Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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concurrentes in
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G
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; dein accedant puncta
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D, B, G,
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ad puncta
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d, b, g,
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ſitque
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J
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interſectio linearum
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BG, AG
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ultimo facta ubi puncta
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D, B
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accedunt uſque ad
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A.
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Manifeſtum eſt quod diſtantia
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GJ
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minor
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eſſe poteſt quam aſſignata quævis. </
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>Eſt autem (ex natura circulorum
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per puncta
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ABG, Abg
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tranſeuntium)
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ABquad.
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æquale
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AGXBD,
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&
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Ab quad.
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æquale
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AgXbd,
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adeoque ratio
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AB quad.
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ad
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Ab quad.
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compo
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nitur ex rationibus
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AG
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ad
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Ag
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&
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BD
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ad
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bd.
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Sed quoniam
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GJ
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aſſumi poteſt minor longitu
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dine quavis aſſignata, fieri poteſt ut ratio
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AG
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ad
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Ag
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minus differat a ratione æqualitatis quam
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pro differentia quavis aſſignata, adeoque ut ra
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tio
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AB quad.
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ad
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Ab quad.
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minus differat a ra
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tione
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BD
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ad
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bd
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quam pro differentia quavis
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aſſignata. </
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<
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>Eſt ergo, per Lemma 1, ratio ultima
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AB quad.
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ad
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Ab quad.
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æqualis rationi ultimæ
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BD
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ad
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bd. </
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E. D.
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Cas.
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2. Inclinetur jam
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BD
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ad
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AD
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in angulo
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quovis dato, & eadem ſemper erit ratio ultima
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BD
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ad
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bd
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quæ
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prius, adeoque eadem ae
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AB quad.
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ad
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Ab quad. </
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E. D.
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Cas.
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3. Et quamvis angulus
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D
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non detur, ſed recta
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BD
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ad da
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tum punctum convergente, vel alia quacunque lege conſtituatur;
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tamen anguli
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D, d
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communi lege conſtituti ad æqualitatem ſemper
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vergent & propius accedent ad invicem quam pro differentia qua
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vis aſſignata, adeoque ultimo æquales erunt, per Lem. I & prop
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terea lineæ
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BD, bd
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ſunt in eadem ratione ad invicem ac prius.
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E. D.
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Corol.
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1. Unde eum tangentes
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AD, Ad,
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arcus
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AB, Ab,
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& eo
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rum ſinus
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BC, bc
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fiant ultimo chordis
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AB, Ab
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æquales; erunt
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etiam illorum quadrata ultimo ut ſubtenſæ
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BD, bd.
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Corol.
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2. Eorundem quadrata ſunt etiam ultimo ut ſunt arcuum
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ſagittæ quæ chordas biſecant & ad datum punctum convergunt. </
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Nam ſagittæ illæ ſunt ut ſubtenſæ
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BD, bd.
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Corol.
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3. Ideoque ſagitta eſt in duplicata ratione temporis quo
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corpus data velocitate deſcribit arcum. </
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Corol.
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4. Triangula rectilinea
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ADB, Adb
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ſunt ultimo in tripli
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cata ratione laterum
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AD, Ad,
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inque ſeſquiplicata laterum
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DB,
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db
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; utpote in compoſita ratione laterum
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AD,
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&
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DB, Ad
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&
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db
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exiſtentia. </
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<
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ABC, Abc
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ſunt ultimo in triplicata
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ratione laterum
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BC, bc.
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Rationem vero Seſquiplicatam voco tri
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plicatæ ſubduplicatam, quæ nempe ex ſimplici & ſubduplicata com
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ponitur, quamque alias Seſquialteram dicunt. </
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