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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 16
# Folio 1v, page id 00068229:0002
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a + b in a -b = aa = bb
a +b +C in a + b - c = aa + bb - cc + 2ab. quod in + aa +bb - cc = 2ab = a + daabb = 4aabb
Ergo a + b +C in a + b - c in aa + bb - cc - 2 ab = a = 2 aabb
aacc
=2 bbec
seu d. in u 7 4b - aac - a cc - 2aab 4+ 2abe) - aab + 2abc = a - raa bb. vao a - 
seu d. in1 4 (4, a bv = bbc - b cc - 24b1
+bct ai d + b+ c d  ap da +2ab
Quod et per se demonstratur nam a in a = a + ab
et a in ab = - a b abc
et amrabe = 2a bc + B abcd. Ergo a in a - ab + 2ab = a ( + a b) - 2 a  =2a bc + dabed
+2a bc)
 habed evanescit,
a
Si jam dentur quaetuur a + b + C+ d tunc promum multoplicetur
a in a ab + rabe prodit a - 2ab + gabed quod runsus multiplicando
iper a - 2a b -. d abed prodit a- 4 ab( + abc)+ 4ab  324bC
idem et su demonshatur:
+ 3abC)
+4a 7bc
4 1d in a d e 1a + rabc AL. 4. D2 +4  + 4. C
dd
+24
644b0d
-32 av0d
a Ad 13K
ut probetur res in nomeris ponumus a  erit  4geb
24 4M Y 4. 8 + 4. 6
32
pusio a 4 a b abC +4a b 1∞abC +X
4 - 4. 12- 4 12
4
pudit a + 2462- 44b - 4 abC + 4ab + 8a bC +24 aeb c
4 +26 - 4. 12 - 4. 12 +4. 6 + 8. 12 +24
6
90
+ 144
seu a-4 ab + 6ab + 4 a bc - 40 abc 
4
64
e
(= 4. 12) +6.6 ( 4. 12, - 40
Fortusse autem tam alte assurgere necesse non fuit
A
Nam a + b+c elevatur ad at - raabb et proinde a tantum
assurgit ad gradum quartum. Cujus rei rationem esse arbitror, quia
si fecissemus aquationem a + b+ c =0 possete  cotunis considerando
ipsas b et c ut irrationales quadraticas, quae sunt ambignae, patet
a habere quatuor valores seu quatuor radices;   liipsius autem
a ambiguntas si nempe a + bAc =0 _ et a + b - c, = 0 et a - b +C =0 et a - b -1 =c
nam ipsius a volor ambiguus, si quis esse ponere at inquies ita tollentur
quodem a assurgente ad a tolli quidem poterunt irrationales b et c,
non veri iratio ipsa a simplex en inationalis. Verum respondetur,
andem prodire equationem sive ponas ta, sive -a quia semper
ab nan b effici potest in aequatione ut unum membrum sit signo +
affectum, omne eadem ergo quae ante manebit varietas, nempe
quae in reliquis tantum esse potest. Nsec opus est respici
Re impe
crratio nalia, sed tantum ad signa
sit a + b + c = 8 ptr et prodeat a quatio in qua haeliterae
non nisi paria signa habebunt paras dimensiones habebunt, ea valebit
Nempe pro
pro omnibus siguorum in his mutationisus.
a + b +=0  quae coincidit cum = a - b - c =0
= a = b + c =0
=a+ b - c =0
=a + b + c =0
a + b = c =0
iaque aequatio prodiens habebit quatuor radices possibiles et
a = b + c =0
a = b - c = 0
proinde assurget ad gradum quartum.
tantum
d
a
ve
eu radsces
b+ c+. dqm
d octavum ut nis
a in a + 21 ab + 32a b + 31 abc2 H abC + 2u abed + muabcde
hes s
eotul non
ces
ad gradum
(aequavis puotus erit 2
dut ab + 4b + 1 a +() 41 ab Cr (2) 32 ab + 32 abC + (3) 311a2bcd + B) 221 a65 + (2) 121 a b2d +. M abidex
2111 abCd
+ 311 4bC
e Ergo it destnantur termini ubi y literae dimensiones omnes parres non habent, fiet
+ 22143bC
+(2) 2111a V cd + cunabider
+ab +  4 + 311 abc 
41 = - 1 e et 32 =0 et 311 = 2 et 221 = - 2 et 2111 = - 6 nm es
se terminus autem v et quia hoc loco e = a poterit et 111I poni =0 adeoque negligi
Ergo a in a - ab +2 a bc - 2 a bc, 5 6 a b cd dobit a - a b - 6 a b c
et assumi deber
Generaliter igitur ut tallantur  duae, tres, quatuor, quinque ec rrationales quadraticae
vide ade tuo
opus est assurgi ad  dnos, quatuor, sex, octo de gradus et generaliter si numerus
prationalium sit B numera dimensibanto aequationis erit in - 2 seu n- 1. 2 hoc est numerus
oodatiarum in aqvatole cassutam urntate montus desfliretur padustum eudt mous dimensionum asuadis ante
1.
nullio uniones braiones thnco
1.
4
54 aebet
ab4
ab + a b6
12
a
ab=
a  b d
144
96
64
160 160
(9  des
644 67242
a +bsC +d
1+ b
AtVC
a 
b+c-
umo videtur
s