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# Catalogue record: 1163 — Quadraturarum investigatio per calculum
# Incipit: Primum omnes figurae simplicis aequationis nempe paraboloides et
# Date: 1680-01 (80-01)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=1163&katnr=58701
# Work: https://leibnizlegible.com/work/00068073
# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 1
# Folios: Bl. 20–21 (from the shelfmark LH 35, 7, 1 Bl. 20-21): 4 pages, canvases 39–42, page ids 00068073:0039 to 00068073:0042
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068073&page=39
# Lines: 252 recognised on 4 of 4 pages, mean confidence 0.79
# 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 20r — 00068073:0039
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 21v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068073/jpgs/default/00000039.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 72 lines, status machine
u de neductione omnum antronaplum qd circ. et Hyp. de
educt: curu ad rationates
de imaginaris
d
salculo per musmenta.de additionibus ordepatata
per constantes Ergo cubiliariae prius atiae Jannar. 1680
Quadraturarum investigatio
per calculum
Ennum omnes figurae simplicis aequationis nempe
est
paraboloides et Hy perbolvides ita quadrantur,
quia differentiae earum si sint progresspionis geometro
cae seut ipsis proportionates proportionis geometricae
dz
H y. dus B LX
Hinc et quadrabiles qua u ex his componuntur.
X
X + X B
+ X2  co.
et faciendo  v. fiet
et X  ro +1 semper datur qua
ratura. Et qua docunque si sit
a
.
datur figurae quadratura, ergo
et haec quadrari potest: X + bx +
X 4 b X
seu
X
X
X
2
VX
y+b - b
 eon
YX +  1
sx
y
fiet X + 2-Y
+
ne
vx H y + b
p- b3 + V Ly-
27
ponatur ae 1C+C
X
V.
X + 2RX + 410.
 1 + X 4F
quadrari potest. Ergo eb 
seu e
X2
4
t eX
x + bp y. X+Xy
SX
2 X +X+y+45
4
+
X
4+y- 4
a
p- 4y -b Habetur
t
ergo quadratura hujus figurae
Yγ. 4y +c, d,
sed quia omnes istae Vrae naps non sunt
plane analyticae, huc alius calculus subinde
a ine adhibitus analisticoteros per
momenta seu multiplicationes
Exempli
27

## Folio 20v — 00068073:0040
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 21r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068073/jpgs/default/00000040.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 52 lines, status machine
d
gratia quaero:
A
1
 XtIP
 X + 1
xt
XH X= 1
s
a
=X +1 pC-.
++X+I pt
X =1
Ergo.s.
X
el- = s
X=1l. X -1
Haec tamen
X+I
X+I
omnia consistunt in simplici
resplutione alicujus ordinatae in alias simpliae
A +t le
L + c
Xta X+b
pone c7,8- 1. fiet
a i+c X + b.
b-a
e
nac
X + bX + ab
+ a X
X+ bXFab quae proinde habetur
ta
ex quadiatura hyperbolae. Hra omnes
haben possent ex quad. Hyp. praeter 
X + b2
quod scimus ex quad. Circuli Eodem modo, in omnib
rationalibus.
tlia
opinor non difficili calcuo
ostendendi
inveniemus modum Omnes curvas rationales
pendere ex quadratura circuli Hyperboleo.
Jam omnes curvaeitationales reduci
possunt ad rationusles aequipollentes mea
mel lodo transmulatoria cum diophran
Jea combinata,, saltem enim
pe
se non licebiticebit addere alia arbi
taria quadrabili, seu jam reducibili ad ratio-
nalem

## Folio 21r — 00068073:0041
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 20v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068073/jpgs/default/00000041.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 63 lines, status machine
ores
sit curvae conicae quaerenda aquatio
7
24
a + b X p y. AbX dx riey dy
bx dxp Va + bx du. seu dus bx dx
a
di 
sum
2
b x ax
a + bX
dy + dx 
a, +
quaentur codo
2
a +bx
ra +bx2
1
1)
codue
sexplicernda est hoc modo ut faat
som u
substituto valore fiat. bX +1 pco
quantitas rationalis.
a2+b+2
fiti, cop V b2x 4a + bx, Pae + bX AC
a + b XC
cujus nimerator vinculo comprehensu debet
esse quadratus. quod fiet si primum
sit: a +bx quadratus  bX + x rby +y
p
fiet: X py a2- y2
e
27sb
et 1a+bx
seu 2a +bx a
y2
2y
Esficiendum jam etiam ut
a- 
at
fier Xp
2
et H a +y at Z
2Z5b +b
 a = +y
2y
+b X + a si
+ b
et Sb x +a
2Z Yb+b
2y5b.
Restat faciauus. XH ay
2yrb
a +Z
H
2Z 1b +b
a = 5t
2
2756+b
quaerenda relatio inter y etz. Sed Eanc ratio
nalem reddere non est in potestase

## Folio 21v — 00068073:0042
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 20r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068073/jpgs/default/00000042.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 65 lines, status machine
si
quia nom nihil aluplius explicade
licet aliaqui indeterminata laberetus
Otiam via videre licet an exstationalibus fierii possint
inationales datae. Nempe si .addamus inter se plure
at
bX + C
C X 1BX + 
+
1. dx 1 co ax
Xe ponendo imationalem videndum an inde oriri
use aequatio data. Sed jam inulilem esse additionem plurum
rationalium cum olnnes reduci possint ad unam. Nam
scribamus:  a + bx + CX  + dx etc. dx B eo ax:
1 + mx +nx 4 soxhele
et pro x sumende assumendo aequationem novam quamlibet.
inter X. V. e (quia eo frustra assumitur eum statum cirimnari
possit pess in deamquean inde prodire possit aequatio data
ponendo dy 11. sed hoc distinctius examinanda. Interi
Considerandum posse adhibendo armaginarias, fae ad huc
ampliora praestari, ut. si ascribamus bx + CX S-1
e
modo postea aliam aequationem ad hibeamus in ua sit.
bX CX F-1 quanquam aliquando ne id quodem necesse
sit ut videmus in cetu radic. aequ. Cub. esse modia
ex primendi veram quantitatem per unam imaginaram
q
es
Notandum hanc met hodum utiliter adhiberi ad revocanda
omnes rationales siquide d commode licet ad Circul.
et Hypero. Nota ndum et cop 
Ergo
a 1b+2
a scribendo a BC. fiet: C01Dvb. unde fiet:
momentum aliud
90 b
aliud
X +c
dob -C+X
quorum summa simul
C A
dabs summom co. jamalinde ad rationales reduci postest
possunt alia hud modi seu habetur cor du habetur et
tdeo. Hinc binis modis haberi potest summa figurae
en dictis. Hinc ipsa dicta et altentili solum.
dum et semper ultra purgi potest ut si sit
IX+1Z Hinc faciendo X1Z. fiet: V
e
unde rursus assurgi potest: per momenta
Sy2 +1, 2Z dZ
F
1
 vel
aequationes.
S.
+1 2
a
dt
saciendo:
+1, EZ du unde novae rursus
1
ordinatae ad co addendae erant. Aliu
rationales ut ex
et pro
poss.