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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 240r, page id 00068196:0475
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Quaeritur Sydx posito y = LaXtL, XEm, Xn) per quadraturam rationalem
Hoc tentavi assequi si possem quod sub bliaculo est reddere quadratice extrahibile.
sid est sint tres quantitates A, B, C; earumque biniones AB, AC, BC
Manifestum tres biniones invicem ductas facere quadratum est enim AB. AC. BC = AABBCC
sit A = t + 11SOD + 1210 + 13SQ in omitto coefficientes, quia ab ipsis O, D, 4 alsorberis pos
B = 2 + 10D + 1OG + 1D9 sitax + L = AB, a, x + m = AC
qXin= BC
Fietat Caixtl x+m, Xtn) _ = AB
C= + 3, 10D +32 10G + 33 1D9 Ergo Xet ax = AB- al = AC = am = BC - an
+∞. ∞h2
100.00ti8
H1. 31 (3D
12. 3200
13. 33DG
-0000 am
000 26
+00t 10D +00t 109 + 0ot 1D coincidentientur AB- al ~ AC- am
+112 ... 1224 -. 
+ 16 .
118....
41319.... 13..
+45. 31t 0D 00zetu Vee
00. 11 .... 00. 128 ..
12. 338 in. 1.337).
13. 329 1,1. 13.319 11
00f... 0051.1.
318F  324 11.
326 ..
315)..
33 . 11. 33D...
137.-.
11C...
00dt DE
00.138.
H1. 326.
13.316..
BC- an ~ AC - am
in irrationalibus ex priore
G(L. 12 +ca 1
-12. 3)-13. 52
9- 00 700t +112
-31 l = 118
+6232 80D. 12
14135Z - 00. 13
 = 00, 40o C +314, : +12. 33 -0032
-31G +13,32 =00.33
D - 00, tootq + 12E9:, +11. 33 -00. W
00b.... er  n a 32
es
= 126. + 13.31 - 00. 13
0 +008 +327 :, +11.33. = 09. 31
permutande oletor 12 - 314
+13,3) =00. 33
C = 00, +00 + 1343:, + 11. 32 - 00. 11
S2d.... Ex Daise e  3)
136 + 12. 31 - 0V. 132
(1 = 00, +006 43347:1 +11. 32 - ov. 31
petuntuli ers
13 =31H +12. 31 - 00. 32
I 0c 37( Denominatores valorum exprimantur compendiose et apte ad permutationes
4 = L 1391 00. 33)
Zprioris (23
 posterioris 223
+00723t  - 11. 223 B
131. 123.. 713,. y: 123 223
-12. 213 6,: 113, 213
F13L1B.
- 13. 2128,: 112. 212
Derioris 113 Oprioris 112. ubi Ultima et penul
posterioris 213
dextras bilicium
coincidentientur hi ipsius 2 valores,
primus secundo, et tertius secundo
coesscientes ipsius  inter se 3 aequando
... 8...
ita orientur qualuor aequationes pro determina
tionibus ipsarum 11, 12, 13; 31, 32, 33
130Z213) 123 23 vel F 00. 113. 213. 24, -31 19 p1G 215 Y31. 13. N _5
in se latet d periuntant. 1 et 02. + 00. 123. 213. 223 - 32. 123, 23, 23 +32. 113. 123. 213
+ vo. 12. 213, 212-32. 112. 213, 22+ 32. 113. 112. 212
 fit exsi Haec aequatio sitp praecedente petntundo el 3 permulando ∞) et 3
it latera permutant 02 et . 3
a (125 (. 113. 23 = +4517
5
23p3 _el +00. 113. 113. 217 - 11. 113, 1a3. 23 L 11. 13. 23. 213 1 2
s
400. 113. 123.2, = 12 13. 123. 213, H2. 123. 23. 2231
C fit ex V permutando, 10 et 30, item, 100 et 200
 in B lapera permutant, 1 et 02
u  13. 2B = Le  18) m. 22 vel tev. 112. 13. 213 1 pui y 213 7 3. 1 3et2 213)-
400. 117. 113, 212 - 12. 12. 113. ER. +12. 112. 212. 2131
Y fit ex 3 permutando , l et. 3
7 fit ex 4 permutando 10 et 3., item, 1.. et 200
in Y latera permutant, 2 et 13
ita omnia sunt palchre harmonica, et pleraque sine calcs et legem justitiae sevant
Posteriora ex prioribus sine calculo per descriptionem parum utartem fieri possunt
Examen comprobutorium habetur, dum idem ex alio et ad huc ex alio formari potest
daa inquationa et pars una aequationis formari potest ex altera
eos =e cxi t Or 
Vidudon en cortetu ilou daritidiper tedeles cn eandi causa, qua bolies rarius dentent quam balites in aopi. ta s als i10
Ex aquatis valoribus
ipsius et fiet:
Ex aequatigs. duolus
valonibus ippos +
Ex aequatistuobus
vatoribus ipsius))
termutando ut e n
Ex aequatis duobus
valoribus ipsius 0
ex proxi
ermutando 02 et
Ex prima aequatione coeffi–
cientium ipsius 
difsrut hatera periu
Ex secunda
400ru3h
+32. 113.
ex praecendente ipsins  valore permut andu ut  n2
e = +00)212E
+00223) 113, 213=
Exprima aequatione coessi
cientium ipsius C
Ex secunda
Notae bilices
ut 12, 21 lexprimunt
Notae rcilis...
1bv 12124)
+33. 112 115
er constaptibus
Res omnis pq ex irrationalibus reducta est ad 499. 1, 2, 3, 4. Vel 5, 6, 7, 8
idem vnlor vel qui aptessimus, substitatur in valoteta ipsoum  D, C, prout aptissimum. Et habebur vatores novi horum per
praeterea  loco Z in Rationalibus substimatur medius valor ipsius 2.
Solas altimas t et
ita jam bis u
antinu
+ (5  =)