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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 7
# Folio 10r, page id 00068004:0019
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 Nunc Demonstrationes genantes, quae retionales
Quunguam autem non semper radices cubicae ex ejusmodi binomiis vel residuis extrahi
patet tamen illis inesse ac pirinde
quamvis
possint, nec proindi P destrutio imaginariarum quantitatum semper palpabilis reddi
quea
non possit per calculum, semper (nisi qui ad sequationem analyticam, semper tamen
o
in ipsa ve natura factam intelligi pac
liceat
irre fragabilibus extra dubitatie penere
 demonstrationieus tamen confirmare operae pretium erit quarum duas in promtu habeo
ex pri quibus prior quae nec rationales nec irrationales, aequationem cubicarum
radices morentur, ex quibus prior huc redit. Formula Cardanica satisfacit aequationi
ut n
ostentam
Cubicae Triradicali, seu tres radices reales habenti, omnis hadix aequationis cubicae Tudadicalis
Ego est ejus radix. Omnis autem radix aequationis cubicae triradicalis est realis, ex hypothesi;
tres enim habe (tres enim habet radices reales exhypothesi, et nulla per nec vero plares quam
tres sive rales sive imag iniginarias habere potest cubicanlla). Ergo. Formula Cardam ea
ex aequationis culica triradicali applicata est ducta est realis. non obstante scilices
imaginuria quae ut a schetem L- q quado Cum autem in omni solaque quantitate
aequatione Cubica Tri-radicali intero  formmeais Cardami imaginaria quantitas 
esta
 popto P esse ingorem quam Y:) interveniat ut a Scholenio ostensum est
n’appendice t de resoratione Tubicarum Cart Com; in Geom. Cartes adjecta:
la
asu in Tuta
4
ia tere
27 aequationu
demper esse realem
sequitur  ergo non. formmulam Cardanicam in qua imaginaria intervenit
Superest ergo ut ostendam satis facere. Sit aequatio X+ = qX + r 110. eu erit
radix (ardanicax  Y + F - 4 + S∞= N -9u. quae satis faciet
12
27
aequationi si in locum irta ipsius X substituto in aequatione hoc ipso ejus vatore omnia
hactunquen antelestum fore
quia X p) A + B, et A, q. ri
a
destruentur. Quod fieri ita ita patetit.
X+
A3 +3 ABA + 3ABB + B et 3AB,191O= + sr. q
Xo15
 X H
gX 1
r
+2262. r + r C3 ABH91 0 -tr q
A + 3 AB) + 3 AB7 + B
4A 4B
sive explicando A et B.
Quod hoc calculo fiet: etsi
treviori fier potuisset:
sive explicando A. et B.
est auem B
3AB F q 1B)
B4 eF B5 13
per BH 1B
iabit B 1 se
(C+  
4
et+
AnB1C
9
PhAB (O= + sY-4
dalit s Z +Y sr 
27
Ut ex hoc calculo
patet.
dote
12
MBM (O
12
-+
47..
se AB
unde
sive 3A
 11
1
2
(62
2
-
ine AN p (3)
0
Vr
aude 3AB ((916 3= Y