LEMMA IX.
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.
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.
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.
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.
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.