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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 184r, page id 00068196:0363
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 dx aqu
 aequ
De Quadraturis omnium figuramm rationatium
per circulam et Hypelbolam adde dimentionet curvaesam
xdu  Sydx aequ Xy
Nam d Xy aequ X dy + y dx
quia X +dX. yrdus = Xy aequ d.xy posito
quod crescentibus X. crescant et y. Sed si crescente uno
alterum decipscat Tunc quando scribitur ex pro uno scribendum
est -dy. vel contra.
2x x dXy aequi
 CgX. y f 9.  Ergo + S a lu 4+ se  aqu. 
eu dato d datur sd et contra
X y aqu X dy + 29X s. Ergo s X sy 4 asyx.  aequ Xy
. exKydy + AsyyX x aqu XXyy
Et
(X dy + 3 6Xyax aequ. X’y.
sit X aequ Xy + ay scribi potent X sy aequ X gaus 4 a yt du
x aequ
ax hxyt -yyx ex +: Axxyy
3Xdx aequex dy 1. Xy G 1 ag y Ergo 3 Xydx aq u Lu Xy ly 1  + 2X dy +2 Xy dy
XLX
X xy
a
2Xcus
Ergo 37dx aequ 3 Ay au + 2x dex f. - 2 Xy. X + s +29 9
Ergo ex data quadratur assaguaae datur:  3   y  xy d
(1
a + bx + cy + d XX + eyyt, fxy +g Aet h_ + R Xy+ lXy + MX ny +p X stquy +YKy eq.
unde fci: b x + cdus f 2d Xdx +2C ydy+ fXdus + 39 X sxt3hitus + KX du El Xytus ra m d  , 1 p du   t Ertisdus 2p
sdivedet
+ fydx
tqu teruguto
x et pro dy scribendo seu pro d
+2Hrydy + lyy dx
seu dividendo per x e pro x scribendo. V. fici:
b + cv + 2dx + 2e yv + f xv + 39 X +3hy v + KX v + El xy v +4mX +  g C t3gxyr   
x+ 9 +erxyy
+2K XIs + l yy restabunt X et x. in aequat.)
epe harum duarum aequationum, tolatr 1. et 3. tonatur
vitur regnula
y: restabunt X. et V
deer data initur curva cujus fi aequatio explicata sit per ordinatam a
jam
natura
et abscissam comp aequationem inter abscissam Xe et ordinatam v. comparetur
aequatio data 5 cum arbitraria 4. quae comparatio si non succedit impossibile est
cuvam,
eofiguram curvilineam datam habere algebraicam quadratricem.
s sed ut per partes eamus, quaeramus primum curvas rationales quadraboles.
est autem corvae ratiovalis quadratri, etiam ratio nalis
X
+
 X aqu. Z X  Ergo ZS_ X sx aqu. X
H Habetur autem quadratura omnium figurarum per aequatio valores rationales
integros explicatarum
ut R. a + bX + CXX + dX etc. Erit suum quadratrix
y aequ ax + b XX + CXY + d X etc.
Habetur et quadratura omnium figurarum
v aequ x vel X vel X vel X etc. vel
215 1
5
 vel L. vel  etc. quorum summae:
put 106
simplicium rationalium denita
1
ubi nota L esse aream spati
ex Hyperbolni
infiniti
X
in Tilimeo