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# Piece: https://leibnizlegible.com/api/records/434/text
# Catalogue record: 434 — [Aufz. zu Differentialgleichungen]
# Incipit: Sit aequatio differentialis primi gradus v. g. dy : dx = (1) xx +
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=434&katnr=58010
# Work: https://leibnizlegible.com/work/00068199
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folios: Bl. 166 (from the shelfmark LH 35, 13, 2c Bl. 166): 2 pages, canvases 331–332, page ids 00068199:0331 to 00068199:0332
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=331
# Lines: 157 recognised on 2 of 2 pages, mean confidence 0.81
# 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 166r — 00068199:0331
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000331.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 125 lines, status machine
primi gradus
sit aequatio differentialis, vig dy: dx = Xxtgy, : aa, d
ctipe:  y y ux Cde dx  2x d dy e: n ae seu dy aa = xxdx + yydx
sit Xxdx = aa dco, fiet dyas _na du a ddus ad = gydex + rydydx
et XX ddx + 2 x dxdx = 0 seu Xddx + rdxdx 20
ergo adue aa = gy  Axdx aadeo + 2ydy dx
rdnox
quidi si faciamus dus sa
ya de. dx fiet et de constans, fiet ddu avx texa 
seu
aadu-adddo a dodi, ved edutr an dvab; vrdxx
y =-aado: vax, fiet actus = (dux) vddo vav: vo = xx dx ta doariou
unde
aa  ddr-aa B de do = 2
seu -at vddv (+ at dedv = vuxx axdx (+ a dvdv seu - a ddv: dp = xx dudx
sed si sit du constans fiet y=  sitque = ab dv: v ix et dus = aeb vdv a bdvdvax =
in aequ. 1. ponatur
Cp dxde d tda abb dodo (: velxd seu -abvde. ldx - a bdy dvax = vuxX dx racbbdrdu 
sit B: d= (tlarara)
2 +l
Atiter k are gia = do: x
sit g: a = 100a + a f + a gg 2a  et, a 0 a M + a  nn + a3 p etc
200.C
y dy: dx = XX+ yy,: aa
et sit f = uOX +1OIV et gg = IROXX + IBIXV + 1020V et h = 13OX + I2IXXyt 112 Xyy+ 103y
m = 210X + 201V et nn =  20XX + ZIXV = 202VV et h3 = 23OX + 221 XCy + 212Xyy + 203y
et ita pono df = d  dex c 3 + des f et d(99) = dx A(9) + d  +99) et ita pore Bm caeteris
ita ex. 9. fiet, dgq = 2. 120x dx + 111bdx et d(99) = nxdo Z o2vdo. Et ita porro
et
exprimi per aequationem rationalem qualem posuimus
Cdx)
dy: a =  (200a + a m, etc) ((a d f + a agg  etc) dx + (a  5 + a  ggene
Et sane si y anlt
2ann + etc) ax + (a am + a Anner) du ) 
(2) (200  + a m + ete
et gy: aa = (2), 100a + a f + et, :, 200 a + _ m + etc)
3 (200 a0 + aet m + et) C(a  +5 + ex) ax + (a  f + etc) dx)
et ex a aady = xxix + gydu, fiet
(Ca dm + et C) dx + (ae Amterc) du)
a 2(100 a + a f tet)
= (2) (200 a + a M + et) XxdX taql 2) (N + a f + et) dv
facollimum est
Ha Hinc apparebit an tali arte tolli possit a, id enim omnium quae fieri explicando possunt,
in
facillimum est. 200 a 20 videamus an omnia reddi possint homogenea ipsi 
dateri
Caeteris destunctis ia erit. sunt autem homogenea hunc a a a dx hododer px + Ap +)
Fggdx +p gdo a f.dum + fann aV)  aa a. a 2. 10g9+0
2
fn. f. n mg. a mgg.
+a a a
200. 200a
aaaa
isi facicur onm, n, seque=0. Et h et seg9.  g. 0. al sit sole
ge
proditi raaa
 non devet. D  et praeterea 2a a 100(4G+  0gdV) qod estinferiu ipso xxa
si non debet assami ultia Xxde eritat m, , P etc =0. ergo debet maneo 
adeoque tolli debet 200 a 3200dggd + dggdv) +  aa. 1b0 . a
et
Si nolumus assurgi ultra gradum indeterminatorum qui in XX dx
trde xedx
ta e2Bdv
ffdo, quod
oportet aute omnia n
+a’a a 2otf ax + a a 20 sdv tad  2oodggdx +a a2200 dggdo r d taa.  ff 
esse aequales nihilo. Deinde 
a  e
IC200. 200bb d
aaaut d
e
tr, et sequentes
+aa a a 2100 f dv
retineo assurgi ad g’av
sed video si ddeui
si gonito
=  200. 200. 262 d Z
non p
sin b
mmxax nec
quod si faciamuse
12. sed literae solum ad sunt
et prioribus 6 tollenia
 yn a 100aa + afd
simul reperientur m
 sed tunc non aderunt sifficientes literae, quia fit debent esse
fractio in valore exax destinetur
3dZ, ergo tres sunt gradytollendi et in qovis gradu duae prima aequ. 2
et fiet xdx = a 26Z160 + b8,  (971 80 +C)
qua i f et dd
sed restat restanit ergo
asserigitur ad gradum FY
sunt in universum ae
ra +FdR + OvaSLGBA2
1oa m  t100
2804: 7218V +C,
9019 (326vel) ar +3dv) et liat 200 aat aam +un-(orrioro
2+ b +5V ft etq orpittamus quod si retine
iri ultra C et q et m
Ergo faciamus. X
licebit
A.
eD
u Bete
restabunt ergo gq et f, fiet q sit- 2tb
e
ene
1 C, 5,9 b, et in
detur
200a4
en
una adi
Aegdz + 20045 4g9 dv
amtun
200a + fdv +200
10 a 4mav. 100454 indZ + 100 a d AN
ama
 1do a tm a p 100100 e 0
5
aa
in secundo 4, in 3tio ae
l200
laque recuo
esperata ideamus:
t upra. Ex aadus xax lgus axsiet

## Folio 166v — 00068199:0332
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000332.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 32 lines, status machine
aadu = Xx dx + yy dy sit y = a  O:  X = a D: 8 dy = a, 7 d0- 0 d E: 98
proestat sic
et dx = a, E d D - Dd4,: 88 et dy: dx = 9dD- ∞d:, pdD - Dde, unde tandem
fiet (3+0- 099d _ aC a 79 auo Cvata
N = 004 dv - 10Dd8 + na DD8dD - aa D DDde
si sit X g= O et _ si esset _ = B   0+96: a et X = D 196: a
fet Ly = dC) 195 + 0 d46): 2196 = 285d0) + d d (28) S.: 2296
et ex similiter ponendo D pro O et dy: dx = 296d0 +0d(46), : 296D+ d (26)
et yy: aa = 0096: a et XX : aa = DD86: a+ sed itaassurgetur ad gradum 6.
Retineamus ergo aqu. in qua assurgitur ad gradum 3tium, si O, d, et e sint
simplices secundum; et V, Erunt tollendi gradus primus ubi aegq 2, 2dus, ubi 4
gradus prums ubi aequs duae, secundus ubi  primus abi aequs quatuor, secundus
id est literae 12 et opus est arbitrariis 12. sed  nobis
ubi, aequationes sex.
pp significat. 22 + . 2u +. YB, et ila de caeteris
sit formula + aZ + an dv + 7opdZ + 2349 av + 50 redz +2 45e do
tantum adsunt
310 d Z + 302d
sit alia formula t 410 sdz + a + a~ (3dZ + 4e )dv + 43 Cdd Z + A03 
20cd
quae multiplicetur per t t, et productum comparetur priore,
fiet 31020 et 302=0 et 410t = m et 401 t =n ergo m, et habere debent
et L5d = +3 et tBB =5
habentur, de ideo oportet ipsas, omnes esse divisitulies per t. Sed
rationem constantem, +β = pp et bs
optimum erit formulam exprimere per omnes radices quibus producitur
nihil lucrpbimur opinor hac arte
si  O et dy = et fiet Tady = OdG +  dC et ay =
ut formullam tertii gradu per C.
sit ay = 03 et ax = 46  it ady = Od3 +Dd0 et adx = t dG + 8dti
et dy. dx fo, aa dy = p yy dx + xxdx siet a0 dD + a DdB = E h  ma) de 19 d  dia
sed ita assurgitur ad gradum stum