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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 7
# Folio 5r, page id 00068004:0009
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Unum adhuc exemplum ad ducere operae pretium erit
cum radii
rationalis in Car irrationalibus Cardanicis latens est falsa
sive negativa; quoniam variatur nonnihil calculus et video raph
Bombellum, analyticum certe doctissimum egregium certe artis
analyticae magistrum, hic haesisse: nam ut supra dixi radiles rationales
veras extrahere potuit, falsas non potuit. Eadem est difficultas
Exemplum esto:  X - 12X - 9  cujus aequationis radix salsa est, a, - 3.
ex regula
erit C. X H
a
a
10 2  N 175 + A9 = Y - 175
et opus ergo est, et ostendamque hanc quantitatem facere = 3. quod ita
fiet resumta formula superiore, et pro b ponendo - 3. ac pro ac
faciamus
Sumendo 2. Ponamus XF - 3 + s - 7
= + - 2 id est
X p - 3. Unde cuband binomium, pariter. 2 + f -g, ac residuum,
- 3 -  - 2, habellamus primum ex binomio: - 27
+3, 3, 21 63
minus; nisi aut immenso labore omnia pertentet, aut singulari quodam et quoniam, ut postea dicam utile est, c de inveniendis ope
1(-343
54
16.4
l-9, 8
irrationalium, rationalibus cubicarum aequationum radicibus agitur, efficis
ut secundus terminus penultimus sit exacte divisibilis per 3. et ultimus
per 2. ideo quod consequemur radices aequationis propositae multipercando
per 3 cum defectus est in penultimo, per 2. cum est in ultimo,
et per 6 cum est in utroque
falsa
Exemplum esto X X - 48X - 72  0. cujus radia est
Xp - 6. erit ex regula
XH 1B 36 + s - 2800 + 16 36 - + - 2800. ostendendum
est ergo harum duarum irrationalium summam nihil aliud facere quam s
. Quem in finem ponemus X , - 3 + f-7 - 3 - + - 7
C
O
B ∞
sine XH N + B, vel Xp, 1C  + 1C B73
Resumta ergo formula superiore qua erit
A, b+ l-ac  8 b - Y- ac f et p
ac nunc pro b substituendo 3 et pro  ac sumendo 7. erit
A.
sive summa inita 
-27
13, 3, 7163
36
343
+Y, + 6, 49, 912646
- 9, 81, 7 r, 5103
+ Y - 2800, ut propositum
eodem modo ostendetur esse B
esse ) 36 -. Y - 2800.
eB
EXBAE
BECL.
EIANAODELS
hufo ex  estar Ergo cum ex regula Cardani sit X
(.4
omon
XH (036 + f.2500 + cum ergo sit X  (B-6p, 1( + se Y,
erit X H 1336 + l- 2800 4 + (5 36 - Y - 2800, ut regula Cardam
praescripserat, quod spendendum proponebatur
- 64 4
fiet ac p
sive ac
31
81
15
1c
+2
73
e
+2
18
+981 1458
378
123
t3C
427
3636
5
4
5103
5446
2646
2600