Newton, Isaac - Philosophia naturalis principia mathematica

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DE MOTU
CORPORUM

SECTIO II.

De motu Corporum quibus reſiſtitur in duplicata ra
tione Velocitatum.

PROPOSITIO V. THEOREMA III.

Si Corpori reſiſiitur in velocitatis ratione duplicata, & idem ſola
vi inſita per Medium ſimilare movetur; tempora vero ſuman
tur in progreſſione Geometrica a minoribus terminis ad majores
pergente: dico quod velocitates initio ſingulorum temporum
ſunt in eadem progreſſione Geometrica inverſe, & quod ſpatia
ſunt æqualia quæ ſingulis temporibus deſcribuntur.

Nam quoniam quadrato velocita

149[Figure 149]
tis proportionalis eſt reſiſtentia Me
dii, & reſiſtentiæ proportionale eſt
decrementum velocitatis; ſi tempus
in particulas innumeras æquales divi
datur, quadrata velocitatum ſingulis
temporum initiis erunt velocitatum
earundem differentiis proportionalia.
Sunto temporis particulæ illæ AK,
KL, LM,&c. in recta CDſumptæ,
& erigantur perpendicula AB, Kk,
Ll, Mm,&c. Hyperbolæ BklmG,
centro CAſymptotis rectangulis CD, CHdeſcriptæ, occurrentia
in B, k, t, m,&c. & erit ABad Kkut CKad CA,& diviſim
AB-Kkad Kkut AKad CA,& viciſſim AB-Kkad AK
ut Kkad CA,adeoque ut ABXKkad ABXCA.Unde, cum
AK& ABXCAdentur, erit AB-Kkut ABXKk; & ultimo,
ubi coeunt AB& Kk,ut ABqueEt ſimili argumento erunt Kk-Ll,
Ll-Mm,&c. ut Kkq, Llq,&c. Linearum igitur AB, Kk, Ll, Mm

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