1000000030

(;90

G E O M

lulO of

1:\C

f,¡.\.,rCI 0:1 113 anJ

is Cqll.J

lhe . Ifo (hy lhc f.nte) DE \\ill

egual 10

Br ;

hUI

TJr.

f'lu,,< on HC.

airo (byan 67. eor. 4') c911al

FC ; Ihlfl'fOlc

C.r.

Jicllce in

, i ~hl."n:kd

Iriangle, Ihe Ilypolhe. •nu

I~C

tobol!., r, or BC, lVi ll be uoubl<of Vr:.

n~fe

"nJ on<

01'

Ihe

lc~s bcin~

!;;ven, lee m"y ealily find

Afm II.e 1.1111<mdnncr ;1Ill'y ue prol'cJ, Ihal if in

Ihe olha, by Idl lr.& ,!,e f'l

u.re

of Ihe g"'ln leg J"rol1l Ihe Ihe

tl ian~l<

AKG, (Iig. 2.) AE UC lak,n equal 10 a

fquHe

Ihe "ypolhenok and Ihe fqu"e rool of Ihe Ihi,d par l of AK, anu lhrough E be d"IVn

~.D,

p.trdIld

(C:n ,inJer II'''! oc Ihe

I~g

rcquireJ.

KG, "nd meeling AG in D; Ihen ED II'dl be "Iual 10

C.r.~. Hen~<,

Ihe

Ic~s

ia a

,,~hl.angled

triangle b!· • IlurJ p..rl of GK, and AD equal

a Ihird

r.trt

ipg gi,en, we mJy fi nJ lile

"JPolhenu ~c,

by uking

Ih~

AG .

lum of

Ih~

L¡

!l.lr<

of Ihe gi"lO

Ic~s,

anJ

extralqin~

Ihe

Likewifc if in any lriansle ABC, (fiS.

3.)

upon Ihe

h]JJrerool of Ihal fumo

fiJe All. be laken,\E, equal 10 one fourth. one lilth, one

70. Huponlhc line All (Iig.

53')

Iherebe drawn afcmi· lixlh,

ce.

of AH, an.l Ihrough E be drawn

I:.V

p,,.lId

eirdeAD!l, IVhnrecelltrclsC••tlIllon lhepoinIC Ih<re bc

BC and mwing ,\e in

Ihen DE I'Idl be one

r"ifed a perpendicu lar 10 Ih. l,ne AB,

viz.

CD; Ihen il fOUllh, one fi(lh, ene fixdl,

6e.

of I3 C, and I\ l) Ihe

is pl.,in Ihe

~re

l)J;s a quaJrant, or eom.lins 90 degre!s; I,ke prtrt 01" AC; and, in general, if in anyIriong'" AIlC,

luppofe the are Dil

be di,ided inlo 9 e'luJI ares, caeh lhm be , n'umed a POiOl

on ooe uf ils fide, AH, and

of IVhich IVill eonlaio

JegrCC!, Ihen on Ihe poinl I:!

Ihrounh Ihal POill1be dWl'n a line ED, pardlld 10 one

raifing DE perpendicular

Ihe hne AH, il \ViII be a of il lidl'SI:!C, and meeling Ihe olher fiJe AC in D ;

u ngenl 10 Ihe eirele in Ihe

p~inl

B, and if 10 erery one Ihen \Vhatever pan

¡l E

is of AB, Ihe fame palllVill

El)

of Ihe u:virtoos of Ihe quaJronl,

viz.

10.

ll 20, B

30,

be of BC, and AD of AC.

40,

oc.

l'0~

draIV the line, I,ngem,

&e.

(as in Ihe

C.r.

Henee il follo\Vs, Ihat if in any triangle ABC.

feheme) IVe

have lhe fine, langeol,

&e.

ro every there be drawo ED, parallel 10 one of liSfides llC, and

leo degrees io Ihe 9uadranl: and Ihe

fam~

way \Ve may meeling Ihe olher I\VO in Ihe poinls E anu D, Ihen

have Ihe fine, tangem,

&e.

10 every fingle degree in Ihe AE' : All :: ED : BC :: AD : AC; Ihal is,

is lO

q'JadraOl, by dividing il inlo 90 equal p.rts

be~inoing

Al:!, as ED is 10 I:!C, and Ihdl as AD 10AC.

Iro:n B, and draIVing the fine, I.ngenl,

&e.

10 all Ihe

?2.lfanyIwo lri.nslesABC, fig 4.abe, lig.

are fimilar,

~res

beginning al lhe fJme point!l . By Ihis mtlhod Ihey or have

~II

Ihe

aD~lcs

of Ihe ooe equal 10 all Ihe . ogle.

draw the lines of fines, u ngenls,

&e.

of a eemin eirele of Ihe olher, eaeh 10 eaeh refpeélively; Ihat is, Ihe

on Ihe feale; for afler draIVing lhemon the eirele, Ihey. the angle CAl:! equal 10 Ihe . ngle cab, and Ihe angle

take Ihe lenglh of Ihem, and fel lhem off in the !ines

ABC

equal

lhe ansle abe, and the angll' ACBe9ual

drawn for thal purpofe. T he f. me way, by fllppofing to Ihe ansle ae b; then Ihe legs oppofile lO Ihe e9ual

the radius of any number of equal partS, (fuppofe

1000,

aogles are proportioned,

viz.

AB : ab :: AC : ac :: and

or 10,000,

&e.)

it is plain Ihe fine, langenl,

CJe.

of e- and A!l: ab :: I:!C : be:: and AC : ae:: BC : be.

very aremun confin orrome number of lhefe equal partS;

AB of lhe larse!! triansle lel off AE equal!

and by eompuling Ihemin partS of Ihe radius, \ye have ab, . nJ through E draIV ED parallel 10 I:!C, meel–

tables of fines, langenls,

cre.

every

are in Ihe

q~a·

ing AC in D; Ihen finee DE and BC are rarallel,

dranl, ealled nalural fi nes, langenls,

&e.

and Ihe loga- andABerofling lhem, Ihe ansle AED will (byart .

;6.)

rilhms of lhere give us \ables of logarilhmie fines, lan· be e9ual 10

I~e

ansle ABC, whieh (by fu ppofilion) is e·

genls,

&e.

See LOGAR ITH MS.

qu.l lo Ihe angle abe, alfo Ihe angle D.'\ E is (by fup-

?l. In any

trian~le,

ABC, (Plate XCIV. fig.

J. )

if pofilion) equal 10 Ihe angle ca b; lo in Ice 111'0 IflAoglts

one of ils fid s, as AC, be bifeéled in E. (and eoofe· AED, abe, Ihe 1\\'0 angles

D.~E

AED of Ihe one,

<¡uently AC double of AE) and t1ltough E be drawn are equal 10 lWO angles cab abe of Ihe olher, eaeh 10

ED, parallel 'lO BC, and meeling AB in D, Ihen!lC eaeh rerpeél:vely, and the included fide AE is (by con–

\ViII be dnuble of ED, and AB double of AD, Ihrough aruélion) e9ual to Ihe ineluded fide ab; Iherefore, (by

draIV DF, pm'ld lO AC, meeling BC in

Jo' :

for art. 61. eor.

J.)

AD is cqual 10 ae, and DE c9ual 10

¡joce, by eonflrutlion. DF is paraIld 10 AC, and DE pa- eb; bU I fi nee, in Ihe Iriangle AI:!C, Ihere is dra\VnDE

ralle! 10 !lC : Iherdore, (by arto

; 6.)

Ihe angle ll FD parallel 10 BC ooe of ils fide5, and meeting IIIt IIVO o·

will be equal

,h.

a~gle

BCA, (and by Ihe fame arti· Iherfidesinlhepolms Dand E, lherefure (bycor.art. 71.)

ele) Ihe angle BCA wiIl be e9ual 10 Ihe angle DEA, Al:!: AE :: AC : AD, and Al:! : AE:: BC : DE, and

confequently Ihe angl. BFD wi ll be equallo Ihe angle AC :

I\ D

!le: DE; and in Ihe Ihree IJ!! proportions,

DEA; alfo, (byart .

36.)

Ihe angle I:!DF \ViII be e- intlead of Ihe !ines AE, DE, and AD, pUlling in Iheir e·

qual 10 Ihe

~ngle

DAE; and fi nee DF is parallel10EC, quals ab, be, and ac, IVe nlaIl have AH : ab:: AC: ae,

aod DB

p.r~I1d

10 Fe, lhe quadril'leral DFCE IVill be and All : ab :: I:!e : ue, aod IdtUy, AC : ae :: De :

parallelo~ram;

and lherlfore, (by .rt.

59.

cor. 4.) be.

DF wiIl be equ,110 EC, whieh, byeonnruélion, is e9ual

73.

The ehord, fine, langenl,

&e.

of any are in one

to AE; fo in Ihe IWO Iriangles BDF DAE, Ihe IIVO eirele, is 10 Ih. ebord, line, 1.lngenl.

&c.

of Ihe

f.me

angles BFD and BDF in lhe one, are equ.t1 tOlhe 1100 are in anolher, as Ihe radius of Ih. one i, 10 Ihc radlO5

angles DllA 2nd DAE in Ihe olher, eaeh 10 caeh re· of Ihe olher, fig. 6, 6. Lel ABD ahd be IIVO eireles,

fpeéli,ely; and Ibc included fidc DF. is equal to Ihe in· BD ud IIVO ares of thefe ci,c!cs, rquallO one anolher.

cludcd fide AF.; Ih'!refore. (lly " l.

1. eor.

t. )

Al) IViIl or eon!lning of Ihe Can1c numbl'r 01

d~t, e,s;

r l)

1.1

equJI lO

DIl,

anJ eonfc1ucn:'. All uouuJc of

Al)

Ihe l.lIlbenl;, BD bJ Ihe checus,

lli:.

b< Ihe Jioes,

fJr.