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# Piece: https://leibnizlegible.com/api/records/59835/text
# Catalogue record: 59835 — [Aufz. zu Differentialgleichungen]
# Incipit: [dy]² = [dx]², aa + bxx:, aa + cxx est aequatio in qua dy est elementu
# Date: 1690
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59835&katnr=57545
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 373 (from the shelfmark LH 35, 13, 1 Bl. 373): 2 pages, canvases 741–742, page ids 00068196:0741 to 00068196:0742
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=741
# Lines: 103 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.
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# 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 373r — 00068196:0741
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000741.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 78 lines, status machine
L2
() 2
Aye = dx aa + bXx:, aa + exx est aequatio in qua du est element
curvae Elliptuae vel Hyperbolicae. fiet aa du + exxdus aad x - bx de
sit y = 10 + 1X + 2 X + 13 X +14X et C et dy = 11 + 2 12X + 3. 13X + 4. 14X etC
fiet audy = aa, + 1. 1. 1. II + 21. 2. 11. 12 X + 2. 1. 3, 11. 13X 4 2. 1. 4. 1. 14+ X f 2. 1. 5. 11. 15X
12. 2.12.12 2123. 12. 13 2.2.4. 12. 14
3, 3. 13
2. 1. 3. 1
J. 1. 11. 11
2. 1. 2. 11. 12
Cxxdu = C,
da
+BXX s
naa
Et inquirendo in destractitias, fit 12L _ 1H - f+.
1. 11. 11 aa = aa =0 seu 11 = +1, et 2, 1. 2. 11. 12=0 seu 12 =0 Et fit
B3 (7) - b - C, : 6
et 1920 ex termino X. videndum an non
et ex termino X fiat 16 =0 nam is ita stabil 2. 1. 6. 11. 16 + 2. 2.5. 12. 15 + 2. 3. 4. 13. 14
ndios
+ 2. 1. 4. 11. 14 +. 2. 2.3. 12.13b ubi patet omnia ubique esse teni
(11)
pares, alioqui omnibus reliquis destructis destrui et 16 seu esse 16 = 0. Rem
ergo reassumamus: dy = 1. 11 + 3.13X + 5. 15X + 7. 17 X etC. Et fiet
+aa dis = a4, 5H. H. 1. + 3. 1. 3. 11. 13 X + 2. 1. 5. 11. 15X 4 2. 1. 7. 11. 17X  + 2. 1.9. 11.19Y
ca
+ 3. 3. 13. B3 2.3.7. 13.15 2. 3.8.13. 17
.... ..... ......-..
.-.-
 -- - -  DH. II. 11
..-. -. 5: 5. 15 15
2. 1. 3. I1. 13
2. 1. 5, I1. 15
3rai ...............  3. 3. 13.13
AbX EX = (1qua + EaV Lo maatiCL thaathaC2  = 31 + BX +3AX ere
0
u
=a
.
ana
 ygRX
114)  ti n 34+ 3 19+C94 130
Est bitu  Hinc quia y = uut 13X + 15X + 17X ete. ideo incipiendo
at X quia a 15 X, quia in aequationis 13 termino 15 continente nempe x incipiunt
omnia procedere uniformiter, fiat 15X3 + 17X + 19X etc = v. Et  ex 
Et fiet y = 11X + 13X + v seu v = y = 11X = 13X3
Assimatur jam alia
aequatio aa dco + Cxx deo- aa dx0 quae nihil differt a priore, nisi quod b = 0
et fial cd = 21X + 23X3 + 25X  + 27 X etc. Constat Dari relationem
inter x et co ex suppositis quad quadraturis Conicis. sit autem 
118)
(181)
25 X + 27X + 29X C = e fietque eo = 21X + 23 X + e seu e = c0 - 21X - 23X
6
ob datas auteri quadraturas Conicas, datur  ex c ex X ergo datur et e ex x.
porro 21 = 1 et25 et 23E0, si absit b. fiet C = co - x et notandum
etia 13, 17, 21, 25 ete. fore = 0 seu debere evanescere, si absit b. quia
quia abest b. Ex his quidem patet e eodem modo deficiri ex a invenim
ex X, a, et 0, ut V invenitur ex X, a, et b. sed ita ex data e per X eto
 non deo indeniri v, quia non apparet quomodo pssit o ingrediatur calculum, id
enim fieri ab infinitis modis, neque adeo apparet quod proo substitui debeat . b.
illud interim si contra data fuisset V ex x, a, b habita fuisset potius e ex
X et a; pro b substituendo 0. nisi forte contingat, ut ora tunc evanescant
illud si sit b = 0. proditum evanescere etiam 23, 27,  etc. et fore
Et
W = WKX +25 X + 29X etc
 uXX
Cd = 21 X 4 25 X + 29 X etc.
ut yx = et ut Y Axt = F ut y - 1X = 13X + 15 X et c. videamus an constans
quaedam lex haberi possit, secundum quam 13 15, etc eodem modo habeantur, quam 25, 29 et
ac
 hoc posito haberetur semsi veluti si 13, 15, etc eodem modo haberentur ex a et b
quo 25, 29 etc. haberentur ex a, et  3a. Jane 13 = b:, 2. 1. 30, 11 (15 = 3C. 1. 1. 11. 1.- 3.3,
3: 2.1. 5. 11a4 et 17 = et et 19 e.
. 1.g. Ina sed video id uta ecedere
sed 25 = fC, 8. =C. 1.t. 11. 11,., 2. 1. 5. 1 au 2  29 = 2E 1. 3 b Bu S l sis
2. 2.

## Folio 373v — 00068196:0742
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000742.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 25 lines, status machine
aata
51t
Ergo su -i x + ccX- ex er
+
a at
a
+ bx - bes + b ex
a
du = 31 + 32Y + 33X + 34X etc. fiet ex dictis
33aa + 31C = 0 seu 23a +31 C =0 et similitem
et 37a 435120
34a + 32C = 0 et 35a +33c =0 et 36a +34C = 0
ergo 35a = 31CC =0 et 37a +31C3 = et lins patet cater posse ductum
4
quid si sic si scribamus aa co + ex deo - aa cx - (bX b X x = d
4
ita enim fiet deo = adx + 1b b. 2a X,: Vaa + CX quae penderet a qud quadratione
Conica. itaque fiet pro 11 12, etc. ad hibendo 21, 22, etc. fiet.
it. 1. N aa = aa = 0 seu n =  +1 B =
cy-
aetcx aerae
s
a+b
atb
- 1a 20