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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 189r, page id 00068196:0373
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Mirabilis res foret ista depressio, descenderet enim ususe d
imum . +Xt. ubi alia ratione haberetur rationalitas
vi. Ex periamum in casu facillimo
habebinus X + 3 P x + 35X + e
+ b
X + m X + ulmx illm
+ 2l X + lX
YN + b. fiat H aego X+ B
X + 2VX + H
∞ + m
+mXX +2lux +um
a X +ul Xx + U.
Cinferendo maego 3e 2l. aequ 3C - l aeqo C +b
C1
2ll
ergo 3el ((2 (B aequ. be (H1)
(2)
it Vyab fiet
X +l V X+m, explicalis
e. lim modo qui sequetur. pusus
faciendo X + m aequ coco fiet:
X aequeco- m. et habebinus
proy3+b habebimus
coco m+l. Co 
valore ipsarum
30 l- l’ aequ  +  Ego ssytab tus ato spro i ed vrulo u  eo
I +e - 2el aqu  el = 2C faciamus: C aequ vl. et ex aeru. 1. fet 0: 3 ul  equ ov ill - EI)
seu 30 +2 aequ 3v. (seu rrgo vv- V aequ (aequ. 26- 2tco rlo
Qui culculu spne miraeiam
seu v - V +  aequ 1 - = aequ -E seu eequ Vaequ. f-E = Ealito rusus pro e
2
pro e substituamus valorem In aequ. 2. fiet : 38p13
restitua
3. (3 
b, et quia notr
et quia habetur V, etus. ope quaeratur lope hujus novissimae aequationis, habebimus ante
e
l’aequ
2b
300 - 1 - p3
ut appareat
se sit
.Itaque res succedit, licet perquatiatur
imaginarias, quod tali casus
vitari non potest.
pideamus et pro radice Tubica ut:
ve
2
pions. reu
excutiemus
itaque credibile est etiam in altioribus rem succedere extra p
sit radice quadralica.
faliamus: y aequ l + mX + HXX + X. ergo loco b + it, habebimus X et C.
ubi tres arbitraicae, sit X+ S 3p XX + 3 pX + p ducenda in
X +  cxx + dX + e, habemus. septem aibitrarias. Ergo res gcedet
sed quonodo 3)X +. reducetur ad rationalem, seu cubum:
X + B.-ogo fiat. N + sXX + gX+h aequ X + 3VXX + 3V x + v, fiet.
X qu. X + g X + h aequ. 0. seu X + d X + 11 g- 3vn aequ . .. . = v - h
v3
ce
C- 3V
3uo
f- 3V
= 3Vy + f-3V
f-3Y
Ergo ex (C9 - 3un, +4 D- h. F. 3 exhahenda radix quadratica.
ed vero prostari potest methodo deprimendi supra dicta
Atque hac quidem methodo non obstantibus imaginariis semper habebrims
desideratum, et proinde absoluta est tandem
Methodus
carationales puros reducendi ad rationales saltem interventur
maginariarum. Quod nihil refert, quia et imaginariurum
interventu in ipses rationalibus indigemqus ut omnes ad quadra
turam absolutam vel circuli et Hyperbolae revocemus. ubi
cum Cirentus ille aut Hyperbilu describi non potest, sates hinc discimis
novam speciem quadraturae, et ad tates species etiam hoc modo
crrationale revocantur
superesset ut ide quoque in aequationibus affectis doceremus
sed raio opus est affectis, quo ad puras non reducantur, neque
enum facile quadraturis qu curvarum, in quibus utraque ordinata
b
quartum gradum transcendat, indigemus. Et alioqui constat omnes
effectus sallem inervento imtpruariam ad purus reduci
invento
ttus, 
et