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# Catalogue record: 544 — Calculus in differentialibus replicatus per semisubstitutiones
# Incipit: dy = (1) ydx. Ergo y = (2) ∫ydx Ponatur pro y sub vinculo eius valor
# Date: 1690-09-11 (90-09-01)
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=544&katnr=58121
# Work: https://leibnizlegible.com/work/00068104
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 9
# Folios: Bl. 4 (from the shelfmark LH 35, 8, 9 Bl. 4): 2 pages, canvases 7–8, page ids 00068104:0007 to 00068104:0008
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068104&page=7
# Lines: 194 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.
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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 4r — 00068104:0007
# Source image: https://digitale-sammlungen.gwlb.de/content/00068104/jpgs/default/00000007.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 141 lines, status machine
1. seplemtr. 1690. Hhc alia
scheda ejusdum diei in
dy = ydx.
Difaquato calculus in differentialiber replicatus per semi substitutiones
cr
= Syax  Ponatur pro es sub vinculo ejus valor sydx
u
Et firt ex aeg. 2. fiet y=Sydxd ubi rarsus pro y sub vinculo ponatur ejus valor ex aeg. 2
fiat
= c (syax dx dx etc. Mn Setessy ax posito ax esse constantem, et e numerum infinit
e
Nempesi HC Hyperbola et AB, y, ft et BC, V
et AH, 1. fit V = 1: y s adeoque dx:du = v
Et tunc X est infinita repraesentans summam omnium BC ad
indo ab A, hoc est spatium e infinitum hyperbolicum FABCF
Et notandum quotiescunque falimus Say= y  ipsam y incipere
faciamus GH = Z seu y = 1 +Z. ita det  rursus incipiat a nihilo
a sipl ecatu
sum ax est fomana- a sidaci ea
tiam infinitem tunc sed summa omnium l: 1+Z sec omnium BC ert sump. a omnium BC inde a GH
sem area spatium HGBCB.  Et fiet, dq= dy adeoque dq = dx + Zdx
et
quia
et Z = X + sZax. Et substimendo Z= X + = XX + sZdx dx ali
4
ordimna Et semisubstituendo in oq. 3. ex 3. fit Z = X + ZXX + sszax dx
X incipit a nihilo
ileque et rursus
eisubstituendo in aeg. 4 ex 0g. 3. sit Z = X + = XX + 1 X+SRax dxax et
1. 23
x
7
datur ergo C= Z
+
ete +
X
34.
constatuintem esse X+
lrqu per 6fd, 6. fit
C
123
ete s)Zax dx dx ete ax
 posi 
Estcantem CB. seu posito esse
Sit fetc. F..
u
1+Z
7
x +)X+  X e
L dx +C
1X
seu
tx
da
1+2
en
(12)
HX-
(XaX = LX
SXdxdxx  LX
(8)
12
posito ex constantem
 ia at  du  log 1-
Ergo fia  X + f. 2+X
+
AX  1o 1C + ex eo i stA
(16)
(dx X
SXdcar = 1 X
Stu
posito esse X a2 seu posito de esse num  cujus loq sit
4
sit sete ssZu dx dx dx et dx = C = Nad 
(e
seu posito X esse log. de 1+Z. seu posito 14Z esse num. cujus las
(13
seu iq fode X
6
6)
ita porro in infinilum continuando, fit
= X + L X + L x F.
ala e
1.2
e
et ut alisas seirbamus
14
iceret
sert
posita a onstante eoc et C numero efinito
in datore ipsius (, aeq. 17. contiunalavere in infinitum supposita dissimulare Z. qua si
sites
ex. sam
evanescentem foret
La  L
7)
etc S) dx dx dx esc dx - HX log 1 - x - 1. adeoque
et valor habesetur insi ititi trunscendentis quantitatis in aeq. 4 superioris calculi, adeoque
absolusi hberetur tabs vator. Videamus in
serie infinta,
sit
CO L = KCC aY + bX+CH + d F + ex et fit  s2= ax + L Wx F. bX erc
sil
so
e
10
q= ax + LBX + L cx erc et
12
c
S.
an
l
lar
1 2.3.44
( 5. 
b etc.
comparataalis seu homogenea
tine quia qanlitas X estordi summa omnim bX 2CX + dx et
ipsa X. scribamus loco ejus mX fiet Z = aX + mX = nx, quanta ad aestimationem homogenertalis
ubi paet selisss fore = infiite parvo seu evanescenti ergo et 0 =3
infinitum Hu
quia Runditio
popio quan
titas  est
ordinara et
ax infinite par
va. Hinc et
 est infini
te para
Er go seu eva
esse, et ehumenim
X
itaq
sedesse
per dinitule a poten in
drvisa intelli
nati ob legem
homosentolu
manit y
quanblas 
constantem it

## Folio 4v — 00068104:0008
# Source image: https://digitale-sammlungen.gwlb.de/content/00068104/jpgs/default/00000008.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 53 lines, status machine
sit dy ydx +Xdx fict Y- fy dx
caeterum nombire est alium adhuc calculum potuisse institui
locoquan
in ag. 4 non substatusen sem substituissemus valorem ipsius y ex aegl.
ad valorem ipsius u ex ipsa aeq. 4, ut in ag. 3. subptis semsubstiferimus
1)
exsagu
valorem ex ipsa aeq. 3.
Nempe faet lobaequaro
y x x+ LXX+  X+ X+
113
et rurjus in aeq. 18 emisubstituendo aeq. 18. fiet
1.28 S.
k. dx
2IX + LXX + F+ 1 X + X + X eu. et ita hocmdo
1.23 b2.38 1. 2. 3, 45 1. 2.3.4.5.
12.
prodit eadem prorsus aequatio quae per viam priorem
umamus aliam aequationem
dus = ydx + XdX. seu v= =X + Sydx. Et semisubstituendo in aeque 2 exaya
fit y = LXX + L X + Sydx dx. Et rursus semisubestiluendo in aeq. 3 ex g. 2. fit
4)
ZXx + L9X+ X +Sydx dxde et ita porio.
1
Ergo habemus problema solutum. Nam quia posita BY X= sdZ
e
quae est aeq. 2
est fera raculi scedentis
ita
(
est per rq. 2 et 9 calculi praecedentis et q hujus vagx C-E+y.
fiet utique y = 7 - x, posulo Eberi bore R ad , ut num
posito 1+Z habere se ad X, ut numerus ad suum logurithmum ob aeg. 5.
Hinc si sit du = Ydx + dx + xax + X dx erc. semper  solvitur problema
dy:dx = y4 N. semper solvitur
per loganthuos. Sed si sit generaliter talis aequalio per Hyperboles quatraterom posito
du a+bxicuFaxCf ec = dx L= mx tnyEppEgru e (esse rtiona
cribere poterimus sic du fnium se dy- D+x. 
per x
sBn
si sic dy = dX + (X dx + X dx erc + yax + m y ex + ny ax ere satasse similitur
serviel methodus Y = ax + bX + L Cx etc + L ax+ bx + L ex + en sed 1es pauto implicatia
sed sorte aliud onmodm  remedium, quaerantur deq et adius, ec. Done ponento
d
ddx =0 perdd sit aeq. dy = ydx + my dx sirt ddus = dus ax +2 mydydx
Ergo log dus = X + 2 in sy ex ab fyax = y - sy ax. Ergo log. su = X + 2mys = s ax
sed hoc parum potius inaegor. An sic det de- adus: N. dus+ Emyay
la
differentio. Differentialis =0
Hnc arte semper effici pt ut ea litera, cujus 
habeatur sola ab una parte sit
ere
iter express