Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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DE MUNDI
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SYSTEMATE</
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LEMMA V.
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Invenire lineam curvam generis Parabolici, quæ per data
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quotcunque puncta tranſibit.
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<
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>Sunto puncta illa
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A, B, C, D, E, F,
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&c. </
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>& ab iiſdem ad rectam
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quamvis poſitione datam
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HN
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demitte perpendicula quotcunque
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AH, BI, CK, DL, EM, FN.
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Caſ.
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1. Si punctorum
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H, I, K, L, M, N
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æqualia ſunt inter
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valla
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HI, IK, KL,
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&c. </
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<
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>collige perpendiculorum
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AH, BI,
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CK,
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&c. </
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<
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>differentias primas
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b,
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2
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b,
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3
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b,
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4
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b,
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5
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b,
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&c. </
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<
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>ſecundas
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c,
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2
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c,
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c,
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4
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c,
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&c. </
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<
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>tertias
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d,
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2
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d,
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3
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d,
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&c. </
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<
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>id eſt, ita ut ſit
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AH-BI=b,
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BI-CK=2b, CK-DL=3b, DL+EM=4b,-EM+FN=5b,
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&c. </
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<
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>dein
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b-2b=c,
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&c.
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<
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>& ſic pergatur ad diffe
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rentiam ultimam quæ hic
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eſt
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f.
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Deinde erecta qua
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cunque perpendiculari
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RS,
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quæ fuerit ordina
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tim applicata ad curvam
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quæſitam: ut inveniatur
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hujus longitudo, pone
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intervalla
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HI, IK, KL,
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LM,
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&c. </
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<
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>unitates eſſe,
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& dic
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AH=a,-HS=p,
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1/2p
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in -
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IS=q, 1/3q
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in
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+
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SK=r, 1/4r
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in +
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SL=s, 1/5s
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in +
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SM=t
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; pergendo videlicet
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ad uſque penultimum perpendiculum
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ME,
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& præponendo ſigna
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negativa terminis
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HS, IS,
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&c. </
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<
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>qui jacent ad partes puncti
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S
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ver
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ſus
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A,
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& ſigna affirmativa terminis
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SK, SL,
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&c. </
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<
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ad alteras partes puncti
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S.
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Et ſignis probe obſervatis, erit
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RS=a+bp+cq+dr+es+ft,
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&c. </
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Caſ.
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2. Quod ſi punctorum
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H, I, K, L,
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&c. </
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<
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>inæqualia ſint inter
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valla
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HI, IK,
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&c. </
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<
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>collige perpendiculorum
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AH, BI, CK,
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&c. </
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<
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differentias primas per intervalla perpendiculorum diviſas
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b,
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2
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b,
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3
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b,
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b,
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b
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; ſecundas per intervalla bina diviſas
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c,
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2
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c,
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3
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c,
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c,
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&c. </
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tertias per intervalla terna diviſas
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d,
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2
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d,
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3
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d,
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&c. </
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<
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