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# Piece: https://leibnizlegible.com/api/records/17574/text
# Catalogue record: 17574 — [Math. Unters.]
# Incipit: Ad invenienam summam seriei 1/1 + 1/4 + 1/9 + 1/16 + 1/25 etc.
# Date: 1696
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=17574&katnr=45234
# Work: https://leibnizlegible.com/work/DE-611-HS-958803
# Title: LBr. 57, 1 · Bernoulli, Joh.
# Folios: Bl. 116–117 (from the shelfmark LBr 57, 1 Bl. 116-117): 4 pages, canvases 233–236, page ids DE-611-HS-958803:0233 to DE-611-HS-958803:0236
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-958803&page=233
# Lines: 97 recognised on 3 of 4 pages, mean confidence 0.76
# 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 116r — DE-611-HS-958803:0233
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 117v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000233.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 94 lines, status machine
Ad inveniendam summam seriet= +
+
et
indigeo inventione hujus quadraturae t Lix dx
+1
Repenio autem Xe lIFX dx = XE lEFX. 1: (et1) - ( (XeF Ax: (14X)) : et 1
itaque generaliter succedit quadratura inbs excepto tamen solo
casu nostro tunc enim cum sit C++ =0 ubi C = - 1 tunc enim fit
C + 1 =0 et subsidia quadrandi evanescunt.
cogitavi  rem aggredi hoc modo d, litX, 1tX, : X = 
aliter
2LIFXdx : X  - LIFX dx: XX ErGo (LEX dx: X = = LITX, 1+ X: BX +iS liX dx: XX
Itaque quaerere jam oportet. (LIFX dx: XX
et
1
d, l1+X. X = e. liFX X dx + LIFX. X dX : 1+X Ergo liFX X dx = LiX. X -2sliEX X
et1 etd 1+X
jam in nosto Ergo desideratur flix xet ex seu in nostro casu, ubi e =-2
desideratur sLIFX dx. jam Lix = A LIFX + LuX. Ergo
X + XX
XIXX
(Litx ax =(l 1t Xdx - sLitx dx
x+XX
+X
1+ X
-
X
Fit ergo
ItX dx = - LiF X + 2slix dx - 2 (litx dx. Quem valorem
substituendo in aeq. Cd fiet.
lirx dx) = lixCx C 
1 2X
1tX
L LiX Ax (LIFXdx - sLIIXdX
X
3 1+X
et fiet (LiX dx = D litx, quod sane per se dudum per sepatet,
quaesitam autem evanescit
d litX 1X = 1 lix ex - lix dx seu (t i  ex  slie  lix x
XX
Xx
LiFx3, 1+X = 3 lirx dx = LiFX x - L iFX dx
XX
 na
XX
X
Qlir aner lx lie ax
AL EX 1X = Elirx dx
, ita im =
A3 liX ax - d. lir . 1X
Hinc si e=n, et n -= 1 fiet liFZ AX. X LIAX IFX. X - X = :s2+I) = st 1 + X dx + nFI SliFX X dx
Sed hoc aliunde patet. e summari posse omnes parses
(li+X dx = Li+ X lIA et
1+X
se
0
e+1
 Litx
+
e- l
1+X,
Li X da semper datur nam si siat 1X - Z Constat dari fit s17 qder
d liFX A = ed x: d C LiX x dx + n Lux  dx
xnest _ B. 4 et
1+X
1+tX
ItX
IFX
Sit B = LIFX et n = - 1 fiet litX = LIFX, li EX unde nihil novi
1+X
XIXX
a LITX IAX, = dx + LiX dx I. d, LiR iH = ax - L. +X dx I i, LiEX ItX = dX - -lIEXdx - liEXAx
d li e nde 4
X
Xx
XX
XX
AX
Hinc sequitum stitadx esse quantitatem infinitum nn
d liex ux = de laliercx d lierita ex Elixeda - l FX Ex. Erqui sponendo
liFX itx el x d lix ex
XX
xx
_X XX
d, X, LiX IHX = XdX F lirdx velux Xdx )d, NX LiEX 1EX = XXdx FElitXAH +3 liE X Ax
lexita li d lux dilx -luxa H = d B liru 1x = de -litude +  14. e   
xt x
eX
x
1tX,
. 12l: d d LIEK x X = e lim : X +n lirX X:. l 1
seu Sli X dx = LIEXE IFX X = e slitx: X dx -nsLIE. X  IFX dx
Su X ax = pr1 X tine -  G X H Cia . I
es

## Folio 116v — DE-611-HS-958803:0234
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 117r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000234.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 2 lines, status machine
TFRIED WILHELM LEIBNIZ BIBLIOTI
NIEDERSACHSISCHE LANDESBIBLIOTHEK

## Folio 117r — DE-611-HS-958803:0235
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 116v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000235.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 1 line, status machine
eeeeeeeeeeeeeeeeeeee

## Folio 117v — DE-611-HS-958803:0236
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 116r); this page is its left half.
# No recognised text on this page (recognized).