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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 185v, page id 00068196:0366
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si
quandocunque nominatoris formula habet omnes radics
possibiles, semper curvam quadrari posse vel absolute
vel ex supposita quad ratura Hyperbolae sed quoniam
hroc modo non semper entatur nominator Resolvi enim poterit  realor
alioqui
in tolidem valores habentes nomnatorem simplicis gradus, ubi
nihil refert quid sit in eorum numeratore, mod o omnia sint
rationalia.
Video tamen et in aliis remedium
succurere. Resumamus D aequ.
faci summmus V C, A+x aequ.
Ergo
a + bx + CXX + X C
1R+ X.
A(+ mX +XX
X + 1 R multiplicemus per X + f fiet X + fX + 1Rf
Hinc v f aequ.  f + X
l+mX+ XX
vD aequ. B.
IR + X
0
aequ
iRtx. Umxtx
seu Oaego D. X+  D
V tamatex aequ
t (4) +2. 1R + R ( XX - mx - C
act bx t CXX + X
+. A X
IR+X
1R+x
lAmXIXK R+X
PP- D aegut XX + fX + 1Af
+IAX
= XX - mX - l
fiat f aequ. im = 1R. et habemus.  & - D aequ 1Rm - 1R - l aeque m - iR+X 1
.a + bX+CXX +X ltmXEXX IRtX
balorem denominatoris cationalis cubici ad interiorem
e
reduxamus ergo ad inferiorem
Cumque in aequatione Cubica qualibet sit una semper radio possibilis
manifestum est. Cubicam semper reduci posse  semper procedere hanc reductionem.
Ladem methodus etiam in altioribus adhibero potest.
Quodsi
Si sic aequi q phominator valoris sit formula quadrato- quadratica
 eaque habeat radicem possibilem vel unicam, semper
reduci poterit ad gradum proxime inferiorem.
si vero omnes
habeat radices impossibiles, tunc saltem dividi poterit per duas. formulas
possibiles.
Verum tunc non tolleturindeterminata
verum quia per superiora omnis formula
in qua est numerator in nominatore deprimi potest, ad proxime
ex numeratore.
inferiorem, saltem adhibita imaginaria. Ut diximus sub finem
paginae ante- praecedentis, hinc Nihil id quidem refert
Si aeu formula sit quinti gradus necessario unam habet radicem possibilem
Si formuta sit sexti gradus
Superest jam ut investigetur quaemodo figurae eatio irrationales
revocentur ad totationales aequipollentes.
a X + bxy + Cy + d  0. est aequatio ad quadrabilem, quascribatur jam lx +  aequ.
+e aequ.
lX + m y + n aequ. X. et tollendo X prodibit aequatio:
it + gln Z9 + 2li Z + X yy + 2nn. y +nn ) aequ. 0. quae formula conferenda
+bl Zy
Abmyy +bny 
sit a + by ex y  e
+ dlZ
+cy
dl
+dmy +dn
cum data selundi gradus
Rradicis extia quadraticae extractio ex qualibet formuem integra, ut Y a + bcaa ad etC aequ. y.
scribatur X aequ l+ n Z +erc
p + 9Z12
etc
eoque valore substituto, pro X, ascendetur ad 7. Imo faciendo
Nut denominator sit quadratus ascendetur ad 912. Comparandus
quadrato, quod quadratum fingatur etiam, a formula sexti
gradus, in qua cum saut Z arbitrariae ea his sex additae dabunt
alieus ad fercle atem opemnde sunt chum serimr