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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 60v, page id 00068197:0120
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Hac metrodo optime habetitur, utrum posibule sit vet ab in eotae equare quatrato, nam quantumunque
alsurgal valor rationalis ipsius , facile deterumpr potest modo, generpla
crev + ac = quadrato
X. X+D, X += quadrato. reducitur ad pab
si nempeponatur ax = vV. Esto _ =R sXF + agF + aah, :, ZZ +q E + Inn
si retab, verac- ssut naGgZ+ raafh Z24
votab=
fiet vv + ab =
Haagusquee r 2 a g
qu (Szt etc) + ab qu (32 +  Zd ma)
qu (ZZ + mq + mn)
ab qu (77+ m Z + mi) L
+pZZ + a92+ aarq.
+f. +ag.. + dah 5
fit aquerrn) a t p1 3 + aqr anri
2+f.. ag. F nahi L
qu (pZZ + a9 2+ aar)B
qu (22. + n Z + mn)
(4pZZ + agZue + aar
- f. = aqu = aah
Ae s+pZZ + a9Z + aar
al45. qagu + aah
et posterius ob raeq.3 erit (5
faq faati bZZ + 2bnZ+ bun = tepi ep Z3 +ae92 ta et
Oo purus fiat eZZ + 2C m Z + emm Etpgue + agZ + aar
aequ. 2 eut,
_ def. facg. qu’et
f 4e=p. aF (4 f = e-p = ep -abie +ag = rem -a9 = aeq- 2 abaje +an tenu s a
ob rem coincitentiationes 61to 2 
Hinc e tabp L eetab,: 2e aq = reem + rabm,: 20 Haar = eemm Fabun, : 2C
Similiter vet ac-
qu (f22+ en) tac quC2ZF  +m
qu (Z2+ m Z + mn)
+5Z2+ a tZ + aacZ in
aC qu (22+ m q + mn) = +f.. ag aat
qu(52 tatz +aaes
qu (ZZ + in Z+ mn)
+522+ atZ + aa 8
a 5.. (4) ag. (4) aah
a  H522 + atqe t aac et   uiqu n) L
ob piem coincidentiam (Il mo) erit (+)5 = i -5. = is - aC,: i (+ ag = _. IM - aF =ait - ZCn, : 1 (+) dhx ip d aig.
a2 +am m
Unde oi et qu (Z tm)
H) fu () ag. f)aah
Hure ((2m) fit Saii tac,:2isat= imtracnc: 2i et aa  = limm tacnn, : 2i
B ex S,t 7 fit (13t0) f = + ee 4 ab, : 2C ag = 4 2eem f rabn, :2e aah = + eemm L abnn,: 2C
et ex 11 a c2fl 14h) f - (4) ii (4) ac, 321 ag = (4) eiim H) racn: 2i aah = (4) iimm ) acnns: 2i
Ex binis valoribus ipsius ag. fit m: n = tabi () Y ace,: e fecip Deii, unde
Habunu min Z mm: an Erjo Mn ta et paatet V anrando
trmratibatm
aabb +a’arce
ar s n cugo
.  SeE.B ant nran
)adbCeieii t ce it +(+) 2013 = nm: nn
pursus ex binis valoribus ipsius aah fit mm. nn = F abi(fce i + eer (4) eii, quas duos ipsius
mra: nn valores aaequando fit qu ( abi () ace) = + abi (+) ace, Feei (4) eii. seu tandem
tabi F) ac  fabi (4) ace = + eei (7) eii seu fabi Z eei. = (4) ace (7.) eii
Sed ex binis valoribus ipsius f, sit ilidem +abi + eej = (+ ace (4) e ii. Ergo quod mirum
res succedit hactenus, et fit  ji. sit 4 1: (4)1 = L 1 et fint ii A 
ii + ZeC + ab
+ e- 2ecab + a abb + C- 2eeab + aabb + 4acee
itaque  hustrantsit hic catcslas nec hac via ad extractionem blaiA.
et i
side quia t pro actitiai
itit
abee
44 ac ee
pars radicalis fiul extrativilius
licet, nam seqe
1 t a
it nam isa
aine adc c 
et fiet aa bb -2be zabee + e = a a cC- 2ac ee + e
par et
=mm
returhim
EErgo vel in =1
vel  n =0
elsicirtu 
sid M =1
olist 
si divelsi vatrrs ipsius
esset _= tab
fret i = fac
vul se ceset
H e fouet 12 bt a
seu ab nec =  act ee, et 2ee = abtac
Sed generalius tentemus aa bb -2abec + + 4ac e = e + 2 ceAaa coco
et firt aabb- 2abee + 4ac es =+ 2 eeaco + aacoeo Sorao = eV + ab - tab pe r. s ent, n ralit
adac
et siet aabbrabee fgacee = 42eV +2ee ab + eevo +Revab + aabb
sin  ace -2abec 20 f.et
et firt  habeF tace= = reco + 2e ab + eu + 2vab.  Divellatur haec aequatio in duas, e q
et brerioris calculi causa fint  ~ - fiet 2a 4 ace = - 2geb et evV +20 ab =0
fi etque et= -2aC sed ita siet etiam e = -2 ab quod non licet. Ergo reliaquamus ambiguitatem +rab
ipsins +. Et fiat e evit 20 ab =0 seu ev = -2ab ut -2 ab + 4 ac =  + 2ev +2ab,
ubi pra  sed ita pro eo substimendo valorem fiit rab + gac   aab + 2ab
et foret et foret b queb b + 2ab  b L 2b = Z ac. sed talem deter-
Ex aeg. X fiat - 2 abe + gace L a  adv= a1v Freev + 2eab + evc + 2vab
minationem inter datas nolume
aliter ua bb = rabee +  + 4ac et = aaboA rabeco + recoco fit rabe te +4ac e = rab etoco
Haec aequatio divelatur in duas = 2abe
Alter, aa bb -  abee + et
 gacee
 qu ( + 228) = _ac + e 88 + hace
et fiet ab = ee = ad 8 a. + et 
si in vabre ipsius