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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 4
# Folio 56v, page id 00068049:0112
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Vedeamus an non transformatione aliqua effici posst
formula in qua duo termini suimm vel altimi tollibiles
sunt qualuor factores
e m +ysa m- ysa m+y m - y. seud in ponanusta
fent mst mhy m-hy ntus m-y. qui a ducti in se invicem
debent producere quadratum vst
La2
et
sit mahus = sv ioZ et m- hy = spo ZCo et nty = + moga t
pro ducentes
. 
et fiet mahius st V  im hy s
2
Engus 2m = stv + stoz
communem
quatrat
3 a. et fiet quadpitum (SsEt vxuroZZCoc
5.: s. ... .. + v  M+ hy = stv m - hy 2 spoZ
u.-
. z
L
se im + y tloco m. 1= vzco
.- co 
(∞)
= FWco + vZco et 2hy = stv - sZ = 2htWco hoq
solvendae ergo ae quationes b,7, 9, ut obtinentur aeq. 8. post illas. Ex 8quo est
(62)
+ + s 2m: lv +WZ= (ex q) M CDONCo, W +VZ: tv+noZ= se 1) to so- v2
v2
E = ubi ut inveniomus s vsi per aeq. lo. resolvamus aeq vationes H et c2. vatores V et 12
idi aquemus cun vatire 10. fit 2m = cotevW + torZ+ thoWZ + VWZZ seu
co=n: trW + tuviot tuW Z+ vuZZ sei c 1d
a + et similiter co m. Hov -W10ZZ: a, tW pqe Exvalire id et 72.
1+ 1
i V ujam duos. ipsins Cd valores  13 et 14 inter se aequando fiet
At
83E 2h. 2v + WZ. tW-BZ = m. +vtoZ. t
700.
 papc es S Cee 2M: evau2 eB    ritou
es
 + et ex aeq7 est 5= Cd, tW + vZ:, +V+ WWZ et ex aq. 9, s 2hco, tW -vZ tv - W Z
11 de 2
(14)
oc = et denique ex 1 et 12 fiet: 2h, tW - vZ tv + WoZ = t W + v7, tv- WZ
de
 es faciendo multiplicationem actu ipso sac 2hFLWB + LHWZ - vozt - vWZZ= FWV-tw Z1 vV, t v1 2
is
SC
seu  2_ AVV
2h - 1, vWv, Et -Z2
C 
2h + 1, +Z, v -w1 ubi saltem eo casu
quo poneretur h= L facilis prodivet solutio, fieret enim vv- W010 = 6, seu
seu
li v = =D = W: quod si v = 1o fieret co = 5 =  2m: B, L +Z
et ∞= per 8 fieret y = 5B, t-. Z  seu y = B 2 mB, t. Z.: fp, L1Z
ergo saltem illo cusu Hyperbolae vel Ellipseos haberetur solutio, et
quadrali latus soret am:  (B E+ Z on tZ(VD seu ammtz:, F +Z2
Sed stemus in generalibus, et sit v = 10X + 1 et W = 20X + 21
et t = 30 X + 31 et  = 40 X + 4) fit vd v0 = 10. 10 X+ 2. 10. UX L 1. 11
et (WWW = 20. 20XX + 2. 20. 21X + 21. 2) et tF = 3a 30 XX +2. 30. 3IX F 31. 31
et _ZZ = 40. 40. XX + 2. 40. 41X + 41. 41
et VW = 10. 2OXX + 10. 2) X = 1. 21 et tZ= 30. 40Xx + 30. 41X + 31. 41
+ 11. 20 X
et tE. Z2 vWV = 10. 20. 30. 30 X F 10. 21 7 3. 30 X3 + 11. 21. 30. 30 XX,
11. 20
2. 10. 20. 30. 31 X) + (10. 2 j2 2.31 xX F 2. 11. 21. 30.31 X
+ 31. 40 X
= 10. 20. 31. 3IXX + 10. 21731. 31 X = 11. 21, 31. 31
ed ante quam in calculi prolixiore pergamus 1.
considerandum est
posse aequationem 16 ita divelli in duas, ut sit 
2h 1, +F- Z2 e2) BtZ
et βVW 2h+ 1, v-wW, quae divulsio
quod summe notandum fit sine suppositione, quia B est quae cunque
ex aeq. 17. fit H- B ZF + B
sit (3: 2h- 1 =B si 3. 1 42 - B + 
Z= ( +Bh-27. seu t R
Eb122
seu 1 - D4Z + = BB, = 472+ 8837. 4 seu fiet
F= LByt 1a2Et 06 seu t e E2t: Z B+ 54+ BB. et eodem modo, si faciamus
22)
MF 3: 9 2h+1 = A. fiet vY D vv- Ay W= WW seu vV - AW P+ SW - WX, A1X
quia antem
seu  v- = AW = 12d 1, 4 +X seu 20: W= S + 14+ M. ubi si β  adeoque et D est
habeauus pro constantibus non opus est ulteriori aliquo pro aequuationitus 21 et 24
his positis v: Wd est constans _ +: Z
copans. fariamus B.I
ii valoribus
1d, 5, et y, aequationum 13, 10, et 8, fiet co= em:  t ; et S Em: it enat, et denique
 erit 2hy = 2m: 1+ N. 1 (. sed ita fiet y constans contra intentionem itaque nihil actum est, nise
sutlo
faciamus (3 esse indeterqusam 3: 2= 6, it .  = iea res in arctum contrusa est