1tatem BIut ſumma omnium AH+BI+CK+DL,in infiNI
tum, ad ſummam omnium BI+CK+DL,&c. Et BIden
ſitas ſecundæ B,eſt ad CKdenſitatem tertiæ C,ut ſumma om
nium BI+CK+DL,&c. ad ſummam omnium CK+DL,&c.
Sunt igitur ſummæ illæ differentiis ſuis AH, BI, CK,&c. pro
portionales, atque adeo continue proportionales, per hujus Lem. I.
proindeQ.E.D.fferentiæ AH, BI, CK,&c. ſummis proportionales,
ſunt etiam continue proportionales. Quare cum denſitates in locis A,
B, C,&c. ſint ut AH, BI, CK,&c. erunt etiam hæ continue propor
tionales. Pergatur per ſaltum, & (ex æquo) in diſtantiis SA, SC,
SEcontinue proportionalibus, erunt denſitates AH, CK, EM
continue proportionales. Et eodem argumento, in diſtantiis qui
buſvis continue proportionalibus SA, SD, SG,denſitates AH, DL,
GOerunt continue proportionales. Coeant jam puncta A, B, C,
D, E,&c. eo ut progreſſio gravitatum ſpecificarum a fundo Aad
ſummitatem Fluidi continua reddatur, & in diſtantiis quibuſvis con
tinue proportionalibus SA, SD, SG,denſitates AH, DL, GO,
ſemper exiſtentes continue proportionales, manebunt etiamnum
continue proportionales. que E. D.
tum, ad ſummam omnium BI+CK+DL,&c. Et BIden
ſitas ſecundæ B,eſt ad CKdenſitatem tertiæ C,ut ſumma om
nium BI+CK+DL,&c. ad ſummam omnium CK+DL,&c.
Sunt igitur ſummæ illæ differentiis ſuis AH, BI, CK,&c. pro
portionales, atque adeo continue proportionales, per hujus Lem. I.
proindeQ.E.D.fferentiæ AH, BI, CK,&c. ſummis proportionales,
ſunt etiam continue proportionales. Quare cum denſitates in locis A,
B, C,&c. ſint ut AH, BI, CK,&c. erunt etiam hæ continue propor
tionales. Pergatur per ſaltum, & (ex æquo) in diſtantiis SA, SC,
SEcontinue proportionalibus, erunt denſitates AH, CK, EM
continue proportionales. Et eodem argumento, in diſtantiis qui
buſvis continue proportionalibus SA, SD, SG,denſitates AH, DL,
GOerunt continue proportionales. Coeant jam puncta A, B, C,
D, E,&c. eo ut progreſſio gravitatum ſpecificarum a fundo Aad
ſummitatem Fluidi continua reddatur, & in diſtantiis quibuſvis con
tinue proportionalibus SA, SD, SG,denſitates AH, DL, GO,
ſemper exiſtentes continue proportionales, manebunt etiamnum
continue proportionales. que E. D.
DE MOTU
CORPORUM
CORPORUM
Corol.Hinc ſi detur denſitas Fluidi in duobus locis, puta A&
E,colligi poteſt ejus denſitas
171[Figure 171]
in alio quovis loco queCentro
S,Aſymptotis rectangulis SQ,
SX,deſcribatur Hyperbola ſe
cans perpendicula AH, EM,
QTin a, e, q,ut & perpendicu
la HX, MY, TZ,ad Aſymp
toton SXdemiſſa, in h, m& t.
Fiat area ZYmtZad aream da
tam YmhXut area data EeqQ
ad aream datam EeaA; & li
nea Ztproducta abſcindet li
neam QTdenſitati proportio
nalem. Namque ſi lineæ SA, SE, SQſunt continue proportiona
les, erunt areæ EeqQ, EeaAæquales, & inde areæ his propor
tionales YmtZ, XhmYetiam æquales, & lineæ SX, SY, SZ,id eſt
AH, EM, QTcontinue proportionales, ut oportet. Et ſi lineæ
SA, SE, SQobtinent alium quemvis ordinem in ſerie continue
proportionalium, lineæ AH, EM, QT,ob proportionales areas
Hyperbolicas, obtinebunt eundem ordinem in alia ſerie quantita
tum continue proportionalium.
E,colligi poteſt ejus denſitas
171[Figure 171]in alio quovis loco queCentro
S,Aſymptotis rectangulis SQ,
SX,deſcribatur Hyperbola ſe
cans perpendicula AH, EM,
QTin a, e, q,ut & perpendicu
la HX, MY, TZ,ad Aſymp
toton SXdemiſſa, in h, m& t.
Fiat area ZYmtZad aream da
tam YmhXut area data EeqQ
ad aream datam EeaA; & li
nea Ztproducta abſcindet li
neam QTdenſitati proportio
nalem. Namque ſi lineæ SA, SE, SQſunt continue proportiona
les, erunt areæ EeqQ, EeaAæquales, & inde areæ his propor
tionales YmtZ, XhmYetiam æquales, & lineæ SX, SY, SZ,id eſt
AH, EM, QTcontinue proportionales, ut oportet. Et ſi lineæ
SA, SE, SQobtinent alium quemvis ordinem in ſerie continue
proportionalium, lineæ AH, EM, QT,ob proportionales areas
Hyperbolicas, obtinebunt eundem ordinem in alia ſerie quantita
tum continue proportionalium.