1recta illa b, PC a, PQ c, CH e& CD o; rectangulum a+o
in c-a-oſeu ac-aa-2ao+co-ooæquale eſt rectangulo
bin DI,adeoque DIæquale (ac-aa/b)+(c-2a/b)o-(oo/b).Jam ſcri
bendus eſſet hujus ſeriei ſecundus terminus (c-2a/b)opro Qo,ter
tius item terminus (oo/b) pro Roo.Cum vero plures non ſint ter
mini, debebit quarti coefficiens S evaneſcere, & propterea quan
titas (S/R√1+QQ) cui Medii denſitas proportionalis eſt, nihil
erit. Nulla igitur Medii denſitate movebitur Projectile in Para
bola, uti olim demonſtravit Galilæus, Q.E.I.
in c-a-oſeu ac-aa-2ao+co-ooæquale eſt rectangulo
bin DI,adeoque DIæquale (ac-aa/b)+(c-2a/b)o-(oo/b).Jam ſcri
bendus eſſet hujus ſeriei ſecundus terminus (c-2a/b)opro Qo,ter
tius item terminus (oo/b) pro Roo.Cum vero plures non ſint ter
mini, debebit quarti coefficiens S evaneſcere, & propterea quan
titas (S/R√1+QQ) cui Medii denſitas proportionalis eſt, nihil
erit. Nulla igitur Medii denſitate movebitur Projectile in Para
bola, uti olim demonſtravit Galilæus, Q.E.I.
Exempl.3. Sit linea AGKHyperbola, Aſymptoton habens
NXplano horizontali AKperpendicularem; & quæratur Medii
denſitas quæ faciat ut Projectile moveatur in hac linea.
NXplano horizontali AKperpendicularem; & quæratur Medii
denſitas quæ faciat ut Projectile moveatur in hac linea.
Sit MXAſymptotos altera, ordinatim applicatæ DGproductæ
occurrens in V,& ex natura Hyperbolæ, rectangulum XVin VG
dabitur. Datur autem ratio DNad VX,& propterea datur etiam
rectangulum DNin VG.Sit illud bb; & completo parallelogrammo
DNXZ,dicatur BN a, BD o, NX c,& ratio data VZad ZX
vel DNponatur eſſe m/n. Et erit DNæqualis a-o, VGæqualis
(bb/a-o), VZæqualis m/n—a-o,& GDſeu NX-VZ-VGæ
qualis c-m/n a+m/n o-(bb/a-o).Reſolvatur terminus (bb/a-o) in ſeriem
convergentem (bb/a)+(bb/aa)o+(bb/a3)oo+(bb/a4)o3&c. & ſiet GDæqua
lis c-m/n a-(bb/a)+m/n o-(bb/aa)o-(bb/a3)o2-(bb/a4)o3&c. Hujus ſeriei termi
nus ſecundus m/no-(bb/aa)ouſurpandus eſt pro Qo,tertius cum ſigno
mutato (bb/a3)o2pro Ro2, & quartus cum ſigno etiam mutato (bb/a4)o1
pro So3, eorumque coefficientes m/n-(bb/aa), (bb/a3)& (bb/a4) ſcribendæ ſunt
in Regula ſuperiore, pro Q, R & S. Quo facto prodit medii denſitas
occurrens in V,& ex natura Hyperbolæ, rectangulum XVin VG
dabitur. Datur autem ratio DNad VX,& propterea datur etiam
rectangulum DNin VG.Sit illud bb; & completo parallelogrammo
DNXZ,dicatur BN a, BD o, NX c,& ratio data VZad ZX
vel DNponatur eſſe m/n. Et erit DNæqualis a-o, VGæqualis
(bb/a-o), VZæqualis m/n—a-o,& GDſeu NX-VZ-VGæ
qualis c-m/n a+m/n o-(bb/a-o).Reſolvatur terminus (bb/a-o) in ſeriem
convergentem (bb/a)+(bb/aa)o+(bb/a3)oo+(bb/a4)o3&c. & ſiet GDæqua
lis c-m/n a-(bb/a)+m/n o-(bb/aa)o-(bb/a3)o2-(bb/a4)o3&c. Hujus ſeriei termi
nus ſecundus m/no-(bb/aa)ouſurpandus eſt pro Qo,tertius cum ſigno
mutato (bb/a3)o2pro Ro2, & quartus cum ſigno etiam mutato (bb/a4)o1
pro So3, eorumque coefficientes m/n-(bb/aa), (bb/a3)& (bb/a4) ſcribendæ ſunt
in Regula ſuperiore, pro Q, R & S. Quo facto prodit medii denſitas