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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 7
# Folio 4v, page id 00068004:0008
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2
by2. b78.
demonstratio generalis de irrationalibus:
sit quantitas
sie aliqua  2b. poterit eadem et sic enuntiari.
b b
E
+ 1 - a + b - s - ac Quanquam eim V- ac sit
nons
quantitas inaeginaria, summa tamen ideo, minus est realis; quoniam
imaginarie destruuntur; Dividatur haec formula in duas partes
binomitem et es b+ Y-ac, et residuum  - Y-ac, et
utriusque separatiu in vestigetur abus erit ipsius bgr-ac cubus hic:
 +b = ac Y- ac et ipsius b - s-ac, cubus ent  + b +ac f- ac
3bac. + b ....
adeoque erit
16) +b3 = ac Y- ac + +5) +b3 +Vac L- ac
sive
-7bac +2b..
-3bac = 3 62....
ibac  B B +Y)  bac 
btac
b + f- ac 4 + b 6 - - ac
-3bac -3V....
e 2b
N 2b
cum residuo suo conjun
quodsi ergo extali binomio emsmodi semper extrahi posset radio
Cubica, quemadmodum exhoc quidem potest utique semper junctis inter
se binonno et risiduo semper hebebitur tolli posset imaginaria
sed quoniam data ejusmodi expressione:
16) = +Yr -93 +- Y- 4 qualem exhibent subicae
4 37
211 4
27
quia Cquantillas data
aequationes, non semper extrahi potest, id est non semper (Y. in b.3bac
data
la rec quantitas    93
in duas atres:
27
aC3 + 6acb- 9btac dispesci potest, a nobis, sine alia
aequatione, aeque ac data, difficili potest; ideo fit ut nalitte
ex quantitatibus realibus. non semper possimus imaginorias eliminite
rationalsibus
Exempla autem in rumeris proponere utile erit
sit
aequatio qua et Atter Altertus Girardus utitur: X X = 13X - 12p0
i cujus radix vera, est 4 constat autem ex regulis
Caud Ex formulis autem scipionis Ferrei sive Cardam erit
XY 15)6 + s- 1225 + (0 +6 - s-mis
7
videamus ergo
an posito quam expressionem rectam esse et realem, et admissendam
sic demonstrabo: Ponatur X ( 2 + -t + 2 - f- 
erit utique X 1, X uti aequatio postulatii videamus nunc an inde derivari
rimu
iu
Cutus  21 Su, erit 18
possit formula cardanica
-37
2 + Y.
faciendoque
b + Y -ac
b H2. et ac 1 2
habebimus pro cubo ipsius bireac formulam hanc:
3, 2, 3 2
fa
+A..
sire 6 Nimirum cubando, et Superiorem formulam
+ b - s- ac huc asplicant
+8
46, 2, 4A sive summando 6 + Y -nc
9. 16, = p 41 296.
Eodem modo cubus a 2 - f. erit 6 -  s = 1225
ac proinde
3) 6+ s = 1225 erit 2 + f - 3. et (36 - s - 1225 erit:
2 = s.2: ac juigendo binomium residuo erit
X sive 13) 6 + s - 1225 + 10 6 - +
Es idemquo quad 2+ s. t +2 - s. 3
id est erit 4. Ut ostendere propositum erat
vel
Ergo
sive ac H = 13
8 - 2, 2 ac p 6.
8 -6 ac.
486
29b