Corol.

Principia Mathematica. 31

rn . . n I- D R xAB RDGT . 1

Lst igitur Kr xqualis -—_: idco-

N

DRxABN

que si producatur RT ad X ut ( h ) sit RX xqualis

id est, si compleatur parallelogrammum ACPY , jungatur DYsecans CP in & producatur RT donec occurrat DY inv . v RDGT a

X ; ent X r aequalis-—-, & propterea tempori propor-

N

tionalis.

Co-

bitur a —Undesity =- a. S.-

t—g y e—x

. a d x

Si sumptis fluxionibus d y —

fv aev bi

re habebitur—;-r —-——4.L.—,

b bg b—v

Sc

a d

g

-- Si ponatur RC five =

e — x ’

, . . adx adz . ,

erit — dx = dz, Si -— — , ideo-

e—x _' z *

c d x dr

que — a. S. - -a. S. — =a.L.z=a.L.e-x

e—x z

( 40 \ Quare erit » = —■ -+-a. L. e — x

+ O^conß. Et quia evanescente y , eva-nescit quoque x , invenitur constans

— — a.L.e j & hinc y — —- a. L. e—x

a x e

—a. L.e — - a.L. -. Est enim L.e—L.e—x

g e—x

— L.-> & signis mutatis E. e —x—L. e

z — a JX -f- a. L ——.8ed (ex demon-

bg L—v

slr.) <? =—> atque ideo ae-agzzfg, &

V—g •

fg

a.L

-g

— ae— —agb a vb—v b

i quare ent etiam z —

-- L.

e—x

17. Ex hac atquatione alia deducitur interD V & V r. Si enim dicantur D V = »& Vr — z, erit ob triangula D C P, DRVsimilia, DP ( 4 ) : D \ (v) = DC ( O •e xi eb — ev

D R (*)=-£-■ & meo e-x- —^-

e b

& -^ c = T~v'> similiter ent DC (e):C P (/) — D R (-^) : VR =£p ideo-

que y — Rr — VR — Vr — —r—z. Qua-

n R v A R

(h) * Ut sit RX aqualis

&c. Hoc enim facto, erit R X ad D R utdata A K ad datam A , ideoque locuspunctorum X linea recta quae transit perpunctum D , ubi evanescente D R evaneGcit quoque R X. Coincidente puncto Rcum A sit R X seu AY:DAc;Al):N,& ( per theor. 4. de hyperb. ) D C ; A C= A B : G D seu Ä Q ; & divisim DC; DA= A B : B Q i per constructionem vero C P:DC —BQ:A, ideoque ex aequo CP:DA = A B;K = A Y:D A, ac proindeAV — C P. Unde si compleatur paralle-logrammum A CP Y, jungaturque D Y se-cans CP in Z, erit DZ linea redta quampunctum X perpetub tangit. QuoniamD R y A B

igitur RX —---, & X r = RX—Rr

D R X A BA'

DR xAB + R DGT

A '

. „ RDGT

erit X r— ——, & propterea, ob da-tam A T , X r est ut area RDGT , ideo-que ut tempus quo corpus ex loco Dpervenis in locum r.

Lib H R

SiCtlKU.SlCTIO I.Prop. IV.Probi.II.

5 7 -

* In.