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# Catalogue record: 34062 — Johann Bernoulli für Leibniz. Demonstratio multisectionis angularis. Beilage zu N. 33
# Incipit: Demonstratio Multisectionis Angularis insertae Act.Erud.Lips.1701 m.Ap
# Date: 1702-06-10
# Sender: Bernoulli, Joh. — Addressee: Leibniz
# Akademie-Ausgabe: AA III,9 N. 034
# Other printings: -Leibnitii et Bernoullii Commercium, 1745, 2 S.74-76; -Gerhardt: Mathematische Schriften, Bd. 1-7, 1849-1863, 3 S.699-702. -Orio de Miguel, Obras de Leibniz, 2011, S. 701-704 (span.);
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=34062&katnr=30364
# AA III,9: the volume in the Repositorium des Leibniz-Archivs (CC BY-NC 4.0): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-bd-iii-9
# In the Akademie-Ausgabe: AA III,9 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/LAA-BdIII9.pdf#nameddest=34
# Work: https://leibnizlegible.com/work/DE-611-HS-3653099
# Title: LBr. 57, 2 · Bernoulli, Joh.
# Folios: Bl. 76–77 (from the shelfmark LBr 57, 2 Bl. 76-77): 4 pages, canvases 151–154, page ids DE-611-HS-3653099:0151 to DE-611-HS-3653099:0154
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-3653099&page=151
# Lines: 134 recognised on 4 of 4 pages, mean confidence 0.87
# 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.
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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 76r — DE-611-HS-3653099:0151
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 77v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-3653099/jpgs/default/00000151.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 30 lines, status machine
Demonstratio Mullisaitionis Angularis in ratione data
sive surda sive numerica, inserta act. erud. Lips.
an. 1701. m- April.
Lemma. L.
Lectangulum sub deagonalibus trape i cujuscunque circulo
inscripti eit aequate suna rectangalorum sut lateribus
oppositis.
Demondiatio Lupo satii nota est et passim habetur
in fuigonoa etricit.
Leama II.
Sica semicuculo ADB, duo habeantur
arcus aequales AC, DE, quorum ille dia,
metio sit contiguus, alter ubicunque sum
Luc, ducanturque subteata CB, DE, EB
ut et CA, DA, EA; duo reitang. sub DB
et CB minus reitang. AC et Ad, fore aequale reitang. sub
AB et EB.
Denoncr. Ob simil. trnaag. AC F, BDA eiit AC. CF:;
BD. AD et AC ia AD = Bd in CF, iteunque ob simil. triang.
p0 
BdA, BEF erit AB. BD:: BF. BE et AB it BE = B in BF
= BD ia BC - Bd in C F = BD in BC - AC in AD. Q. E.
Lemma. III.
lositis iidea duo fore reitang. suib BD et AC plus rertanque
sub BC et Ad aequale redanque. sub AB et AE.
demonitr. Conjungantur punita C, D per rectam
CD qua parallela erit ipsi AE, et ducatur DG parallela
ipsi AC, qua etian pii AC vel DE erit aequalis adeoque
duuto radio oD, erit trang. OBD sinile Lrcaas. Gd2, ob
aagulos 00d et BGC eidem arcur visistentes aequates. Htcus.

## Folio 76v — DE-611-HS-3653099:0152
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 77r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-3653099/jpgs/default/00000152.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 27 lines, status machine
OB. DB::dG. GE et OB in G2 = DB in DG, et sumendo utii
uigue duplum AB in GE = DB in AC plus DB in AC, addeado
utriques AB in AG vel CD, erit AB in AE = DBin AC plus
DB in AC plus AB inCD, est autem  lem. 1. DB in AC plus
AB in CD aequale BC in Ad, ergo AB in AE = DB in AC
plus BC in AD. Q. E. D.
Demonctratio fornulae sectionis anqularis,
firce pramissir, res ita peragtur: Eito dianeter AB = 1
subterta AC = X, adeoque BC = Si- xX = (brevitalis gr.
y subteasa AE = a, adeoqe BE = Vi-aa  (breo. gr.
Ponantur primo arcus AC, DE contigui, evarescente
intermedio CD, ita ut arcus AE sit duplus ippus AC;
quo casu verta AD, AC ut et BD, BC coincidunt, adeoque
per temma 2. erit yy - XX = b. et per bm. 3.. 277 = a;
ponatur nunc arcus iatermeduis CD aequalis ipsi AC vel
CD, ita ut arcus AE sit triplus arcus AC: ergo per praced
casum erit yy -x = BD, et 2Xy = AD, hinc per lem. 2.
y3- 3Xx y = b, et per lemm. 3. 3Xyy -X3- a. Si nunc ponal
arcus intermedius Cd aequalis duobus arcubus AC, ila utar
cus AE fiat quadruplus iprurs AC; ergo per secundum
casum erit 53-3∞X Y = BD, et 3X7y - X = AD. hinc
per lem. 2. y Cxxyy + XF = , et  let. 3. 4X7 -4Xy
=a. Porro Li arcus intemedins CD aequalis Aaruatur
tuplici arcur AC, ita ut arcus AE evadat quialuslus
ipsius AC; ergo per tertium cassumm erit yt. 6xxyy + xt-
BD, et 4xy -4X Y = AD. hinc  lemm. 2. y= 10XXy +5Xty =b
et per lemm. 3. 4Xy -10X37y +XF =a. et ita quontque libetfa

## Folio 77r — DE-611-HS-3653099:0153
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 76v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-3653099/jpgs/default/00000153.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 45 lines, status machine
cile continteabitur: ubi Lac elegans progrescioni genera
lio conspicitur: scilicet sumta potectate binomioy+ X
cujus exionens sit numerur quo avus AE continet auua
Ac, erit illius potectatis terminus primus minui tertio
plus quinto niaus septimo plus nonoR. = b, et terminus
secuadus minus quarto plus septo minus oitavo plus deumo
Hi. =a. Quandoquidem igitur sumto numero potectatis
indetiaito sive fraito sive integro, sive surdo sive
rationali, erit ut olim orteadimus et nunc satis notua
si
n
s +X = y + ny
es
n- A
X ∞ R. n-C nor V4 n. n. 1. n-2, h- 3 3, n. h. 1. n2n
a
1.2
1. 2. 7
1. 2.814
+L. ergo patet sit AE sit ad AC ut n ad 1: haberi
pro indetinita seciione angulam hanc progrestione
universalistimamu, b= s
=h. h - 1hor 2
p. M- 1. n- 2. n 3. n - 4. n-5 n. 6 b
1. 2.3. 4.5. 6
n. R - 1. R -2, M- 3 7
1-2.3
+
1. 2
n. R - 1. R - 2 n - 7. R -
1. 2. 3.. 52
d
n-5
n. n. 1. n -2. n 3 n 4 
X
1. 2:3.
n
k-1
X
Hi. qua duae
progressiones omnes aquationes finitas possibiles complulum
tur pro puvois numero n integro sive pari sive impari
hitca ia casibus enim abrumpuntur semper progrestione
in reliquis autem abevat ia intinitum. Hinc paret qua
in aitit dipinue de lac materia.

## Folio 77v — DE-611-HS-3653099:0154
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 76r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-3653099/jpgs/default/00000154.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 32 lines, status machine
Problemes.
1. Trouver une combe algebraique avec laquelle un arc de cercle
donne pris ensemble soit rectificable.
2. Trociver une coube algebraique, dont les aus soiert propor
tionels aux segnens de cercle.
3. Les combes paraboliques et lyperboliques de quelque degres
qu’elles soient transfamer et d’autres courbes algebraigue
ensorte que les ars des unes soient egaux aux aris des autres.
. Ayant donne un polyaome elevé a quelque puissance que ce
soit par ex. un trnome eleve a la virgtieme puissance abre
a+ b+c20 on demande son plus grandterme.
5. Ayant donne cne progrescion des nombres natures eleve
a gurelque puissance que ce soit par ex. 1+2+3+4+5 . .- 100, que
est la progression de 12+3+4. .. + 100 continuée jusques au
ceatieme terme, et dont naque le me est eleve a la septiene
puistance, on ea demande la soine sans bequious de calcul.
X
. Ayant donne une quantité differentielle
.
jentens par
p’el y des grandeurs rationelles fornées de x et de conitantes
de quelque mahiere qu’on puisse concevoir) il en faut hrou
ver l’integrale, ou du moins le reduire a la quadrature du
cenle ou de l’hyperbole, car l’un ou l’auhe le peut toujours.
c’ert la dedant que conficte le grand mystere des quadratures
et des reclifications, et des reduitions aux espaces circulaires
et hyperboligues, c’est ce qu a été desiré de tout tempt.
ay resolu tous ces problences la, mais en voyey un qui paroit
leger, qui fait pourlant de la peche.
7. Deux nombres étant donnes metn. trouver un quarre
XX qlui fasse avec chacua un quarre parfait; scavoi
M+ XK = a un quarre, et h+xX = a cin autre quatre.