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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 806933011:0851
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III.
Gent oF the Anele ca d (1. c. the FanSCMC OI HIdIF
the Sum ot opposire Angles;) and in che licele Tri
L L I
arxesad. 8 d Will be Tangenc ot che Ansle z a
ins proporrional, eb: ea : : c d : d 5: PHaE 15, 11
NVords, TTalS the Sum of ibe Lecs, is toss Ber
Difference:: as the Tangent of hals theSum of the
opposite angle, is to the Pangent of hals their Diffe-
rence; bur Wholes are as cheir Hlalres: Where.
Fore the Sum oF the Legs is to their Diference:
25 the Pansent oF hals the Sum oF che opposi
(1. C. The FAnOCHE OF haIF the DifferenCC OF OpPO-
sice Angles.) Bur the Segments oF rtre Legs ol
any Friangle cur by Lines Parallelco che Ba(C, BC-
As che Sum os che L add  le siven, is 1
their Difference:: Sois the Tansent os hals il.
hals their Difference.
beins Whacche siren nsle Was  100 olo
and their Lifterence is non found; add TheretOr
their halF Sum, and half Difterence rogether, ai
ic sires rou che greater os the tuo Aneles souht
and half che Difference subrracted trom ihe hal
Sum, leares the lesser Angle Cought. And chus
havins Found che Angles; il the Side oppolice to
ehe former siren Anele be Poughe, ic will be tound
casily by Pardies axiom, Thar che Sides areas the
Sines of the Angles.
TheDemonstrarion oF rhe Third Axiso, ls VILCL
Ix chus.
their ViHerence: aS FC LaHSCHE OL HAIL MMC SUII
ot their oppolice Aneles, isro che Tansent oF halt
their Difference.
required.
 10 20
solred.
Sum os the ocer o Aneles, to the Tansent
Non che Sum ol c L. ex tvo Angles isknonn
Arisles, is to the TalSe et Ma CCM CLCMCL
Wrhence che ruo followins Cases ut caus l
llay che Sum of the Legs ot any Angle a iS re
Giren FI, O and a. Required c, b3
FOr HI - 1 - O, IT -O:: 25 F, halEZ Opposite
Angles, is to T, half X opposice Angles, and ille
alEZ. ) halF X = b: and haCL.
Demonstration.
Produce O, one ot the siren L egs oF che Angle
siven, eill as become equal iO FY Or C a and che
Drav a d bohich wiil De Lespeet s (2.
16.) and draw d e, wrhich Wvill be Parallel 10cb:
Siven FF O, and a. Required Bd
First sind che Angles by che Former Case, anc
bisect bf in e, join c f, and bisect ic alo in d: chen 5, b: HI: : S, a: B: OFS, 6: O:: S, A: B:
ed Ede Debbired addtorito
(6. 92.) Then wwill che Anele c a d == d a f: L. C. tC
ehe halt oFc a f, Which external Angle c a f-
I b: Tharis, eo che Sum oC rhe oppolice Ansles
A X IO M 1I
The Base is to che sum os che L eps:: as ehe Dise
rence ef the Lers is rothe Difference of the Serment
os the Base made by a Perpendicuat let sall fron
the Ansle opposite ro the Base.
Forthere is allo anorter Case, in Plain Oblique-
Triangles, wtich requires 2 parriculat LLX IMM l
solre ic; and thac is, Where aII chree Sides are BI
ven ro tind che Angles. Flere lec Falt d Perpena
cula trom any AnslerOis  ide ds d P:
Oram then 5 a Parallel ro cb; (0 will the Angle
5ac, bc equal roche alcernare one c. Andictrom
halE esle Sum oEche opposie Angles, ron calre che
sesser Angle: 1. 0. IE trom c a d, You cate 8c
chere will remain the Ansle s a d, equal ro hal
theDitterence OL u Opposte AIIBICS.
and then (ay, A5 the Side d ois 10 d a- ac, th
Sum ot ine oetet l silles.. Soisthe Ditference
FTaIF oF wrtich addca W HaIF a c,
halF d c, ic will leare the Mher Sesment pc. AnC
ate cay had thus: d a : Radius :: d. P. C0.
And (0 alto, il trom b e, halt che Sum oF thie sinc orthe Ansle C.
Legs, rou rake, O the lesser Leg, there will te
main a c equal ro halt che Ditterence ot the Le9s.
And ihen since the Triansle c a d is Righr-anglec
it a d be made Radius, cd Will be che Tan.
ofthole ruo sidesu ao a tourch Number
fires rou cho
Scoment ot the Bale ut P, and l Lubrracted Fron
Wotien etose Segmienes ale cluo sou, che Anoles
sine oF rhe Ansled: And a c: Radi:: pC: C0.
ahis.
The Demonstratibu bt Woie la r
Demonstracion.