# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068197:0263
# Permalink to this reading: https://leibnizlegible.com/page/00068197:0263?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 132r, page id 00068197:0263
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 133v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=263
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000263.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 76 lines, status machine
# Lines: 76 recognised, mean confidence 0.82
# 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.

Hic est quadratmra mnihi ab Hugem
pro curta X+.
nxy
par3 sem eo
pagua
co
a une
Iem prbenta Bgammo propontt sili Cartescos toluo per serie ndus supponati pro habenda sere
Hugemrus in literis 12 januar 1693. ad me datis proposint mihi
quadraturam curvae cujus aequalio sit X+y= Xyn. posito n esse
constantem, et x, y esse coordinatas
Hanc sic inquiro pro
(4)  pB: 1+ p3 qunit jams quoritut a
1
ita res est reducta ad quadraturas rationales, 
quaeritur BX videamqus igi loco quisit. sxto
 pat v = e -n. pet v + n= e- 3ne + 37 e et x = n:, ce- methun, -
n:; e-ynet 3ne. quae partes sunt tractabiliores, sed placet redire ad aequ. qus
ubi p+V prodit ex = n+ v  n - nv + v0 et n - no+ ve  rursus resolvi potest
nam fiagatur vv - nV + na =0 fiet VV = nD + onn: 4 = un: 4, = na seu V - n : 2 = n f- 3: 2
seu v =  n: 2, 1+ S. 3 Ergo fit nn - nv + vDZ v - n: 2 +  n. 2 F. 3, v- n. 2 - n. 2F3
Ergo denique n + v = VAn, v- n: 2 + n. 2 F3, v- n: 2 = m. 2 F3 Sed at generalius pro
cedamus faciamus per cpendrm x =) nV: , vtn, VAP, vtq et  videam,
quomodo tale quid componi possit ex Tribus: X= b: vin, + C; vtp,, + f; vt g. quae addendo
in urum fiet: bvo + bptbgV apq + CBD+ CUrCqbenq + fDDt Sn+ BAsp,: ven vAp vtq X quae
aequatio ut conferetur aequationi I1 fiet b+ c+ f =0 seu b=O -c- f et rursus fiet lpa + cng + sup  0
et pro b substibuendo valorem, fiet ex aeq. 14 fiet: - epa - fp9 + Cnq + fnp = 0 jam quia conferendo
4, 10, 11 est p = - n: 2 + n. 2 F3 et GC =n: 2 - n: 2Y 3 fiet p9(e nn: 4 1t S.7 - 5-3 + 3 nn
et exaeq. 1b dividendo per nn, fiet - c - f  -c: 2 + c: 2 F.3 D - f: 26 f: 2 f.3 = 0 seu fiet,
C: f = - 34 53,: +3 - 5.3. Ergo fiet cif = + seu = - f sed ita sel b 6 quod satis
mirum est. Exequamur tamen: _fietque X   S:, vi in f:, v +  B- 1-, n.0
X= -f:, v +  S. 3 - 1n. 2, 45., v- F.) rr R.E (P- vf + 5i 2tiB +s f ibi  rat un
25
=fnsus, u n nn seu BC: f = (9 - 6F- 3- 3:, 9- 3=) 1 - (-3 seu C = - f+ f.3F. Ergo per 25 et 19.
fiet b (5 = f3f -f =) - Bif fit ergo X = -fs. :, vt n ∞-f + SE3, :, v+ (ag - 1B. 2, + f1, v- 123H M. 2
Unde sequetur legarichanam ipsius X esse lugatithocum ipsiusvanc e
e
eece
A= elog -fF. 3 log vin  -f. +f f. 3 log v + 53. rn. 2 + f log, v - F3+IN. 2 per compendum
n
30)
si sit XZ -10 log 11 + 12 log 13 + 14 log 15 fiet X- b 13 15: H X. = log, 132 151 11 foc
Eg
est paradoxum valde nitabile, qu u Nempe X: 5 erit log3, D + 5 . v- P3tihiL . v
seu Num; p.s   seu x - log, 13. . 1 15: 118. posito 10 = s. 3
eusit  coenu:v=
A Idem in fallor contingit in aliis sit XX + yy = xyn
Ergo ex aeg. 2
Examinemus rem prius in simplicioribus Xx +y = Exy haec equatio est ad rectam
fiat y = _ _ XV: n fiet nnXX +XXWD = hn XXV. et evanescentibus ex siet nutvo-bno
tyebolat
Faciamus X + yn = Xy it y - pv. n fiet: Xn + xon = XV seu nt vn =xV
 X+  = nXy sit y = XV:n ergo X n +X v = nXB seu X n + XV = nv. seu X = nv; n+V
Hujus ergo quadratura pendet ex aliquo alture qual ocurrit in curva Ellipseos.
si velim _xris servare fiet x + = n al X+y= nxq vel X   = rg .n vel xjus =Xy. n
quae estia 
si sit X + pX ∞q - 0 fit X =   + Y 4. 2 + 49. , + p er
0+ 2 = q5. l. 1+ 2.1 - 3 yy = nX - X: y
Mlemini olim ebiam incidere quadrabilia, ut y-Aptex: Saa m
4 nam doaa - Xx = a - 2x ax: Iaa - xx Hoc igitur quadrabile est: jam
FE = Y 93
sit aa - XX = a Trax +XX Wv.net a - X = a +Xx vra seu a - aex = a vv +XoV seu X - a - avv; v
et Saa = XY = a + X V. a jam a + X = (aB) + a + a  ( avD,: votaa, et sit areo
Vaa - xx = 2vaa: vo+ aa. Erso fit y = ba - (: 2va. seu sx - vrtaa. ∞ 2avdv - a- avp 2Vdg
(2votaa seu dx = ( 2av du - 2avav - 2ae vdv (= 2aV dv: vv +a seu dx - 4 a4v: (3) a + 
Ergo vax  seu bpdx: taa = 2baa aanvo: aav Hoc ergo summabile, utileque
sit  aa - ax = coco aa aa: taa- ae = y. siat
eut reducere ad lohanithmos
Paa - ax = coco eribque X =  - a - eco: a et Ax = - 2codco:a Ergo fiet
Aniter saa- ax =
ydx = ax 20 - 2 cindeo: c = -1a deo
LraL ta Taa - ax: laa tax = 7: a
 av; aator = y a sit
derum redeamus ad aeg. 4. et consideremus non quaeri a nobis
Id ergo quaeramus pag. seu
xdo, sed potius e y ax.