# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068196:0035
# Permalink to this reading: https://leibnizlegible.com/page/00068196:0035?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 18r, page id 00068196:0035
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 19v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=35
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000035.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 81 lines, status machine
# Lines: 81 recognised, mean confidence 0.88
# 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.

Ex diffeentiatis aequationibus is propositis, quaeramus quae nascantur non nisi tangen
tium conditione data
3 + m = 0 dat dy + dm = c
hy + m =  dat h dy + y dh + dm = 0 si sit data langentialis dy + bydx + cdx =0
Erit m = sedx et h = 1 sed ita fieret b =0. Ergo datam tangentialem multiplicemus per h
fit h dy + bhy dx + Ch dx = 0 fit, Cm = schdx et bh dX = dh seu (bax = sdh: h
itaque data aequatio tangentialis habetur soluta, si in proposita fuisset doo
M constans prodiset hdis + ydh qu =0 seu fdy: y + sch: h = 0 quae est ad rectam
yy + my + n =0 dat 2ydy + m dy + y dm + dn =0 sit data 2ydy + 2bdy +cyax
+ 2 e+ =  fit n = 2Sedx m - 2b dm = 2cdX. Ergo fdb = Zcdx seu b = sc ax
quae conditio debet esse in data Tangentiali quod an praestari semper possit, aut quatenus
videamus: sit vdo + β de + J dx + BdZ = 0 sit v = fiyt et de = fdy + yd ft 
et exaeque data fet ffydy + fgy 48 + fydg + gfdus + gyd5+ gdg  erga ut
yis tollatar fiat et sconstans i prdel vel potius f- 1 et sit gdis + gd  + βfdy + Bydf
+(3+4 + V fy dZ + Y9 dZ + Dd7 =0 ubi ut gy tollatur, pato
quod si jam Explicemus 0 per ). si faciamus v = fy + g et sit f variabilis, in vdy
seu potius quia nit refert
asseergemus ad 7 contra scopum praesentem, ergo fiat fconstans
f, et do = fdy +d9. Et ex aequ. data b d b + (dv etc. fiet: fe
ffydy + f9 dy + fydg + 9dg
+ β d9
f (3 dy 
s  sit  Z- x et f9+ fβ sit b
et fdg + fYde sit cdx, debet
esse 59. + f(B = sg + f5V4x
seu AB = CVdx.
sed ita nihil lutratisum
Z= X + h et fiet f9+fB = fde fg + f sVax + fssah seu fit 
+ B da
β ∞ f. dZ dβ3 - 7 dqCe,: y = dh ergo h = sdβ:7 - X, et habebitur desiteralum
fapiamus simpliciter v = y et Z da tosu
Ergo faciamus
et ex aequ. data solvenda vdv+ Bdr ete fiet ydy + B dy + Gty ax + olx =0
vs dh
ad eoque
et debet esse dβ = Vx + Vdh, et quaeritur h et fiet saB: 7 - X = R
ydy +B dy + X () -1) _ ydx + (B - 1) dx =0 eritque  quam comparando
supra prodata assumtae ydy + bxe bdys etc. fiet  b = β et c = 1- 1 + (d B. 7)
+(+B. )) ydx + B (d β: 7) ax
et C = B-1 + D (d β: ) adeoque data aequatione  solvenda ydy + βdy +) ydz+ eaZ-
fiet yy + 2βy + 2X + 258dx F 2 spa (3.) = 0 vel pro X ponendo valorem z-
runsus sed hic nova prodit incon novum prodit inconveniens dum volumque
omnia reducere ad datam. Nam tollendo X simul ullimus invenimus du = d (3: V
quod est contra hypothess generalitatem. Idque poberamus praevidere ex dB= )x+ Vahi
Nondum ergo oquationem initata 
ydy + bdy + eydx + e x =0 sic transformare licet, ut fiat relatio inter duos
fit enim dβ:) = dx + dh = dZ
quid sit si At data vdr + βdv + Yvd Z + Dd7 =0
ut ante siqe trans formetur ut fiat y = f+ (9: y) erit dv = d54 , dg  -9
terminos medios quae o siveratur
Aliter ergorem  ad huc tentemus
calitit ergo aequationem partib quo in gradu Maecedenti sit aquatio
data sit aequatio  V dv + (3d0 +  dx + B dx = 0 et v = 5ytg (f. a) ly + 9
sed ita assurretur ad Y dx
dv = (f: a) dy + (y: a) af + d9, et fiet
ff: aa) ydy + g(f: a) dy +(f: aa) yy df +9: a)  df +
fa) y dg + gdg
+(β: a) y df +Bd9
maneatque fvariabilis.
+ Bff. at Hy
in hac aequatione
tollamus terminos
ab dy et ydx
et fiet g - - (3
et fiet g= - β eta sed ita rurscis in  termino ydx coanescit
+) 5. aa y dx 18ia gex
B+ 3(f. a) dy a)
etiam f et fit dβ = P: a) dx rursus ut ante, quod non est in potestate
si aequatio data essetR y + Am y tan =0. et fieiet
+R1.a) yy dx + ydm + a du:
fiet 2aK = sf ol cet yay et obyydx fiet adk = fdf, quod consentit priori nam fit 2adK = 25 a
hic consersus boni est ominis. jam m = (9+ 3)f: a et n =  g9 +)(3 dg + (V.: a) gdx + Bdx) superest
2(X: a) Hdy + m dys
novus valoe, ipsius d am aequandus priori, et fit gdf + fdg + β df +fd B = 9 df + f d9 + Bdf + Vf: a) dx
in aequ. (85: aa jydy erc
implices fiet β =  g - - β et B= 3): a quae est nova determinatio
sit +
v Aliter
sit sed ita rursus assumtitiae evanescunt
tollamus ternui rentis
prosto silam