Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

ta recta BD; curvam verò tangat recta BT; ſitque BP rectæ BD
particula indefinitè parva; ducatúrque recta POad DTparallela,
curvam ſecans ad N; dico PNad NOrationem habere majorem quâ-
vis deſignabili, puta quàm R ad S.

46.1.

Fig. 174.

Nam ſit DE. ET: : RS; connexaque recta BEcurvam ſecet in
G, rectam POin K; per G verò ducatur FHad DAparallela. quoniam igitur BP ponitur indefinitè parva, eſt BP & lt; BF; adeóq; PK & lt; PN (nam ſubtenſa BGintra curvam tota cadit). ergo PN. NO & gt; PK. KO: : DE. ET: : R. S.

IV. Hinc, ſi baſis DBin partes ſecetur indeſinitè multas ad puncta
Z; & per hæc ducantur rectæ ad DAparallelæ curvam ſecantes pun-
ctis E, F, G; per hæc verò ducantur _Tangentes_ BQ, ER, FS, GT
parallelis ZE, ZF, ZG, DA occurrentes punctis Q, R, S, T; habebit recta ADad omnes interceptas EQ, FR, GS, AT(ſi-
mul ſumptas) rationem quàvis aſſignabili majorem.

Nam ducantur rectæ EY, FX, GV ad BD parallelæ. Habent
igitur rectæ ZE, YF, XG, VA ad rectas EQ, FR, GS, AT (ſin-
gulæ ad ſingulas ſibi in directum poſitas reſpectivè) rationem deſigna-
bili quâcunque majorem. ergò ſimul omnes iſtæ ad has ſimul omnes
_rationem_ habent deſignabili quâvis _majorem;_ hoc eſt recta AD ad EQ
+ FR + GS + AT ejuſmodi rationem habet.

46.1.

Fig. 175.

V. Hinc inter computandum, omnes EQ, FR, GS, AT ſimul ac-
ceptæ nihilo æquivalent; ſeu rectæ ZE, ZQ; & ZF, YR, & c. æ-
quantur; item tangentium particulæ BQ, ER, & c. reſpectivis _curvœ_
portiunculis BE, EF, & c. pares, & quaſi coincidentes haberi poſſunt. quin & adſumere tutò licet, quæ evidentèr his cohærent.

VI. Sit porrò _curva_ quævis AB, cujus _Axis_ AD, & ad hunc
applicata DB; æquiſecetur autem DB in partes indefinitè multas ad
puncta Z, per quæ ducantur rectæ ad AD parallelæ, curvam AB
interſecantes punctis X; quibus occurrant per ipſa X ductæ ad BD
parallelæ rectæ ME, NF, OG, PH; ſit autem ſegmento ADB
(rectis AD, DB, & curvâ AB comprehenſo) _circumſcripta ſigura_
ADBMXNXOXPXRA major _ſpatio_ quodam S; dico _ſegmentum_
ADB non eſſe minus quàm S.

46.1.

Fig. 176.

Nam ſi ſieripoteſt ſit ADB minus quàm S exceſſu _rectangulaum_
ADLKadæquante, & quoniam AReſt indefinitè parva, adeóque
minor quàm AK, liquet rectangulum ADZRminus eſſe _rectangulo_

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