# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068000:0021
# Permalink to this reading: https://leibnizlegible.com/page/00068000:0021?run=73
# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 3
# Folio 11r, page id 00068000:0021
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068000&page=21
# Source image: https://digitale-sammlungen.gwlb.de/content/00068000/jpgs/default/00000021.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 111 lines, status machine
# Lines: 111 recognised, mean confidence 0.72
# 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.

res Numeritales ut summa duorum quorumtibet, et diffecenti
datu
Tres Numeri quaeruntur tales, ut summa et differentia dorum quorumlibet sit quadratus
sint Numeri A, B, C, sit A+ B = Sf A+ C = gg B+C = hh A- B = l A- C = um B- C = nn
Sit 2A = ff +l = gg+ mm 2B = fF - I = hitr. + An 2 = gg - mm = hh -nn
a
paetera A = ss -B = g9= C, sit B = C +  - g9 = hh -C, fit 2C = hh + gg - s = gg - mm, fit f5 - hh tm
qua valo  substituto in valore ipsius 2A duplici, fit hb t gg =  ht + IL et ex 2B omplicifit mm = U+ nn Ita
siet hh = (1) = g9- U hh = (2) = ff - mn un = (3) = n -l Et has tres aequationes solvere
sel paestare oportet; fiat h = 4 = g- l, p: 4. Ergo ex aequ. 1. fiet g- l sp: 49 = 5= gHl. Pursus sit l f m, r25
Rursus sit h= 6 = f -m, r: 5, fiet et ex aeq. 2 fiet f - m r. S = fOm 2= f+m Et ex aq4. 16 fil guli pos- fan, 15
+- l, p: 9 = 8 = f - m, r: s Tandem ex  4 sit fiat n =, M - l, . rest ex aeq. 3 fiet m -l, H. v = 10 = m +L
Ex aeq. 5 fit 9pp = 1 = 994+ (44 seu g il = 11 = PP +44, ; pp- 44. E eodem modo ex r9. ) sit f: m = 12 = 1+ SB, :, rY S
Et ex aeq. 10 fit m. l =  + W, , L m: l = 13 = et FBH, : H = vV. jam in aeq. 8, explicemus L per g ope aq. 11
et m per 5ope aeq. 12, et fet P g = p - 49g - f L, f - rra ss f, seu 3 = _ _ (r. 5), vr +Se 1Y + 1s e
seu d = 14 =  Y4. pet qu
rrass.
pP +94
pp- 44
Ep: 4), pp tB - pp +99) rrs
Ergo fiet d =15 = PP +49 fL- WV, 1Y - 51 =  r9. pp+ 4q seu tt. ∞V, rr - 55
pa ertss
ep - 9q H+WV rY+ AS
p4 rr+ss
pp -49 = h pus pp pq F bt 49 a bih + 1, 19. si + 4 quadrito
pp - 99, H+ vi
Rursus rr = ss = Htvo h es seu rr - U+ vo YA (4 + B+ 2 vV hh s.
ee
H- on e
ec
sine divulsionefiet.
xp YY+
LpP, tt+ vD
Itvi
H e
H - yV, pq
Y5 = 55
A+ v Ettrr, aee
+tth BA 1
++Vraxv, ree
posset haec aequatio develli in duas r = 5
be posterius to ob oq. 52
sunt duo quadrati
 sati aequandi qto si b abesset multipligatio per Haun, et H v
esset confecta se nam appia
p
Aaxrg aX y exxgus exsuponentia quadratum
(99 pp lt + vv)   = (H. vv, p9) (55 rY) : r5
Hin P Y qu (4 - pB)  qu Crem ss, qu qu s tlave
aurss
live
qua
epu
2r5 At+vo
Ad  a esser H- vv -
es; 2re  t, fint rY+
Finganus fidt NS() H E s I- 00, (, Y(1) S - rd
olatur Unitas atrobique
= (16 = 25 r
pd
+F+ vF 2HDP, 4el,
C
et 1q= p (nec spo
Debet esse  quC 94 - pp, tF + Wv) + 4 ppgq. qu Hl. vr quadralus
dut due aequantisto
AppGI
RG
un
c Jagamas
dua Heve (a-
fiat r4s = (B:))S, si si Lt. W, 1+5, Y 5, Y.
al
a
BV, pa
+Se
sit r- 5g: p. 9 = ((3 (1) 9: 5 se 45 = 55 =  x9 - 99. (3. 1))
+2
r+s s tlvo r+5 - 1V.C
IY H- vV, rtC
u (H. vo)
i ppaqu, Uv p4
= x
seu (pg= 9q) (B: γ) +5s, : 5 = (H- vo)ps,:, (p LH + p VVL 9Ht + quo (EptD + poe
seu (p - 9) q (3. ) + 5s, :s = (tt - vV) p5: 2pov + q(H+ vV). Fiat F
Fiunt tandem - 99 (3:Y) +55, is = 0
seu 5 = +9(B: ) et superest (Pa (33. y) = (1t - vr) p B (9B: V): 2p vo + g (H + b0)
Finut tandem  et s, in certie rationibus ad B
seu 2pro (+ __ (9+G, + quV = + () (+t9 - vvg, sit P = 4 sed hoc nolim
Aliter videtur procedi posse in ultima aequatione licita , fiat 2p∞V + quo -0
fial p. 9 = (β3. V)5 Redeamus ad aeq. 16, fiat 9= (B. V), 1+5, et 5 = (: n) ptqu et fiet
ergo fit S = R. () p + ( (1) (B. 1) Y + (A: p) (3.1) 5 Ego shabetur per p et Y
et rq = V: a) ps siet r = βCp +4) = a5,: a = p5(: a
Lt - vV, r2S =
V. C
pa: p
HC+ve, f9
et q etiam per P et
seu V β pat β99 - a5 = Yp5 dibidatur haec aequatio in duas β9-)5 et β9 = a5
si esset r+s = (B. a) r +5(p +9
fit V = a, ita modo ) ponatur a, res per se procedit sine divulsione, si scilicet rq = pS
nam aeqqualio βp9 + (399  pS(F aqS dividi petest per (4- a5, et prodit P + 9, sed hoc
postemius puto non posse fieriato ergo prius et ex aq. 1b fit Et- vo rS  B = 1
a
posito ergo X5  β9. a, sint r = (β3p+B9 - B9: a
ser r= βP: a. et sa et siet A re,
U+V
A+ vV p-9
ex -B4, seu H- vo ad egges suesiva so)
+vn
0
ins usue  ko snna ol dat siml ob