# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/632/text
# Catalogue record: 632 — [Aufz. zu Kurvengleichungen]
# Incipit: Sit aequatio finita indefinita pro curva, nempe N = (1)
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=632&katnr=58209
# Work: https://leibnizlegible.com/work/00068201
# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folios: Bl. 260 (from the shelfmark LH 35, 14, 1 Bl. 260): 2 pages, canvases 519–520, page ids 00068201:0519 to 00068201:0520
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=519
# Lines: 198 recognised on 2 of 2 pages, mean confidence 0.80
# 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.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# 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 260r — 00068201:0519
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000519.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 91 lines, status machine
paginu
possamus etiam non universalissimo catala ut in inraecedent. sodquli
scalarit
L
ter pradus seu
Aualysin. sic
u
er
sanones progredi. Synthetice, ut tabulae fiant, quarum regressus del
sit 01 + 1y + 2y = N, ubi  0, 1, 2, de signant coefficiertes
rationales integros ut cunque ex X formatos; unde quaeratur series infinita, seu valor ipsius y
rationalis per seriem infinitam. Idem fiat, in aeq. cubica pro y in quadrato. Quadratica etu
sed difficultas est examinanda, utrum possit tali modo exhiberi series infinitq ut certum ac
non esse possibilo aliam seriem infinitum dari, quae non sub eruta contineatur. intelligo integia
rationalem, cujus terminus nullus habeat  coefficientes infinitos ut
 (0 + 1Y + 2V
IL od + (2) 0117+ 027y
+ 11.
a
e
FDry+y
ubi faciendo yt y y, etc =0
000H (3) ) (3002 (1003 (3)004 pp.
e
3)o
ac
1) ou k
s
∞d
nam si habenta pro unitate
invenientur 12, 119
.Verum haes E
et  Mechodus non est proba optima, nam non tantum
i integram nam Y primus et secundus fermni
snt, habent coefficientes infinitos praestat emo
sit fa ex primetur u per fractionem cuius et nominator, et
cumerator sint infiniti, tunde tamen postremo erni potent serres
si divisio instituatur.
sedita directe venietur ad ordinariam
0+ 19 +
d.
L 
oe hoc modo y non habetur per seriem rationarem
e2
infimita ordinate,
e
4
a
seu +0 + 1y + 29t +39, +4Y +5y e = N
N.
11. D + H. 1 +H1. 2Y +_. 3 4 F11.4
+120 +12. 1 +12. ZY+12. 3
+13.0 +13. 1 + 13. 2)
14, 0 +14. P
5
et perinde est ac si aequationem datam multiplicassemus per un + iy + 1y + 13 y et
adeoque per serem infinitam urbitrariam, qui est generalissimus procedendi modus. et ganerati
ter operae pretium est eniri calculum, quasi et data esset infinilo vel saltem indefinia
fit n = - 1:0 ct (2∞,20 4 1.1.0.  et B fit bl_ P12 =- 2:0  - 11. 1: 0
ot 13 = -3: 9 - 11. 2:0 + 2: 6 + 11. 1.f. 8 et 14 = =4. 0 = 1. 5. 0 4 2. 2:0 4 H. 1. 2: 04538-
seu 13 = =3:0 4 1. 2:6 -2:0+ 1. 1311 et 14 = -40 + 2.2:6+ 31:8-1. 1. 2: 0-9: 0+ 1.2. 0.
et 15 = -5: 0 + 2. 3: 8= 1. 2. 2:8, +, 4∞: 6-,2.2. 0 - 1. 1.3: 024 1. 1.1. 2:6
1.t 16
4:0 +1. 3.0 422.0- 1. 1.2.8, +B3: 6 = 112: 0
~1.1.2.6
+ 1. t.lit. C
Hinc si respectire faciamus N = 12, vel 13, vel 19, e. vl 15 et fiet
1 21 03+ 1itsnc 6.d lit e
11-
∞0.7 41. 2
-0. 4 +0.340. 1.3-182
+0.317 +0.3 = 6.1. 2- 0. 1. 2+ 13
1.2:0
-035 +0, 2. 3+0, 2.3 +6. 1. 4 0. 1.8-0. 1.2-0. 1. 2 +1.2
+034 - 6. 1. 2 -6. 12. 2- 0. 1. 3 +0. 1. 2+0. 132+0. 1.51
Et quo longius procedetur, hoc exactius habebitur valor ipsius 
id eoque et ipsius y quia y = 0+ D. 11,: =
a mem potentiae ipsius O idem habant
progressionem autem dabio ipse calculandi modus. ubi cunque omnia et us
signum ounus montol numeratua uno gradu dimensionis excest nominatorem. ed quolibet membuo
qua pint potenlrae inti Tat cunt literae eed si gradus ipsius o detrahatur a gradu membri, residuas
et tume literarum concurrentium cum o. signa alternant cum podentus ipsius 9, sed siqua nomina
porisppisita numeraloris signis. Illud superest investigandum cur aliqui termini aliquoties occurrant.
t 6.1.2. bis. et ∞  C+ ~ ter. et 8 1. 2 bis, et 0, 2.3 bis, et 012 ter. Observo id
totandum aequationem C divisibilim esse per aeu  Et semper series infiucta divisibilis est per
C tantum in literis sibi immediatis pr
1
finitam identam jam se finita sit indvitibilis rationaliter ex hypi nonest
divisor communis. Desidisotur ergo invenire divisorem fiditum sertet infinit
quationis
in

## Folio 260v — 00068201:0520
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000520.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 107 lines, status machine
sit aequalio finita indefinita  procurva, nempe N= R10 + 1X + 12X + 13X etC Y +
= 20+21X + 22 X + 23X3 7. y  30+31X + 32X + 33X etc y + eta
ponatur eadem haberi per feriem infinitam y a + bX + CX + dX + ex etc
fiet  = a + 2ab X + 2aCX + 2ad X + 2aCX + 2as X C+ Zag X trah x +ZaKX + 2alX
+ bb X + 2bcX + 2bd + abC +2bB 12bu +2bh + 2bL
+ Cc +2 cd trae
= 6+ IX + 2X + 3X+ 4X + SX + 6X +7X+8X + qX en
dd
Ect 1c rctr
2de
y60+ 2 0.1X+ 210.2X + (20. 3X + (2)04X + (2)0. 5 X + (2)0. 6 X + (2(0. 7X
+() 1. 1. (2)1. 2 (2) 1.3
(1) 2. 2
(2) 1. 4 (2]. 1. 5
(2) 2.3
(2) 2.4
(1). 3. 3
(2) 1. 6
6)2.5
(2) 3. 4
=01 + (3)0DIX 1 002X +) 403X. +3) 004X 15(005X + (3)006 X +007 47
+(3) C. I. 1
(1) 2HI
(3) 022
(3) 112.
4
20000+ 4 000(  + (1, 000, X + ()0003 x + ()0004X
+(619311 ( (6)0022
47
Rogula coefficientium combinatoriae
membre idefficientis
β) H. 3
(3) 122
14V0
IHt ( IIZ
(3) 033
6) 114
(1) 222
44000
1610053
(4(0222
(4 1113
quantibales (6) 1122
in quavis potentia ipsius X non ad hibentur nisi literae boifficientes binae
Summa literarum caefficientium numerice expressarum aequatur potentiae
sub
6
B)115
(3) 17
(3) 223
+(4bo
(4) 1114
(6
(4)1222
si
2df 209
ec 2ef
+(2)0. 8X ex
2) 1. 7
(2) 2.6
(2) 3.5
(1) 4. 4
)
4  D. 8 X Hs ogl
3)044
1 110
(2)214
3)2.39
1400047
(6)0044
(4) 1115
(6)1133
t
3) 117
(3) 144
(3)325
(1)333
ovqe
e in oix pabet 0td + 2=4
Gradus membri aequatur gradui potentiae ipsius y., verb. g. 1122 X Occurrit im
ciens 1122  seu l E est 4 graduum, quia 2+2 = 4
In
225 29 palet 2+2+5 ∞9
Ex his jum seribi possunt Termini sine calculo, v.g. in V termini rb, et ideo membri coeffi–
totus coefficiens ecribetur
exhibendo ordine omnia membra ex binis quantitaetibus composita, quae sint 4 dimensionum ita non
nissunt alia prodere quam 0006 0055, O222 IH3 HEL et conslant C, neu
e
non possunt aliu prodire quam 000b 0033. 0222 1113 1122. Ubi nota linarem
corollarium in eodem coefficiente potentiam eandem pjusdem literae non nisi semel occurrere posse
Lineam in edis potentiai unimam esse sive sateo in sectuda sullum unum intertia tresete. in quarta sine
ipsius potenta me
cursuus inq
(2m erc. Si potentia ipsius y esset 5, in linea 5t, 10no ema
Nmncte praefigeodi comnbinatorii inveniuntur lege alibi traditu
Et ita porro. Sisam in aeq. 1. pro y, y, pert substituantur valores ex aeq. 3. 3. 4, ere
tunc aeq. inde producta 5. fiet identica, si quidem aeg. 1. est finita. Et pronde si data sit aeg. 2
infinita, sed quaesita . 1. finita, explicandae coesticientes, 10, 11, etc 20021 erc. 303) er, ut
fimitio succedat quod autandum petgendo de gradus in peadum primum enip aeq. 5 finiemus
in X deide in et dem in B erE. et ita pergendo prodibit quaedam calculandi series
sive progressio, quae dicebit, quid in progresso sperandum. dignus promoveri calculus, nam rursus
certa lex in explicatione aequatio nis 1 per aequationes 2, 3,4, et. servabitur
si in aeg. 5. quae identica esse debiet quaeratur 0, 12 etc. quantur seres
infiuta ex data finita, sin. datis 0, 1,2, 3ele quaerantur 10, 11, itC; 20, 2), et
quaeritur series finila ex data infinita, quod est difficilius quia mesemper possibile
11333
(8) 1116