( 79 ), 

likcwifc the 2d and 4th by what is an equal Component 
in each ; and we havc^ '^ 

5,Zfr,xT|Z<rr. S, X fr xT fXcr; ;Tf |P, 
T^|XZX,Bafe. 

Therefore (^hy ZemmigJ.) 
'^f *Rfr.S^|Xfr ::T^f P.TffXXZ,Bafe. 
And therefore the Roots of theie Q^ianticies are P/o* 
portionaU 

S,|Zfr.S |X cr: : rf P.T |XZX^Bafe Q.E.D. 

DEMONSTRATION It 

By Rule ^. S, Zcr. S, X rr ; : t f P. T ^ X XX, Vertex, 
Rule7.TiZrr.TiX^::T|X:XX, Baf^ 
T, f^ZXX,Bafe. 

Kule8. T|XXX,Bafe,T|XXX,Vertex;;TAP. 
TfZXX, Bafc. ^ 

The;refore, Multiplying the Homologous Terms into 
each other, and Dividing the 1 ft and 3d Produ^s ; as 
lil^ewiief he 2d and 4th hy what is an equal Component 
in each, we have^ 

• 

S^Zrr. xtj Zfr. S,Xtf r XT I X O"; .• Tf f P. 
TffZXX, Bafe. 

Therefore ( by Lemma II. ) 
5g,|Zrr.;g5|Xrr;;TffP.TffZXX,Bafc 

Therefore, 
SfZrr.g |X^:;T|P.T|ZXX,Bafe. Q.E.D* 

ThisDemonftration fuppofeth P =: Z XX, at Vertex, 
tl^t is, that the Perpendicular falls withm the Triangle; 
\]nx the (amd Proportions will afifbi if it falls withQi^t. 

L 2 PROB. 



r 8«) 

PROBLEM IL 

A any Sph^ic Triangle kt there he ^iven tvs^ 
Angles J wth ths inter jacem B^fe ( or/de 
mlvM between tbqfe -<%/#/ j 7i Jfn4 tkfi 
0tbe¥ tm Siiesy and the wmatMng Angle. 

S,fZiX.S|Xi;Xj;tiBafe. T,|Xrr. 
g|ZXX.g|XiX;i T|feafe. %\tcr. 

The Demonilration may be perfbrm'd by the lame 
Ztf^mf^s and Rules that the iu^:ceding w^s^ but mil 
^e ticedlefs, if yoif pbange tl^ laft Tnsatgh into its 
Supplemental, 

R U L E I3C. 

In any Sphjpric Triauglp liavinE let fall § Perpendi- 
cular Arc ftpm the Vertex to thp 0afe^ it tyill be 

I'IZSegm.oftheBafe. TfZrr;; T|X^r. Tf 
X Segm. of the Safe. 

For, by tbe ficmd Rule, 

H BIM-H DR g BP ^^ DP : : g BA-|-S 0A 
- • ^ pA {A^ QA« 

T i Zfr. T ^. X 'r^jTtZ^egat T f X §p|pi. 
t!f X*. Ti^ffcgm. ■(: t Titer. t^ZSmqi. > 

PROB. 



(8i) 

PROBLEM ra; 

Suffofe there are gfvett the Three Sides of 
a Spherical Trian^e^ to find, au An^e^ 
Hwiftg imnj^itfd «f Feff$nd$t*d^ let 
fall to make two Right-angled Triangles, 

,T|Bafe.TfZfr:;T|Xfr.T,F. 

F ^1 Safe, tbc Perpendicular &H»| wi£'|tlK 
Triangle. 

Tbea, fJ, ▼ Jl P ; ; T, B A. S B, 
R.tPPj:T,PA,Sd. 






f I /^ / «. 



C82> 



r-r 



O F 



3tereographic ProjeiSion 



O F T HE 



SPHERE 



^^Xehrgrafbi€ ProjeBtcn is sl Branch of Per/peSivfy 
$J vfmch teacheth us how to draw the Appearances of 
the Circles of the Globe, upon a Plane pafling thro* 
the Centre fheteof ; fuppofing the Eye jin Surface of the 
Globe in the Pole of the Circle oh which the Proje6Hon 
is to be made. And here the Jird thing that ought to be 
don^ is to prove that the Reprefent^tion of all Circles 
deicribed upon the Surface of the Globe ( whether great- 
er or lefler ) ihall be Circles upon the Plane of the Pro* 
je6tioD» lo order to which I premiie this 

Z£MMJ. 

Ifa Scakne Con^ ABC, who& Bafe 
1$ a CiTcle> be cut by a Plane pafllng thro' Plate VIL 
the Vertex A, and Center pf the Ciycle i%.I. 
of the Bafe ; making the Triangle* A B C, 
( the Plane of which Triangle being perpendicular to 
the Plane of the Bafe of the Cone. I fey, That if thi^ 
Cone be fuppofed to be cut by tiro other Planes ( per- 
pendicular to the Plane of the Triangle ) the one parallel 

to 



4 



I 
d 



ta the Bafe as F Gy tbe other E D, (b as to make tb^ 
Triangle A £'D> fiimlar to the Triangle A C B. the 
CurvitiesF H G, EH D, fhallbe Circles, as is proved 
hy AfoUmimy Mjgdorgim^zxA other Writers of 0«/rx» 

P «(> P. I. 

If from the Eye at A, placed in the GlobeiVt Superfc 
cies, there he infinite Right Lines imagined to be pn>* 
dttced to the Periphery of any Circle ( fmall or great ) 
B C, defcribed on it, thereby making a co- 
nic Superficies. I lay, the common Interfe- Tlate VIT. 
6tionof the Surface of this Cone, with the f^.II,IIL 
Plane of the Circle E G ( in ivhofe Pole the 
Eye is placed) or any other Plane DQ parallel to ity 
ihall be a Circle. 

DEMONSTlLATIOir. 

The Triangle A B C, is the Seftion of the Cone thr<f 
its Vertex and Centre of the Bale. Now if we prove 
that the Triangles A B C, and A C j3i are fimilar, the 
Propofition mil be manifeft. 

Now the Arc D A rrC A. therefore the Angle AB C 
=r AC 3, by the 20th of the third of Euclid ; and 
the Angle A, common to both Triangles, therefore the 
Triangles ABC, and A C ft are fimilar. And therefore 
by the Lemmay the common Se^^ums, C f^ and c ^ 
which is parallel to it, Ihall be Circles. 

C O R O I^ I. 

From hence 'tis evident. That if the Circle to be pro- 
je^bd, be parallel to the Plane on which the Proje^ion is 
draiyn, its Reprefentation will be a Circle ; for the Cone 
1^ dut by a Plane parallel to its Bafe. 

II. If it be any way Oblique to th« PUnc of the Pre-' 



( 84 ) 

witl be Alt ucmxiiih^ tothe Memfk04 

Ilf. But if fthft l^j^ be in th« Plm^ dTany Clrde^ lift 

Reprdentation will be sl risht Line^ the Lines <}rawh 
from the Eye tothe' P<f|jpr)rf b4if in the {saxte Piarte^ 
and theoDmmon Se^ion of twp Planes is a right Line* 

iVi The Lbe h »'^ |hepM|elM DiMMter of the 
Gtrtk B& ther LlM E G is wkkh Ift the pr^e^ied^ J)i^ 
*mttery I call wi(h MtV Qugf^^ tk« ^/ft# ^fMufufH. 

PROP. IL 

I 

tf « gveflrt Cl»^ ^ to be pareje^ed u|Mk 
f /« ftf Vtl. t&e l>/tf^ of another gf eat Cirole, the V^*^ 
Itgm IV. of the prqjeted Circle fhall lie in the Zbte pf 
J^<^r«ry;^di(^t}f«0ni the Center of the pri- 
mitiire Circle ( or Circle on which the ProfBim is 
4Miri^} 13QI $h^ T|i)g0tit ttf the Elevataka or ^ fiud 
Ci«de «b6^«b» rir«i» of the PrajiOm. 

• I>£MOirsTAAtlQN. 
Suppoft tte Sy^«t A> pnl^^^ ^ 1?^^ Cifcfe-B Q 

Upon-tQQ pV^ or tWCirok $ 6 v the Point 6^ appear^ 
in th^lAKM M^i^e^ at ^. ib thit F i is the Tan^^nf of 
i Ihe^ Afigk H F B^ the dift^^nce of the ^oint Bs ff om the 
Vertex, by the ioT* of the Mir/ of Euclidy and G^ tTie 
other Extremity of thei?/<M»f ^^r of the Circle fi C, ap- 
pears in the Zine ofMeaftns ^ ^ fe that the DUbance 
Fr, is the Tangent of off the Angle HFC> the Dili- 
anoeof t^ Poi^t C> fponi the Vertei^ the Line (f che-^ 
ing biiieik^d in iys it the Centsy. 

Now the rigHCoU^ed Trungk i A r, is divided by 
the perpendicular A F,icto two right-angled Triangles^ 
S AFyAcF^ iimilar to each other by the 8tS of the 6th 
Book ofMtgciiiL 

Therefore the-iA A F=f BFHssAf F=:|A;&, 
Therefore the Angle BF H ;= A i ^» and confequently 

"" ' their 



their 0)mpleinents are equals vi^. BFE— FA /. There- 
fore F i, the DiftaBcc rf dteGente^f^^of tfee proje£Wl Cir- 
cle from the Center of tlie primitive Circle is equal to 
tho Tangent of the Ardi E B, th© fijev^tion of the 6ir- 
cTe to be projec^bd tdr tbe Eddiut A B of the Priihitiire 

Circle, f* ^* ^v • .; *i 

Mr* uugfiirei dotf riiiinui^ only I the Appearance oT 

the Extremity of the tXiaafcter above &e Plane of tibt 

Prqje^oi^ gives this Riile for dfe^cnhing the Circle. 

From ft thePpfeofthe LineofMearures makeH m 
nBE— «? », and draw the Lines km\ k n^ p Ihall be 
the prmedted Pd« of B C and i its Cenicft Tbe, l^adim 
beiiffi ibzLi A, wiuch he proves thus^the Angle i bkzx 

B^fe the Angle i b k—b BF+BF * that is F A ^ 
4- F A ^, for the Afc H ^.=: 2 B Bby Conftruaion. 
Hende F /» is evidently the Tangent of 1^ E. and A I, its 
Secant^i u 

C O R O U 

Hence the Secant of th^ Circfes Elevation fct off in 
tliB tine of Meafiires from b^ the jprqjeaed Extremity 
of the Efiameter above the Plane ot the Projeflion will 
give / its Center, as well as the Tangent of that Ele- 
vation from die Centes; 

i^ROP. nr. 

If a kffer Circle whofe TohB lie ia the 
PUme ef the ProjeSufn wet e to be project- PhuVJIL 
ed ; tfiejCenter of its Reprefentation on the JF/f . L 
Plane fhall be in the Une of Mcafures diftant 
from the Center of the Primitive Circle by the Secant ot 
that kffer Circles diftance' from its*P<?/^ and its Semidi- 
ametcrihallfae equal tatheTanyutof thavDiftanea 



DEMOJrstRJTtOK 

Suppofe the lefTer Ciidt pliE^g through the Feints 
By C, equally diflant ftotn the Fole G, were to be proje- 
£ted upon tlie Plofie EG, to the Eye at A. Draw the 
Lines A B, AC, interfe^n^ the Line of Meafures in B 
and €y the Reprefentatibn of ^he ttvo Extremities of the 
Dianieter, biffeft theLin«r*<?}ti /, then is' ithe Center^ 
Upon the Point i mtb the Difiance. i b, deferibe the 
Circle iibC Cy draw the Lines H B ^,F B> and i B. 

The t^vo Triangles A H B, and ^ H F, having the 
Angle at H Cbinmon, and the Angles at B and F right, 
are iiniilar. . ... 

Therefore XHAB rr ZHe F 
butXHAB = |HFB 
andXHcF=:|iiB. 

Therefore Z H F fi = X ^ i B. Therefore the Tri- 
. At^le F. B s is right ancled at B. and confequent- 
ly S i is the Tangent ofthe Arc, B G the DiOance 
of the leffer Circle firotti its PoU^ and Fi its Se- 
cant, Q. E. D. 

Mr. Ougbtred only makes irfe of A the Reprefenta- 
tion of B^ and then diie^s to dra>v the Radius F B> and 
B / perpendicular to it, touching the Circle B G. proves 
i to be the Center,tbe Radius being < £ =: i R thus 

tbeXi^B-XiBA 
For the X ^ =: F B ^-l-B F b tharis r: i B C+C B ;- 

bBi., ' "" 

PROP. IV. 

If^vo Circl«CBF, and ABI, interfeaeach other, 
the Angle F B I, form'd by them at their Intcrfeaion." 

M' ^T^} ^ ^^ A"8^ C B A, made by the Radius 
C B, A B, drawn to the point of their Intcrfcaion B. 

DEmir- 



f 




(87) 

DEMONSTRATION. 

• • • 

' To the* Point of rnterfeftion B, draw the* 

Line D B. a Tangent to the Circle C B F, TlaUWlh 

and E B a Tangent to the Circle A B I. f/j. II. 

Now becaufe the infinitely fmall Portions 

of the Circles C B F, aud A B I do coincide with the 

Tangents D B, and B E, and confequently have the fame 

Dire^Hon, therefore the Curve lined Angle FBI — 

to the right lined Angle formed by the Tangents DBF, 

Now the Angle CBD=L==ABE,and taking away 

£ B C which is common to both, there will reniaiH 

CBA=DBEiff=:FBI. f. c. ^ 

PROP, V. 

In Stereograpbic ProjeShn the Angles made by the 
Circles on the Surface of the Sphere are equal to thf 
Angles made by their Reprefentatiyes on the Plane of 
the Projeilion* ' . 

DEMONS TRATIO N. 

Suppofe the Eye at A, proJ3<aing the 
Angle R B S, upon the Plane £ F : Let Plate VIIL 
BD, be a Tangent to the Circle B S and Frg, III. 
B C, to tfe Circle B R, to the common 
Point B : The Plane E F, and likewifc the Plane 
made by the Tangents B D, and B C, are both per- 
pendicular to the Plane of the Circle B Q^A, and their 
common Se^ftion DC, will therefore be perpendicular to 
the Line E F, produced.' 

The Eye at A, projefts the Tangent B D in F D, and 
the Tangent B C in P C^ 

I am now to prove that fhc Angle C BD, made by 
the TaT^ents to thefe Circle^ on the Sphere, is equal 
to the Angle CFD, made by the proje<aed Tangent 
on the Plane of the ProySlm Draw frC^ pai:allel to 
ftF^and joynAQj 

M 2 I 



r 



(8?) 

(a) By 32. 3, For the Angle DB Ar= A QB, = 

£uciid. (B) CO (d) 

(^) 21. 3. ABa=BFD=:DBF. Theref. 
CO 29. I. CO 

(i)^x. I. BD=:DF. 
CO 6. I. Then in the Triangfc? CDF, 

0^)4.1. CDB, CD, DF=:CD,DB>^ 

the AnglesCD F, and CD ft i»th 

Right. Therefore CFD=:CBCK y. e. i. 

^nether Dcmofiflrafim hj Mr. Halfcy* 

Let E Bf Ljbe any great Circle of the 
Tlate IX. Sphere t the tyep^cedinits Circumfe- 
F'rg. I. rence, t5 its Center, P any point thereof ; 
and let F C O he fw|iDied a FUnc ercjfted 
at Right Angles to the Circle E B P t, 4?n which F CQ 
the Sphere is to be projefted. Dra%y E P crpflEi^n this 
P/tfwtfFCOin/?, and fftM be the Repre(entatw)n of 
the point P ; to the point P, draw the Tangent A P G, 
and on any_point thereof, as A, ereft a Perpendicular 
AD, at Rigkt A^ls^ totbe WUme^UVl^ and draw 
theLincsPD,AC,DC: TbentheAncJe APD ftall 
be equal to the Spherical Angle contained between th^e 
JPlanes A P C, D P C ; draw alfq, A E, D E, interfea- 
ing the P/<werFCO, in thepoinjbtf and d, which will 
be the Reprefentatioh of the pointy A, an4 D ; jpin ^ 4> 
fA I fay, by theTri^iffile ^ dp^ isfimiUir tp tfip Trianr 
gle A D P, and the Angle apd, equal to thp Angle A P D# 
Draw PL, AI^., paralld toFO, and by reajim of 
the Parallels a f» ad :: AK. A D. but ( by the 3 2^ 
of thetbird oi EucHd) in the Triangle A K- F, the 
AngfeAKPntpE-APkiEPG: thprefoce 
theSideAIJ. — AP.andtf^. ad :: Ap. APt Hence 
( the Angles D AP, d a j>, being Right) the Aiude 
APD= i^pij that i^ The S^rieal Apdl? onfibe 
Surface, is ecuual to that pn thp Vrok&iQXk ft ^« ^ 

' ^PRQF. 



( «9 ) 

p n o p. vi; 

All lefTer Circles whofe Ftks lie not in die plane oit 

wljich the ProjeBion is to be made, are projeAe4 by 
laying off in the Line of Mea&res from tlu^ Center^ 
the Tangent of half the Diftance of each Extremity of the 
Diameter fronulitt Bde ( of the Cirde on which the 
ProfeSim is made ) oppofite to the Eye, and the Didance 
ofthefe two ycojiQfkea Bxtremilies bii^aodwill be the 
Center of the Circle in the Projc^on. 

For Example. 

Ib PMff VIL fig. n. Sap^e BCtohef^chlefier 
Circle; Lay cffFi^equal to the Tangent of half the 
At A H B» and F i equal to the Tangmt of half the Ascti 
HC ; dbis Diftanoe he heii^Hifeacd, fliall give die 
Center of that Cutek in tiie IwfgSwh The Rcafon of 
this is erident; from wh|t hath been Aod in the Dcmau-^ 
fimtim oftiufprfi Fipofcfitk 



P R O P- VII. 

All ftmt Cixcfe^ ( a$FMe ¥21. Fig. IV. ) p£»g 
througb the Point i, fliall have their Centers in the 
Right lins rLpwrfaced i which Right limi I^ is 
at Right Angl^ to the line of Meafiuss EG> pa^i^ 
diVQWgh i» the Cejoter of d» Circk A ^ 1|^ which 
CBts&line £ G^ oritsR^u^fentatire Sefflicixcle» at 
Right Ai^kfti 
For all great Circles pafling through ^, fince they will 

cut again ata Semidrcks diftanoe, muft pais thro' 

the proje<fled oppofite point, that its through r ; 

tiieK&TC 'tis evi<&nc their Centers wifi he in the 

linsi iiL produced, if need bew 

PR OP. VIIL 
If it ifore mfiaxoi p ixiu/r^ a Cirde which ihgU 



(.90) 

paift througb the point b, and nuke anj given Angle with- 
the Circle A A H, fuppofe of 20 ieg. Lay off i ky equal to 
the Tangent of 20 dcg. to the ^dius ib\ I iay, the 
poineiis the Goiter^ required. 

. For draw the Line k by and defcribe the Circle bKc. 

. Thenhy the fourfb Prop. 

ThcAnglc A^R = «^t. = 2o*^. 

CO ROU 

Since it has been proved in the fifth Tropcfitlmy that 
the Angles made by the Circles on the PUme of the 
ProfS&imy are equal to the Angles nude by their Re- 
prefentatives on the Surface of the Sphere : It will there* 
fore follow^ thatif ^ was the projeftcd Pole of the World, 
A 6 H» the Pro je^Hon of the Mx a Clock Hour«Circle^ 
^and it ivere required to proje^ all the other Hour- 
Circles> which make Angles of 1 5, 30* 45 deg* Bcc with 
that SiJC aClock Hour«Circle ; *tis but putting off in the 
Line i L» from i, on both iides the Tangent of 15, 30,45) 
^ 60y 75. to the Radius i by and you will have the (inters 
of a]ltho&Hour*CircIes in fuch Horizontal Projeftion. 

From what hath been demonftrated in thefe Eight 
Prcpcfitimsy 'twill be eafy to proje^ the Sphere in 
any Pofition •* I will now proceed to fliew how to 
meafure any Arch pro]eded,whether Great or Smal^ 
in order to which I premiie th^ Lemmas. 

LEMMA I. 

If.a Plane ps^k through the Poles of two great Cir? 
des £ Q^ and H 0» its common Se^on with the Sur- 
face of the Sphere will be reprefentecl by the great 
Circle Z P N H ; and if the tworemoteft 
Phte IX. Poles, P and N, be conneaed by the right 
Pfg. II. Line P N, and a Plane be fuppofed to re- 
volve 



i. 



f 



'•n7!> 



•/._>-, 



f 



»^ ♦ 



( 91 ) 

volveon'thiiLiheFNas an Axis ; ahd ktthe conmion 
Se£Hpnof this revolving FUnc with the Sur&oe of the 
Sphere be repreiented bythe Circle PDKN. Ifay, 
that this revolving Plane fhall always cut off* equal Ar- 
ches, upon thoie Circles ACL H O, whofe Pol» are 
thus CQnneded between it felt PDK N, and tUe great 
CirdeZPNH*, that is»t|ieArchDO=: ArchKQ. 
And likewiie from any two parallel Circles, b o^ and « 
5, equally diftant from the iaid Poles N and P.diati^ 

dozz i^f» 

ForPO=:NQj,andtht Angle O PD =: QJN K^ 

it being the Angle of the Inclination of the two Planes^ 
and the portion of the ]efrerCirclePD=:NK:There^ 
ibxetheie two Triangles being in all cafes qualified alike^ 
willbeequal; andtberefbre DO=:KQj and for the 
(ame Rea(bn the Triangles oT dsmd fHk^xe equa^ 
andooniequently^0:::tf. Q. £• D* 

LEMMA n. 

If a PUme pais through the Poles of two PUae IX* 
great Circles, £ Q^ and H 0> its coihmon J%. III. 
Section with the Sur&oe of the Sphere will 
be reprefented by the great Circle ZPNH; and if 
the two neareft Polesytisoidpy be conneAedby the right 
LineN^ and a Phuie beim^in'd to revolve on&s 
Line Np^ as an Axis, 'tis plain that its common Seftion 
with the Surface of the Sphere will be a Circle, and is 
reprefented by the Circle /»,/,r, 5, f,/, «!,», ^,N* I iay 
this plane fhall always cut off equal Arches in tbofe great 
Circles £ Qj> H O, whofe Poles are thus conneded ; 
and from any other Parallels to them, -«f, b #, equally 
remote from py and N, between it klfpSy tyWytf Nv and 
the great Circle Z P N A ; but on the contrary fide, 

that is, £5=: MO. mt=:fo*brzzn ^•HizztQs, f^' 
the Triangles p£S and NOM> p^tixA N«/, pb 
randNfi»,f Hi andNQ^f, may, as before, be fhfiwn 
to be exaftly qualified alike. Andthexefore,Cy^. Q.ED. 
I now proceed to give fome Inftances in the A^al 
Proje^ion of (he Globe upon the plane of the Horizon, 

and 



( 9a ) 

Pbte TL Sfhttt apm tbc fUm ^ Ibt Htri^mf fir 

Dttfinftddtt Cirdk M fi S W,. yepidbwing tb^ Ho- 
rizon ; through Z, the Center, draw the LiaeS Z N^ 
zai EZ W> JLt ligheAfiglcvf ffaefi^k rt^pfefaidng the 
MbiidUs of tiK jplaee^ or Kdtk and Sovitb AxiauUli) 
tfe othef I die piune VeMkri iDt Azutmth of Eaft mI 
¥ML Nb^rbeetttietbNbrtfKftiP^ofifaBWorUk 
dWant Amn the SSeoith ( oroppofite Me to whidi the 
l^isplacid) ^S J^» opnmh La^oflfthe'ii^atgeiieQf 

fe)mZto^; then will p be the Reprefentation of the 
Northern Pole of the Worlds bj'th* Eighth Pr(fp<fition^ 
and the Circles which paifes through the points £ p W^ 
wfflhBftkftRqMMTcaiutioA «f tlae Hour-Circk o£ Six ; 
and its Efevadon above; the fhtne of the pr^edilDn be- 
ing equal to tte Ekvatioil df the Fole^ iwf • 51 deg, 
^a mn^ ha^ offin die lineof Meafur^s^ Z T, rae Tao- 

S of thai! Cirele's £]dvadfiir ; Tben will T be the 
erof that Carde, by tbtfkoitd Ffopifptiof^\ then 
de&ribrtlK CinrleJEdW. 

Now to deferiJ^ the Eqiiafiory t^fe £)Uha(» fcom 
thtZenkbis ^ideg.'^emin* Takethe Taifif^t of its 
kaif, vif. li'fdeg. 45 fnm. to the Radilus Z W ( whtdi 
you ms^cafl^hayel^the help of aSe6h>r ) and let k 
otf in tjieLineof Meafusres fromZtotf^rafid theEqu^ 
tos fliall'p&tluoiigh the Meridian N S>in the point if. 

Ncnr die Elevatton of the Equator bedng sdways e^yml 
t^ tfab CotspfexncBt o£ the Latkude, whieb tn this Cafe 
is 9^ d^m 30 Mriiw Take the Tangent of its EkratiOBf 
ai»onrit odFin tbe Lineof Xieafims from 2 toX Then 
ftall }i be the Center of tho EqiiaM^. T belt deCbtibe 
the CireleEtf W, which yoik may con^leat in am occult 
Aacby fi)ra U&bereafcor to be itaeatiaiird. 

Now 



(90 

INoiv for drawing the Tropic of Cancer. Take the 
Tangent of halfitsneareft and greateft Diftance from 
the Zenith^ and put it olFin the Line of Meafures, jFh>m 
Zto»y and fromZtor; the Diftance i>r being bifleftedy 
the point of Bilfeaion fhallbe the Center, by the ^x(b 
JPfoptfitUnm Then defcribe the Circle 9n^y which 
compkat in an occult Arch for farther Ufe. 
• After the fame manner defcribe yfs}ftj the Tropic of 
dpricffrn : And the like Method ufe for the Delcrip- 
lion of any other parallels to the Equator. 

Now to draw the Hour-circles which make Angles 
of X5i 30»45, Vc. with the Hour-circle of Six, it hath 
been fheived in the Or. M Froftfitimj that their Cen- 
ters mil all lie in the Line PT P, which is perpendi^ 
cular to the Line of Meafures paffing through the Cen* 
ter of the Hour^rcfe of Six : And that the Center of 
any Circle paffing thro' the Point py and making an Angle 
therewith, ivas to be found by laying off in the Line T P, 
the Tangentof that Angle, to the Radius^ T. There- 
fore put ofFin the Line T P on both fides the Tangent of 
15> 30, 45, CT^* and you*ll have the Centers of all the 
Hour'-drcles. But to avoid confufion of Lines in the 
Scheme, I haveonly putofF thcTangent of 30 deg. from 
T to ^, and thereby defcribed one of them, vi^. the Hour« 
circle of Four in the Afternoon* 

Now if 'twere required to divide W ^ which reprefents 
the Quadrant of the Equator, into its Degrees : Lay 
your Ruler upon ^ the projected Northern Pole c^ the 
Equator, and the equal Diviiions of the Qiiadrantof the 
Horizon W S* for inftanoes, upon 30 and 60 deg. and the 
Ruler ihall cut the Quadrant of the Equator W ^r, in the 
points 30 and 6o* So that « 30, reprefents lodeg. a 60^ 
reprefents 60 dcgm And the Uce for any intermediate 
D^rees. 

. Or a Ruler laid upon ( or a Line drawn from) t, the 
projected Southern Pole of the World, and the equal Di' 
yifuHis of the Qumdrant of the Horizon W N, fhall pafs 
through the fame Points 30^ 60, Cf r. in « W, the Repro- 
ftntatiYeofthcWefternQ^draQt of the Equator. 

N Not 



(?4) 

Kowihe Rnfto of tbni ^^B be cirideat, Arm Ais 
PrcjcaMatik£}« » fiippofedin tte andetf-Fok of die 
Hofko^Qt )^£^ «Mla Uw dxscwnhom the Eje to 
db Kwthert Peto of tk WgH4 pifo ti^ 
)forufti> $ N ift j^ tke Re|il«ii^ 
Piok^. Let tfa» iaB3to»rx Line wfakk jpatfe&iraii ths 
Eye to the real Nortfaem l^k^betfaeUmtbafcoaBnofts 
the- tWQieimtcAPofcsckf die fiquaneraadHBiini^ 9C^ 
cefdii^tirfbQJir^XaffilMTaixliitti^gm f»« 

voire upon that Am ; tWs J^iMr fludi aqewrij^g tin dik 
XeiwiM^ttet offeqiftftl Aldae»«idMre(ieat CSrdbswkofe 
iNilesjrefktiftenwe^bdhetvrcen it felfaxxl dscMeri* 
diafi N $^ wtttch tafEes dtfough tfaii two P9ks.of tte 
H^naei^and (nMi Poks of theEquator^ Now^dv Eyo a 
in the Nadir i» itt the P/«k^ of this ievolYii« Hme Y 
JW)d thcye&we bjr the D&/>i Orolkrf of the j^d Troftf^ 
iiiUy the RGfKtfai(akip» «f that Cirdie^Yhidtri&die coii»> 
nwa S^aioa ofthKkrevo&ning PkM? with d» Sttr&oe of 
m Sphere, wiB be rqwefaited to the Eye m dwcomnntt 
S«ftiiMi«f thtt revolhris^ ¥Utney voA die n«M of the 
Pmjedionv d«tit» ( if d«» i^^tae paifes dtfongh ^ in 
the Qjiadisint & W ) htthoLioe^^ifil^iniodivo' 
^ 4(;« ifl die Q|UMbaiBt $ Vr> the Seakm » f^ dew diev^ 
&fe tf-^cv iom £tttac«r>.it cqniralenll toS gc^itatfae 
Horizon, &'r. Por thi». imrriving nmt akrays cuts 
the Liaeof Mttiiues ia;^ the Pn^ftiea of the Northern 
Pola 

Bm. wheo } ^ride the QUfMAnxka W^ bjr Uses drawn 
fron 11^ tbtprejeaKmof theSotxthem Po^ to the equal 
XMirifieBS of t\xi Qoadtaet W N. The Urn upoo 
which the Rcf^dntioii iif the Ffmm is aode^isangW 
Liaedra^froaiithsEye&fipofidin theNsrii^ t»tbe 
Sottthnm Poley,aiid peodnoed till h meet the UneofMea-* 
fures N S produced in t, the Reprefentatiott of dM 
Scmthara Pole ; hathff&stfann^Lcmmmy aJTOMr that 
ttenrebr€4«poi» this L^outsofFeqoalAidMffi the Q»^ 
4»aat of tibe £<)natDr^ W, aodQ^adranc oftike Hoeiioo 
W N. Aod cborefhoe a £ioe disym fvim 9*001 ^ ia 
the Qp^ant N V^ Aali'^tfanM^ ^iDeheftne- 
fcntation of the Quadrant of the Equator <c W» 

Now 







^' /^ 



/ 



Htm t^dittdk the Tiopite ffOmcer^ 
Pir5, If I would dp it by means of the pfojcftedNoftb 
Pole ft «fi«pfe tfeeTwipie V)f' Omctr is dfift.-nrt ffom the 
K<rtifc-I\>ie^#^,3omf».frotti Ae iNorth-Ptele theie 
ifMft heaiVirxife/to tJw H^rtzcm projei^lrcd, tfcat is 66 
ifcy* 30 «ifip»« ^liibflt ffom Ae uwier Ppjetjf the Ho- 
riiion, in which the fiye is pflaced, its Diftanee from th^ 
tipper mie of the Horizon is' 113 rfej. 50 «?^. Tate 
<!ie TtiJgent ^ its hdf, v«^. 56 d. 45 w. and jwtt ft 
on theLineofMeaftifes-from^to A, Z A ftall Ic the 
^Semidiamecer of t!ns PiordBel ; or the point A may 
|je 4etertn«fd by putting AeChord cf 66 <t }o ««)». 
•from E to MP, a Line drawn from E iteoogh w, tvBI meet 
tJieLtiieof!Nfeafttt«intiiefamepotflt A. Defcribethtty 
the Circle A BCD,a Rufcr laid from y, to the equal Di- 
vffions of tins P«r«ffe/, ftia!! t?ttt the correfpcmKien^ ftwits 
tir De^wes in the Tropic of Cij»tfer, For inftance: A 
linedfawnfrom^to 30. intbe Quadrant A D,lhifl cut 
the portion of the Tropic of C«mtr n ^, in 30 : Sp 
that« 5Q,fkallreprefeii ybAg.n6o^'&hs& i tyidfe ut 60 
in the Tropin 0»r. 

SicmMfy !f I wmld diviae*i$ Tropic Irt' the jneans 
iftf the pixj e feA Southern-Pole ir\ 1 confider that tbe 
Tropic qS Cancer is diftant from tbeStmC^cm«-J*bte of 
theJEquator 113 i£ 30«r. 

jf therefore prcjjed a parilld to the Horfeoon as fe 
dHbnt from the under ri?)feoFtJieHoriton , and confc- 
qtientJy from the upper fdte rf' Ae Horrioirvor Kenltii, 
€6d,Y>jft. Therefore I put ijffthe'Tat^ajtofitshaJj^ 
v/\. 33. ^ 1$ i9r« in the jjat trf* Mctfores, from i?^ »> 
«; or ptttthe C3iord of 66, 3^, from Wto w, a lii* 
Axuwn ftomCtoiv, fhall eut tlie Lh^ bf Mcafures Jb 
the!ainep«[ftt«: Dd^cribetbe<!5^deAf#f^i, foppofetJlp 
fljradrant c 4, dividcS, a Ruler led4 w ;» tine dtttwn 
from ;^ tti ^ 60, ^ iwthe Quadrant e^ "flwll cut tfc 
pottion ot the Ttopic cfCamcf m the jioiflts 50, 6p, 50. 

Nowthd^ f^hmh A RCth -a f c iy \iiAiA ^y«« 
dmt^ for the Girifion oF the Tnw t)f diwc^i tvMl iH- 
|riafc|!jc Ttoffctjf pipfTdiri^ t>f^, iijnct dnWTi W)(nT 

N 2 



(96) 

to the eqiul DvnGaes of tbe Scaaiaxde a b c d, vnSl 
divide the Tropic Vf s)ff. 

Or Lines drawn from Tf to theeqiud Divifions of 
the Arch FCH9 will do the ikine. I have omitted draw- 
ing them, left I fhould too much perplex the Scheme* 
Tnefameis to be underftood of any other two Parsl- 
bis equally diftant from the /Equator, or any other Cir- 
cle. For having iirft drawn two Paralkls to the Hori- 
zon, or other Circle on which the Proje&im is nude, 
by the fore-mentioned Rules, that are neceflary to 
J meafureone of your parallel Circles, by Lines drawn 
1 from both Poles : The ikme two TaraJkls will fcrve 
to meafure or divide the other parallel Circle at an e- 
qual diftanoe. Only here obferve ( a^ in the Divifion of 
the Tropic of Cafricom ) that that Pivr^i/cf/ to the H&- 
jizon^ which ferved to divide the firft Parallel from the 
proje£l:ed North-Pok^ muft be ufed to divide the Para/*' 
M at an equal Diftance from the proje^ed Soutk-Pole* 

The Equator W tf , befides the former Ways given for 
its Diviflon, may be thus divided : Suppofe the Semi- 
circle FGH, was divided into 180 equal Degrees, a 
Ruler laid upon any of thefe Divifions, and tbe proje* 
^ed Northern^Pole ; as fuppoie from 60^ or v, in the 
Q(iadrant GH, to^, it ihall cut the Equator >e W^in 
^o, which I thus demonftrate. 

Let tbe Lines 60 p 60, 60 X, Z 60, be dra^vn, as 
.airoXE,and|>E* Now becaufethat ZX, isthe Tan- 
gent of the Elevation of the Equator above the Horizon, 
,i//:f. 98, 30. and Z;, the Tangent of half that Angle : 
Therefore in the Triangle E ZX, the line Ep, divides 
the Angle Z E X, into two equal parts. Therefore by 
the^&V^ofthe/x^i&of EucHOy the Line Z X is divided 
in the point f. So that Zp. p X.: : Z E. E X. but 
"LE — Zjy andEX = Xv.ThereforeZ£.t»X : :Zjf. 
X V. but the Angle v^X=:Z|»y^ Therefore by 
the fiventb Pi^^ofition of tl» fixtb of Euclid^ the two ' 




.W> ^re e^ual ; Ap4 con(cquen(ly their CompHn^nts 



(97 ) 

GX v> andY Z S are equal : And thertfbre thisMe* 
thod gives the iame Point 60y in the Equator « W; as 
if a Line had been drawn from the Tokp, to the point 6oy 
or jy in the Quadrant of the Horizon S W. Or draw 
Lines from any of the Divifions in the Semicircle F G H* 
to Ty the prqje&ed Southern Pole, ihall divide the £- 
quator. Forinftance^ Lines drawn from 30, 60, 90^ 
in the Quiadrant F G> to t» pals through the Points 30^ 
60, 90^ in the Quadrant of the Equator a W, as you 
may fee in the Scheme, and may be demonftrated by a 
Demonftration more eafy than the former. 

TheTropic of C4»rfr may be divided likewiie as the E- 
quator was,^/^. by dividing the Semicircle a r 1^ intoiSo 
Deg.and drawing Lines from thoielHints thro' the Point p, 
and they will divide theXropic of Cancer. Or Lines drawn 
from thole Points to. t, the Southern-Pde will do the 
fame ; as you may fee in the Scheme, Lines drawn from 
the Points 30, 60, 90, in'the Quadrant r g to t, pafs 
thro' the Points 30, 60, 90, in the Tro^c of Comer. 
The fame Method may be ufed for the Divifion of any 
Parallels 7 TheDemonftration being like the former, I 
therefore omit it. 



OF 



(9^8) 



O F T H E 



Orthcgraphic Projedion 



O F T HE 



SPHERE 



OMArgrOfbk Pn)je6Hsn fuppolies the Eye (cipsA- 
diouiartoA PlaaepaJlii^ thrcHigh tfe den- 
ier oftlse Sphere, pofiftediB a Peipeodioibr 
tottut Flaas ewfted mm ifae Center, but at utk^imte 
jDiftance ; or ^vhich comes to the fame thing, iii^po&s 
that there were Perpendiculars let fall from each Pointy 
to the Circles to be proje6bed to the Plane, on which 
the Proje^on is to be made : Thefe projcfted Points 
will fall in Right-Lines on the Plane, when the Circles 
( great or fmall ) to be projefted, are perpendicular to 
the Plane of the ProjeBicn , but i^Oblifue^ in EUipfesm So 
that vou may have as great Variety of thefe iorts of 
frcjeSionSy as in the Stereograpfne.^ But they being 
more Troubleibme^ by rcafon of the Oblique Circles 
being proje^d inEUipfes ; Mathematicians have confined 
themlelves to the mod fimple of thefe kind ; vi{» up- 
on the Plane of the SoUHcial Colure, commonly called 
the Pianifpbereyor j^nakntmoy which is admirably ufc- 
ful and eafv, I ihall here give you a fliort Account of 

In 



i*.rf 



1 



• . 



! 

■ 



v) 



M 



'4. 






r 99 ) 

InjPJkf^iUh, 1%. ift>tbfe Cirute ZKNOi *pw- 
jents tbe SorKUml Colure, or "Mtridvkn pai&eg tbi^uf^ 
the ift Pf^ts. of ^ afld y^, tbe Ccntef A H]^^(sd^ tie 
Eaft or Well: Fointof the Horizosw 

H AO tfa^ Horizor^ itst A^eis Z A N, mpnbpt$ the 
AzizBKCh of £aft or Wcft^ or 90th Azimuths 

lEAmn theJB^uinoftiA], its Axis FA|, iseffefaM 
the 6th Hour Crick. 

^ A \9' re|»re6nts the Ecliptk;^ Y" A s!$i ks Aii% rc« 
prtfenting a Circie paifiDg thxo' the E^vtBollulPmiits 
T and jfiu 

O F the heighlh of the Pole, whofe ComptoKot His 
r= Z P9 i5the heightof the E^uinoAiaL 

Sufpdfe the Sttn's place to bein JLc deg* 9r Sly od^ 
iMT in ^ o di^. or in ;is: ode^*t\i6ktbinUr.9'4\ jtftant 
fiom the E(|iiiQo^kial Pointo ( both which are psajfAti 
in A) 60 /i?^. therefore take the Sine of 60 ^. aiid pat 
^ott intheLxncA s,a]!hdA YT, ffoniAt hoAWayito 
^» ® » ^^f<^ Points ikevv the. place of tto $00 for 
the giiret timts. Kii^. Thsftva the Q^dzsmt^ A ^ fiur 
the Summer, and that in the Quadrart, A )^ for the 
Winter; and through the Point @> draw a PasaiHet to 
the fiquMMxftkl Dd,<^S, ivhkhis the Projection of the 
Sun's Par aUel fcnr that Da)& 

Hence it .ippears, that the Sun that Day rifes or fets in 
^PoifitM or m, CuhBinatesatNoon ita D ovi/, ooaies 
tQthe6diEteur ioT^is. dueEaft or WeftmQ^thece* 
&seA)^or A«(istbe Sineof the Snn'sAmpUtdb^T 
® istheHhe lUsht A&eniioo of tlie Sttn> thati^ T 
coctaimas jtaaty D^oees as the Bight A&enfiof^ ha^ It 
hees oeaiaredin the JEquator,. and therefore 2 like Aaeb 
MDorM/ die Setntdiutnal Arch oftheSuiH^M Tor 
/» t the Aiocsiiooa]: Di^Serenc^ to be jsumbred from tkr 
next Cardinal Point backward or forward. 

Suppofe the Sun's height were taken by a Quadrant^ 
or other laftrument, put that Altitude from H to L^ 
fond ixixsLX / Parallel to the Horizon, cuttbg the Pa- 
rallel of Declination D^inS, and the Axis of the Ho- 
rizon in X, S will be the place of the Sun in the Ana- 
lemma for that inftant* Wherefore S T fliall be t^ 



I 



( 100 ) 

)ikeDi(bmceoftfaefSun from the 6th Hour Circi^SX: 
its like DiArn from the 90th Azimuth, whence may be 
£ouiid intheiePanilklSy both the Hour at which the Ob- 
ier?ation was made, atid Azimuth from the Meridiaxu 

Thofe Arches of IdOfer Circles ( which I call like Arch* 
es) may prqportionally be transferred to their greateft 
Fsurallel in this manner : As the Semidiameter of the 
kdler Parallel, is to the like diftance : So is the Semi- 
diameter of theCireleZHNOyto the Sineofthe true 
Diftance from the Centnr. For Inftanc^ in the like A(^ 
feenfional Difference TMto the ml of the Parallel 
T M 4 ^niw a Smndiameter A if^and within the Trian* 
gleAT4 drawfromM^ the Right Line M t, parallel 
to AT; Thenl^iTM:: AitA^, makeAFriA^, 
then is A F the Sine df the Afcenfiona] Dit&reno^ by 
which the Semidiurnal Arch is more or Icfs than Gx 
Hours. 

Henceitis eafy to deferibe the EB^ifthat ihall pais 
through the given Point M, in xnyKath of the Semi* 
tranfrerfe A O) to the Semiconjugate A M« Thus, 

MakeAO. AM :: X/.XG: :f P.#r, the Curve 
Line^ dratvn with an even Hand tnro the Points 
Z> G M, ihall be the EBi^s required. 

Or thus by your SeAor ; Take A O in your Compan- 
ies:, and open your Line of Sines to it, then take A M and 
apply it to your Line of Sine%andobferve what Angle 
tis the Sine of; then draw as many Occult Parallels to 
A O as you pleaie, aud making each of them Rad« fuc« 
eeffively, lay off the Sine of the Angle you obferved A M 
to be the Sine of^ and do fo fucceflively, a Line dratvn 
through thefe Points (hall be what was deiired* 



Stereo- 



( loi ) 



Stereographic Problems, 

In order to the Proje<Slion of the 
federal Cafes in Spheric Trian- 
gles. 

P R O B. L 

To find the Pok of any great Circle Projected. 

T^ rO T E that the Pole of any great Circle is the 
I^^J repreientationof that point of the Surface of 
JL ^ the Globe ( on the Plane of the Projed^ion ) 
'^ivrnch is 90 dsg. diftant from that Circle. 
Thefe Circles are of three kinds, either Plate XL 

I. The Primitive Circle A B C D, the Tig. IL 
PlaneofwhichisthePlaiieofthe ProjeSion* 

2« A Bigbt Circle ( or the projected Reprefentation of 
a great Circle perpeiidicular to the Plane of the Pro- 
je^ion ) the Eye oeing in the Plane of which, mil on 
the proje^h be repicfentad by a Diameter^ as B D or 
AC ^ 

9* An Oblique Grcky (or the proje^ed Rcprefenta* 
tion of a great Circle not perpendicular to the Plane of 
the Proje^n ) which will be always reprefented 
by an Arch^ fuch as B F D ; the points of interfedion 
of this Arch, with the primitive Circle (being conne* 
aed ) will always pa& through the Center of tli^ primi^ 
tiv€ Circle. 

O For 



< #o^ > 

For this Arch B F D is the Reprefentation of a Setni- 
circle Elcyated a b a ye thePJaaerf <he P wj oAi e« <)• «iidl 
iince t^vo £reat Circles cut one another at a Scmicir* 
cVsdTftancef their common Sedionwill beaDiameter, 
and here repfeii^ntecU)}^ the*Diameter B D. 

m 

C AS E i. 
Tofnd tbe ftk if tieiprjtid^ Clrilet A B>CD.< . 

Find its Center E, by the ift Prop, of the 3d Book 
of Euclidy which point E, is the Pole. 

C A S E II. 

%fni iBe %U of a l^ghi circU% fuppofe BD. 

RULE. 

Cro6 that Diameter wiA another Diameter A C 
at Right Angles^ the two Extremities A and Care the 
Poles of the right Circle BD. Or the Chord of 90 
dtg. fet dFfrom B both ways^ will determine tlip Fcnncs 
A and C« 

CASE m. 

lio fni Oe F4fk rfm^ OhUfUC Crcje. 

Note, that the Pole of an Oblique Circle is in that 
[toneter of the primitive CiroH vAiA paffcs 
through the Center of the ptcyo6toi ObHqtte 

Circle. 

JS Xji MP Z £• 

A B C D the •primitive 7 p. . ,^ C E 7 ,. . p„ 
B F D theObUquc ^ Circle and J ^, ^ thdrCen- 

'^rs given; X« 



( lej ) 

T»f»i tbe^ fdte of tbt Oili^eGrck ;& F p. 

r 

1. Through E and C, draw the Diametei:' A E C, 
cuttiBg the primitive Circle io A and<i> a^i^the Ob- 
liqueCirde in P. , 

2. A Ruler laid on D and F, will cut the primitive 
Circk in 4. 

3« Take ^O'deg. from a Schale of Chords^ and put it 
off* in the primitive Circle from 4 to ^. 

4. A Ruler laid on D, and this point by wiU cut the 
piameter A E C in G, the Pole required ; which poiat 
G is the projefted Pole th^t lye§ above the Plane af the 
Projeftion, but if the projeftcd under Pole haid hrei? re- 
quired, lay off the Chord of 90 deg. from 4, the contrary 
way to Cy a Ruler laid from p to c ,.will cut the Diameter 
A £ C produced in the point yy the proje6^ed under Pole. 

P R O B. II. 

Tp lof iffwfi m the TroicBim anf Angk requin4* 

DEFlJriTJOK. 

A Spherical Angle is made by the Interfe(ftion of 2 
great Circles on the Surface of the Sphere, and is Mea- 
lured by that Arch of a great Circle wliich is 90 deg. 
from this point of Interfciftioiv intercepted between the 
Legs or Arches that conftitute the Angle. It hath been 
proved in the 5th Propaiition of Stereographio Proje- 
4ion, that the Angles of the pliane of the Pi^jp^ion.are 
equal' to the Angk^they repre&nt 00 the Surrace^ of the 
Globe. 

Jn M Problem there ivt three Cafes. Either^ 

X. Where the angalai* Point is at the Center of the 
primitive Circle. 

O 2 ' ;2. Where 



(104) 

2* Where the Angular Point is in the Periphery of 
the prinntive Circle. * 

3* Where the Angular Point is in any point given 
ivithinthe Periphery, and not in tbe Center^ 

CASE L 

To make an JngJe r»bere the Angular Point if in the 
Center cftbe Primitive Grfkm 

RULE. 

Such Angle is imde in allrefpe^s like a plane An- 
gle, v/f. by drawing 2 Lines from the Center, fo that 
the Arch of the primitive Circle intercepted between 
them, be equal to tbe Degrees of the Angle pfopofed«' 

CASE II. 

To ntdke an Jngky that the Angular Point mojf be m 

a Point given in the Periphery. 

Plate XL For Example to make the Angle HAD, 

Pig* IL having its angular Point in the Periphery 

at A of 36^. oof* 

R UL& 

The Primitive Circle A B C D boingdefcribed* 

!• Lay a Rule on E, the primitive Circle's Center, 
and the given Point A, which will likewife cu| 
the primitive Circle in C* 

2. With the Secant of the given Angle 36 deg* an4 

one Foot of the Compa&s in thegiven pomf A^ 
defcribe the Arch X. 

3. With the fame opening of tl^e Compaflfes, and one 

Foot in C, croft thtf former Arch, X in X. X is 

|be Center of the ObUque Circle A HQ np- 

* ' ' ' king 



( 105 ) 

king an Angle with the primitive Circle of 36 
deg. as is evident by the CoroUm 2 Frcpofition of ^ 
Stereograpbic ProjeBion. 

CASE III. 

To make an A^le wb^re the Ofigular Point is in anf Point 
given within the Peripberfybut mt in the Center. 



^ 



ZX A MP L 



k 



A B C E the primitive Circle 

£ itsCenter ^ l^ven. 

G cheFoint 

Ta iepnribe an AngU at the Point G^ cf\^ deg. 

RULE. 

I. Through GandEy draw the Diameter AEGC. 

2« Crofs it at Right Angles with the Diameter B D« 

3» Lay a Ruler on D and G, and it will cut the pri- 
mitive Circle in ^. 

4* Take the Tangent of the Arch CA> and put itoff* 
in the Line A Q from E to A. 

5* Ere^ a Perpendicular A ^ continued. 

6*. Lay off the Tangent of the Compliment of the 
Angle requir*d from A, cither way to gy which 
Compliment is in thisCaie 45 isg. 

7, Upon the Pointy with the diftance,^ G, defzribe 
the Circle Qa\ the Angle A G <•> is the Angle re- 
quired* 

This foUo^vs from the %tb Propofition o£Stereog. Pro jcSm 

P R O B. IIL 

To draro a great Circle through any given Point j making 
pith the Primitive Circle any given Angle. 

^^ * RULE. 



w 



(Id6) 

RULE. 

1. With the Tangent of the given Angle, and one 
Foot in the Center delmhe an Arch. 

2. With hsc Secant of the (ame Angle, and one Foot 
is the ^ren Peiot, (bike ao Arch eroding the 
fbnaer; the Jntgrkitum of which Arcbe$ is the 
Center of the Circle required. ^ 

EXAMPLE. 

Pl^Oe XT. A B C D the primitive Cif cfe7 
Fi^» IL E the Center thereof > Given. 

Ithe Point 3 

Through the Point I, 'tis required to draw an Ob- 
lique Circle, (b that it itta}^ make an Angle at the primi- 
tive Circle of 36 deg, 

1. With tfaeTatigenc of ^ deg.Toions Foot in the 
Center;^ defcrihe the ArchX. 

2. With the Secant of this Angle, and one Foot in 
the given Pointy I make an Arch, cuttinjgthe Arch 
X inX, X is the Center of the Oblique Circle 
AH Q maldng with the primitive Qrcle the An* 
gle required. 

The leafim of which is evident bcm the ad Propofition 
ofStin^m PnfeSiiam 

P ROB. IV. 

3> draw a great GrcJep^^lks thrptighanf 2 Pomts given 
tntbhi tbt Ptrtfoety tjtic Prfwitivc Grclc^ 

RULE. 

J. Through any one of the given Points and the pri- 
mitive Circle's Center, draw a Diameter produced 
beyond the Periphery. 

?. Crofi this Kameter at Right Angles. 

3. Through 



9* Through the Point montion'dy draw a hvot to the 

Extremity of this fccond Diameter. 
4. At the end of this Line in the Peri|ihery of the 

primitive Circk, ereA a Perpffldioular cutting the 

firft Diameter in a third Point. 
5* Through the two P(ant8gi»eivand this third Point 

ftrike a Circle,' and it Ihall be the great Circle re- 

quir'd) by Prob. 8. oi GeomeU TrobU ^ 

EXAMPLE. 

A B C D the primitive Circle^ Plate XIL 

E its Center p- Giyem Jsjg; L 

FandGthetwoPdnts \ 

Jbrgugb 'SandGHkrefflrtdtoiraw a grtat Cirdc. 

I. Through EandFdrawtlieDiameisr APECT* 
12. Crofsit at Right Angles with thelKsuneter BD« 
3* Draw aline either fromP to£^ or frpmP tp D* 
4. To either ofwHch lines F B orFD, at the Ex- 
tremities BorD^ereA the Perpendiculars BI or 
1,interiefting the Diameter A C produced in T, 
the gd Point. 
5* Through thefe 3 Points P,G, and I^ de£briliedie 
Orcle KF G H^and'ti$ d»ne; 

We hece;liave nodux^to prove, bvt €hat the Line 
111^ that Gonne&s the 2 IntexieAions with the prim- 
tive Circle^ pafles through the Center ^ (or (ban 'tis 
the Repidentationof a great Circle. 

JDEMOKST:RjiTION: 

TE> EB:: EB,EI, the Triangles i?E Band BET, 
being funilar by the iib Prop, of the 6a of Et^liJ, 
and H E continued will fall upon K> the other interfe* 

aioni^ fiu^ ififspoffibktodoi)theMi&^Jet'^^^ 

«n 



( io8 ) 

onM, and produce HEM to Ly then by the 95th of 
the 3d of Euclid. £ F. EH : : EL. E I, and conie- 
queotlyEHxEL=zEBfr:EHxEK=EHx£M» 
and by Confeqiienoe E M=E L ; that is, the part equal 
to the whole \ which is abfiird. ^ S. I?. 

P R O B. V. 

T0 ir^m Agrc^ Circk pcrpenSculdr U a given great 

Circle. 

General Rule. 

• Draw a great Circle through the Pok of the given 
gitetCircfe, and it will be jperpendicular to a a great 
Circle given. In this Problem are 4 Cafes, either 

' I. Perpendicular to the primitive Circle. 
2« A R^ht Circle perpendicular to a Right Circle. 
9« An Oblique Circle perpendicular to a Right 

' 4« An Oblique Circle perpendicular to an Oblique 
iCirde. 

CASE I. 

To draw a Circle ferpendicular to the primitive 0rcle 
liven. 

RULE. 

• • ^ 

Through the Center of the primitive Circle draw a 
Diameter and 'tis done ; for the Center of the primitive 
Circle is its Pole. 

C A S E II. 

Xo draw a Bigbt Circle perfendkular to aj^gbtdrclc 
given. :' 

ROLE. 



4. -t . 



This is done by drawing aDiameter, pcyferi d ib tfl ar 
"0 the Diameter or Right Circle given. Vti- ' 






, 



1*5 «•* 



\^' 



>: 



-Hr^^ 



-■4 



^409) 

C A S E ni. 

T0dram an Oblique Circle perpendifutar t^ a Right Cir* 

cUjnven* 

RULE, 

t. Find the Poles of the g^iven Right Circle^ by Prob 

the l(^ Cafe the lA^oi Stereog. ?rob. 
2. Draw a Circle thw>ugh thofe two Poks. 

EXAMPLE. 

A B C D the primitive Circled TUte Xf. 

E its Center > Given. %• IL 

B ED aRight Circle 3 

'Tis required to draw an Oblique Circle pcrpendi-f 
cular to the Right Circle BE D given. 

I. Croft the Right Circle given BED, with the 
Diameter A C, or lay the Chord of 90 dsg. from 
B to AandC, which are the 2 Poles oftheright 
Circle BED. 

2« With any Diftance, and one Foot in A and C^ 
ftrike 2 Arches croffing one another in X* X fhall 
be the Center, and A H C> the Circle required. 

But here obferve, That if the Oblique Circle required 
to be drawn, were fo limited as to make a given 
Angle with the piimitive ADC* 

Then take the Secant of that Angle in your Com* 
paifes, and placing your Compalfes in A and C, make 2 
Arches croffing one another in X;:X fhall be the Cen* 
ter. Or take the Tangent of that Ai^le and fet it ofF 
in the Right Circle ( or Diameter ) B D, from the Cen- 
ter E to X, and 'tis done. 

But if the Point H be given, through which this 
Oblique Circle fliould pafs, without any relation to 

P the 



( "o ) 

the Angle it fliould make with the primitive Cifele, 
(though that be Virtually given) a Circle ftruck thro' 
the 3 Points A, H» andQ %v91 inrwer the Demand* 

C A S !i IV. 

Tcdraw an ObliqueCirck perpendicular to agivenOblifuc 
Circle^ 

RULE. 

1. Find the Pole of the given Oblique Circle^ by 

Prob. I. Cafe the 3d. o( SterCig* Problems* 

2. Through that Pote draw a great Circle, or which 

is the lame, draw fuch an Arch as tmy paischro' 
the Pole fo found, and may interlecfl the primi" 
ttve Circle in Points diametrically oppouCet 

fJateXa. tXAMPLH. 

Fig. 11. 

AHCGtheprimiiiv«airde> . ^ Kl tiioAt 
A B C an Ob^ucCindc ? ^^^^ ^ Y SCfenters. 
*Tis re<pus«d todiAw anOhlitlue Cifde pet{>endi- 
<5ular to the Oblique Circle ABC given. 

«. To liAd E the Pole of the Oblique Circle given 

hyFftib^ lit Cafe 3d. 
^. Lay a Rukrover the Center I, any-wife^ cutting 

itefMiiftttivcCiroleliiD and G) ft Circle ftntck 

<Chn£r^tlietliMePiiiifftiD)EatidG, fhall be 

the gieat Cirde reipttF^« 

fiat hBtt clt&r^ That if the 96int K fin the given 
OUiqueCirde ) ie given, Aendrtw a great Circle 
tkni* the t'Podits £ and K, 1^ the 4^ Problem. 
And if St be required tliat ^ Circle t»ffi% thro' 
the Pole*, ihaa alfb make a civwi Angle with the 
priinitive Circle, it may he done by the sLFrat. 

PROB. 



( "X ) 

PRO B. WL 

To l^^ ^ fuantltf tf Degrees m agreat Cmk^ , 
In this Probkm «fe three Calb. 

1 • On the priti^ ve Circle* 
7f On a Right Circle. 
3. On an Oblique Circle. 

CASE L 

T« % 9f my number ef Degrees m the frt^itlve 0rele* 

R IJI.& 

Take from your Line of Chords C to^ljat R#4itts) 
ehe number of Degrees you would U/ oft» #ind tritn^ 
fcr it to the primitive Circle, and 'tis *one« Th^s needs 
no Example* 

CASE a 

JR'/j^ ^€ty timber of Degrees m l^git (^ir^fff 

RULE 

Teethe Tangent <>f half the Angle yc^ would lav 
off, and But it off in the Right Cincle horn thie Center, 
^nd 'tis none. 

rig. 11. 
A B C D the primitive Circle, E its C^nt^r 7 ^. . 
A E Cthe Right Circle J 8*^^- 

On this Right Cifck A E C, 'tis required to lay off 
from the Center E 45 deg» 

Take the Tangent of its halfj vz^. 22 d, 30^, and lay 
it off fjrom E to F, thep Ihall E F reprcfent 45 dfg. 

P? pr 



( I" ) 

Or from the Scale of half Tangents take 45 dtg* anl 
It will reach from E to F« 

Or take the Chord of 4.^ ieg. and lay it off in the 
Semicircle ABC, from B to tf, a Ruler laid from D to 
« will cut the Right Circle in F> fo that £F ihall re- 
prefent the Arc a B> vi{ • 45 deg. 

C A S £ III. 

To l^g off anj number cf Degrees upon an Oblique GrcU. 

RULE. 

J. Find the Pole of the given Oblique Circle by 
Trob. 1. Cafe 3. 

2* Lay the given Number of Degrees on the primi- 
tive Circle, hyCafe the ift of this Problem. 

3. Lay a Ruler to the extremiti^ of the Arch laid 
down in the primitire Circle from the Pole^ and it 
^ will cut off a like Arch in the Oblique Circle. 

J?late XT. E }C A MT Z E. 

Fig. IT. 

A B C D tbe primitive Circle ? their Centers C E 
BFD the Oblique Circle 3 are \C ^ 

T'is required in the Oblique Circle B F D, to lay oif 

an Angle of 45 dag. from F. 

J. Through C and E the two Centers, draw tbe 
Diameter C E F A, and therein find G the Pole 
of the Oblique Circle given, by the ift Prob. 
Cafe 3. 
'2. From your Line ofChords, put 45 deg. off in the 
Primitive Circle, from A. to tf. 

3. Lay a Ruler to Gandtf, and.it fhall cut the Ob- 
lique Circte in «, fo tjaat F « reprefcnts 45 deg. 



* I 



Tl»5 



K-v' 



» *t 



The like may be done for any other Nuinljer ofDe- 
gree8> and from any Point, 

Or thus, Layoflr45 deg» in the oppofite Semicirde 
BC D,fromC to^> a Ruler laid from ^ the proje^d 
unisr PolCy of the Oblique Circle B F D to ^, will cut 
the Oblique Circle given B F D in the fame Point « • 
So that any Arch or Number of Degrees, may be laid off 
in any great Circk given, by the means of either Pole ; 
thebemonftration ofthisis already given in the Theo* 
lie part of Stereog. Proje^lcn* 

P R O B. VII. 

Tq Meafurt awf part of a great Circk prokBed. hi tbk 
Problem are three Cafesy and are but toe Converje of 
tie former* 

C A SET. 

To AUafure atij Part of the primitive Circle* 

RULE. 

Take the Arch required . to be meafuxed in youf 
CompajfeSy and apply it to your Scale of Chords^ and 
it ihews you the Degrees > this xieeds no Exampkv 

C AS E n. 

; To meafure any part of a Ptigbt Circle. 
RULE. 

J. Lay a Ruler from the Pole of the Right Circle, 
to the 2 Extremities of the Portion of the Right 
Circles, whole meafure you?d know, and it will 
cut the primitive Circle in two Points. 

?. Take the EHftance of thofe t^vo Points in your 
Compares, and apply it to your Line of Chords, 
and it will ihcw } ou its lyfcaliirc. EXAM- 



("4) 

Flate XT. EXAMPLE. 

Fig. II. 

I w^d kaow the Mcafare of the Portion ef the 
Right Circle F E, a Ruler laid from its Pole D t« E, 
cuts. the primitive Cinrle in B ; and laid from D to F, 
cuts the prifldtire Circleln a; Take « B in yotir Com- 
faSBssy and apply it to your Line of Chords, and it fhc\va 
Toci tfaemraberof Degrees^ th$t£ F reprelbnts, or its 
Meafure. 

The like for any other Portion great or finally whe- 
ther taken from die Center or not. 

CASE IK- 

T§ Meafure wf Portim of an Oblique Circle. . 

RULE. 



1. Find the Pole of the given Ohlique Circle. 

2. Lay a Ruler on the Pole to the two Extremities of 

die Portion, whDie Mtofiire is defii^. 
3* The Ruler ^ laid, will cut the Primitive in two 
Pmnts^ whole Diftance Meafiired on the Line of 
Chords» ^res what is deiired. 
I tamxAfc ft needier to give an Ea^mpl^ the Rub 
beit^ib plain. 

PR OB. VHL 

T0 meafure my Sfheric At^U. 

General Rule. 

A Spheric Ai^ ill the Projcftion, is meafured by 
jncafuriiig that AiA ef a gjcat Cincte, which is inter- 
oqtfed bttwoen die two Arches that c^tSAH the An- 
gle at a C^ia«l»8Cs Diftanee from die Ai^bir Point ; 
^hatis^the Angttfar Point being the Pok of the Arch, 
iKttchan Archhcddfcrihtd in the Projeaion', If not 
'^afuic die EHfiamce between the two Poles of the 

Circles 



( "5 ) 

Circles containing the Angle; and tliat is the meafare 
of the Angle required. « 

In this Problem are three C^fis. 

u When the Angular Pcto is in the Center of the 

primitive Circle. 
2. When the Angular Point is in the Periphery of 

the primitive Circle. 
2. When tbe Angular Point (s vffduik 4ie Wgrk&ft 

Circle, butnotintfae Center. 

C A S £ !• 

TemeafureamAtgleytohentbe Aigt^sr JNim ft mtbo 
Center. 

RULE* 

Such in Angle is meaCured by takiiig tfae Arch it (iib- 
tends in your Cooipaifts, atid applying it to >«ur Line 
of Chords. 

CASE IL 

To meafwre 4m Ai^h rokenAeAngtd^t P4mt km tb$ 
Peri^rjf iff the frmkive Circle. 

M JL A M PZ E. naeXU. 

Fig. mi 

ABCDthePrimithre CSpcJc? .^^„CE 7 their 
BP D an Oblique Circle 5 &^^^i X 5 Centers. 

'Tis required to Mea^ire the Afigle K B H. 

Through X and £, draw the Duunecer A C. 

In this Diaoieter find x the Pole <£ the Oblique 
Circle BP D, the Dillatiee E x or A P, meafored by 
aScecfe rf hftif Tangents is tlie Meafim of the Angle 
required. 

After the faieeBuinner P Eor C x^ is the }fe«^tie of 

theAnglePBE. 

CASE 



y^- 



(ii6) 



CASE nr. 



Tt meafiire an AngJe^ Vfben the Angular Point it wUlmi 
tbc frumtivc Orclc^ ^ut not in the Center* 

Z X AM? L E. 



ABCD thePrintitiVe^ ^ ^ 7 ^^"^ Cen-: 

DPGBan ObHqne >Cijrclc,aiid^X > ters gi- 

LOGF an Oblique 3 6^ 3 ^«n- 

Tis required to meafure the Angle F G B. 

1. Through £ and Y> and alio through £ and X, draW 

% Diameters^ and in them findjr and x, the Poles 
of the two Circles conflituting the Angle, by 
Trob. I. Cafe lU. 

2. Lay a Ruler from G, the Angular Point to x and 

y^ which ihall cut the primitive Circle in n and 
«v, the DUlanoe n m taken in the Chords, is the 
meafuie of the Angle required. 

The iame Method is to be taken, if ie were requi^^ 
xed to Meafure the Angle B H K, conftituted by'a Right 
Circle and an Oblique^ vi\m A Uuler laid from H, to F 
and X, the length of the Arch in the Primitive ' Circle, 
intercepted between the Pdnts cut by the Ruler, ihall 
be the Meafiue of the Angle. 

PR O B. IX. 

To dram a ParaEel Circle. 

DEFINITION. 

Parallel Circles on the Glob^ arefucfa which divide 
the Surface of the Globe into t^vo unequal- Parts, and 
are always Parallel to {onfe great Circle. Parallels on 
th; Projeaion arc the Reprelcntation of thefe. 

In 



/" 



("7) 



In. tbU frob. are three Cafes $ dther^ 

1. Parallel totte Prinfittive Circfei 

2. Parallel to a Right Circle. 

3« Parallel to an ObHque Circles 

C A S E L 

To draw a Circle parallel to the frimitiveClrcle. 

RULE. 

: This is done by taking the Tangent of half the Pa- 
rallel's Diftanoe from the Pole, and therewith ( one 
Foot being placed in the Center ) defcribe a Circle and 
*tis done* 

EXAMPLE. PlateKlL 

•J%.IV. 
A B C D» the primitive Circle^ E its Center. 
'Tis requir^todrawa Circle parallel to the primi- 
tive^ and 30 deg. diftance from it, that is, 60 deg. from 
the Pole, 

Take E F equal to the Tangent of 30 deg. and with 
that diftance^ one foot being placed in E, de&ribe the 
Circle F G H M, and 'tis done. 

C A S E II. 

'" - 
To draw a Circle parallel to a Eight Circle. 

RULE. 

1. From the Chords, lay off the Parallel's Diftance 

from the Pole of the Right Circle given, both 
ways, marking the Points. 

2. Take the Tangent of that Parallel's Diftnace from 
•the Pole, in your Compaffes, and put one Foot 
in the two Marks betore, Noted fucceffively, 

Q; \ ftrikc 



( "8 ) 

ftrike 2 Arches cutting bne another; the Point of 
Ii)tBrfBftio%iull te tfaeOeninr mfOat ParalkL 

Ortbmy Croft the KightCigdegj»!Ptt i ndi> Diame* 
ter atRi^ht Angles, vnadi let he pfodafioA; take the 
Secant ofthe Paralkfe IMEbmoe JR«n tin fblst aad fet 
it off in this Diameter, from the Center on that fide 
the Circle ihould bedrasrn, and it (hall give the Cen- 
ter required. 

rtg. IV- 

ABCD the primitiveCiick^ Bits Center, 

BED a Ri^ht Circle. 

nrisreqiured to driwaCiidepai^fel toiotifaelUg 
CircileBE P> at 40 deg. diftanoe from it, or 50 asg. 
feomits Pole C* 

1. 1^7 off* 50 deg. frfiinC, to K. and' L» 
21. T^e the Tangent of 50 deg. and placing the foot 
of die Cei]q»&s>oBen'iltDthatdi&ance9 in Land 
IL, sad ftrike 2 AxcbescHttii^ one ansdKria Y. 

Or the Secant of 50 dsg. ( the Parallels diftanoe fsom 
the Pole ) let off from £ in de Diameter AC } will 
stach to Y, theCenter re^r'd. 

CASE iir. 

To dravo a CtrcU faralkl to an oblique Circle^ 

RULE. 

From the Scale of half Tangents, lay off the Pa- 
laM Diftanoe from the Pole of the <%lBque Circle 
^ven, hoth-wa)^) in diat Diameter of 4A)e Primitive 
Circle, which paifesthflBH^h the Pole of the given Ob- 
lique Circle^ then Note thoie Marks \ the Diftance ^fdiefe 

Points 



( 119 > 

Points liiVs&ci, gives the Centei^ of the Parallel, 

^latfXlU EXAMPLE. 

ABCDthePriimtlve2^^. 5EZ^u«^.4- ^ 
DFBanObUque | Qrcfe, J ^ J ^^^^^ Centers. 

IW refuiredto dram a Parallel to tie Obliaue Circle^ 
DFB, 40 deg^diflaMfrom ity or $o de^.dijiant from 
its Pole. 

I 

u FindG, the Pofe of the given Oblique Circle 
D F B, by ProB. I. Cafe III. 

2» Meafure £ Gr by a Scale of half Tangents ( or by 
a Scale of Tangents, doabling the Angle you find 
it) and fuppofeit ta be 94 i?^. then take the 34 
id'^,+50^^.=84, and lay it down by a Scale 
of half Tai^ents, firom E to n^ and take "yOdegm 
--34^6g,=:i6; lay it off in the Lin& AC, from 
E to fft,m»h the Diameter of thisParalH bikeft 
jw n in (j^ (? is the Center required. Or the Tan^ 
gent of half 8^/:f, 42 fct off from E, will reach 
to», and the Tangent of 8 deg. put off from E, 
will reach tam. Then with the Ccat&to^ and 
Radios d/«y deieribethe ParatleL 

PROB. X. 

To meafure anjt Aeeb of a Parallel proj&Sid. 

In tbit Problem there are three Cafes. 

u When 'tis Parallel to the Primitive Circle/ 
2. When 'tis Parallel to a Right Circle, 
3» Yrhen *tis Parallel toan ObEquc Circle. 

0,2 CASE 



( ISO > 

CASE I. 

T« metfttrt toy ^cb tfa Pandkl to tbefrlmithe Circk* 

RULE. 

A Ruler laid from the Pole ( or Center of the pri- 
mitive Circle ) to the equal Divifions in the Periphery 
of the primitive Circle^ fhall cut the corrd^pondent De* 
greesinthe Parallel 

C A S E II* 

To meafure amy Arch of a ParaOel to a "Bi^t 0rcU* 

Suppoie I were to Meafure or Divi^ the ParaM 
. I L, which is 50 ^ 
the Right Circle D £ 

RULE. 



K I L, which is 50 dxg* diftant from C, the Pole of 



1. Dcfcribe a Crn7e parallel to the primHwe CheJey 
as far diilant from the PoU in which the Eye is^ 
as the parallel K I L is from Q the Pole of the 
Rtgitt Circle D £B> 50 ^g. and oonfequently di- 
ftant from the oppdite Pole, in which the Eye is 
juppofed 1 30 deg. Take the Semi-tangent of 1 3a 
/fe^. C or which is the feme) the Tangent of 65 deg. 
at^ with your Compaffes open'd to that Diftanoe, 
and one Foot in the Center E, deicribe a Circle* 

2. A Ruler laid from Q to the equal Divifions of 
that half of this Circle^ which is parallel to the 
Semicircle BAD, ihall cut the Parallel K IL> in- 

. to its correfpondent Degrees. 

Or 



(^ 121 > 

Or a Ruler laid from A9 the other Pole of the right 
Circle DEB, to the equal Degrees of that Semicircle 
which is parallel to B C D, fluU likewife divide, (^ 
meafure the parallel KI (a 

CASE m. 

To Mcafureany Arch^ cr Parallel to an Oblifue Circkm 

EXAMPLE. PlateXlU 

f/f.V. 

Suppofe I were to meafure, or divide the Parallel 
mKtty parallel to the Ohlique Circle D F 6» and di- 
ftant from its Pole G, 50 deg. A Parallel to the Ho^ 
rizon mud he defcribed, as much diftant from the Point 
or Pole in vMch the Eye is, vi^. 50 ^g. and confe- 
quently diftant from the oppofite Pole, in which the 
Eye is 1 30 deg. taketherefore the Tangent of 65 degw-^r 
Semi-tangent of 130 degt and defcribc fuch Circles as 
in the 2d Cafe. A Ruler laid to G, and the equal Di- 
vifions of this Circle ( parallel to the Primitive ) fo de- 
fcribed, fhall divide the Parallel ntKn, into corref- 
pondent Degrees or Parts. 

Other Methods for doing the (ame thing, you'll find 
in what I have laid in the Theoretic part of i't^rw^ro- 
pbic Pro jejuni to which I refer you. 

XTote^ I have omitted the drawing of thefe Paral 
lels, mentioned in the 2d and 3d Cafe, fince they 
would run beyond the Limits of the refpedive 
Fijgures, and becaufe it is fo very cafy to con- 
ceivethem as drawn. 



THE 



( t22 ) 



*^H*i 



THE 



Stereograf^c Projedion 



O F T HE 



Several Cafes in Ri^An^i 
SpHEiac Triangles. 



^mmim 



FROB. L 

The two Legs given^ tofindtbi two Obli^ Af^ks and 

EXAMPLE. 

Given B A=:|OA J Required S ^"8^ {p 
PA=25 5 ^ >HypothcnufeBP. 
C A S E I. 

Suppofe the hsg A B belaid off in the Primitive Cir- 
4Jte, then thro* A dniw a great Circle at Right Angles to 
AB, which will be a Rjight Circle or Diameter j then 
with the Semi-tangent of 65 itg. the compli- 
jnent of P A ( or which is the (ame thine, TUte XIIK 
with the Tangent ofits halt vi^. 32 d. 30' ) Tig. L 
one Foot of the Compares being placed in Cafe L 
^he center, def^ribe a Circle pardlel to the 
primitive> cutting the perpendicular A F in P. Laft- 
ly thro* B and P draw a great Circle, and 'tis done. 

To 



( "3 ) 

Tv Nki^itn tbe tbmff rifmrtJL 

The f SSit^l'^l^-^^^ ^^ P'^'^f S 

C A S E IL 

Sttjffffi tbe Zeg BA, ImdJmmmaJXamcUry crJSgbt: 
Circk^ tbe F»nt ^bmgttt the CSemter. 

The ^primitive Ciide beiiudiairx^ and Ukrwile the 
right Circl^ in whichviott Zsf^g^to ia^ 
downBA; from theCenter B> take the PlateXllL 
Tangent of 15 deg* at Sopd-^ai^ent of F/g. L 
joaeg. and lay it in tbe Diameter from CafelL 
Bid A; then draw a g«sat'CuPc]^ F^fiV 
thro' the Foks «^ tbe AjaeB A,aadllK foiBt A, indiich 
will therefore be perpendicular to B A : In this Cir- 
cle lay oiF A P=2$ dig. l^ mtaosofitsPole. Laftly^^ 
draw a Diameter thro' P iad B, and *tis done. 

• 

To Ma^itn ^ Ahtgs reytureJL 

The i?s^rSiV}^««^»>y^'^ {I 

C AS E nL 

Supfofe tbe Leg A B, Uiddawh in a JDtametery or Migbf 
Cirffle^rbepamtBhekgmtm^ Center^ but M mgr 
point alftgned in tbat Diameter* • 

1. By thedhtflp 4>fa Scrienffcalf Tan- i^Mr XOL 
gent6y4a^.cf B J^ I0 ffftrefbot 30 f/^* !• 
deg. CdfelU. 

}• Thendm' tltf .Boltt rof the Right Oxck gWen, 
and tbe oaint iiiadeSaahe a gneat Circle, which, 
is therefore perpmflinAir to B A, and therein 
meafure AP=:2< de^ by means of its P^/e. 

' ^ ' 3-Laftly 



■ 
r 



( "4) 

g. Laftlf) a great Circle (buck through the Poines 
B and P» anfwers the Probkfti, 

The Hypotbenufe and Angles PandB^are meafured 
by thejtfa and i^Prob. 

Note^ That the Leg A B, may be laid down in any 
Obliqoe Circ1e> either from its Interferon vnAk 
the prittutive Circle> or from any point given 
therein^ which I forbear to infift upon> not doubt- 
ing but the Reader, knowing what was done be- 
fore will eaiily apprehend. 

P R O B. XL y 

The Hypotbenufcy and one of the acute Atgks givcn^ to 
fndtbe other Oblique jAtigle^ and two Legs. 

EXAMPLE. 



iiyen,yB=6oi?^41„, 



AnEle^t =<o C ^ Angle B. 



There are fcveral Varieties, I Ihall only Inftance ip Two. 

CASE I. 

Where the A^k Fghen^ is placed in the Periphery of the 
primitive Circk. 

Plate Xin. 1. From the print P, dcfcribe a Paral- 
Fig. II. lei to a Right Circle at 60 Degrees di- 

Cafi L fiance. 

2. Draw a great Circle through P, making an Angle 
of 50 dsg. with the primitive Circle, and cutting 
the parallel in B# 

3. Through 



\ 



h 



1 1 



^ 



C ^2 j ) 

5« Through the Ccntaer of the primitive and point 
B, draw a Diameter, cutting the pritnitive Cir« 
ck in A, and 'tis done. 

• ' . 

C A S E II' 

I 

• % 

Where the Point P it given at fUeafure. 

I* Through P, and the Center of the TlateJilU* 
primitive Circle> draw a Diameter, Fig» II. 
or Right Circk. Cafo IL 

• a. By Prob. 2d of Stereog. Vroh. ftrike a great Cir- 

cle pailing through the point P, and making an 

Angle with the Diameter drawn through P, of 

^od£g. \ „ 

g. By the help of the Pole of this Circle, meafure 

PB=6o. 
4. Through xht Voles of the Diameter paffing thro* 

P and the Point B, ftrike a great Circk cutting 

the Diameter in A, and *tis done. 
The parts required, may be meafured hy theheljf 
of the 7 th andith Vroh. of Stereog, VroL 

PRO B. III. 

A Leg roHb its adjacent AtigU given to find the refi* 

EXAMPLE. 

Given PA. 50^.7^ , Sr?' f'^'m"^ 

^ ^ Z Angle B. Cafe J. 

I fi)aU inflanie only hi two Cafey. 

1. Where the Leg P A, is projeaed in the primitive 
Circle do thus; 

I. FromP, lay off PA=:5© d:!g. in the primitive 
Circle. 

• 2. ThrottglPP^ draw a great Circle^ making an An- 

gle of 36 ^e^. ' 

y Thibugh-A, draw'' a Diameter or Ri^ht Circle, 
^Uid^'tis done, 

k 2. Wh^^er 



Pto<r XIV. 1. Draw ai Kameter, and from the 

Ftf . I. Center! lay iJffthft Twffx^ of 25 kg. or 

Sanitangcnt of 50> from the center P to A« 

«• Dravr^inother Diaoi^eiy ma*iii& w the given 

^. Thrb' the Pbfef df the giVentt^ftiter, ifi JyWdv 
P AB pi«feakiiaidPiftAjttta*lgr6^^ 
«ttii<the 2d Dianieter irf B, theti ii P.B A tjK 

Tfiangfe tcqttirtid, tv&fe partsi ^^1* luteafuftd 
l>y Ph>fc 7 and 8, rfSiPeHrflf;Ph*. . 

P R O. B. IV* . 

34e Hyf«tbemtp!,mdme iegffveth ttJJwItfte «ff. 

J. Wteii»Btsfnthe P*ri^A<frf drtI«^r/«»iHite*w*» 
2. Wherc'tis in the Center. , 

C A 5 E r. 

1. Defcribe a Parallel to a Rigftt Grfclc, whofe Vole, 
B> is diitant from B^ it^i«g* 

2. Defcribe a Parallel to the pimitire Qrcite, diftant 
from the Centei-, by the Complttoent of P A, ^.. 
53 idrg. which may cut the former Parallel in P, 

3r Through the Gcnter of the primitive Qr^^, and 

the point P, draw a Diameter euttii|g the priwi- 

tive Circle in A. 

4. Laftly, Thrcugh B and P, defcribe a fircat Cirde, 

^ nd 'tis done, jGASE 



C J«7 ) 

» • » * s 

C AS E fl. 

%. Draw a Diameter. P/a^eXIV^ 

2« Pn the Cepter B^ with the Seml-taj)- f^. IL 

-gfnt of 46 i&f. Tfefcnfea Circle pa- Cafe II. 

rallei to the primitive Circle. 

3. Defcribe a Pord/el. to the fisft Diameter, diftaht 
from its Pole by the diftance of the Complemeot 
oPP A, vAfich jyiU imarfca *e P^r^ife/ to 4je 
primitive Circle P. 

4. Through ibs Center of the primitive Circfe and 
point P, draw a Diameter. 

5. Tfaroogfa the P^^olr the firftpiameter^ an4 point 
P, draw a great Circ^y and you ihall fa^ve thp 
Triangle B PA, 

Whofe unknown parte may be meaiurecji by Pr^ltp 7, 8, 
4)fSterfig^ Prok - - ' ~ • . 

P R O B. V. 

\A Leg with Its opf^fite Jingle glven^ to find the re^. 

EXAMPLE. 



Plate XIV. Given Angle 8-24^ 40^ Z S'S B P 
f^j.III. PA=;i6. i7Ci>An.R 



1. Draw an Oblique Circle, making an Angle with 
the primitive, at the point B, of 24^! 40 ^ by Prob* 
2. 01 Sterecgn Prob. 

2. With the Semi-tangent of 73^ 40^. the Comple- 
ment of P A, one Foot being in the Center, draw 

^Circle parallel to the primitive, cutting the Ob- 
lique Circle in P. 

3. Through thfe Center of the primitive Circle, and 
point P, draw a Diameter, or Right Circle, cutting 
fhe primitive Circle in A, and 'tiidons. 



(128) 

The parts required, are meafured by the 7th and Sth 
Trob. of Stereog^ Prob. 

P R O B. VI. 

Both Oblique Jfiglesjgtve$h ^0 fi^ ^*^ ^^J^* 
S X A MV L E. 

mteUm Gi7enB=46*^. ^Reqd. 5?^ 

1. By Vrijb^ 2. ttiabe the Angle B P A=: 59 Atg* bjr 
drawing the Oblique. Circle P A. 

2. Find the ?oUf^ of the Oblique Circle P A, by Vrob. 
U Cafc^^o^Sterecg. frob* 

3» Through/^, the PoUy draw a great Circle, fo that 
it may make with the primitive, an Angle of 46 
deg. which will cut the ObUquc Circle PA in A, 
and 'tis done.* 

The three things required, w^. A B, B P, P A> are 
jueafured by the 7th Prob. oiStereog* Prob. 



■PiPHM^MHMMH^ 



THE 



( 129) 



THE 



PROJECTION 



O F 



Ohlique Spherical T fiances. 



^pi 



p R o B. vn. 

Ttuo S^ieSy and an Angle oppojhe to me of them 

given^ tofr<>ject the Triangk, and find the 

reft. 

E XAM P L £. 

Given BP=r34i $0^9 CP> 

DP=25 30 >Reqd. <D, ^lateXXV. 

B=:30 73 ^BD. F/^.V, 

A2rf^, That this Cafe is Axnbigaous> tbere bdiig two 
Anfwers, but by the Projedion, you may give both. 

1. Make the Angle P B D=z 30^ 7', by Prob. 2. Cafe 
2. by drawing the Oblique Circle BD. 

2. On the priinitive Circle, lav ofiFB P, 34* deg. 50. 
3^ Draw a parallel Circle diftant from P, 25 A 30jif. 

cutting tne Oblique Circle in two poinCs» D. IX 
4. Through P, and each of the points D* D. <lraw 
great Circles^ and you have two Triangks B P D» 
B P D, in one of which the Angle D is acute^ intfae 
^thcrObtufc^O'c. The 



CUO> 

The pftsti required, may bemeafured by Prob. 7 ta 

p.^OB^ vnr. 

Tw9 Angles^ 4»d A Side opfofite to one of tbem given, to 

EXAMPLE. 

Given BP=:6e)^ 10' 7 C^D, i 

TUU XV. B =40 00 >Rcqd. ^ P D, * 

fig^h ^D=;50 00 \ /p. 

I. By ^rrf/2. of Stereo^. Trob. make Ihe Angle P B D 
rr to 4odeg. by drawing the Oblique Circle B Pv 

S. Draw a parallel to a Right Chrtl^ dlftant from B^ 
60 d^. PM^Qg tb&01)^ue Cix^lein P,by Prob, 
9. otS^teog.Prob. 

J- Through P, draw a great Circle, fo as to make 

-akAiiil9Pf io4^ ^jgl^F; ai4 '(ti^ da!9^5 

^yi0J^^ ^'^^F^ .*^ «neaft«ed by Prci^^ and 
o, oF ihrtf ^^. Proh • - ' 

P R O B. IX- 

l[9i^^!Sgiv>itbdn^l$^Uincly4ddi to prop9 the Tria^- 
x^ 4ttdfn4tb&reft.' 

« 

C3ivcpBp5^6Di 7 CP^> 

//i^irtf XV. B=4o > Reqd. J P, 

Jl » JM^W JM» 01>lique Cij^f BD> iKt^u« f» Angle 
with !tlii|irwniiaye Cjarcie:;4oir^, by Pr<^. ^. of 

2> Make 



i. >fa>e BP =:0O4fef,. and by help of tU Tcl^ 
' df tte Oblique Glrcte B S, «i«*^*D±»7 ^*. : 
' J» fhftxigb P and D> di«^ a gmi dli% ^and *tis 

4* Tfae {Alts reqmrM) tie6 ftfci^M %y fr4^ 7 and 



• r 




[ TSriof^n mid }nd tbt r^^ 



(Sven B 46^ 

BP6op>>^ lb. Fig.llL 

%k 'mU BPr:6bi%i i6^ fi^ i9#l^'d, O^ |. of 
^ereog. Prok ... 

2. Make the Angle BPD5 1 X.Sie^.o^, by drawing 
the Oblique CfrcW F Df aid at the pi^nt ^ 
make the Angle PBD== 4P 4^. hy^ ffvk '%^ 
#fai^ twt>Obl^ Cift^fesWili interfe^ iiTS% 
and vis ddnlb* 

TBe parts re<{ttirtd are in^iiii^ hfPrti.ftiAi.(£ 

P R O B. XI. 

Jtree Side's Iking ghefh, h frojeS fBe tyiat^U^ t^^i^^ 
theAngkSn 

E XAM P L S. 

Given B Vzi6od icf 'I Reqd. C ft 

DP=4<5. i%> the ^P, Phae Xf^ 
B Dsg87 ^ ^o iAniHes^lO. /«. IV, 

0» 



( 1^2 ) 

!• On die prinudre Circle^ make BD^zSyi. 2cf 
the greater given Sid^ by help of a Line of Chords^ 

2. Diiw amrallel Circle at 60^ lo' dKbanoe from 
B» anaKkewife in another at 4^^ 12' diftance 
fxomD> cutting each other in the point P. 

3* Tfarodgh Band P, and likewife through D and P, 
drak a gnat Circle and 'tis done. 

VcU^ That in tbe Triangle H I K, made by the Cir* 
cle*s paffing through the Poles of the fides B D> 
BP, and DP ; H i is the meafure of the Angle 
B; HR the meafure of the Angle D> K I the 
Sttpplensentoftbe meafure of the Angle P^ which 
nay be meafiued by Frob. j.ofStcreog. Prob. 

P R O B. XII. 

nree Angles behig ghcny to make a Triiutgle, and sm* 
fsjucntlj to meafure tbe Sides. 

» 

R U L E. 

Plate XV. 

Fig.l\. Wiriithe three Angles given, as if they 

were fide^ projeA the Triangle B DP, 
Tbea meafiire the fides of the Triangle H I K, made 
by three Arcs paffing through the Poles of tbe fides 
DB. BP and DP, and you have the Angles of the 
Triangle required ; tsc^c that the greateft fide is the 
Supplement of the greatcft Angle in the Triangle 
B D P, and coniequently of tbe greateft fide in tbe. 
Triapgle re<][uired» 



F I N I S, 



' 




-! 



1 



% 



v.' 



I. 



