# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068199:0278
# Permalink to this reading: https://leibnizlegible.com/page/00068199:0278?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 139v, page id 00068199:0278
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 138r); this page is its left half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=278
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000278.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 81 lines, status machine
# Lines: 81 recognised, mean confidence 0.85
# 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.

X+l, X+m, X+n ~ AB, BC, AC
X  +bYS A =  uNq + 1218
B = 2 1G + 2218
AB = 11. 21t + 12. 22 8 +  (11. 2 + 12. 21) 1t8 = X +l
fiat X = 11. 21G + 12. 12 8, -+ (11. 2P + 12. 2) f 45
X = 11. 31t + n. 32F - m 4 (14. 34 + 12. 31) +46
et denique X = 21. 31t + 22. 328 - n + (21. 32 + 22. 31) Yh8
Ex constantibus seu irrationalibus, fit 1 11. 22 + 12. 21 = 11. 32 + 12. 31 = 21. 3+ 22. 31
ope harum duarut aequationum quaeramus 21 et 22, Ex priore fit 22 = 11. 32 + c2. 31 - 12. 21;
six posteriore fit 22 = 11. 32 + 12.31  = 21.329 :, 31. quos duos valores eet quando fiet
11. 31. 32 4 12. 31.31 - 12. 21. 31 = 11. 11. 32 +11. 12.31 - 11. 21.32. adeoque fiet
21 2 +11. 11. 32 + 11. 12. 317 i, 11.38 = 12. 31 seu 212 H. 1. 32 131. 31-
2 = 11. 32 +(2. 31 - 11.227 :12 = 1. 32 +12. 31 - 22.317:,32, fiet
=12. 31.31 = 11. 31.32
11. 32. 32 + 12. 31.32 = 11. 22.32 = 11. 12.32+ 12. 12.31 = 12. 22.31, unde
eF eodem modo
22 +12. 12. 31 4 11, 12.3272, 12.31 = 12132
 11. 32.32 = 12. 31.32
Ex Dariantibus seu rationalibus fit:
Nec differunt valores
nisi a.. 1 et. .2 permutando
11. 215 + 12. 226 - l = 11. 31G + 12. 326 - m = 21. 31t + 22. 316 - n
Ex ag. 5 fit 6= (11. 21G - 11. 31( G + m -l, :, 12. 32 - 12. 22
Ex aeg. 6 fit 6 = (2. 31- 131)  6-(H. 21 -2. 31) + n -m
6= (21. 31 - 11. 31) G +rn = n . 12. 32 - 22.32
Hi duo valores ipsius  sunt coincidentiandi fiet gnos
(12. 32m) - 12. 32l = 2213 Em + 22, 22 L = (12. 3em 2. 3. H - 1 rn + 12. 22 n
m.l =
et (10m0) 12. 32 = 12. 32
32l (12 - 22) 4 22m (32 -12) + 12n (22 -32)
12n (32 - 22) + 22 m (12 - 32) = 32l (22 - 12) =
unde patet permu-
sabua aequatione, quod
comprobat calculum
11.12, 2. 32 (=11. 12. 31.32 H. 2. 2237 + 15. 22. 31. 3C = 12. 3. 51. 32 (=11. 12. 31. 32L 12. 21. 22. 31 + 1. 12. 2. 31
Ex aeq. 9. fil 17. 21. 12. g, - H. 31. 12. 32 -
ob aeq. 11 fit  22 = (13t0) = 12 3t 1 grn  12 3EL a n  n),: 2 l 12R + (12- 3e)m
quam aeg. aquando aequationi 4 _ habebitur _9. sine 2K et 22. Ex aeq. 12 habetur etiam valor psius
22 = ( 151) = 11. 12 4. 32 - 12. 21. 31. 32 7:, 11. 12. 3(- 12. 2931 =11. 31. 32 + 11. 21. 32. Unde aeg. 13 et
11. 12. 31 -12. 21. 31 - 11. 31.32+ 11. 21.3212. 32 (l+1) = 21. (11. 12.32-12. 31.32) (3ub un Fe
unde tandem fit 2= (1. 12. 31 = 11. 31. 32) + ) 12. 32 (len:, (12. 31- 11. 32) (F (an) 4 (1. 12. 32- 12. X 32) botur
sed compendia oneo: Ex aequio B 9 12. 32. 12527 : 12. 32 22. 32 = m ly :, m - n = 12 2L- 1. 31, :, 21. 31 - 11. 31
15 conjungendo fiat
quani aeq. jungendo aequationi 3, fiet aequ. (17ma) sine 21 et 2
ex a4. 11 fit 12. 32(l - n) ,:, m (12- 32) (3et 1n +32l - 12n = (12 mo) 22
Conjungendo aequ. 9 et 10, fit 22 = (13 tro) = 11. 21 = 11. 31, :, 21. 31 - 11. 31 = m . 1,:, m-n. seu fit
11: 21 m (= I1. 3Cm - 11. 21n + 11. 3In = 21. 31m (= 11. 31m, - 21. 31l + 11. 31l. Unde prodibit
prodibit (1450) 21 = 1 1F 3t (L ) 11. 3C, L-n, :, (11 - 31) m + 31l -1R
pursusque non differunt valores 21
 ut et.2 permutando
cedi) -un i 
 31. 32G. E1. 11.313225 N1. 12 31 51((1 297125, 5)3∞3 cL ( 3 3131 l (2 11. 12 - 14. 32 = 12. 31) = 11. Un (2. 31. 7 2 - 1. 32 - 12. 31)
Aequemus duos valores ipsius 21 ex a99. 3 et 14 fiet
(B. 1. 31. 2L = H. 12. 31. 31) (l- 2 (11 -31) m 11i 1 h . 3 H. 31. 36
(6) 32. 32 l (2. 11. 12 - 12. 31 - 1. 35) + 12. 12n (2. 31. 32 - 11. 32 - 12. 31)
Et fiet (11+ 3)m (F ILIL. 32 = 1. 2. 31)
Et pari jure ex aqg. 12 et 4 ae quando valores ipsius 22, oportet fieri
solae ergo supersunt praestandae a94. 15 et 16. Ex aeg. 15. habetur 32: 12 aequatione simplice per 31: H ad Cubum ascendenter
(-12+32) M (+ 12. 12. 31 + iutr. 3
 ex aaeq. 16 habetur 32: 12 aequationes cubica e per 31: 11 simplicem.
 so 3l. H ponamus X: a, et pro 32: 12 ponguns 
vivis
Ut res reddatur simplicior, pro 2E3 poramus X et
1 - ed 1Craa ay _ ax ram l 
Xpyte Xgg e exaeq. 16 fi a
ex aeq. 25. fiet (-a +X) 
=(6) = Gyl (2aa- ac
(22y- af - ay)
unde vel pet =m contra hypothesin, vel fiet Zty-
neque ita A et  unt proportionales, A et C fimil proportionales
ipnfi, ra-9 - 3
a qu a g y, quae aequatio est identica
seu 32: 12 - 51: 11 vl 32:31 = - 12: 1. quod procedit;
Nam A = 1S + 1210, C = 31 SG - (12. 31: 11) C6, unde
Ergo quaesitom non procedis unde
licit inam fieret l-n uam i isue didindto st Hid sut jes tutt non serer sa  t h.  a 1 = st, 1
C = 311G + 3216
Ab eg. 15 subtrahendo 4g. 16. videtur aliquid prodire aliquid
(X - a) (y+ X) (aa - aX) M
iet =aa = 2aZ - Z7 + aa - 2 ay F97, l, 2 d = 2a
(y- a) (y+ (a4 - a)