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# Title: [Ps. 1] (shelfmarks Leibn. Marg. 178, Stück 1; Leibn. Marg. 178)
# Folio 140, page id 175905688X:0162
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F L. OR TM O HDE DE BE AMNL
BIna PTOPONUNEuE: exhibenda tantum restat demonstratio radi-
cum, qua ex ipsis, secundum Cardani regulas, quas superhac re
in medium aftert Capite secundo libri ejus, quem de Arre Masna
Seu Regulis Algebraicis inlcripsIc, educuntur. Cum hoc iplum
difficulcarem Force non exisuam parere polseriis, qui in eundem
locum aliquando inciderent, quippe qui a Speciola AISCbFX, e
mucux incer Arichmeticam e Ceometriam relationis atque con-
renientix ignaris, non facile percipiatui. Quocirca ut veritas
exrractionis harum radicum expendatur, demonstrabimus pri-
mun sequens
14
ascendune, reduei posse ad aliquam formam carum, quae hac pa-
L E M M A
CFctâuteunque linea AC
Din B, Ostendendum Css
Cubum linca AB, una cum
cubo lineae BC, _ triplo
producto lincarum AC.
BC, AB, Cumul aquari cubo linex AC.
Sit AB ∞ a, B C ∞ b, erieque A C ∞ a F b. Productum li-
nEaTUI AC, BC, AB, eric ba a F bb a, cujus triplum 3 b a a
+3 bb a. FIuic C addantur cubi lincarum AB, BC, FCE a3 L 3b a a
.3 bb a- L b3. Ec manifectum est, Cummam hanc aqualem esse
cubolinea AC.
Demonstrato itaque hoc Lemmate, habebitur primo Loco
VC. 1 :9ut V = 49. ] :, P - V C. - 29.  V +94 + 1, P3.
Ilinc in tigura adjecta supponendo binomium
VC. + :q.  V 294 . + , p3 aequale lineae AC, _ residuum
Y C. - = g Y V = 99-L, p3 aqualelinca BC, erit corum
ditterentia V C. + : q ) V = 99- L ,, P
VC.  =qu V +99.  , P3 aqualis linea AB. Tam vero
Ctacuendo A B ∞ Z crit difterentia Cuborum ex lis radicibus
nimirum ditterentia inter cubum L = g.  V = 99 - ] 2 P3, et
cubum - = 4- = V = 49.   P3, Causerendo hunc ab ill0) aqua-
lis g. Oux propterca aqualis eric differentix inter cubum li-
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