# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/16204/text
# Catalogue record: 16204 — Joh. Bernoullii problema Act. jun. 1696
# Incipit: Invenire Curvam ABC
# Date: 1697
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=16204&katnr=45251
# Work: https://leibnizlegible.com/work/DE-611-HS-958803
# Title: LBr. 57, 1 · Bernoulli, Joh.
# Folios: Bl. 160 (from the shelfmark LBr 57, 1 Bl. 160): 2 pages, canvases 321–322, page ids DE-611-HS-958803:0321 to DE-611-HS-958803:0322
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-958803&page=321
# Lines: 105 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.
# 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 160r — DE-611-HS-958803:0321
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000321.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 35 lines, status machine
e edinchdemus
am
seu rut
Redinchormus debet esse DC qui HC) = ~. DC - XXtyy
et DCC)  (XX) + (9y). Ergol XXFgy 3 (xX) + (yy) = a
jam ob  C (C) rectam fiet (2d0) X2y = (X) : (y)
quaeritur relatio inter X et y. Fit to
v. xLD.a. posito V dari ex X
uo semper elicet aliquando ircationaliter) atineri potest. Ergo per 2, utique V
debet vdari eodemmodo ex X, at ut X, ut ex (X)t () (Y) x) = D: a
jam exa 3 aeq. 1. est t) a: XX(X) = 1+ Gy. XX, C  Y1 + (1y) XX. ergo
fiet?  jam eto seu per 3 et + hit P ab: t (X) = 1+ 2 ∞ V: ax + v. at, 1+ Wb: aa
+
da
o Hinc sequitur s (X)
p
compuuma: X (X)
Temper esse
tem sed  ex
etiam est est e commnnis Ergo yt siet communis, quod etabsurdu t
 vidtur hocbilema non esse pollibiler Et dato anculo EDC
do
eon
22. 35 op mhgns
2o) Eguo op anorsuos
nosuon
datur DF AB(F,
se
si sit DC. DC aqualis constanti osu Exxtgu  ,
aa
mad
d
fipt 1 +g.x, sit D = ad: X () -. Ergo X (X) semper
est commuris, et dalo angulo DFC datur DF. D(F)
pton

## Folio 160v — DE-611-HS-958803:0322
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000322.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 70 lines, status machine
Joh. Pernonllii problema. Act. Jun. 1695
invenire Luoao ABC ju naturae propretatis, ut durcta ducta a
puncto dato d recta de utcunque DEC, silidum sab
uno jegmento de, et quadrato ulterique DE sit.
plineam
vel fig. 3, ut fe s ) F ut qoadratum
sales R solidum sub GB et quadrato FB sit
estanti dato aequale; prius ex his considerare placet
bi
sit 
DE dx+v
e
ntot
FE y. 1) ((B) (C)
e
Xx+gu(XX)(yy) L a
Waec Hoc problema non est nimium,
seu non continet proprietatem pura
hopau, nec revocari potest ab tan-
gentium inversam. Nega enim  tam
ate accipitur, ut punctum unun cuivis alteri possit applicari
video posse assami vam quameunque arcumcurvae
HL, dende posse centro
s seu DC describi arcum irculi LM, et in ea
sun electo puncto L. junctaque recta IL, applicetur
curva assumta HL, ad L, sic D iptam et deguat
deinde sumta junctu DE secante arcum HEL i puncto
quocunque E quaeratur EC, quae erit a: DE.
eita etsi HC sit recta, tamen invenietur  curva LC
si D= tangat C lineam HL, jam melius coibunt in anum HL, et LC
et potent inveniri aequatio ipsis communis, ducendo ectum aequationes in
se invicem sed si una cuva aequalio selisfaciat utrique, s quae
sit in deprimibilis habebitur desideratum
Aequationi gu, (XX gy) = a auedet hoec
nae
2da) X: y = (X). (7) sisit et (tio y∞ V. et fiet
i sit y - x. F et ex 2 fiet X = (X) quod non procedit, nam fierent
X et H) aequales. () Si y - bx + X fiet e, (y)- 
L
b(X) testis
Ergo ex 2 per 3 etq fiet: (1t t ib. . C =b(  (ugnaL
quod si jam tottatur ex jam ex t ex cg. 1. tollantur y et (y) per
aeqque. 8 et 4, prodibit aequalio inter X et () coincidens aequatione 5.
Hinc poraeq pro aeq. 5 3 assumi posset formula generilis
Ex pn 2 fieril potest (X x+ gy, : gy = (XX + (yy),: (9y)
bo
ce
Forte utile erit adhiteri et tungentes. Nam ex aequ. 1.
fiet ( X X + gy. (xx  (y ly (: ( (X) A (yy)) + V( X) H(y) 2 Xdx + 2yly = 0
et cursq a ydx - xdy, : yy = (y) dx) = (X )dy), : (y))
sed itac non poterimus magnopere lucrari
Ex aequ. 1. frit E bX + Cx + X, Sb(X) = CX) + (X = a
o)
et tollendo urationalem fiet aequatio (10) quae concrdere
debet cum aeq. h, id est ducta aequatio in arbchanam
debet pnduce d a equ. 1 coinie t seu saltem
debent habere divisorem commmunem, vel potius sumponendo
ag. 5 esse indivisibilem debet aeq. 5 ducta in artitiariam
1gto)
Generatissime )y: X = X: a pro ag. 3. et (y). (X) = (X): a
produlere aequ. 10.
fiet. Erqu t X: a = (X): a Hini s si subessent cretronalie
sublatis illis et dividendo per X- (X) fieret aeq. ubi X et () se habent
Ex aeq. 1. fiet
eodem modo ida. Ergu jac x et guim si
(6b) a3: XL X X. (X (B = 1+ X: a, 1+ (Y): a Ergo fie
(7mo a. et : () =B A  + X: a. seus q  : XYa
et pro X. a ponendo vieraxa
sutor seu fll. XXG a eX: au B Vit Fa a 3. v B rvia
al. 2