Huygens, Christiaan
,
Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica
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ſive etiam - 3y
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+ ayx eſſe affirmatam, ac proinde 3y
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- ayx
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eſſe negatam: </
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<
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xml:space
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">aut quando 3xx - ay fuerit affirmata, tunc
<
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- 3y
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+ ayx eſſe negatam; </
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<
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xml:space
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">ac proinde 3y
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- ayx eſſe affir-
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matam. </
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<
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xml:space
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">Per hæc itaque apparet ex quantitatibus per regulam in-
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ventis, quæ erant {3y
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- ayx/3xx - ay} = z judiciari poſſe ad utrum ca-
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ſum conſtructio tangentis pertineat; </
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<
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xml:space
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">nempe excomperta diſſi-
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militudine affectionis in diviſore & </
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<
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">dividendo, ſequi ad prio-
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rem caſum eam pertinere, hoc eſt z, ſive F E, accipiendam
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eſſe verſus A: </
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<
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">ex ſimilitudine vero eorum affectionis ſequi ad
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contrariam partem ſumendam.</
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fig. 6.</
note
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">Poteſt autem quantitas z ſive FE per regulam inventa, non-
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nunquam ad ſimpliciores terminos reduci ope æquationis datæ,
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quæ naturam curvæ continet: </
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">velut in hac curva A C, axem
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habente A D, verticem A, cujuſque ea eſt proprietas ut, ſi à
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puncto C in eâ ſumpta, applicetur ordinatim C D, fiat pro-
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ductum ex cubo B D (eſt autem B punctum in axe extra cur-
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vam datam) in quadratum D A æquale cubo quadrato DC. </
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ve ponendo B A = a, B D = x, D C = y, fiat æquatio cur-
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væ naturam continens, iſta x
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- 2ax
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+ aax
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- y
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= o. </
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<
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ponendo CG eſſe tangentem, quæ occurrat axi in G, vocan-
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doque GD, z, fit ſecundum regulam z = {- 5y
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/5x
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- 8ax
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+ 3aaxx}.
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</
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<
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">Quia autem ex datâ æquatione eſt y
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= x
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- 2ax
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+ aax
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, re-
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ſtituendo pro 5y id quod ipſi æquale eſt, fiet z =
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{- 5x
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+ 10ax
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- 5aax
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/5x
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- 8ax
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+ 3aaxx}; </
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<
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xml:space
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">ſive dividendo per xx, erit z =
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{- 5x
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+ 10axx - 5aax/5xx - 8ax + 3aa,}. </
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<
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xml:space
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">Et rurſus, dividendo hanc fractionem
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per x - a, habebitur z = {-5xx + 5ax/5x - 3a}. </
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<
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xml:space
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">Quod ſignificat
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faciendum ut ſicut B D quinquies ſumpta minus B A ter, ſive
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ut B A bis unà cum A D quinquies ad AD quinquies, ita BD
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ad D G; </
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