Newton, Isaac - Philosophia naturalis principia mathematica

Si rectaAE & curvaABC poſitione datæ ſe mutuo ſecent in
angulo datoA, & ad rectam illam in alio dato angulo ordina
tim applicenturBD, CE, curvæ occurrentes inB, C; dein
punctaB, C ſimul accedant ad punctumA: dico quod areæ tri
angulorumABD, ACE erunt ultimo ad invicem in duplicata
ratione laterum.

Etenim dum puncta B, Cacce

11[Figure 11]
dunt ad punctum A,intelligatur
ſemper ADproduci ad puncta lon
ginqua d& e,ut ſint Ad, Aeip
ſis AD, AEproportionales, & e
rigantur ordinatæ db, ecordina
tis DB, ECparallelæ quæ occur
rant ipſis AB, ACproductis in
b& c.Duci intelligatur, tum curva
Abcipſi ABCſimilis, tum recta
Ag,quæ tangat curvam utramque
in A,& ſecet ordinatim applica
tas DB, EC, db, ecin F, G, f, g.
Tum manente longitudine Aecoeant puncta B, Ccum puncto A;
& angulo cAgevaneſcente, coincident areæ curvilineæ Abd, Ace
cum rectilineis Afd, Age:adeoque (per Lemma v) erunt in dupli
cata ratione laterum Ad, Ae:Sed his areis proportionales ſemper
ſunt areæ ABD, ACE,& his lateribus latera AD, AE.Ergo &
areæ ABD, ACEſunt ultimo in duplicata ratione laterum AD,
AE. que E. D.

LEMMA X.

Spatia quæ corpus urgente quacunque Vi finita deſcribit, five Vis
illa determinata & immutabilis ſit, five eadem continuo auge
atur vel continuo diminuatur, ſunt ipſo motus initio in duplica
ta ratione Temporum.

Exponantur tempora per lineas AD, AE,& velocitates genitæ
per ordinatas DB, EC; & ſpatia his velocitatibus deſcripta, erunt
ut areæ ABD, ACEhis ordinatis deſcriptæ, hoc eſt, ipſo motus
initio (per Lemma IX) in duplicata ratione temporum AD, AE.
que E. D.

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