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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 42r, page id 00068198:0083
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traestat ergo retinere gereralitatem aquationis D faceere
bb + 11. H + 31. 31 + 2.7131 = 4 a (11+31) + 1131 - (11+ 31)  B, 11 1135,-
49aa
aquandum quadrato s ubi seu bb - F - 2 (11+ 5131) +6. 1131 -B b (1139
36aa 3a
seu (1 -- 3t - qu - 3 qu (1 - 31) = b (11431) + bb - 12 fa  quod
35ga
aequandum quadrato. sed idem licet aequare nihilo. Quo facto
= in duas divellemus aequationes, unam -2 (1 - 31) + 3 bb ee, seu
11 -31 = +bria alteram. 3 bbee - b (11+31) + bb
a a
seu 27 27  (27 ee + 1) bb - 12 a5-
 36aa
4 aa
3taa ha
(11 +31) ubi e est arbi
taria
et n ita solutio habetur rationalis, modo calculus sit probus. Ubi
e arbilaria ita assumi pt, ut numeri fiant quam minimi
2 ex. g. si e esset 3 foret 11 - 31 = + b: 3a et 11+ 31 = 6 bb - 3af. 2
9ab
 et 2. 1 = + 3bb + 2bb- 6af. subi si  + = - fiet 1 = - bb - 6af
 et et 6af = 3bb -6ac e Ergo 1 = - +bb +, 6ac
et 1(- BR = 31 = 11 +
2.9ab
2.9 ab seu 1 = = 2bb + 3ac
da Ergo 31 = - 2bb + 3ac + 3bb = bb + 3a5 -3
 Et 21 = 114 31 - ha = 2bac - bb - 3 bb - 6ac - 4bb
itaque 11 = 3aC- 2bb 21 = bac 4bb
gab
gab
ab
gab
31 = 3ac L bb 9ab
at vad
10 = 20 = 30 = 1
attimativa, incideretur
in imaginarias quaener quae cujus que haberetur 11- 31
+0
et huic emtando posset fier. 3 qu (l-3) - 12fa + b. aequari 0)
36aa
et restaret - b (11+31 + bb
- b.. -0 ita b non inpedietur
vaaem expo divilo, nominatorem in valore ipsius 11+3)
36aa
pplicatione facta ad Ellipsia et Hyperbolam repo videbinus quae
sit conicae contricae aequatio 2aX + (a: L) XX = yy sit erit adx +(a:b) Xdxcydus
sumtio optima
seu dy: dx = a + A: b) X, :y et dydu: dxdx = aa + 2(a a : b) X + (a a 1bb) XX, :yy
et dydy + dx dx = aa + 2(aa : b) x + (a a : bb) XX + yy,: yy, seu fiet
aa + 2(ac: b) + (aa: bb) XX + 2aX + (a: bGXX , per Z ax + (a cbx. fiat
aa tatb: V)ju :gu  + X = Z + C, et fiet 2ax + (a: b XX;
seu aa + (a rbs: bY
(a:6) ZZ + 2 (a: b) cZ + (:b) cc + 2aZ + 2ac s, destinantur terminus in quo
simplex Z, et fiet 2(a : b) C + 2a = 0 seu C = - b adeoque X = Z - b et fiet
a: b)Z3 + ab = 2ab = gy seu (a : b) 33 - a b = yy seu Z2- bb = b yy
Ergo tandem dydy +dedx,: deax= Caa) B aa) 8 ab H(a a. bb) ra : b)) ZZ.:  ( B) (a: 6) 2Z - a b
 at b, 5b) . L) = (a + b a : b1Z3 - ab,: (a: b) ZZ - ab
b: a). et dzdz: dydus = (yy. Z2) bb 
et dzdz + dydy  dy dy-(99 4 (bb: a4), + Z2,: , et fit ((bb : aa) + (b = a )) 2y + bb,: (b. a) 12 + bb
ex eodem valore ipsius aZ. ay dy: dz habuissemus dydu +dzdque: d dzdzue hoc modo, examinis causa
dy: dzue -Z. 9)(a: b) aydys: dz dze - (27: 9y) (ad: bb) et aydy+ dz dz. : dedZ = 23 (ac .b6) + yy,: yy, seu
(aa: bb) +la: b1) ZZ = ab: (a: b) 22 - ab, ut ante
Mutemus d et in Y, et b in g. quoniam ipsus literas a et bsupra aliter sumsimus, et
TCi Fertarquu +rY9q (r; et q)
reducta re ad atque ordinata ad scopum nostrum fiet
(est atr4q2Z + 1r44 4 C4, r + 4)
= dy -
yy +rq,  Ver:, r+4)
itaque nitaque ut irrationalis ducta in dZ vel altera ducta in du, divellatur in  irrationales simpliciores Methodus
-94 (4, r+
lys dus + de d 2) = c
Nota in hllipsi e semper majus quamr, circulo excepto, ubi aequantu
C in priore semper est affirmativa, in posteribre est affirtua
acaso ettes pepr in ces ae tur e su i put e aste
n Ellissi lonst n aserntra rationali destrui bZZ vel b0
ZZ (b, aarab) 3, Z
hastenus explicata