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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 4
# Folio 18r, page id 00068049:0035
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Genculiss
Sint quatuor radues yt a
debet aequari quadrato. fiat y=  +
et radices seu factores erunt. PX + 44 pX + 99
snfactore,
y + b
X + 99
bx + br
ax + ar
 8 X + 99+b.
p+ b
otc ytd unde productum
e PX + 9
a X + ar)
px+ 99 px + g9
ex + eY Cdx + dY
X+ 49ter, X= V rd
+c
ubi duo sit bis  fit duas radices inter se aequari, factum aequatur quadrato; ergo
Ci
fiat
99+ aret q9+br. (a seu (99p t 99b + arp  arb = (9907 9ga + brp + bra
p+a=
seu 99 a- b = r pr. a- b. seu 99: r = p  seu 99 pr
p+b
Quo valore substituto in aeq. fiet  ex + PY hoc est X = p quod est
yde est indeterminata
et erit yta = a XX+ asxrat isXX + Sx + +t
contra posit propositum, nam 
PXX + 99X + Fh
XXY SX + +F
itaque faciamus; y =
(2) pXX + 99X + p
et eodem modo  habetur y + b=... tantame in 2 pro a ponendo  b
et  si volumus duo quaedesyta et y+b, inso nvicem ducta
et Nominatores s divisi per vefficientes ipsius  XX erunt
XX + 99tas X + r3+ aH
Xx + 19+bE x F  + bH  etc
P+b
etb
quos duos valores si aequemus fiet: (990 + 99b + a 5p f asb= (p G9 + b5p + aq9  ab5)
eta
pta
e et 99(a - b) = sp. (a = b) et  fit  19= sp  superest jam:
Y+ at = 1, bF. seu (V∞ + rb + atB E att b) = (P)+ Va+ bEp Abta
seu p = V: 16)
p+a
et fit y=
P+ b
PxX + spx + Ep seu y = pB quod est absurdum
X + Sx + e
99+b5 r+bF
49tas x + V + at et _ + p+b prb
+a
Ergo non debent ad inter se aequales poni XX +
pra)
sid fantum efficiendum est, ut divisorem habeant communem, nempe opsidere
ei considerando Xx f f9+ aC X + 1+ QF.
+a
pta
ut = 0. etsi revera non sit, fit
99+ a X + D _ + 299a5+ hass -
XX + L
pta
seu X = 
 et factor est
X
4 pp + 8 pa+ 4aa
g+ 29gast aass= pr pat. p 
4pp + 8 p a +4aa
9 a5 + 49+a- ap ra. rtact.
2p + 2 a
49 + a5  - f 94tas - a pta. r ratt
2p +a
et alius tadior
et alius factor X + 99+ a5 +  4quas = 4 pra. r + att  ib - 20 2C = 1d
et factor
factor.
et alius factor
2p+2a,
Et simiciter secundum b, fiet
99 +b5  -  49+b5 - 4 p + b Y3+ b1t
2p + 2b
X + 44+65 + S99+ b5 - 4 p+ 6. Y +bF.
Et tales adhuc aliu qvatnor dantur secundum cx et d.
2p + 2b
et unum secundum a, aequemus cum cogreo alio secundum b,
Brevius res conficeretur si possemus facere, ut XX + 99 + a5 X + v +at
seu ulsit sit
4 + 299 a5 + aa s5 () r+ at p+a
p +a C
sit quadratus
+qaatt
. adeoque 9 +ggas t aass = 4 pr. +. 4 ApC 140
4 p p + 8pa + 4aa p +a (3)
et similiter est : q + 299b5 + bb 5s = 4pr + 4 apF4 4ar4 qua t
2)pF + 4 13 + 4a F et auferendo invicem)
et residum dio cldendo per a -b. fit
aquationes 4 et 5 fit acb 5s 4 C - b f
seu s= 2t
+atC = p+ a+C F ex ag. 5 dividaber eq. 4 per a + b. fiet
et r3D e4+
obe  aeq. 2 t., 9+ 499at + 4aalt- 4pr a I9t. pHt +lta a
seu Y = 49t - ptt seu pt. (a9ua 4aa  = 4 pst ( 4 agqb opptl napte attaat alta
ruis
 ergo ex
99 = 2p +
exx + EptX  pF
4O. ser ex r9. 1. fit ly -
seu set qua AppHF 9- 4 p+ 9d + 4 pp et L
= PF et B seu 3 = P. quod abtusunt 2tX + F
facroies ia 1b 2b
ca = ib 2a = 1c
1a, 2a, 16, 26, 1c, 2C, 1d cd