# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068196:0020
# Permalink to this reading: https://leibnizlegible.com/page/00068196:0020?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 10v, page id 00068196:0020
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 11r); this page is its left half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=20
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000020.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 94 lines, status machine
# Lines: 94 recognised, mean confidence 0.79
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

Redeamus ergo ad X3 = h + Ist, h- T = A d + 7 sA, d - 2sM, 5I+ 7, 51- 3In
Si sn = ∞ fi et 5V7 =  DD. qui ha B et + non possunt esse simplices, saut secundi
per spaginam pracedentem, sint secundi gradus, ut amplius resolvi possunt, et quia
et ex trahanda est raix ex h + 7Il, G- Tst, et ex 51 + Hn, 51- In,
ita ut alter sit quadratus, alter divisibilis per quadratum sit h = 10XX + 11aX + 12 aa
et 4 - 20XX + Z1 ax + 22aa V debet eb t2o XX HL + 21 ax H + 22aa Hl quadratus,
10XX = H1aX - 12 aa
et 120Xx N(+ 2 ax Il + 22 aa Sl
+ 10 XX + 1 ax + 12 aa
etiam quadratus; vel oportet ut
nam si radices eorum sint AB, CD, fut ABCd sit quadratus necesse est aut utrumque esse
quadratum, aut, quod non licet aequari quodd variuntia inter se. Est ergo
XX + 2 a sC - 11d aX + 22  Il Indaa ~ Xx + 2bX + bb, et fiet
20 l- 10
20+l - 10
24 b = 21 Hl - 1_ et bb -
21. 21( - 2. 11. 21Yl + 11. 11
4, 20. 20l - 2. 10, 20 fl- 1 10. 10
2, 205l10
ut - 211. 21 ANC + 1 = 4, (20. 22( - 12. 20Yl = 10. 12
225l- 12. pt fiet
4, 20l - 10
Et parijure  21. 2 sl C  sed ut alter etiam sit quadhalus oporset tantum
mutari signa in 10, 1, 12, et siet
- 10. 21 Yl
21 l + 2. 11. 21 Il +11 = 4, 20. ul + 12. 20 Sl 4 10. 12
Fn, 1 - n. 
-
in
A
+ 10. 22
(3
-l: In
2= Tstanin
sIn
Cnhendo fit 4. H. 2,  4 4. 11. 21 NV = 8, 12. 20 + 10. 22
Ex priote aeq est Ex posterdore aqu Ex aaq. 2 est 2t Al + 1 2 (2, 1220410.24. A pari jure
C + 12. 2en CC
e
l
+ 24
si a etiam 51+3In, et 51 - 7 sn debeant esse quadrati, redit adhuc semel aequ. 3, et
Ex auua 2 fit 21V +1
prodit aequo 2 et pro aeq. nova prodib 2tn +H = 2 v( Id. In
10. 22 s
hendo aeq. + aeq. 5 fint (2) 2(s (2) (... l..) - (2) 1(.. . n ...). Et si dividatur aeq. 4 per Al, et
aequ. 5 per In, et detrahatur noc a illa proveniens ab illo fiet ce
1H: M = A M(... l.. .) = +(.. . n. ..). Sed in aeq. 6, pro 21 substituendu
in
valorem ex aequ da fint 2. p. 20d 2 a (3) tC. . l. .) - .1 (.. M..) Ergo ex aegg. 7. et 8, invicem
2-1 Su
Av. 2IM
(9
fiet
duolis latere dextro in deptudu ctisinistro in secundam,
(..L B +.
(1.. B. ..)  21 f(... l.. .7 1.. n. ..)
In
sin
Compendio sit, 10. 22 + 12. 28 = 202 et 12 2 + 10 20 10. 12 +20. 22 = 104, et 2- ll+ nin = 103,
2510. 2 (28 Ist +nin) L
3
quae est mere cognita, fiet 2. 102 2 1ento 2 = (... l...) = 1oblF 201 sL et -. A (. . M. ..) = 1IB+ 20. In
d
fiel
sIn
212. 203
et denique +4. 100. 207 +1, 101l, +101n L 10. il 10c In  - 4. 100. 201
1205: l 420P: n (22) 201.  201: )
201 : 1 ~ wortR
101:  201: Mn
2 post cosu
Cortn+ 2o1 (2) 201: 1l 201. rn
quae aequatio silsulsacit dab tiet nisi i axbitrariae evanescant, dab. l 201: 101 aequatione
2.101.201 2.101. 201
quadratica. seu 10. 22+ 12. 203 :, 10. 12 4 20. 22, adeque adeoque H: 20. 10 tantumque cavendum ne simul
ne sit to  4. 4 =  Hc tantum cavendum ne fiat 10 20 : 10 = 22: 12 = 21: 12; ia
Sed videe si potius procedendum s  a exit tam si M =0, posumus vel X facere quadra
enim constans ratio foret ipsorum
tum, vel saltem formulam qparti gradus factam ex quadratoet formula secundi gradus priore
casu X+ L, exirre debent esse quadrati (ec    sed posteriore altent
alteruter tantum. sit ergo x X = NAX ses AB NaBa Si N Z7. 1 +
Xa et X+ I = Sux 7N_ SV5V Z +l seu Suse 87 a + 77 it A L, : 27 1 71
7377 et py Zat + 27 A Nal debes a quari quabrato quisitaah3) X3 7 aa =
a- et 77 , it a t = aGB + I 1I  a 3B - Z it sal = 5eg 7 aa. Videamus
X = NN ZGII
an liceat facere hPa t 2it Vel = susa et hBa= +p sal = s Ta idem cal culus institui potent pro
aa hGBr
sit s ~ eo GB = 10 N 2X pMa F = 20X 121a
sit 7 = 10X + na 5V = 20 k + 2Ia 4 7 = 30X + 314 BF. -KaXX + frax + 42 3B = 60XX + 6IaX +62 aa
 ∞5I
L pouendo n, et pro G1 ponendu
fiet  que oX na) tee    aua ilxrna=
Et fiet