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# Piece: https://leibnizlegible.com/api/records/18129/text
# Catalogue record: 18129 — [Math. Unters.]
# Incipit: Omnis qvadratura seu Sb_ac
# Date: 1695
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=18129&katnr=45413
# Work: https://leibnizlegible.com/work/DE-611-HS-855907
# Title: LBr. 560 · L'Hospital
# Folios: Bl. 32 (from the shelfmark LBr 560 Bl. 32 (=45412)): 2 pages, canvases 63–64, page ids DE-611-HS-855907:0063 to DE-611-HS-855907:0064
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-855907&page=63
# Lines: 90 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 32r — DE-611-HS-855907:0063
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-855907/jpgs/default/00000063.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 40 lines, status machine
omnis quadratura termini quem ingreditur 
M potest recua  (posito M dari rationaliter integre ex X) reduci
ie M.
sit ergo dy = X Im fiat y = n Im. Ex 2 fit du = dusm +ndm
vel  dus = 2m an +ndin Im : 2m. Ergo ex aeg. 1. et 4, fit
2Ym
X = dn + nam. Ergo si possibilis est solutio in ordinariis, oportet
e 
n sumi talem pro n sumi tatem torulam per X,ut nam
2M
ob datam m per X, saliter n ass talem pro n assumi formulam ex X
utn nam sit integer in indeterminatis, et quidem qui sit X.
agdp- xdq + pgam
ut an + A
am
fit n =  fit dn = ag dp -pd4: 49e fint X=
ap
Xe = 2amgdp -2ampdq + apgam, : 2 m ga
oportet apgam dividi posse per m et rampaq rapdq dividi posse per q
et 2a dp - 2apdg e apam nursus dividi posse per q. quod vx
quod non facile fiet, nisi p sit div. per 4, quo casu n erit integer
 si P et con sint
m
sane videndum an rapdq dividi possit per 4, nisi
primi inter se. video id esse video id esse impossibite, in indetermi
P et q esse primos inter se ob n = at
matis: itaque jam supponunu et R
ergo sequitur n esse integrum.
itaque nam esst integer
am
  dv =ds _b_ + G, dus  dv = XV + nH) + pdy, fiet
M
9+edu = v + n y + 
dy fiet Lalb. y + bdy + dedyt cddy =
=de +ndy + any + padu + apdys. Ergo tandem fiet
db y + bdus + dedy + Cddy = b y + c dy + ndy + du y + p ddy + ap dy
et res est reducta ad unicam literam
e
drggmodt
omosgur gel sup 5, deb Lngu jum dlnplsu heo

## Folio 32v — DE-611-HS-855907:0064
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-855907/jpgs/default/00000064.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 50 lines, status machine
Omnis quadratura seu
sit dy- SXsC
e
(b sec. reduci et ad PX tac
sit y= msC fict du Cam se + mde: tc seu
 = dm sc + inde sc: C =) dm C + m de se: c
ei
 a se. Ergo fiet
ve E dm c + mde, : C. ergo  mdc: c = (integro) X-dm.
e
1
Y fiet de = R. f
Y
VR + Y. f. g K + f g dk e
v
sit C2 f
Et fict de = +fF  K + FGY g K + dk: K
.Et fier
ce seu firt f( m. f. a mg + d m. f: ds + m. g : d9 + m K: a
2
Xe - dm
vel hm: gK df =rm fKdg + m fgdk : fgK = X  - dm
lunt autem h,h et Y constantes) et 5, d, h datae t per X; ergohine facile
1
h Y Y ergo de = h gHf + Y f. k. d9 + fug. dK,: fgK
habetur m
quod est theorema salis memorabile, quia respectu indeterminalarum
non differt ac non variatur, quaecunque sit potentia ipsarum fg ercdior
t
e c = f
Jorum. ejus quod et aliunde patet ex natura logarithinctum
porro suppono f, et g et k esse primos inter se. ergo
ut rvisco sit  M, hgKdf + rfKdg + fgdK: fgk sit integer
e
seu X. am necsario debet m dividi posse per fgk. sit ergo (fgndq
M = F9Kn et fiet n, hykds + rfKdg +fgdK = X = df gKn - f dgHn  gaKN
pro p po
seni ex QE _ sit h+ 1 = (h) r+ = (1) et fiet
(h) gKn df + () fkn dg + 2fgn dK + fgK du = X.
pro K pone p9. fiet dK = pdque + 9dp et fiet
hE (h) opq R df + (r) fpgM du + 2fgRpdque + 25gM9dp + fgte aM = X
posito jam constare, quod F, d, , sint simplices, et (ut 2X + 3 posito 2 et 3 esse
se constantes seu ex fua ( fat d ExF5 
q
debeat
constantes) et df, dg, dk, dp, dq est
assargere R, ut perveniat ad A quia f,9 , 9 seu f, g, R, sunt date
et C = fgK, fiet (h) = 2 = (Y) et fiet
 2gKn df + 2f9n dgs +25gR dk + fgK M = X
sit p = 1. et K = 9.