Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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LIBER
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PRIMUS.</
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LEMMA XXV.
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Si parallelogrammi latera quatuor infinite producta tangant Sectio
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nem quamcunque Conicam, & abſcindantur ad tangentem quamvis
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quintam; ſumantur autem laterum quorumvis duorum contermi
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norum abſciſſæ terminatæ ad angulos oppoſitos parallelogrammi:
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dico quod abſciſſa alterutra ſit ad latus illud a quo est abſciſſa, ut
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pars lateris alterius contermini inter punctum contactus & latus
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tertium, est ad abſciſſarum alteram.
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<
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>Tangant parallelogrammi
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MLIK
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latera quatuor
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ML, IK, KL,
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MI
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ſectionem Conicam in
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A, B, C, D,
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& ſecet tangens quinta
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FQ
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hæc latera in
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F, Q, H
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&
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E
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; ſumantur autem
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laterum
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MI, KI
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ab
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ſciſſæ
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ME, KQ,
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vel
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laterum
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KL, ML
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ab
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ſciſſæ
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KH, MF:
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di
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co quod ſit
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ME
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ad
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MI
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ut
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BK
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ad
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KQ
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;
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&
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KH
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ad
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KL
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ut
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AM
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ad
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MF.
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Nam
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per Corollarium ſe
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cundum Lemmatis ſuperioris, eſt
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ME
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ad
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EI
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ut (
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AM
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ſeu)
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BK
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ad
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BQ,
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& componendo
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ME
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ad
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MI
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ut
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BK
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ad
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Q.E.D.
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Item
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KH
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ad
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HL
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ut (
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BK
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ſeu)
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AM
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ad
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AF,
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& dividendo
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KH
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ad
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KL
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ut
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AM
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ad
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MF. Q.E.D.
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Corol.
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1. Hinc ſi datur parallelogramum
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IKLM,
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circa datam Sec
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tionem Conicam deſeriptum, dabitur rectangulum
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KQXME,
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ut
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& huic æquale rectangulum
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KHXMF.
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Corol.
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2. Et ſi ſexta ducatur tangens
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eq
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tangentibus
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KI, MI
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occurrens in
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q
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&
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e
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; rectangulum
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KQXME
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æquabitur rectan
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gulo
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KqXMe
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; eritque
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KQ
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ad
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Me
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ut
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Kq
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ad
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ME,
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& diviſim ut
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Qq
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ad
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Ee.
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Corol.
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3. Unde etiam ſi
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Eq, eQ
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jungantur & biſecentur, & recta
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per puncta biſectionum agatur, tranſibit hæc per centrum Sectio
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nis Conicæ. </
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Qq
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ad
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Ee
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ut
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KQ
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ad
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Me,
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tranſibit ea-</
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