# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/59834/text
# Catalogue record: 59834 — [Technische Zeichnung]
# Date: 1690
# Akademie-Ausgabe: AA VIII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59834&katnr=57544
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 369–370 (from the shelfmark LH 35, 13, 1 Bl. 369-370): 4 pages, canvases 733–736, page ids 00068196:0733 to 00068196:0736
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=733
# Lines: 162 recognised on 3 of 4 pages, mean confidence 0.83
# 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 369r — 00068196:0733
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 370v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000733.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 104 lines, status machine
1)
Ly: dx = a. Xy + X. y seu dy: dx = a i +  n. fiat seu dy = . M dx + n dy
e
fiat M dx = X deo e et sit du constans, fiet du = a X deo +ndu
per 6
exatt m dx. X = deo seu d m: dx. X =0. rurrus ottys-nix: do a
3)
et fiet d u ndx d X dy-ndx 2d  0 ped nihil sed hinc video
non posse X et y. simul tolli, nisi ad hibita aci ex aeg. 7 Janc si faciamus
praetetea a = du - ndx:  x et disferentiomus, nihil prodibat
novi 1 1
illud video, si sublata una litera verb. gr. X
in aequ. differentio differentiali nihilominus alia, at a maneret
foret unius gradus, tunc hac divulsa in duns evanescere posse
sed dubrum an hoc liceadi
(1t
q
EF. posito f dari ex x, 9, a. differentietur
5g. 2. ut tollatur a. talum equatio lni qua f, df, x, dr y, dy Hanc differenti
opeaequs 1 et 3 lollatur f fiet aeg, (4) in qua ds, x, dx, y, dy. Hanc
Herentiando tollatur By et fiet ut aeq. in qua dfd, X, dx sdx, dus, adus
sed nihil hinc
a dy. +x = C_ B xet F 1oR + 1X + 2y4 HXg + 1X + 22y, :, 3 + 31X + ry eu.
12
10 Xy dx +, X ax = 1 X dy 12ay dy sit rydx constans = a deo
fiet aco= Xe2ay  - 2 11. XF. cursus Xydex + 2Xydx + xay dx = 0
denique - 11 Xddx 3. 11X x 4 12a y Idy +. 2. 12 aydy 20 Nihil
10 X dx + n V dy 4 a X dy fit: 10x ddx +910X x 4 11 1 ddy + 3. 11 y dus = a Xddy+ a dxdus
a = ul
b - Utm
10X dus ddx + (3, 10 X dx dus ( 11 1 X dyaldy. .+ 3. 11y X dis = 10X deddus (110 X . dyse in) aerctus
n
n
mAn A p
c~ l FEmtM np
mtnto
pra
d = liymtsnip nepe
intonBpq
e = Uant burrest
+uy detus
Methodus haec videtur adhibenda: inventa reguta
progessim supdatur d ex 2y, a. inveni
pountur dy. ex dx, X, y, a. ponatur dlx = a. unde
et evanescent differe ptiae reliquae, et quae ratur ddque
dadus etc. ex iisdem postremoa inveniaterlex
a a- l aubam abte n ansbesenduq
b= 1m c= lem  +m +n seu ea lt im+n et
d lemintp. Et ita porro
1 sit aequatio (1) inter HiX, _, dx, dy:
i aequo. (2) inter ax adp ddy.
aeq. (3) inter y, X d X, ddx dadus
aea (4.) inter y, X, ax, dax dddx, ddddx, dalus
eteere
ien dantur per 1 valor dy
in
per E valor daque
per 3 valor, dady
etc
Hinc psy, day, ddaus esc sumendo pro m, n, Bet
et
, (4) (y) et sumendo pro a, b, c et
poferimus per regulam scriplam ipsas (y), Ay1) etc.
formare per ipsas dus, day, erc leu per earum
valores, aB est per y, X,  dx dix, erc. accedente.
confrante a forsan constante a.
Hinc ssi y X primas assumamus abdete non ut
quaslibet seu constantes, seu in dataratione ad a
et assumamus etiam certam velationem inter
Xde, dax vel solas vel cum interventu constantis
vitendum est an simplex quaedam velatio inter
progediendi in inde horum ope inveniantur
(9), ((7)) ((7.)) etc done inveniatur lex propedi
y, ex primis X, y a. priaque ita primae X, y, aity (9) et (yD haberi posset progressio obtineni
posit valorum ipsarum Runilo   ex ak
et de. seu inter a, et
endi. ita habelitur modus habendi infinitesimae
revera eruat constantes et prout assumuntur defer
minabitur punctum per quod linea transire debet
Raper ita reperitur valor generalis cujuscunque y
parvam cotantem irccedentibus solum numeris. Quil
et iasi ceu certo modo revecari potcun
ad (deo, seu ad co, seu ad numerum
ubi reperta generali relatione, haberetar aequad
ex constantibus assumta distantia, a prima, quae
veterminandi ordinatam. Haec lex si non est reducibilus, altem transcendent. ipsi numeri coefficientes habens
tanscet
in valoribus  b, C e. b, C, d, e (tollegolas
latet simper in eoefficientibus. Et ita habebitur novus modi
ad aequationem ne transcendentam quidem noenipatur ex
ne es as de abrit patatur inter de combii atun eeta  s
senu
camnova
n  a aad bl s et  inter se co
a
bac
e
a-2b+c=n
a -3b +3C0d
b. utd b. 3c +id. C
c- 1d+e
4b +6c - 4d + C29
c
cad

## Folio 369v — 00068196:0734
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 370r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000734.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 36 lines, status machine
Exemplo uutamur sit
ac XIX+ odus ad fd = ady. sit e
siet du = Xdx tyAX,: a sit ddx = 0 fiet ddus = dxt F du lx,: a seu ddus = ad x + x dx +yx: aa seu
ddy = dx. atxry: aa et dddy = dx. dx tdu: aa = cx, a + X ry 3: a. et de d4y = dx axtya
jam (y) = y + dy fiet (y) = ay +   Xty dx, : a et (y) = y + 2dy rddy, et fiet
(Y)) = y, + 2dx Xty: a, + dx. atxty: a = dy + 2 dx. xiga + ax utxtys,: a
Alia esse potest calculi ratio, ut A y, quaeramus per  x ax,  et una
 diffeaentian ab y. quam longussime remotam fiet
V = adu - g xax,: dx  aa ddus = atxde,: dx = aldus - atx dx,: d. etc.
Et hoc calculo utcunque continuato, cujm novissima dado dy, sit aequalis nihilo, seu possit
negligi fiet  s ax et dy= dx et fiet ty Hoc posito videamus an sit
dy = Xdx + gdx,: a. seu ady = xax + ydx. nempe pro du, et y substituendo rolores,
fiet: = a dx = xdx  =adx = x dx Et ita res sucedit
Ldy B 9 - x x r tadus 1 27   35  + ade + Bady :
Generalius: ulamur 100 a dx +200a dy + 100 Xdx + 210 X dy + 101 ydx + 201 ydy  0
toncns dd x o  difecrtiando fies A  4200 a ddy + 110 dx + 210 xldytuo se di urd + 201 yadys + 201 dus =o
intor, ordinanium; cum res redire debeat
simpliciter; patel sec erus
fet. Ita sic ingenitur verum, sed plecumque non invenietu 101
pleruuus ad seriem inficulam, ut quando y datur non nisi
salorem hic haberi rationaliter, ergo velat per seriet appropinquandu non exact
=adus aedes aipldus = x ex tad a sit ddx =0 fint adius x NX + rydy dX, a et pro _ y  substutmer de valorem
fiet: deys = 2xdV a +7. y x +2d, dx a a y= a y - 2Y a dx,:, p te r 2lx
cursus cddus = 2 dx ae + 4yxde + EK dya + 2a dy dx3: a3. seu = rdx a + ~G d + 2K dx Y e daytra t traiay, ea
Ergo 9 = a tddus b dx 2a 14a  +2aex: aux trar t aaHXta
aectus- X dx 3: adx sit LdX = 0. fiet dy = a ddy = 2Xdx,: adx quo valore substituto in aeq. 1.
fiet :  = a day - 2a  x - Ax x,: a d seu ddy = a ydr + 2a xdx + a X dx : a
 inagnastat Naa
Erpo addy = a dxdy +2a x + 2aXdx: a et tollendo dus fit: dldus -  adxaydx ra da x dx + 1a a 2a xost; a
Ergo y - adu - x (X- 24 dx - 2aXdx,: adx. underursus dy = a d: y - 2X dx- 2a dx: adx - ydxo pdx: a
et sit y = ad. G LaXAxt- XK AX: adx et dy = a d’y - 2a dx = 2X dx: 4t X = ay x +X Ax, : a0.
et fit y = aL. y - 2a  - 2aX du - x x3: adxs et  2 a dy- 2a dx - 2X tx: a d = aydx + X dx, ia a
et sit y = adey - 2a dx - 2aX dx X dx3: a dx Cumque sic prodedatur in infinituu ergo ft
tenis y= 3a  2aX + XX, : a unde dy= 2adX + 2Xax; a. quos valores substituendo in aeq. B1. fit
alo i a  a7a rahidx Exel + X ex: aa qua
adio ves pucrectit

## Folio 370r — 00068196:0735
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 369v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000735.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 22 lines, status machine
sit y= axdus  exae, rate su dy = agax - 5Xdx,
sit ay  b dy +  dx seu d se sit ddx o et X vocent t, et  dx actimm at ex
s
et dd X dat du; et ita porro. dd d dddx AddE dx, et ia porro.
tit u b ddy + at x: asgo = ay t. dx, : b fit y = b dlus + b at tx + a t ax,: a ax
et potito od oferibi potest
v = bly + BX sit F = X d=dt ax = d X ddt sx = dd X, dddt x = ddd X, et tuprrr
iam sit y = b dy + + dx, : dx. et dy = 3 dx - + dx: b et g
dy = bddy + dt dx, : dx. et y = bb ddy + b dt x + F dx,: x
dy = b d. y + b ddt dx F dt Mx: dx et y = b d.y + b dit dx +b de x + t dx,: dx
dy = BFy + b. d. F dx + b dat dx t + de dxt, : x et y= bd yr b d Fdx  b datdx t bat da + tdx: du
ergo sit, y- + + b dt + b dat  + b det  + B L dr b l. F + b d. F, etc.
Unde patet si F sit quantitas inte rationaliter integre determinata ex, x evanescere alicubi
aliquando differenti alium seriem, et posse inveniri abso late ralorem y.
sit e  sit y = edg + t dxsi dx posito tam e, quam t, dari ex X, fiet dys  ydx = t dx. _ 
1  fit dy = e ddy + dedy + dt dx,: d X, Eppo ex 2 et 3 sit ee ditys sen dy e sit cettus de da L + edtdx,,
fis y -I ee day - d edx + e dt dX3: dx - dedx
ita 4 Tantus differculla ado, et du inde natam comparando cumvatore ex or. 1. rursus novas habebitur
valorem
x. cdx.
Gax. tax
vilor ipsius y. sed non semper ita copendrose reducibilis

## Folio 370v — 00068196:0736
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 369r); this page is its left half.
# No recognised text on this page (recognized).