Oeuvres complètes. Tome IX. Correspondance 1685-1690
(1901)–Christiaan Huygens– Auteursrecht onbekendNo 2612.
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I.ACB est semicirculus. GEF curva ejusmodi ut semper ED perp. AB sit aequalis duabus AC, CB (vel earum differentiae)Ga naar voetnoot2), hujus aequatio (posita HA = a, HD = x, DE = y) est √2aa + 2ax + √2aa - 2ax = y 4aa + 2√.√. = yy 2√.√. = yy - 4aa y4 - 8aayy + 16aaxx = 0 VO tangens. Ergo ex regula TO = = 4y4 - 16aayy / 32aaxGa naar voetnoot3) subt. sive y4 - 4aayy / 8aaxsedy4 = 8aayy - 16aaxx. Ergo TO = 4aayy - 16aaxx/8aax sive yy - 4xx / 2x sive yy/2x - 2x. Hinc Leibnitio curvae naturam inquirendam proposui in epistola d. 24 Aug. 1690. | |
[pagina 474]
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Spatium AGEFB compositum est ex duabus semiparabolis AFB, BGA, ideoque = 4/3 qu. AGFB unde spat. GEF = ⅓ qu. ejusdem. | |
II.BC parallela AL et perpend. in AB. Curvae AE proprietas est ut, datâ EAC, quadratum AE sit aequale rectangulo ex AB, BC. | |
[pagina 475]
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x - y - a | ay/x aay/x = xx + yy aay = x3 + xyy 0 = xyy - aay + x3 aequatio curvae. SO tangens. Ex regula 2xyy - aay / yy + 3xxGa naar voetnoot4) xx = aay/x - yy 2xyy - aay / yy + 3aay/x - 3yy | 2xyy - aay / 3aay/x - 2yy | 2xxy - aax / 3aa - 2xy OT.
Hinc curvam inveniendam proposui Leibnitio 24 Aug. 1690Ga naar voetnoot5). Si x sit cum signo -, debebit et y esse cum signo -, unde liquet curvam EA descendere sub rectam DAB, et ad alteram partem rectae LAM. Rectang. AT, TS = xy semper minus erit qu.o AB, sed quamlibet prope accedit minuendo x. |
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