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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 49r, page id 00068198:0097
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pro quad raturis
Qvaeritur resolutio irectionels Radicts ex altiore in partes quae sint radices
solvendaquantitas
ex inferiore. lab  lil Cum nempe consiturate dEbetur
dque A(Z + abzZ + ac). Ha okonem ponatur Z = C +  +C. Et sit
O = Sod (tnaa) D =, L(2a  t21aa) et e = s(30a+ 31a). C
ponatur porrosqu (Z + abzZ + ac) = aV  = i OD + 130Q + 23 DG + ar
Et quia Z + ab ZZ + aC = aa WV, ideo explicatis Z et v, prodibit
2+ abzzt aeC = 9 + 6003 +40007 + 1980D + ab + 2ab05
+ ab (30
+ac
et
12OCW +2ar OD +2. 13,13696
Ee
ete
X = a +b+C
XA = aa + 2ab
x= a + 42b + aaabe
Caublb 4.2. aab
terminos
quos duos valores concidentios A et quidem ita ut primusm tollumus conseramus  ubi irrationales
et
l 80arra n
+ab6.d+
qualis O) faciamus (2ar (4 (0C) +  DD + QG) +3ab
et 8 = 2. 13 23 2. 13. 23 =8 seu 13. 23 = 4 sed pari jure 12. 13 = 4, = 12. 23.. Ttaque
23 = 4: 13 = 4: 12. Ergo 12 = 13 et pari jure = 23. Quia ergo 12. 13 = 12. 12 = 4. Crit ACg 2- 12-13-
Ad Conferendos tesminos rationiles pro vat in valore ipsius aaur pro aarr substituauus valore
inventur fint ac = 3C +500D + 3 ab 00 Seu ( + 200DD X abO + - dabb-
se) extratendo ntrobique 10 + s ab  u aabb a rea 00 + i ab = 4 H(9bb + nac)
Seu 04 + 100 + ab 9 + aabb = = ac -  aabb. Et extiaibnt 0O +  ab  a (12ac - 3bb)
 esmion matou
Ergo ut res sit possibilis sine imaginariis oportet c esse affirmativam et gac non esse minorem quame
porro in valore curvae conicae, qui huc prodest, et abi C est affirmativa
due dy+ dzdZ) = dz N(2 - 24+  19 22 + g (9: rr4)) erit ab = (g1r, ., r44) q9 et ac - 4 (9. rq
4dc =(4: rr4) aque aabb =(9: 1+ 9) (249+ 44r + rY) (M
22- 94, s(9:, r+4)
et go q non oldetest mind quam ot   riet seu eli  t  d aen db et es Esempe 
ergo 4q non soctesse minor quam +9q+ q psterius Excedit per tit ritaque is ita positis valot ipsius ad e temper est imagnatum
Atque hoc quidem nos nun moratur; sed videndum tamen esset an non evitari possit
Sed hoc est absurdum quia semper est minor, nam
Ponamus Q esse 0 et 13 et 23 esse 0. Manebit 12. 12 quantitas ad huc indeterminata, et
fiet 02 3 _ ac = 3O +  (12. 12 +2) 0O DD + 3ab 00 + aa bb
Sed hinc si extractionem molimur eodem ducimur, ut 1212 sit 4 seu ut 1212 +2 sit b.
verum videamus an non liceat invenire divisorem alium (quan adicem) aequationis hujus, nempe
1 + + 12. 1+ 003B + ab 1C + aabb - ac =0 Divita affumitus sit ∞ t ap e cujus quadratum 
enjus quadratum OA + 2aM OO aamm unde fiet (V (1 2t 2y 1) n +2 geam 5+ + 2005 + nam 0 t aamm
 +( 1B. 5) (r 43. 5)OODD. sed prius du explicari debent 0b et D)) ut omma
a reducantur ad unam variabltem 1. Et fiet
S+p = 10 1   + 2. 10. na V + 11. 1. a a)
20. 21
21. 21
A- 15 MCD = mocol + n. 10 2 a + n. 1 2jaa
al (d0 +2)
am - bb 2ac am =
1. 2
10 bi + 1. ab
20 b 21 ab
am
10. 10 + n. 10. 20 = Ann. 268. 20 = ( nn = 1) 20. 20
biH est
seu fit 10 = Ln. 20 = 420 LInn - 4)
quaesita aequatio hanc dividens esto f1 + ag =
ubi f et 9 sunt 0, haec exaltata dabit quadrata
dabit fWV + 2f 2afqV + aagg = b iquae multiplices
per 10428 + n. 20 et Aeq. D per H, et alte
una ab altera subtrahatur, fietue
+2. 10. 11 a ff2 + 11. 11 a45f
+ n. 10. 21 ... - +n. 11. 21..
= 1. 10. 102fg1 - . 1. 10 aagu
Seu 10: 20 = 2 V (nn - 4) = = n nn - 4 = 12 49. 12 +  8. , (V. 10 20
eti o.o
plaestat sumere oripore
seu 1124 +4. 12 - 32, : 9 Z nn - 4. ubi si 12 = 2 sit  un fit nn-4 = Travicem in
Rursus Exa est f10: 20 =  -, 2. 21 + n. 11 + b, :, 2. 11at n. 21 + b = 2 s(nn - 4)-
et eodem modo
si esst nn  _ nN - 16 - 9 seu AH = 2 fiet n = 5 = 12 +c. seu 1 =
g et nn = 2 et n = = 4
48. 4 100 10. 4 = 576. 4.  seu nn = 676. 4 et n = 26. 2 = 1 +2
nN = nu 16
60
dd ergo 12  26] - 2 = 26 - 10 = 16 et 12 = 4: 15.
10
100
4. 3b = 64. 4. finn = 336+64, + et n = 10. 2 = 5 = 12 + 2 nond
10c
HM -
113
nn - 4. 64 = 225. 4. seu nn = 4. 287 et n = 2. 17 = 12 + 2 firt 13 = -
Et 12 =B quod succedit rationalitar, et fit fa L(nn - 4) -
un-
ide haec quia dnusi  12+2 dividi per, 3 ut fial n
=
X = - 2 fit -B = e2 +
ctante ootdinatarm 
pro ibypeibola aegulisera ad Asemploton relala t positis otentia A fit du i
si
X DIff13 = 10a1+ uda iett
ear
125
2d - 2
 10 1 fit 20 = 1 sit 1 2 1 sit  a = f
29 119
239
Ergo b =0 et ac = f et 30+ DD = Y A12aC =
 ili   66 eso  asis quo ae a  ed as tea    idenua
10=, 2 20 = -10 et 21aa = f13 - 11aa
 u ulcami quantibas BC et 3 sit alti sed  tom anat i  d feun pro cndes porito
qui non sit ipse aequam
atque adeo sum
adtes
pt
acilo
orc aloru
ve sed sum
est di
id
mionem
it
1d
55 vc
mumur 1
deprtendi2
us
 ut
X. yy2
ZyX
24∞10
57b X
00
76
i
150
n
aavo=
13