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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folio 13r, page id 00068072:0025
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seu dv: dZ  Ed Z d - 2 d idq, :,  de d + 24 Z dZ.. quo valore sobistituto in 1 - rdr. a2
19
- 2 dvav: dedz, fiet dG + 4. l des dz d z + 4d3 C542- 4 (d7 d9 - dz 2) (d  de + 2d7 d 2)
- 8 qu (dZ de - dzdZ
 qu (d9dg +27d5)
seu ex exponendo,
e
d92 + 4 de dis did al ag3=g2 dgo az a deaus + 4t 4 d   + 8 d2
sad
(=8 dZ dz d9 de) +6 dZ3dG= 8 dz
s
1 - 2dv idq -2
3: ququ dde + 2dZd2
dlae 247Z seu do
44ue d03+ 8 d3 de +4 d732
 eiAEiqu.
itaque 5.= 4 dp: deo debet aiqual quodrato est aut Ergo per 13, debet quadvato aequam
divitione facta p
(A (+ . dvidque  2 duav: +8. 1( 4, quadrato aequari (17) quadrato ab 1-dvd8: dzaz
-2 = 2 dv: dZ,  2 d : dZ + do dedg. d3 seu dv: d 2 deg: dez - 2, :, d Gde: djdz + 2
seu dv: dz (7)  231 d21:, da des + 2d7 dZ. quo valore substituto in
seu qulcato
1 - 2 dv: dz - 2 ardv: d3 dz fint dc) A ade d c dz d2) + 4 dy 7 - 4 d Z de3  8 d 23 dg  4d2 du desdes
(+ 8 d. 5) = 8dZ d7 d d + (16) 47 d9= 8. l8
:, () (ade + 242 d2) adeoque
erit 1 - 2 dudzue - 2drdv: d7 dZ= (2 X (d ep des - 2d7d2 3:, des de + 2 dZ d Z
A
seu
quod et patet substituendo vatorem ipsius dvidque ex aequ. c9 substituendo in 1- dvage aza)
nam fit 1Zd + 2d7dZ C 2dZ454 + 2des d3 d7 9: R. de d8 12 d7d3 (2d2a8
8 -d dc + 2d7a3 (deqas  d
a
2  21)49-2d3=
q+ 2d7aZ
dcdes +2d2d2
2dZdg dZ4Z ag -48d9 + 2d7 aZ - (3) d8 de + 2 d Z des
se A Z dd) F 2d3 (2 d dgds + 2d
qu Hoc et comprobatur si in insin  -d i: deo) valore invento 1 - dvde  ddque por av: d ponamus
dade +2d7 dZ Z
valorem, nam prodit 182d 2d2 ax de = dZd7 (+ d4 dD- (2d4 d8 + 2 dZd9 ut ante
1C 222
dz dzu +des des
quia ergo de: dZ = 2d+ dd et ita possumus obtinere quadraturam rationalem.
d2a3+ d5de d
 sed prius absolvamus  jam obpiam ex aequ. 16.
Iddo = ( dx +  axay + 1 +  (1 - 2dy - 2 aodo)
d3d2
fice eta  Ex Hoeri est
duar (+do +1 Ao =  - 2dx -2 dudo
d2 t2
22
Unde fil ddZ (22) =d (dv: d3
sed hinc jam video methodum istam procevere non posse, ita enim Cum enim
de
dZ
dzdz, d2
per initio posita vit ddz: d7 = dx: a, et d(dv:d Z) = dxta dy: a. Ergo
ex aequatione 22 haberetur du (ddz: dZ): d (dv:dz) seu du: dx. sed
jam ante haberetur dy: dx ob aequationem tangentialem datam. Ergo si haec methodis
procederet, nimium praestaretur ham duidx habertur bis, atque adeo relata haberetur
inter X: a et y: a sine quadraturis quod non licet.
An ergo licet facere X: a = (1) (ddz: do et y: a ( )4Z: dv et dyr a = ddz: dx
et dy: _ = (4) dvddzue - dz ddv,:, dudo, et dy: dx = d vddz - d Z ddv, : do dv) : ddz (: d v et
= d B daqu - dZ ddv,: d ddzue  du: dx (1 - dz ddv: du ddzue. sit jam aequatio
data dy: dx = XX + yygiaa seu dy: dx =D qu 5ddz: dv + d2dz: dudo et multiplicata, aequa
tione per de, fiat du dy: dx seu dv = a Z ddp: daz = dz dz: do = (9) quav qu sdaz: d0.= deo
d (dv) 2(5cdZ: d0) ddz (: d v) + ddvies aaz: dv =0 et ita forel
2 ddze + ddo (ddz: dv 20 sed vereor ut hoc liceat, nam ita fieret et X: a = - 2 dZ: dV sed ita
data aequatione unde
dco, sd  Et erit
per compendium; et fiet
per 6, daretur relutao dy: dx ex y: a et x: a ante data, et ruit
nimium haberetur. superest Ergo potius facipium +dZdav: daque + azdz: du
dp - deo ~) dv qu sadz: do et dferentiando posita, et differentiando ddv =(12) A dd: dx) age
4 dav qu sdaz. do sed ita video similiter  saz: ax hat seua seu xia haberi dav: daz vel
viceversa. A adeoque haberi quaesitum ante quadraturas, Ae proinde redire oportet
ad methodum ubi X: a  Sdaz:  et H: a (2) Sddz: dZ. unde dy: dx = B) dv: d2
cui melhodo nondum habeo quod objicere possum. Etsi suspectum sit, quod non
possum fabricare exempla apta