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.