Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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LEMMA IX.
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Si recta
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AE
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& curva
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ABC
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poſitione datæ ſe mutuo ſecent in
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angulo dato
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A,
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& ad rectam illam in alio dato angulo ordina
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tim applicentur
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BD, CE,
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curvæ occurrentes in
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B, C;
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dein
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puncta
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B, C
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ſimul accedant ad punctum
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A:
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dico quod areæ tri
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angulorum
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ABD, ACE
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erunt ultimo ad invicem in duplicata
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ratione laterum.
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<
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>Etenim dum puncta
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B, C
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acce
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dunt ad punctum
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A,
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intelligatur
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ſemper
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AD
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produci ad puncta lon
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ginqua
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d
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&
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e,
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ut ſint
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Ad, Ae
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ip
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ſis
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AD, AE
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proportionales, & e
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rigantur ordinatæ
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db, ec
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ordina
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tis
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DB, EC
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parallelæ quæ occur
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rant ipſis
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AB, AC
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productis in
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b
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&
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c.
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Duci intelligatur, tum curva
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Abc
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ipſi
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ABC
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ſimilis, tum recta
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Ag,
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quæ tangat curvam utramque
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in
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A,
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& ſecet ordinatim applica
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tas
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DB, EC, db, ec
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in
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F, G, f, g.
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Tum manente longitudine
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Ae
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coeant puncta
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B, C
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cum puncto
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A
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;
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& angulo
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cAg
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evaneſcente, coincident areæ curvilineæ
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Abd, Ace
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cum rectilineis
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Afd, Age:
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adeoque (per Lemma v) erunt in dupli
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cata ratione laterum
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Ad, Ae:
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Sed his areis proportionales ſemper
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ſunt areæ
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ABD, ACE,
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& his lateribus latera
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AD, AE.
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Ergo &
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areæ
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ABD, ACE
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ſunt ultimo in duplicata ratione laterum
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AD,
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AE.
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E. D.
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LEMMA X.
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Spatia quæ corpus urgente quacunque Vi finita deſcribit, five Vis
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illa determinata & immutabilis ſit, five eadem continuo auge
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atur vel continuo diminuatur, ſunt ipſo motus initio in duplica
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ta ratione Temporum.
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<
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AD, AE,
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& velocitates genitæ
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per ordinatas
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DB, EC
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; & ſpatia his velocitatibus deſcripta, erunt
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ut areæ
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ABD, ACE
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his ordinatis deſcriptæ, hoc eſt, ipſo motus
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initio (per Lemma IX) in duplicata ratione temporum
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AD, AE.
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E. D.
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