# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068000:0022
# Permalink to this reading: https://leibnizlegible.com/page/00068000:0022?run=73
# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 3
# Folio 11v, page id 00068000:0022
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068000&page=22
# Source image: https://digitale-sammlungen.gwlb.de/content/00068000/jpgs/default/00000022.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 135 lines, status machine
# Lines: 135 recognised, mean confidence 0.80
# 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.

tres Numeri tales, ut duorum quorum libet summa, et duorum quorum libet differentia
sint quadrati. sint numeri
10 nx yy + IUxy + itOXX + 102yy
A A+B = ex I AD A = gu B I 200 X + 202y4 2 UXy + 22o XX + 202 yy
Sint nundri A, B
BFC =L
A BaC = nu
+100Ct 1t0X + 102y +II2 Xy + WOXK + 1027y ~.
1E00 210
40 Xx 202 1
A= led 4 10 + 11X + 1XX + 13X ∞1 + 15 X + 16 X t erc
3=
20 + 21 X + a2XX +23X 124X 125 x 126 X + et
30 x31X + 32XX F 33X + yX +35 X 136 X + et
ponendo A majorem, B mediam) Cminorem
ponamus X esse 2
U X
ad qontata
40 + 4IX + 42 XX + 43X  X 145 X 146 XE p et pp = 40 + 2. 40. _IX + 2. 40
50 + 51 X
60 + 61X
70+71X
80 + 8l X
qu+91X
GG = 10+2. 10. 51 X + .-
HH = 60+ 2.60.614 +..
EL = 70∞ 2.70. 71X+.
Mm =H, 2. 40. 81XF..
NN = 9d +2. qu. 91 X +..
12p2 = 2. 40. 42+ 1. 41. 41. 13423 = 2. 40.41 +2. 41.42 14129 = 7. 10. 13t2. 11. 43+ 1. F2. 12
Htinc ob s aequationes AB A+B = YF_ AYBagu BrC-ht, fiet
s
10, 10+20 = 4040 00, 11 +21 = 2. 40. 41
vel compendio 18, 10+ 20 = Bo  11t 2 = id1 12+ 22 = 14C 13+ 23 = 13 14+24 - 13+ C4
130 =40. 131 = 1. 70. H
4
132 L 2. 40. 424
10+30 = 140
140 50. 50. 14 - 2501) 4 +30, 2, (50
(4d = 2. 5c. st + 1. 51 51 10 -20 = 230
et il porro
10-30 2 240
11+3I = 141 (2+32 = 192
21131 = 151
19- 2 =271
H 31 = 24
21-31= 251
22132 = 152
12122 = 2)2
12 32 = 292
22. 32 2351
13133 =143 141st = H4 = 4
23133 = 153
B23 = 273
13 33 =243
24134 = 154
14 24 = 234
14 34 = 244
2333=253. 2434-254
FF
20-30 = 250
ita tollendo ope 6 aequationum literas 10, 2000, vel 11, 21,31; vel 12, 22, 32. vel 13, 23, 13 vel 14,4, 34, et
Habebimus pet qualibet aequatione tres aequationes, quobus mantonatu in quibus solae erunt 40, at, ek ; c. A, etc bobiera
latione
Aliter sollantur prius ete A, B, C, et supererunt tres aequationes, ram
A + B + A- B = 2A = 2A - f5+l = 49+nm 2B = hh + nn  (f - II 2C = g9 -mm = hh - un
lssitrum fil nm= I1+ nn
Rursus ex aequ
sibatum fit lin ZU =94 (5)
A sit h - n = A  9- m, B: 9 Et Aet g+m( q - m = h +. H, (g - m p: 9 -
fit ff -g9 = mm -l, se  sit m -1= f-g, Br: 5. Unde ex pracedente fiet
Cum ope aequ. 4 et 5
f+9 (f -g = m+ l,  f- g, r. 5. Restat E aeq. 2.
possimus habere valores ipsarum 4 et m, quae non exstant in aeeq. 2. supererugt
Ex 4 et5 fiet , m hq nq Epogus + est i m= pq op Dpha + nq, :p
aequs est M Ex aeq. 4 est 9 = hq Lnq + mp, : p. Ex 5est g = lip +  p mq.. 
quos duos valores aequando fit h99 - ngu tmpus = app + Rpp = mpg, seu M = h(p. 99) An (Bp+99): epq
valor ipsius mD substituatur in aq. 8, fiet g = (2f94-(ngq + lipp (tiquen pp + nqq: 2p4 vel valor ipsius inque substi-
tuatur inag. g. fiet g= (2) pp + Em pp C hpD + hqu  ApP ngq: 2pq qui duo valores ipsius u conse nttunt
eadem modo quaerantur m et g ex agq. 6 et 7, per f et l
Y et 5. Ex duobus valorbus ipsius g habetims et ipsius m, poterimus tollere h, et _, et L per S et l, quos
et fit g = h (pp +99) + n Epp - 44),: 2pq
valores substituendo in aeq. 2, habebimas ff. H, oper P, 9, Y, s, ubi id agitur, ut valoem aliquem simplicem obtineamus
itaque hic solutio non usque adeo in potestate est
Brevius videntur rem conficere hae tres aequationes gg-hh+l min-llann ff - hh+ min
seu ubi g semet et u semet  h, l, m, bis. hh = gg- H nn  mm- II nn sa hh= in f- mm
Hinc ex aeq. 1 et 4 fit 9-l, p: q gH. Ex 3 et 5 sit f- m, r: 5 = f+m Ex 2 ut 6 sit m- 1, +. v = m + l
h=  X g B, 9- l, p: 9. = f = m, Y:5 R = m m - l, t: v
Ex ag. 7 datur g: l, ex aeq. 8 datur fem, ex. aeq. 9 datur m: l
Ex eq . (q. 5 explicatis m per f, et 1 percg per g, datur 8  f g: f. sed ita g: f datur bis; adeoque
+: V, quae an tractabilis dubito. videanns
() p+q:p-q. Ex 8 fit f: m  + 5, :, r-5 Ex c9n sit m: 1 = t1V,: t. v
Ergo res redit ad ag9. 5, 7, 8, 9
Ex 5 per 10 a u, fit g - p. 99, p = f - 5  7 seu (91+d(A ( tgrq (F Wy + fist snsre
tandem prodit aequatio, in qua p: 4
fit 20
4
r+S
ptq
13
peu g.5
sed g. f adhuc semel inveniri potest,
ergo fil. 9= ete
124
25
tiv ris
4
Y, p+4y:  B, rS
tm e m = 
=
Y+5, 5
= X, 14. Et firt t- v, r- 5 Y quae aequatib est solubilis
1
P
r15
p- 4. t +V
mo puto ese errre
calcuili
si nullus
rationaliter
Hic sola litra e ab adest semet
A= H e -B = gg - C
B = C + A gg = hh = c
sit 2C =  gg -s +hk
et dt
pet 55=
pono etiam 40, 4, 42, ek; Goyst, 52, et, 60,61, 62 et
70,11172, etc. 30,S, etc, 9019192, eM esse
pa
72. 41. 42 51. γ1. 43,. 
est
GG
est eror in calculo pulctro
habetur quaesitum
verte