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# Title: Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences (shelfmarks Leibn. Marg. 71; ZEN Leibn. Marg. 71)
# Folio - , page id 806933011:0864
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eAt L
The difference of the versed Sines of tmo Arts m
olied by hals the Radius, is equal to the Sine of ha
the dum of tbose artis multiplied by the vine. 
hals che aisselence os ehe Arks.
Suppose, A F, AL Ly CHC EVO ArKS; the ditte.
rence OF the versed Sines is A C - A IL = C F
BF, and che Sine oF halt the Sum oF the ArK
iS A D, FOF F P = F A, the Sine oF halF the dit.
Terence OF the AFkS is FO. Nou che Triangles
NCD, F F B, are Similar, (For che Frianoles a CC
P oaare Similar, cherefore the Angle C equalrothe
AnolC F) TheretOre, aS A C: AD:: F P: F B. OF
AC: AD:: Z F F (1. c. F 0): F B. cheretore
AC XFB = F O X AD. FOT SINCC A C. A
But AF: AC:: bL: B M.
AndmF: ac :: B m: y a.
Therelore mulcipirins he Correlpondent terms oE
boch Proportions,
A EXmF: ACKaC:: (BF. X Bm: B mX7a
25 bL : y a.
C AS F XI.
: F E, PB. cherctore AC X PB = AD X F F.The Three Sides os any Spherick Friansle beins siven,
and consequencty : A C X FB = = AD X F P: 1o find an Anele.
ahar is = ADX L F F. = ADXOF. P. L. D.
AXIOM NV.
Ihe Rectanete os che Pines os che LeeS, is 10 che
Square osthe Radius:: as the Sine of ? Base more
diff. of the Lees multiplied by the sine of= Base
The Rectangle or Product oE the Sines oEche less = diff. os be Less is 1o ihe Square os ibe Sine os
Less, is to the Square oF the Radius: : a5 the hals the ansle required.
difserence of the versed Sines oF the Bale, and ol
che disterence ot ihe Leso, 10 ihie versed Sine of che
verricat Ansie.
Demonstration.
Resume che Foresaid Circles oF Pastboard, ane
suppose chere, B the Arole required, B A, B P.
sequal B M) its LCGS, P A che Basero be any way
Oblique rothe Plane B A, and not Perpendicular a5
besore. Then vill B P a be un oblique Pri
anste.
Lec Fall P B perpendicular to B m, and Xy per
pendicular t0 C a, and P b rob A, and in L. perpen-
dicular 1o0 A; cheretore B b is perpendicular 10
e A, and bL equat bA = L A equal E0 the versec
Sine of the Bale, lels the rersed Sine of the Disse.
rence. OF che L. C85.
Demonstration.
BL X2L : y R X E R. (1, e. bY cheforesoe
Temman and che tirst in Mr. CasivellS Lrisono.
etEY:)
AF X m F: R g :: S, = BalC X = dif. CY X S,
= Base = 3 dif. or 10Sg = Angle.
Example in Numbers.
Suppose B Pequal 60 d. 10, D Pequal 46d. 42
BD equa
equired.
87 d. 30 vere given, and clic Anele P
2 che Bale is
 the DiH. OF che SIdeS 15
EtheBase-) Z Dif. OF the Sides is
the Base = = DiH. OF the Sidesis
Rad. 59.
20, 0000000
S, = bale :t = disE. Cr. X S, = bale 2 19. 6669325
Sum
E diff.
Subrract IhCLOL IAC SICOI CHC L. C8513, 6062354
Remainder = Sg. OF the Sine OF = = P 19, 866679F
The = OF Wwtich is che Sine oF 59d. 03 9. 9333395
That iS = the Angle required.
Therelore beins doubledis 118 4. 061 =7.
A FX m F: R9; :: (BY A~. 4.) bL: y
43.
50, 29
37. OF
39, 6669325