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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 175r, page id 00068199:0349
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Nihil produt novi
Dimensio curvae Conicae consistit in sequenti quadratura.
dylys. 1 tydu de a +(b. axX, :, aa + (C: a) xx sit s addatur utrobique B, et
fiet d dydus tdxd = aa +bXX + Ba_ HL(BC: a a) XX
ax dX
aa + (c: a) XX
et fiet dydy - b: c) dxdx (3) (aa : c), c - b
sit +
c =0
ac
aa + (C: a) XX
seu β = - ab : c
aa + Ba = aa - aab: c - ab
comprobatio dydy- bi estndx = aa (+ B X∞ _- aba (= (ab: c)c: acipX)
dxdx
ac + (C: a) XX
dx dx
seu dx dx : dydy - (b: c) dxdx = aa + (C.4) XX, :, (aa : c), b - c
dy- dx, c: a, XX, quanbitas consta
seu dxdx. (5 Idy - deaa
+
Ay-di (C. a) Xx. sit l a
+
aa: C, b =c
Hbic
et differentiando fiet C (dy + dx Hb. c) 2d X ddx + dx dx (d dys + ddx +b c) L Fddus - ddx) aa
aa:C; b-c
dy + dxVbic  (a4 : C), b - c
qu. dy +dx Sb:c
quae res reducta est ad quadraturas rationales, quia daus:dax dantur ratio
163
naliter ex ipsis dy: dx nempe fit t 2(dy + dx sbi c) dxddx + dx dx (adyst dex  )- b - c =
C. qu (dy + dx 1bC) (d dy - da X Hb: c)  seu ddy : dex = (2 (dy + ax 1b: C) dx + dxdx 1b. c 3b-C) +C quiqus rebe)
 eut na
uus
d nve
B (d(dy: dx): Cdy:dx; 1) - duidx videndum autem on ita
per dynx
naent
a Ax.d perdetoda
nat mi  aum
evocar
et jam (ddx: du 
et videtur cnon qudem fortasse generaso ratione, sed tamen artifiio et aectura rei suppedulato
uadla
per
neralis, omnesque
N e niri taudon possit logaus- logilx
ad rationales. nam  aqt in quae edg formula
ra ergo
x gabeturit
rationalt aequalis nihilo in qua reperiantur 
potentiae, hunc differentiando ri erit aequatio, in qua du
t
dx,
potertici
di
et ad
c consgia qualis 
assumatur jam formula disscex nu X, a, dy  multi dx, dus   qualis
p
e constanti differentiasi primi gradus et hanc differentiando, ut Aco evanescat,
dlad
habebitur aequatiolim qua dyni dax, dayu, dx, dy, X, a, itaque ose
 quae sit talis ut in ipsa X si ponamas jam talem assumitatem
ut ejus ope in aeq. 3. ut ex aequ.  1 et 2, possint haberi valores
et a. etsideo ascendat utcunque ad potentias.
:dx
rationates ipsarum.
Aequationem, differentiando habebitur aequ (2) in qua ad (dyide) datur
ratio naliter
assumatur xia per dy. du et dx. deo
rationaliter per xia, Assumatur alia aequation, in qua xia habentur
et per da
per d rationatiter per du. da et per dy. , et Lidco, atque
dy: Axde
et hanc differentiando habebitur (4to) dxia rationaliter per dy. deo
et ddus: deo et fdxia ): Tady: deo) rationaliter per dy: deo Ex aequ.
2 tollatur x aeer vatorem rationalem invensum postum in aequ. 3.
in qua a d(dy: dx): dx habetur rationaliter per
.ope hujus tollatur x: a ex aequ. 1 et ex aeg. 2.
et habebitur aq. (4) ex aequ. 1. ubei dus. d, deo. Haec aequatio differentietur
u
et habebitur aequatio
ded dy. de. id est ibi
in aeq. 5
Rursus in
exaeg. 2 tollatur x: a, et habebitur aeq(6.) in qua _a ddgs. dx): dx
et habebitur rationaliter day: ddx per dy, dx 
p per dy: deo, et dx. deo et dyx: et ddx
sed hoc aequatro quadratoria reduci potest ad eam quae careat,
per x: a
et quia assumsimus hy-dx, xx  a
H Et sit habetur x(day: ddx) ex
XX + 2dyA 2XBx ax 2Xduderio poset hunc tolli. sublata hinc du per aeq.)
ipsis dax: day. Nam dydx habetur =0
constantem fit (duus - ddx) XX + 2dy X du
et ddue per equ E. habebitur
anX, et sed si d dx assurgat ad quadratum
ddx
Quod totum varon inde dantur quadraturae. Verum quia ex equ. habenus quadcaloru
tax: dx
e dix 
.l poilpere
dxV: ddx =(daus: ddx) X X + XX,:, 2xdy + 2Xdx et pro ddy: ddx
dax. x substituendo varo dabitus 1. 2 et pro dy valorem ex Equ. 1. (sed rifignorem
c, et x: a, sed non rationaliter
tenta pgnoa
aventa sgno 51