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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 3
# Folio 39r, page id 00068000:0077
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videamus an formula generalis secundi gradus uniue indeterminatae quadrato
ut sit
aequari possit. Ver Nempe fxx + agX + aah =, a yy. fiat y= a qu (= h + aa)
2V
et fie F SXX + agX + aah = ae avvth + a c + Lah, v vel fxX + ag+ = avotih + aa - d aat
qua
quo
vel fxx + agx 3 = a qu (+Dh + a4) fiat X = 4 tr ( vh 
1
v
a
2a
jaa
quu
1V
). Et ex aeg. 3 siet
valores aequando fiet frrvoh frra = aorh - a + 20grv. Hic habetur
2V
aa+ ia
solutio si h sit quadratus, faciendo frroph - fHa = 0 nam fiet v0 = a h, et
avvh -a: 4 + 2a9v = Y Habetur et solutio si f sit quadratus,
faciendo frrvuh = aveh seu rr = ae: f et frra frra- a;:  + 2agr = v
Idem aliter. si fsit quadratus fiat frrae= a et frrvh = aevh + 22gr, seu v
+agr
aE- srrk
vel si hsit quadatue fiat ( avvh - a6 = 0 et Erveh - fra = = 2ag, seu r- + 2ae9v
af. frphi
sed si nenter, fh sit quadratus videndum an non res procedere possit faciendo f quadiatum
quo casu et = Fih eut quadratus (nam = fh: th = - f. h). fut ergo fh = ee et eerr DV = a, seu
ero =ae, suoust R+ fra afrr tahWV = + aega: e, se parr+ aah. f) vv- t dg:
H h. f = - ee: tih ergo fit aarr Qua -(eetiti) WB = + 24g: ef, sed ita explicandoter
fint aa (a: ee) = (ee iht) aa v f 2a6gv0: aef. sed hoc solutu difficilius
XX + a X + aagu = a yy - aah + aa 9). Ergo
43
4fau
quadu
4afyy- 4aath + augg ilet esse
delet ponatur esse 0 et res succedet si 4aa fh- gg,: 4af sit quadratus
sit data formula inZZ + agq + aa anZ + aap aequanda quadhato fiat Z - qx tar,: a
fiet mgqXX + 2mgarX + m aarr ~ fxx + agx + aat. Ergo
da
aa
nqx
a
 nar C 4a mguse (merinartaap)- aqu (2onqu + n 4)
aa
+ aap Ergo
na
Aamgq
debet esse quadratus e
sid in hoc valore quadrati H, evanescit q et Y, et ila frustra ad hibuta est haec transformatio
quid si facissemus  - qu tar,: x
e firiet + in gq XX + Emgarx + maarr) fint
lada
an E Xx aX
+ a ap XX
(
4a maarr (mqus tanguaax)  aqu (2mar +a nY)
4mqq  anquaaep 
dd. Xx sed et sic destmitur r, et in numeritore
cvanescit etiam 9, sed nominator redit ad
formutam datam
quid ergo si faciamus Z = q + ar, :, X + 5-f huore
ratorem substituamus in mZz + anz + aap seu m00 + an OD + aap DD,: VV
m gq XX + 2 m agr X + aarr
an que + an 95X + anars l: qu (X+5,
 an arX
+aa XX + 2 aap sx
aap 1)
Et numerator est
aafxx + agX + ah
aequandus qto
Ergo 4a C mgue tangt aapsma atr Fararst aax ) - a qu (em aqu F angs Fanar + raaps) = L F
2
1) Aa
W
Aa (mgus + anq + aap), qu (Lf)
damgumaam + 4amqus anani tamque aapss (t 4a angmaatr)  Gnganars qaagnu aaps igap naeistaris su
Aan
a cmmanii ( aqm agran qu at Magragar, aamagrraaps - a aannqu ss 2a anussanar( quanquuapt aanar
et fiet
qu anarar
+am +aempgus) + 2ann qus + aemprr ga impqus aungus  aennir = H
4a (mqq + an 7 + aap)
in numeratore semper simul sunt 95 fcit
sed nondum hinc apparet aupilium
er q) = aK
poteststoni
920