# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068196:0737
# Permalink to this reading: https://leibnizlegible.com/page/00068196:0737?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 371r, page id 00068196:0737
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 372v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=737
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000737.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 108 lines, status machine
# Lines: 108 recognised, mean confidence 0.79
# 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.

21 sestetbil: 1693.
nserviunt series ad Tangentium inversas d peras redu
et per eas reducitur problema ad aliud faciliis. Nam cum seriei infinitae
aequatione composi Nam quando y habet pro valore seriem infinilam per X, et ita
repraesentantur ejus affectiones vel per se vel in ipsis e affectiones ductae, componitur inde
series infinita; et si jam addatur terminus qui constat ex solis affectionibus
aliu
is lac
explcata
aliquois non habet valorem infinitum, nec procude in seriem infinitam compositam influit, ut
V. g. si esset ax dx, negligendo de potest intelligi secundo gradui serici infinilae compositae ascubi
ax. I seu seniem re licebit  sed ut nos ax evilemus, summatim seriem tantum consideratimus
post  X, inciprendo ab X3, et ita quaeremus coefficientitis opeserici incipientis ab Xh quasi
nihil antecederet. Quod si jam series haec a composita ab X tendens in infinitum est
summabilis, utique linea quaesita este stdinaria. Furmmabilis autem est si omissa quantilile
a X dx aequao assumta ad curvam esset dabilis finite ita enim Y hoc series. Itaque et
A
pus
vi
tunc erit summabilis liaet accedat ax
pro aydx ponam bydu et pro
esaus et distrnctus uorem
sit ergo data aequalio differentialis aadus + Xydy +xy dx taq dx = axdx
singamus aadus ehydus, Zydx + aa d F a faciamus aa do + xVdo +Xrdx + ax dx = o
gaes
 ypes
schedicusmat dicrsse resolaesi resolui queat
a
sed si jam quaeramus X per _ v, sequitu sequi videtur hanc aequationem posse sumiliter
sed hinc 
Porro similithr suitur
resolod, ponendo, XV av +XoAX + avdX = 0 seu ZdV + ZdZ + a d3 = 0. unde sit
v = Z_. alog. Z. sit ergo Z = 10B + 11V + 12 v + 13 v3
(et adz -= a, 1 +, 12V +3BV + 4. 14v + 515V +6. 16 V ex)
7d0 = 10 + 11 + 12 + 13 + 14. + 15. ex
7dZ =. 21 72. 2 + 3.23 +4. 24 +5. 25 +. b. 26 et
(6
v =c
12= 20 +2IV + 2IV + 23V et
per con pend pro 10r utoripec
Erga f V = Z alogs Z t si sit. Erco ex terminis v, v, V, etc. v, et, V, V, etc. quaeratur
series exprimens valorem ipsius Z per V, seu valor ipsam 11, 12, 13 etc. nam 10 manet arbitiaria
porro sed ponamus haec jam ut inventa.
Et ja pergamus ad, aequ. 2, quae et fiet
etfier X et fal    30 +31V + 32 VV + 33V + 24 V et c. et jur quod poo
tiet + apde-  A . V + 2. 32V, 333V + 4. 34V 4 5. 35Y
29(30 +31 +32 +33 +34
X
vx = + B) (41 +2 p. 13 4 14p 15 45
+VYdo=
(8)
= 0
2 Xdx - 21964g X r41
Ansc
ZX- Gut bosex +St VISiPst t C
itaque ope aeg. 8 quaerendo sed quod morum videtur haec aequatio esse impossibilis
aado = ee tu
qua quoscunque terminos assignes ipsi D, nit reperdes quod ipsi a aequetur
caellies teminiscascrbas
Vi si quater
A = 29 + tube sde + 30 +31V. fort nam ita Vxdo dat h
X = 29 + 2. 29(d0 e 2.29. 30 + 2. 29. 31 +
 itaque quia ha haereo nec possum X exprimere
sed puto non scadere quia
fit _ ax - 13 24 - 3. 12, 4. 3+ er Chepdx = a1 242 d 1 eu
fiet: +a Vdx = a 7 X - 131 - 2 12 - 3. 12 - 4. 34 ere
pe
X var = (5 3.1) 31 + 32 + 33 + 24 +35 et
+yxdx =Z,
(Le) . 242 4 3.97 - 4. 41
=a dv - -aa
1 30 2 0  40=0 a1 -
fiet ad Z = a X 49 H  1 3t - 25+ - 3. 5) eu
Zv- 40r5s,  2 + st + 5 +54 ey
Idq  (. 1b v. G 4 -161 - 2 5r 2 1. 87
Sa hac est infinita
ad
sit 238 4 51t 51 +53 ei  00+ e p x
st 
ab = a, X X - 1 - 251 - 3. s 
2d- (59+ st + 3 +  +, et et
Z  (61)- 2.61 9.69 er
nonatur 5o= Bliq et 61 = 0 et 60 2 non iitspai
sit Z. 490+ 501 516152 etc 20- 59049v +60451 vr6ib aeu
Hmo patet omassit temminis uti
vi
0 se
su
 ac ficret aquat i possibilis
40= 30 4 = 2.303, 42 - 2.3032 +31. 31 437 - 2. 30. 33 + 2. 31. 32
44 = 2. 30. 34 + 2. 31. 33 + 32.32 et ila porio
52 +2.50.51
62 = 2.49,43 +2. 50.52 +51. 51 63 = 2. 49, 54 +2, 50. 53 + 2. 51. 52
64- 2. 49. 55. +2. 50. 54 + 2. 51.53 452. 52
sed indeo tamen nec sie rem posse u ferri
-
ostenres
seu cntrobius)eodem modo. inveniri
hunc sinterdum
coefcientes. Et hoc est artificiuqusvo naa
 tar erti e
non esset
Jam posset eli expumus
54V + 9
uo
it = et
et 3) determina
ne curva