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.
LIBER
SECUNDUS.
SECUNDUS.
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
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