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# Catalogue record: 20909 — [X]: [Lösung des Problems der Brachystochrone] Excerptum ex Transactionibus Londinensibus mensis Januarii 1696/97. Ep...
# Incipit: Accepi, Vir amplissime, hesterus die
# Date: 1697-03-30 (97-01-30)
# Akademie-Ausgabe: AA III (assigned, not yet published)
# Other printings: Gerhardt: Mathematische Schriften, Bd. 1-7, 1849-1863, 31 S.388-389
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=20909&katnr=30295
# Work: https://leibnizlegible.com/work/DE-611-HS-958803
# Title: LBr. 57, 1 · Bernoulli, Joh.
# Folios: Bl. 168–169 (from the shelfmark LBr 57, 1 Bl. 168-169): 4 pages, canvases 337–340, page ids DE-611-HS-958803:0337 to DE-611-HS-958803:0340
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-958803&page=337
# Lines: 80 recognised on 3 of 4 pages, mean confidence 0.90
# 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 168r — DE-611-HS-958803:0337
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 169v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000337.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 23 lines, status machine
Excerpta ex Transactionibus Londinen,
sibus mensis sanvari 169 e.
pistola missa ad praenobilem virum D. Carolum Mountaare
Armigerum, Scaccari Regii apud Anglos sancellarium, et
Societatis Regie draesidem, in qua solvuntur duo problemaque
ta Mathematica à Johanne Bdernoullo Mathematico cele-
berrimo proposita.
Tan: 30. 169.
Accepi, ir amplissime, hesterno dic duo Problematum a lo-
hanne ernoullo Mathematicorum acutissimo propositorum exen
plaria, Groningae edita in haec verba:
Onseratur Frogramma Bernoullianum.
Scalenus adetndunus. Problematum vero solutiones sunt hujus-
Pro
l. 1
Investiganda est curva Linea
A Dq
L
D, in qua grave a dato
quovis puncto A ad datum quod-
vis punctum B vi gravitatis suce
citissime descendet.
modi:

## Folio 168v — DE-611-HS-958803:0338
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 169r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000338.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 35 lines, status machine
Solutio.
A dato puncto A ducatur recta infinita AP CZ hori-
Zonti parallela et super eadem recta describatur tun Cyclois
5
quaecunque A25 rectae AB (duitae et si opus est productae
occurrens in puncto Q tum Cyclois alia ABC cujus basis et alti-
fudo sit ad prioris basen et altitudinem respeltive ut AB ad
AQ. Et haec Cyclois novissima transibit per punctum a et erit
Curva illa linea in qua grave a puncto A ad punctum B vi gravi-
tatis suae cilissime perveniet. QC3.
Trobl. I.
Lroblema alterum, si neclé intellexi, (nam quae in Actis Lips.
ab Auclore citantur ad id spectantia nondum vidi sic proponi potest.
Luaeritur Curva KJL ea lege ut si recta PK L a dato quodan
puncto P, ceu solo utcunque ducatur, et eiden curvae in punctis
duobus k et Loccurrat, potestates duorum ejus segmentorum Pk
et AL a dato illo puncto 
ad occursus illos duitorum, si sint
eque altre sid est vel quadrata
vel cubi vel qua=drato qua-
drata etc:) datam summam 
Rg.So
S (cub.
L9 vel  Kcub. 1
etc: (in omni rccla illius positione) conficiant.
solutio
Ter datum quodvis punctun A ducatur recta quaevis infinita po-
sitione data ADB necla mobili KL occurrens in D, et nomi-
pentur ADX et PK vel PLy. sintque L et R quantitates ex
C
quantitatibus quibuscunque datis et quantitate x quomedo cunque
constantes et relatio inter x et y definiatur perhanc aequatio-
nem VY 1
EY + R = 0. Et si R sit quantitas data, rectan-
gulum

## Folio 169r — DE-611-HS-958803:0339
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 168v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000339.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 22 lines, status machine
5
gulum sub segmentis PK et PL dabitur. Si L sit quantitas data
summa segmentorum illorum (sub signis proprus conjunctorum) dabitur.
Si 22= 2R datur, summa quadratorum (8Kg + PL9) dabitur.
Si 2
3 QZ data sit quantitas, summa cuborum (P k cub. +  scub.)
dabitur. Si 2 4
C
QR = 2 R R data sit quantitas, summa qua-
drato = quadratorum (Pk qg.  P Lg9) dabitur. Et sic deinceps in
infinitum efficiatur itaque ut R, 2. 2Q = 2R,
data sint quantitates et protlema solvetur. L. E. C)
d
LR, etc:
Ad eunden modum Curvae inveniri possunt quae tria vel plura abscindent
segmenta similes proprietates habentia. Sit aequatio Y3 + Lyy +
Ry- + 5=0 ubi Q, R et S, quantitates significant ex quantitatibus
quibuscungue datis et quantitate x utcunque constantes. et Curva ab-
scindet segmenta tria. Et si 3 data sit quantitas contentum solidum illo-
rum triun dabitur. Si
.-2h sit data quantitas, summa quadraqe
torum ex tribus ilis dabitur.

## Folio 169v — DE-611-HS-958803:0340
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 168r); this page is its left half.
# No recognised text on this page (recognized).