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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 3r, page id 00068196:0005
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qua Quaentur (dx V(XX +bXx + ac). sit xe fiat () Sydx
fiat y -(x + Z et XX + bX + ac = (β3tio) XX + 2x Z + Z2 Unde erit
X = (4to,, Z2 ac, 8 b - 2Z et dx = (5to) 2 du (:, l- 22) + (22 ac, X4 ( (b. 22)). 
et 9 = 22- 4c + bZ - 27Z, :, b - 2Z seu y - ∞t), -ac - 22, :, b -27
et denique ydz erit (mo), = 2d Z(,  qu eb- 22)) acZ - (3 0 aacc
b- 22
contrahendi causa, faciamus ) b- 2Z =  2B, fiet dque  seu eque - v
et ZL ib V  geosguo) Z= 2V et  (10 mo) Z dc t V + : b
Et ex aeq. Ellisse et
Quaeritur se
si quaeratur dx x + bx +ac) reddendo extrahibile quod sub oinculus
est per extractionem ipsius X, non comprehendemus casum ellipseos
vel circuli, nisi interventu imaginariarum multiplicando scilicet quod
provenit per Y-1, idque facere sane curiosae disquisitionis erit, nunc
ut hoc evitemus scribatur (ayB, faxSex VabXX + aa Ex+ a f) ubi
drimo)
video duplicem rationem occurrere, unum assumendo ad obtinendam
extrahibilitatem, unam si destruatur bxx quod f si b sit affirmativa;
alteram si destruatur af quod fietis ipsa f sit affirmativa. sed
potrst
vellem quod si utraque sit asfirmativa adhiberi etiam u alterntia  potest
via, sed si nentra riffirmativa sit non procedit extrahibilitas nisi
praevia trans formatiore. Quam ut itaque ad ad calculum generale
habendum oporteret facere transformationem etiam in generali formuta
Sed nunc utamur eo modo quo tollitur a s. It aq
sit ergo 3ge (ado) gy 9 f2di)-9
pro ae f scribamus aagg, et fiet
(primo) sagdx = Sdx Y (abXX + aacx + aag9) et 2d0 frat
Caaque
(2d0)  3)=  GCay = X Z rg, unde (3tio) abxx +aue Z tangus a  ze rng t
unde (3tio) abxx + aa CX + aagg = X XZZ+ 2agXZ + aa gg, 
et (4to) x = 2agZ - aac, : ab = Z, et y = 2a97Z aa c Z + aabg a92 27
ab -22 seu (5t0) ag = agZZ - aa cZ + aabg,: ab Z2, Et dx
Et (510) Et dx = dZ, rag (: ab - Z2) 1 (2agZ- aac, ∞:, qu (ab - 22)) seu
(7mo) dx = 2dZ(:, qu (ab - 22)), aabg = a9ZZ + (2) a9ZZ - aa cZ. Et
tandem (880) (aydx = 2s, dx, qu (a972- aa cZ F aabus) :  cub (ab 22)- quae
aequatio rationalis evolvi potest, modo explicato a me explicato in
sit Sed Nunc tentamus et alteram rationem, et pro abxx
actis ejuditorum
scribumus Hxx, nam video calculum ita minus assurgere. CC
et sit mimo) sydx = Sdx A(XX + bx + aC) unde si pro XX stet - ZE, oporlabit
pro X substituere ZY- 1. fiat (2d0) y = x +Z, unde (3tro) XX + bx + ac
XX + 2XZ +ZZ seu (4to) X = Z7 -ac, :, b - 2Z compendii causa pro
fiat (5to) b - 27 = - 2v seu  et pro b fint 2f (6to) b = 2f, et habebimus
(mo) Z = v + f et (8n0) x = v0 + 250+ 5f - ac,: - 2V, et (q y - vv + 2fv + ff
. fiat a 1G.  = ac - 20V = 2vf, : - 2V seu (9n0) y = vV + aC, B, : 2V et (uma) per
itt compendium  faciendo (10mo) ac- f = ag, fiet, X = ag - vv- 2fV, - 2V
 slotoro et = (12 mo) Ry = a9 + vV,: 2V, vel (13t0) 2X = ag ) ag(:V) - v- 2f
l
t1 et
et (14to) 2y = ag (.V) +V, et 2x (15to) 2dx = dv, _ ag (vo) t
66t0)- 4ydx = ex do ag (.) + v trag (.v) + ag (L1) seu donique
(ymol ydx ed 
rag(VV) tnagu(: V4) et si ab sit
rit (1800) ve=hato, fliet (19no) ydx =ad, 129 +d
aduq si hiaec ad Evcndum abplicentur ant ellipsin