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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 4
# Folio 54r, page id 00068049:0107
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9 ay + by + Cy + d aequandus quadrato, fiat y = 9u+ lo: hv+m, et habebitur equanta quadrato
fractio tag V3 + 319 l v + 3a glV + al3
obgh. + 2b ghl. F bh.
+egh.
+dh.
ygm. + 2bglm.
Cglm.
+ chL.
+dhm.
+cgm.
+2c hlm.
+3dhni.
=ben 3: hV + 3 tm V + 3ton V + M
+c ln
e
Explicentur, (si possibile est) assumtitiae sic ut
 numerator et nominator habeat divisorem.
communem hv+n, ita enim divisione
fecta, nominator per se aequabitur quadrato, et numeratorerit
depressus ad. v +. v +. v etc. et ita faclius aequandus gto.
m igitut frmula aequationis 3 ntimepator, comprendiose expressus erit px+ qu + rV + S, qui sit dividendus pe
v+ seu per v +t, instituatur divisio
pP+ qu’+ rv + 5+ pov + 9q V + p
v
+E (D + A
pte
9VV =c
131
rq
ro
ergo fiet + a G M - 3agl M h + 3agl Mfi 2 - aB3
explicando
ag. 7. per
qu. 4
4 bgR.
Fcgh.
+dh).
Ebgh).
bggm).
Ecgum..
CChhD
(3 dm).
bhFF
C2bgm:
Comm.
CEchlm..
3ohmi.
+
ergo resi
(5
12 quie et quotiens aq
bP.
Celm.
(dm)
(6) V+F
et residuus S- r + + q+- pt aequari debet nihilo
seu  quia t =m: h, fiet:
SEh - xtim + 9tn - pn Z0
h
A
e
C
quo laequatio e videtur
non incommode  alias subdividi
posse, quibus destrnantur i
placet s3, et h 93 g2
dC et 9 et qu
ts
o destmendo g  et  fiet: (CH n 2b bn 2Cm + 3 a lnh + 2bImx  Cm B 
e
40) quae aequatio dividi potest, per m, adeoque haberi pt plane per 1M, vel per I
3H
pro cestrundo a. fet m clni 3dm+ b lnd+ rc 2clnst 3dn al ( bent cns
in 48
ita residuum adiviteretur pervgu et habertur g9. sed ipsa aeq. 10 estraed hu soldia sed hoc novlicet ia fiece aB3 = 0M
Sed si eset d = o, quod praeforcuando semper obtineri potest tunc aequ. 10
rarsquse utiliter divelletur in duas, a + blm+ m  et - cen + bl + 2Cm  a
atque ita quaesitem habetur plane
ergo fet sed ita mirifico naturae effugio, in ag. 8_ eouus scunt
omni omnia data, et fit gm - Ih 2)0 ergo m= Ih: 9, quem
valorem substituendo in aeq. 2.
fit y = 9u + l9:, hv + th: g seu = Bgv tlg: hvtlM
seu y = g9v + gl: gha h, et divisa numeratore ac nominatore per qu+ l, fiet
y = g: n
Redeundo ergo ad aeq.3. videnmus an liceat facere terminos ubi V, et V aequales nihilo
ita enim haberetur reductio: aq3 + bgh + Cgh + d h3o ita o evanescet v
et l bggru et . l: m = bgg F 2cgh + 3 dhh, :, 3agg + 2bgh + chh,
Si sit d =0, uti certe effici semper pt, et in casu conicae ad quadraturam rationa lem
reducendae jam tum contingit ex aequ. 10 firt: ag + bgh + ch =0 ita per
aeg. 11 et 1E reducitur formuta aequationis 3. quadrato aequanda ad... V +. . V: 
adeoque ... v +.. v: tx+m  aequanda est quadrato, quod est in pte. in aego.
Ut rem actu jaso, faciamus, no gg substituatur valor eni H in aequ. 12. subslituatur abi
ex aeq. 12. fiit = a l: m =- bb gs bchf + 2aCgh + gadhh,:, B)bg  (dcht abgs h  Chh)
jan ex 12 e g sit g: h = K fiit aa kk + abk + ac = 0 et aF + b ak + = bb = L bb - ac seu
aK =  bb - ac - 2 b
si Hac methodus praebet speu successu. Si proba esset, otineretur in ea reductio depressio 
formulae quadrato aequandirt ad inferiores, seu tractabiliorem
ov