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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 185r, page id 00068196:0365
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primo valores
Sufficit ergo ut eos consi deremus rationales ordinataum
Curvarum quadrandarum, in quibus solus nominator est
indeterminatus, ut
a+bX+CxY F AX
erc
a+ bx a+ bx+DXX
sumamus primum aquo jam divisores ipsius a+ bx + XX seu radices sunt
a + bx + XXX
X + b + ist -a.
et X + b - Y.
seu per compendium X + 1R et X+2R. .. fiet ergo
aequ
a+bX+ XY
aequ. v. am Ergo V X+IR aequ -
XFIR. XtZR.
S
X+ZR
et v XtrK aeqo R
 IR - 2R V aequ. -
et Ergo 
Habemus ergo.
1
R- 2R aequ.
a + bX +XX
X+2R, X+1R
X+2R X+ 1k
Cumque  R et ER sint quantitates constantes, habenius omnes Curvas
rationales quarum denominator est formula secundi gradus, reductas
ad eas quarum formula est primi casu, excepto casu, quanto
1R vel 2R est quantitas impossibilis seu imagnaria, sive quando
a est majus quam B a est quantitas positiva major quam . tunc
positiva
enim Yb-a est quantitas impossibilis. Excipitur etiam casus.
quando 1R et 2R sunt aequales. Sed eo casu jam aliunde habemus
quadraturam, et quidem absolutam, est, enim valor rationa
Hinc patet cum semper pendeat
a quadratura Hyperbolae hinc omnes curvas rationales simpti
lunc valor simplex.
valores ordinatarum rationales quorum nominator est secundi
gradus, et habet radices possibiles reduci posse ad quadraturam
Hyperbolae. Caeteri  qui radices possibiles non habent
valores
possunt solo indeterminatae augmento vel diminutione multiplicatione
et divisione per constantem reduci ad hunc , qui pendet ex
1+, X X
me est osteusum
Itaque
omnes hujusmodi valores erationales secundi gradus vel absolute
(1
nominatori
vel ex supposita circuli aut Hyperbolae quadratura quadrari
possunt.
aequ v. Et ponatur formula nominator produci
posse ex duabus formulis in se invicem
si sit a+bxtcax
ductis: l+ mX+ BXX et 1R + (X. Erit v aequ IEmXHH. IR+ X
 et V Hma tXx aequ
IR+X
fier.  IREx aequ
l+mx +XX
lam XLX IR+X 1 +mX+ XV
Ergo: Ergo XX + m X + q aequ, f
Sed Quia hoc modo  XX _ et X non evanescunt in nu-
a + bx + CxX + X
meratore, utendum esset tribus radicibus sigillatim, ut
supra duobus et haberentur tres formulae, duaeque arbitrariae
multiplicantes duas ex formulis tertiae addendis, quarum ope
tolli peit psent na a t.
Hinc patet eiam etiam