# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068197:0273
# Permalink to this reading: https://leibnizlegible.com/page/00068197:0273?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 137r, page id 00068197:0273
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 138v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=273
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000273.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 111 lines, status machine
# Lines: 111 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.

(1
y = 10 LX = m + 11 Ex+n
gy = 1010m + 1. LK = 10. 10 F 11. LIX +2 01S, X+ m. X+n.
sit v OX+m +2 EXEn, et VV = 202om + 20. 20 X + 2. 20. 21 Tx + m, X + n
31. 21n 21. 21.
Et  qL 30 sxam + 3) IX+n e ZZ 303o m 4 30. 30 X + 2. 30. 31 b, x+m, Xtu
31. 31 2,, +31. 31.
sit jam a+bgy Zv0 et etRyg = 22
ubi a, b, c, f sunt data
Ex aeq. 7 per 2 et 4, ut omnia fiant identica seu X maneat in determinata fier
a + 10bm + 11 bn 2 26m + 21n et  10b4 11b  20 +21 et f 10. 11b = 20. 21
Et ex oq. 8 per 2 et 6 similiter fiet:
(14)
(+ 16fm + 1fn = 36 m +31a et 10f+15F  30 +3 et 10. 1f = 30. 31
2
Ex 1 fit 21= 10. 1 b: 20 unde ex consit a FYSbm + I1bn= 2omet 10. 11B 20. Et ex 1opr 15
Et ex 10 peris  (2t: 10b + 1 b Z 2d + 1 10. 12P 20. Aequatio 17 multiplicata
e
per =n addatur ad aeq. 16 fiet: a + 18bm-n= 28 m-n osimiliter
aeg. 17 multiphcarta per - m addatur ad aeq. 1b. fiet a + 12 b n- m L 10Vb n- m
21
i a18 fit 20= a:mn + 10b et ex 19fit = a: man + n b = 10. 1 b. 20
(22)
Cergo per 720. 21. fit = a: m. n. 4 11b= 10. 11b: a :mon  10b d
(24)
sit  a iman (per conpend) = p et fiet - pp + pb. 112 - pb 10 (+ b 10. 11) = (10. 1 b
23
63
a est duarum
num
1d
Pari jure et H - 10 0 = C. f Sed pari jure 11. 11 = 10. 10 = C: fm- fn. Ergo fit a: b= c: f
eris
i sunt vianeta
quod est absurdum, et contra hypothesin, quia a, b, c, f sunt datae imo in easu talis proportionis evanesceret
indeterminata. Ergo haec via succedere non potest.
interim notandum hoc theorema: si  pro y sic assumi debeat
Si pro y sie atsumi debeat 10YX +m + 1 extn ut a fbyu aequetur quadrato
et
unde patet non posse ta aliam
a, ab(a 20X+m +2HXIn) fore _11. 11 - 10. 10 M - n = a : b
el aequari tali anto, velut C+ fyy aeq. cgto (a 30 Trtm +g1VXFN)
20. 20 =, g; m.n + 1010b et 21 fore 10. 11 b: 20
nam fieret a: b = c: f. Ergo si formula C+ fyy seu  af. b + fyy, coincidit priori a+ byus
m
talem formulam ad hui
martiplicatae tantum per  f: b
a +bgy aequari conto a 20 exam +21 sx+n,
et et Lun aequari onto a 30 sxtm +21 tran
indefinite,
potuissent absolute aequam itri vationati
a numeris rationalibus; quia X+m et X +n
simul possunt aequari quadratis indefinite.
Video igitur confugiendum esse
nanc nt
ad  compositam ex tribus irrationalibus
sed quae in se invicem ductae nunquam sub vinculo
contineant plus quam duas dimensiones. Ita enim
idem fiet in lateribus quadratorum 
quo facto ilti na et entractiones
si res processistet, seu si simul potuisset
quod sufficit ad scopum quadra turarum. Nam
formula summandam constabit ex Hc partibus,
varum quaelibet per se quasi sola esset revo-
eabilis est ad summationem  quantitatis.
IX = 1C L. _X F. 1
yy = 1eM 10X
possit explicari. Rem ergo tenta
Y I. X AR
- 210. 
Or
3x
Da Rorkum +D. Exin, ut pauso antet
_ a 30 l.. +3) f... ut dite
3c
 +byy = VD, 32C+ Z2 fX 1-2C
22c +28b10m (I1. n
= 2 0 I + 2CZ
re + 22(1d 1 + 32f10r2 1,
22b(0 + 2261 a 262
rationalis, etsi tota formula composita per nt r
rationale non aequetur indefinita non
bimus pagina sequent
s
12 p A It X
p 23
A
sed video 2
22 et 72
quae considerali
possunt t
22b. Z. 10. 11 fF. 20 2, Bluna
32 f. Y. 10. 11.= X- 30. 31
nhil
Adese
Ex 5 sit 2Y- L2b1c.1 e 20 CF tinc p7
AE
e
fit _ B10  b11 AEB + D 1Y b: 20
Ex 7 etst sit
3
C R + bloME L bi n Dom + 10= 1, VH. 20
multiglis etut _ 9. .  P  = 1. et pred iunstio
addetur ad aegq
+cetL 21. et, et
AD
n V10 + a +b10 M = - n
 20m Vt nubetur q es
ractio. 10 h per A et 9 habi tur
alter 20.
abc abd, acd bed