Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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recta illa
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b, PC a, PQ c, CH e
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&
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CD o
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; rectangulum
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a+o
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in
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c-a-o
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ſeu
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ac-aa-2ao+co-oo
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æquale eſt rectangulo
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b
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in
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DI,
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adeoque
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DI
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æquale
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(ac-aa/b)+(c-2a/b)o-(oo/b).
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Jam ſcri
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bendus eſſet hujus ſeriei ſecundus terminus
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(c-2a/b)o
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pro Q
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o,
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ter
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tius item terminus (
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oo/b
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) pro R
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oo.
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Cum vero plures non ſint ter
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mini, debebit quarti coefficiens S evaneſcere, & propterea quan
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titas (S/R√1+QQ) cui Medii denſitas proportionalis eſt, nihil
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erit. </
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>Nulla igitur Medii denſitate movebitur Projectile in Para
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bola, uti olim demonſtravit
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Galilæus, Q.E.I.
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LIBER
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SECUNDUS.</
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Exempl.
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3. Sit linea
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AGK
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Hyperbola, Aſymptoton habens
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NX
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plano horizontali
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AK
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perpendicularem; & quæratur Medii
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denſitas quæ faciat ut Projectile moveatur in hac linea. </
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<
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>Sit
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MX
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Aſymptotos altera, ordinatim applicatæ
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DG
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productæ
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occurrens in
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V,
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& ex natura Hyperbolæ, rectangulum
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XV
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in
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VG
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dabitur. </
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<
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>Datur autem ratio
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DN
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ad
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VX,
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& propterea datur etiam
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rectangulum
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DN
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in
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VG.
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Sit illud
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bb
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; & completo parallelogrammo
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DNXZ,
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dicatur
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BN a, BD o, NX c,
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& ratio data
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VZ
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ad
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ZX
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vel
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DN
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ponatur eſſe
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m/n
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. </
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<
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>Et erit
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DN
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æqualis
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a-o, VG
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æqualis
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(bb/a-o), VZ
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æqualis
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m/n—a-o,
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&
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GD
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ſeu
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NX-VZ-VG
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æ
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qualis
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c-m/n a+m/n o-(bb/a-o).
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Reſolvatur terminus (
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bb/a-o
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) in ſeriem
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convergentem
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(bb/a)+(bb/aa)o+(bb/a
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3
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)oo+(bb/a
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4
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)o
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3
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&c. </
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>& ſiet
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GD
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æqua
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lis
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c-m/n a-(bb/a)+m/n o-(bb/aa)o-(bb/a
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3
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)o
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2
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-(bb/a
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4
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)o
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3
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&c. </
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<
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>Hujus ſeriei termi
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nus ſecundus
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m/no-(bb/aa)o
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uſurpandus eſt pro Q
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o,
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tertius cum ſigno
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mutato
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(bb/a
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3
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)o
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2
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pro R
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o
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2
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, & quartus cum ſigno etiam mutato
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(bb/a
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4
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)o
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1
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pro S
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o
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3
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, eorumque coefficientes
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m/n-(bb/aa), (bb/a
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3
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)
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& (
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bb/a
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4
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) ſcribendæ ſunt
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in Regula ſuperiore, pro Q, R & S. </
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<
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>Quo facto prodit medii denſitas </
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