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# Piece: https://leibnizlegible.com/api/records/827/text
# Catalogue record: 827 — [Aufz. zu Exponentialfunktionen]
# Incipit: Sit curvae aequatio xx aeq. y. ut 11 aeq. 1 2² aeq. 4. 3³ aeq 27 44
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=827&katnr=58408
# Work: https://leibnizlegible.com/work/00068041
# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 23
# Folios: Bl. 33–34 (from the shelfmark LH 35, 5, 23 Bl. 33-34): 4 pages, canvases 65–68, page ids 00068041:0065 to 00068041:0068
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068041&page=65
# Lines: 190 recognised on 3 of 4 pages, mean confidence 0.70
# 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 33r — 00068041:0065
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 34v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068041/jpgs/default/00000065.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 136 lines, status machine
quatio
sicuma CX aeq. y.
. ut 1 aequ. 1 2 rq. 4 3 cqu 27
pro liustangentibus: XB
4 aqu. 256 et
irere
 
1 4 9 1b
ge
8 27 64
1 16 81256
(X  X+ aequ. X + x +s X + x seu = X + X Xd 4 la Xd  sed hocnilit prodest
Utamur ergo via nostra generaliore, erit:
+ dx  1X  d +C xlx aequ. Ly. jam lx aq. sdx. fier Xs  aequ. Idv
r
Ergo Adx + dx S x aequi dy. C Cd C seu dX + dx 1X aqu dy seu  44 itlX. y
ale  sedu aeq. vel quia sa eq.  
quod si
ponamus esse X progressionis Geometricae, erit dx etiam progressionis Geonaetricae ergo ducrit
a quanttas constans, ergo 22 vel  erit progressionis arithmetics. Ergo posito X
progressionis geometricae, du erit progressionis arithmeticae in Geometricam ductae
quia YX aequ X hinc  erit progoessionis arithmeticae itaque dffeenti
ge si numeri terminorum pon
ten ad
ifferentiae cortie terminam mnberite indnlarum) in proportionales numeris acithmeticis
act productis ex numeris anthmeticis numero terminorum et termino
EAx
lge. X aq. d
 aeq    adys yt +  d ye dy + cays y er + dy dyde f ptsite
59
7y e dx
fet denique dx yy aequ 4 say ex yx + di  du X + ddx ay x
a aequ y.
=a + a aequ.  a. 1Fa rqu. dy. Ergo ay aqu. 1+ a B
XIdX
jum aluude Xla iequ ly seu xla aeq. ftg. seu dx la q. dy Ergo sit denqbato aquie  1+ a quod
in aliis utilis ert ad, in definite parvas exponentibus e liminandas.
Qquod
in ita
est e etiam generaliter ess q la aequ. - 1 + a seu fore diff. inter ad et ejus logarich
ila  aequalem, 1. quod tamen falsum est. Hoc ergid tantum succedit in infinite parvis sed haec
accuratte discutienda
In aqu. a aqu. y. Si sint y progressionis  andimeticae, sunt y cometueae
E vidcudum jam quomodo demonstretur  hauc esse naturam pro-
gressons Geometiiae, ut sint termini ipsis differentiis propationales
 y. y veccu  a a aris
siltit, X. X. b
at 2b
XIb,  ag. (ay) + xb
xrio 
seu fiet:
se +
a+ . 
v6
p.
a eatae
Nempe
 a i) a (12)
X+ ibdissecntiae lycque X AXB aqu. - X + Xt  x ab su b aequ. L
X Fb eq dy + X  ib o (e B
X+B
X+b
aequ
(dy) x+ q cE opgir
Cdy)A +
X+26
df2y  t y
X + b aq. (X)  dy +Y
X+2b
Cdy) +dy +y
ib
aequ
es
a
Xib ut + xtib q 
fiec eq 
X + du
xardx
dy + 2 X hy + X idy) tdy+ X
X+
X+26
ita b seu dx sustulim ex exponente
nota quia adhibui 2x sumposai X esse progressionis autemeticae
X Pa eq e ita jam nihuis limitavi itaque aljud ad hibendum
non licet
ix 26 nempe b+e
a talbus orpationibus
in lalibus oble iibus non licet obserrare legesn homog nectum cronscendentulem
 et
raga sit
p2 9) 2so 
+ Z b2 aq. bv bx +b aq. v
eo
eundem vertilemi
opitisem ee
rgy. fit axaequ. bx + bl. seu fit l
spabl: ab si requ ubl. a b +c Hinc si bpa 3
13 + Y aeg. + sax + sb.
dulo, si duarm
El. talle is est impossibii
rabolarus
 p
altera
a alterem
bet ad.
alteram
curiit, necesse est
es ca
tera
rectum, a maru call reoto tatus hoc soco tujus tatus rectam
eam
habere catus
 hncqu. B) dx: dy: ag. dx dx. A :: _ dz: dv.  Est autem dx. dy:
Ax: ag:: 29: L 4q. cls dqplodr. dv d q. ( n aequ. Z (NA ) + X (AF) fiet d Au: dvg. dytdx. dv
jam dque aequ _co, et habemus cd + dx: do : 2y: h est  seu no
seu nostro cas du: do: 20: . sami jam potest otatur intro lo ian sen e ad n
et inter X et h sumatur ergo X aequlo Z et h, erso hina habitur latio cd idr seu e ad n
Xag. Gy: h. yaeq B. Ergo Z+ vrik aq. h. quoe aequatio accommodanda ad
duaseadices duviles variant autem  et n at quia paubota endem ideo Z et a manet idem, ideo ft
evanesente Z fiet.
. sed cutus un
na et
1+ DBitZ aequ
ido:: 2b: B ut ante. Si prius div
jam naeq. 2+X. Con
lant
in parlora
s
emus per Z
habuissemus
deremm jam et h variari quod
adeoque variari e
antea disimulaveram se quia tun peberum
novan relatiorem assumere pplimetamer inter n et h
itaque possum
ale 

## Folio 33v — 00068041:0066
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 34r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068041/jpgs/default/00000066.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 4 lines, status machine
la
RIED WILHELM LEIBNIZ
L1
t 

## Folio 34r — 00068041:0067
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 33v); this page is its right half.
# No recognised text on this page (recognized).

## Folio 34v — 00068041:0068
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 33r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068041/jpgs/default/00000068.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 50 lines, status machine
813.
e
S
ote
3
2.
28187
E
ev 2
03
dX
2
8:
4∞
ue
.2
e
3 + = u unc
5
E 2
+ 2on
d.
de c  ce  dui
:5 y  un
  2 53
2
3
Z + X C X 2n 3
+ = = 
2s
5  6-
d
3∞
2
C L
X. 5. 3
e
3
5e
8 C
di
et
2
33, +d-
1 2
e
e
be
e
e3