Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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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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