Newton, Isaac - Philosophia naturalis principia mathematica

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1evaneſcent, & rationem ultimam habebunt æqualitatis. Q.E.D.

DE MOTU
CORPORUM

Corol.1. Unde ſi per Bducatur tangenti parallela BF,rectam
quamvis AFper Atranſe

9[Figure 9]
untem perpetuo ſecans in F,
hæc BFultimo ad arcum e
vaneſcentem ABrationem
habebit æqualitatis, eo quod
completo parallelogrammo AFBDrationem ſemper habet æqua
litatis ad AD.

Corol.2. Et ſi per B& Aducantur plures rectæ BE, BD, AF,
AG,ſecantes tangentem AD& ipſius parallelam BF; ratio ulti
ma abſciſſarum omnium AD, AE, BF, BG,chordæque & ar
cus ABad invicem erit ratio æqualitatis.

Corol.3. Et propterea hæ omnes lineæ, in omni de rationibus ul
timis argumentatione, pro ſe invicem uſurpari poſſunt.

LEMMA VIII.

Si rectæ datæAR, BR cum arcuAB, chordaAB & tangente
AD, triangula triaARB, ARB, ARD conſtituunt, dein
punctaA, B accedunt ad invicem: dico quod ultima forma
triangulorum evaneſcentium est ſimilitudinis, & ultima ratio
æqualitatis.

Nam dum punctum Bad punctum A

10[Figure 10]
accedit, intelligantur ſemper AB, AD, AR
ad puncta longinqua b, d& rproduci,
ipſique RDparallela agi rbd,& arcui
ABſimilis ſemper ſit arcus Ab.Et coe
untibus punctis A, B,angulus bAdeva
neſcet, & propterea triangula tria ſemper
finita rAb, rAb, rAdcoincident, ſunt
que eo nomine ſimilia & æqualia. Unde
& hiſce ſemper ſimilia & proportionalia
RAB, RAB, RADſient ultimo ſibi
invicem ſimilia & æqualia. que E. D.

Corol.Et hinc triangula illa, in omni de rationibus ultimis argu
mentatione, pro ſe invicem uſurpari poſſunt.

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