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# Piece: https://leibnizlegible.com/api/records/242/text
# Catalogue record: 242 — [Aufz. zur Zahlentheorie]
# Incipit: Invenire tres numeros ita ut duorum quorumlibet summa, et differentia
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=242&katnr=59210
# Work: https://leibnizlegible.com/work/00068243
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 30
# Folios: Bl. 4 (from the shelfmark LH 35, 3 A 30 Bl. 4): 2 pages, canvases 7–8, page ids 00068243:0007 to 00068243:0008
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068243&page=7
# Lines: 153 recognised on 2 of 2 pages, mean confidence 0.75
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
# 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.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 4r — 00068243:0007
# Source image: https://digitale-sammlungen.gwlb.de/content/00068243/jpgs/default/00000007.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 109 lines, status machine
Invenire tres numeros ita ut duorum quorumaibet summa, et diffe
rentia sit quadratus.
2
(4
(5)
1
sumeri a. b. c. Conditiones a+ b ( a - b p m a +Cpu a-c F pr b1c. 1. 5
a(tB) +a CB) F) l’ + M  Ergo a FI lL m. a( d + aCD ,, n + p, Erx aF n+ P. Quos duos ipsius
1
a valores aequando erit 1 + m p, n +p. Eodem modo b +c + b - C B q +Y. Ergo bps e r
a- b + b +c p7 m + st ergo a +c m +q. et idem aequ. n. per 3. Ergo n Fi m + g.
(14
a+c + b - C ( n +Y. Ergo a +b, n +X. Ergo per 1. ae erit P f  + 1
si in aequ. 9. pro
1 substituas ejus valorem aequ. 14 fiet (∞+1 + m  ( + p. seu pm +r. Utamur
ergo his tribus aequationibus 12. 19. 15. Quippe simplicissimis
Et primum in aqu. 14. V_ n + N.
faciumus (F r+ e. fiet: C (Varre + e p n LB seu rei n -e et 10) n 4C Eodem a mod
2c
resolvamus aequationem s ponendo Gp n. f fiet (D) A n AX- 2nf 4 p. seu nfi m+ f. et m-flG
se d utile erit facere frhe. et mti hgc: erit n, h. g  + tc
rhe
C2)
vilche the. ex
es vitix-4e. Nunc
seu r
ergo solam quae restat aequabionem , 5 absolvamus  explicando n et r in ipsa habebimus:
p  lge + a v he - 8vhe 4 16e
4
2
scribatur: 6 P p, 649 si + V ht - 8VH
=XV s
+ 4 X
110
seu 64P 5 64tig + v+ h2 - 8Vh + 16.
+ A16
+ XVH - 4X quae aequatio dissolvatur in duas:
64 P1 8. es 6 8pG Yt - 4 + x
59125 XV - XVCX, 9X -∞, 10. sive
nat
fiet 64gco X - Vc X + 4 - x f16 s? scribatur ruisus: siet inde no
648 co X - 4G x 4 - GX  YCo X + 4 + 4GX G
35
645c6 48   X,70 seu X p64918- 48 3
6
epe
(32t  onis
Vel et aliter 6co X - Y Co X + VX + 4 - x - 4 X. 7. aequivalet aequationi 31
seu X645co + 
seu   16
144
et
fiet: 649 c5 + P + 6. 645co 4 + 4 p  16v c0
(36)
vog
Quos duos valores inter  aequando
Ponamus vsesse cognitam
quantitatem, q. scilicet pro arbitrio sumta, et faciamus geo 1 γ:sid: eo
tut turbamur quod _ g. +1 et fiet: 649c0 +1 +6491c0 4  +44 169 + 39 + 168
649  se 2 + 69 c0 4 p+ 40 f 1692  510 + 1610
dividatur haec aequatio induas ap +64g 54 et 4  Ponatur 97
+o g fiet: i 164 i,  ac2A 16 64+ dco
9H.
Y+ 649  44  + H B (g c -64 9 1o + 16o seu Bf 165t-149  +rc d 5-
scribatur 1 +649c0 41 C  A 4AY N   + ansutu
seu
+ or
et fui:
sit  (3Co et fic
Blo B∞ +649to +(3 + 4 at  1692 ct-g  + .C coV
ficit birg Eo β ( 1694-649+ 16. et B + ap opper se  p
64q
ergo int oveut non et
1+ 4bc0  et 
oaedis e  et aB F o
aliter Fo 169t - 6 g + 16 et 644  + AB. rusus absurdium
64g co + (3 +64 g c0 4 β + a (Co p 169t co + est g eo + 16co
2co 1 169 0+ 329 co + 16Co
aco
se 5+6  p ihos  (329 + f. seu _ d( 1 49 - 4
et β + 649c5 4B + aB C 5  co p. seu 1 +649 co 4 + 4βco - s βa p7 0.
249
BF
1
77 e 
et rusus ext 4 fiet: veok p 2 + gX. seu
(39
divellatur in duaf. 649 co - vceX +  X po. et 4 -  - 1x 0
50)
et et 649 k - X rH + 4X - X 57 0.
Faciamus hp, cox. fiet. 64  siveltopt 10d
Div ellatur in duas
61
110
con
C
19S 482co p, S 4 64 d g co . sit S5, Bco
fret: 49  co- 4 pi cot - 1b f + 16 ( du + 64 nga
269+16
ged - et rursus = 1691 4+ 16464 9 c6. seu (1 649 5
2 4c g o1 16 
6 4 (nco4+ n co + 16
50Y 2+ 156
4e
256
+
89r+90
5 -29 + 110
297-9-1

## Folio 4v — 00068243:0008
# Source image: https://digitale-sammlungen.gwlb.de/content/00068243/jpgs/default/00000008.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 44 lines, status machine
(5V
quatio 31. nempe 6464g 0 ∞ - vco X + 4- X p ita commodissime
(37) 
(33, 
divelli poterit in duas: 649c0 X - X 70. et -vco X +A0. fiet ex priore
+
64g c0 A4. seu 8ga seu  H 169. et exposteriore seu 33. fiet vcox fh. seu Xp
(+2
et.
seu ex 35 substituendo valorem ipsius eo ex 34 fiet: X)(
E)) agth2d. hu, et vh  (+) 2 sed udc sit r, 0 quod esse non debet
649 pX  16. Ergo 88 p4. et 29 Y g +, inde ge 9r1)
+9+1
veo
. seu X B9. et quia ve 9e1. fiet
64g c5 X - Gco X - 2 g10 x = 1o X 1 4 - x 70
64 9c5 X - v 10 X + 4 - X (0. i 6ag A _ XH 169 5 )) 4VC
b4 g co X - g co X - 2 g es X = co X + 4- X +gF) 2v. Ergo aVB 2g. p5 +1
10.
seu faciamus 64go
I + 2Co Xg B + 2CoXβ + ( - 64g cs X - 4
X
fiat: 2c0g β + 21o Aβ - 648 c0 +  Yv. seu βF
ex (3- 4 sit quadmtus.
3+2 sit p N C. et B. 2 p Apst
1649c6 -
2c09 42c0
seu BaE sit p ()- n A
seu β1 Nn - 2. F Ap +2. et AM - A M p, a
sit A ) N. et A -   . fiei Ap 2 + p Y) N
et Zn + pi g, N. seu  sep 3( fiet ++ e p, 3
seu np 35. fier 3BV + D N B 1 et Np 1= 7 3
et n p 1-23 et A
12
12
3
Ergo 1- 13 - , - 213+ 7 
145)3 seu 433 ,3. 43, + 12 3 . j
seu a3  2 - Z 3 sita
(P. 4 ⊓  (B+ ep 2 βs Ergo Brjo- 
25Y seu 649 est 2N = L s p) 2C09 N = 8 Cog 4 rco - 810
seu 648c5  N- 4
2C09 + 1c0 H
= 4