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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 235v, page id 00068201:0470
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+ + cX3 + dX + CX
H
- b .. _be. = bd - be
Ponanus b. esse irrationalem quadraticam. Productam aequationem esse
detere rationalem, Erit ergo be rationalis. Ergo quia b irrationalis
quadratica, erit et e irrationalis quadratica. Aliequin enim ex irationali
in rationalem fieret irrationalis quod foret absundum. Sit jam b irrationalis
sumplex id est cujus quadhatum rationale quo posito necessario et C erit
Arationalis, quoniam -V + C. dibet esse quantitas rationalis. Ergo bc est quantitas irrationalis, et
et quia bc + d est quantitas rationalis, ideo d debet esse quantitas irrationalis.  Habemus
ergo unam rationalem c, tres irrationales b. d. . Debet autem be esse rationalis, quod ita
reproesentabimus. b  β + O+V
et e dB - sB). Factum enin extris duabus esst rationale
et cntinet haec formula omnes casus, quibus duaequantitates quae binomium non excedunt, i se ductae sunt
rationales. Potest enim et, ( pini p0. qui est casus simplicissimus
Sed Supersunt  duae adrus
Sed si β3. est quantitas realis tunc C non erit rationalis, nisi certis casibus cum sciliiet Iad simpliem
irrationalem reduci potest. Itaque nondum quicquam determinavimus de quantitate C. Retineamus interim d, et e
unde necessario
quales praescripsimus. et erit - bp β  - 2β6 5γ. quid si penamus C⊓ 2β8+) + ut sclicet - b4 c sit
- D27
et quia -bc +d
rationalis. et -bc p -+ a = .D sV,  + d C. debet esse rationalis, euta recessano, de 2. 
= 2βDγ = AB..
Unaae jam superest implenda conditio e - bd quantitas rationalis, id est Psd autem eb d p
8β - 56s) - bdp a2 β 6 1Y = e β
Sive s
+22y
=BAB..
 et..
2β3 - βA - .
=XO2.
+2βd19
+ A B..
 et est dβ- sB 1V, et debet esse
esse - 107 - B B - n _ B - d B p0
quo facto habelimus obtentum, et fiet:
 +B X3 +  X2 + 2B B1V X + HY
+0 sy.
er
1D..
 d.
234
177
X + bX + x - dX +C
X
a01
17
+ 2.
+
2D..
361V.
= AdB..1
a (2 P2y.
2B27
g
310
2362)
bi  +
n +e
obhmn
ubi artitrariae
sunt, (.V. D. A. B. sed O. est inutilis, ett
sumi potest pro unitate. unde restant
aibitrarica numem 4. (3. 2. )M
ex quibus sola omnium β3 ad altioaem solit
omnes praeter β.  ad nullam oscenditus
pote tater
Venea +  3bc i +6Vcn + C
3be2
1291
AB V
Sit jam  BB b m AC Y dnt t en +5
A Bgm +h   l n vn .Z
 aBcn +ve n
annm suum
6mumt
4mun
+ 4bct +C
ina
Arbatrariae sunt 10. b. crd. e. f. 9. h. l. v. Z.. at aequationibus subsidiatis opus fore hnlus, 2. 5. 4. 4. sp. 13.
nihil ergo confert haec methodus.
se