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# Piece: https://leibnizlegible.com/api/records/36274/text
# Catalogue record: 36274 — De radicibus et seriebus summandis
# Incipit: Esto quadratum abcd, in eo quadrans abcea triangulum semiquadratum adb
# Date: 1673-04
# Akademie-Ausgabe: AA VII,3 N. 018
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=36274&katnr=Cc2+615
# AA VII,3: 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-vii-3
# In the Akademie-Ausgabe: AA VII,3, p. 261 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/VII3A.pdf#page=261
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 206–207 (from the shelfmark LH 35, 12, 2 Bl. 206-207): 4 pages, canvases 411–414, page ids 00068195:0411 to 00068195:0414
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=411
# Lines: 275 recognised on 4 of 4 pages, mean confidence 0.82
# 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 206r — 00068195:0411
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 207v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000411.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 49 lines, status machine
taa = k4 b.
e
q a + 2 a +2b. = L, Aiu n9
+ H, R4 a + R4b..
2a + 2b = 2 a + 2b + Rg ab - Rg ab.
Esto quadratum abcd. in po quadrans abcea Trian
gulum semiquadratum adb. volvantur caircaaxem
ab. Triangulum producet conum, quadeans
Remisphaerium, quadratum cylindrum, et cong
erit tertia pars colindri, Hemisphaerium
duplum coni sintella poducta a trilineo id
edoe erit aequalis cono. Sintella enim com
ponitur ex zonis crcularibus, quarum aequalia
arculo comi coordinato, quod demonstatur
eadem fere methodo, qua lunulam quadra vit
Hippocrates. Uet adeo ment dimensionem
phaerae et reglindri Hrppocinti ex parte quada
debeat Arhimedes. Ergo ut ct circuli
compositores coni, ita ona composituces sintellae
crescunt in ut altitudinum quadrata. Recta
porro componentes triangulum, crescunt, ut eorum
quadratorum radices ad eo ut allitudines, at
non rectae componentis Ti lineam, quae
quomodo cre se ant videamus: sunt scilicet
 ut radii onarum. Sed hap methodo
rem investigar difficile iat iem tentemus:
vicae
Scimus summam superficici conindrica, id est cn
aequalem, summae superficiei cylindricae, id est circulum
duplum majoris circuli in sphaera. Erga sum.
Cum ergo summa peripheriarum constituat hunc
Circulum rationis esa sammam radioram, constiturre
quadratum, quod sit ad eam ciroutum, unt radius
ad peripheriam. Saed cum alibi opcensum sit
quadratum generans cycindium esse ad superficie
cylindricam vel sphaericam seu circulum diagonalem
quadrati ut radius ad conumferentia; ideo sequeret
quadratum hoc. cesse quadcanti aequate, totum
partiquod est absurdum. Hinc demonstratur,
peri nnon posse componi radios, ut peripherias
hocet coordinatas.
Circuli in hemisphaerio crescuut ut applicale
in semiparabola. finus in quadiantes screscunt
ut radices ejus mensirae qua crescunt applicalae
in parabola
Rem ad numeros redigamus aut fiatlucidior
aperiatur fortasse via per arittimetica
infinitorum.
Math. X1, 2

## Folio 206v — 00068195:0412
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 207r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000412.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 87 lines, status machine
4 ab + eq cd =   + RaXa
RqN + Ra ra
Rgab teqed - B4 XX +na +  eq u
En regulam Greviorem; quadratis addatur
factus ex radicibus duplicatus, producti.
Radix erit radicum summa.
Nota si summa radicum ducatur in differ 2
Regula alia memorabili si summa qua 3
ideo sufficit omnea multiplicari seu affici
A438 a
a A917 R457: a
1468
45
rentiam productum est differentia quadrator.
Cylinder.
a Rg1 Bqa
a 960
eonu
erentiam quadralorum
in diridatur per differentiam productumtts
mine resiu
Radices diperentiarum et rehonum sutentur in bis
pr radium a. sEd ita fiet:
a
a 19 36
vel
Hemisphaerim
ua
pan 16C49 Xa 3349 (2aq X
324
Xa 9~ 49 Xe40 49 2a au 4
Xa 4V49 xa 4
49 2a24 Xa 450
Xa 21C 49. Xa 48. a9) 2aequa 8
Ena. ra, summa igno
Quadhans circuli
2a R4 Xa 13,96, R4 a 48, 49
2a eeq uae 24, 437 A4  48 89
2a 4Xa 3049V
2a (147 a +S 29)-
2a Rg4.
Summa ignoratur
aeterum omnia in infinitas partes divise
eogitanda sunt: 
Txa
sed cum 7 debea intelligi= 49.
B
n
25
Hemsphaer
evachans,
1917
X est ratio radii ad latus quadrati circulo aequalis
Sed non opus est utu isto X itermino ignoto
Cum enim sint tadii ut radices circulorum
+
+C
si suum ne veles 2 3 6 18 1 2  2 6 4 eti.
habes 1. 1. 1. 1. 1. 1. et praeterea:
- = + 2 5)
qlu 14 20
2 4) 142 613
Ergo 3
8 2
28(7. 35(7
2 Z Mk. + 2. 3. 4 ek
012
Jm haec 3. 5 2 rusus resolvuntu et fiunit:
1. 2. 3. etc. +  = = et. Habemus ergo
Rammam hujus seriei infinitae iventam.
(2a A4 2
2a 19
2a 29 
2a pg 
24st4
humma ignoratur
in ergo quadrantem irculi ()
tun
iner nosius exhi
iss rr
nits haer
(xa 2
Xa
xa
18
us
na ac

## Folio 207r — 00068195:0413
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 206v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000413.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 111 lines, status machine
Jam tentemus hoc quoque summare:
tote
96.
6 10 15 21 28 36 45
8
10 11.
e 19 26 34
6
B. de Residuo a teulenti dutiso)
10 16 23
10
11
Quadians foc non aeque facile alia remratione
na 3bV+ Na 12 4 (2axq ua 1 40 aggrediamur. Ajo in summari posse
aau nat a 24v hanc seriem:
ratic. e 
5
1
e 7
25 1 20 27
35 et
91 11 H = 121
Es 3. 14 a + 14B 40 B 8.6g
Rsto agb- naa +b, a ib
Ex his exemplis
muni di colligo Theorema
 maximi momenti ad
. Achalysun et omnem
men asilsmetiet
chanino Arithmeticam
Guometriam.
Radices quadratas
dinsivicem addere (subtiotuas
addantur quadrata
Ergo si simmari posset siquens, hublemosummus    etci ploducdum dupolicetu
2 18 25
8  9
33
10
42 1
12 18 25 33 4V
8 9 10
2
2.
facile summaremus.
serem est
34 58-64 + 22-4.
A a + b + ab
2 2 5 2 etc. it patet nums ogressionis Geo
Si esset
2
10
3
7
3669
n L 12
10
5i
9 10 11) 12 4
Quam ita summabimus:
6 8.  2 4
B
20
64 10
10 12
64 100
2.
Ergo summa: 4: 36. 64. 100
Eodem modo altrum 8. 7
aen
prodrctum productum
pu ab + Ba ed= Aq  excedet
a tnaxa
Haec aequatio non
vadralum
differentiae.
nisi arste nostra a surdis in garipotest. Ecce metlodi
generalem purgandi omnem aequationem a numdrs
urdis terro u Non ae crrvi sei nuluus es
Cum in nnfinilum facta supponatur
subdiorsio uit refert 48 pro 49. disfe
Eunt enim non nis puncto, id est nihilo
35 Eetas -2
5.
16
Fi 3381.
hoc verum sgon est, nisi
quando pedices sunt
rationem reperire. Quadcancius ad fa ut
circuli quadraturam. Nota isti Numeri
differenta autem inter praecetentem e hanc 13. 14. 48t. 49. ete sigrifrcant lineas. Et
ex.
dias
quod ad usum calculi. Si quis ergo
reperta est Hemusphaerii ad Xa. haberemus
in 14 pa 4. significat lincams estnumerus
Alia focu transpositio potest hoc medo:
stantum opus est reperiri methonu
ista summandi:
Ra 1B, 49- 36- 13
R4 24
33
R44
R45
Rq
10 12
100 12
Qui rationem  summandi
eg 4g
repecerit, universalem is nobis
uadraturam (irculi exhibuent

## Folio 207v — 00068195:0414
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 206r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000414.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 28 lines, status machine
Hinc nihi tanden quaesita diu se ratio offere
summas ineundi fractionum quadrabiearum
cubicarum etc
16
b: a
grestiones habentium invenibe latione tosolum
2. 0
+ 2 + 4 etc. facit Infinitum = 1 + inb  inf- 2
62u 20= 2 35
sina tisri. 4
vita 2 quis. a2 unta 4
Sinif - 12 puf. 4- 26 iif- 
2, a 4 Rg b
a + b
2.
6 b 12 12
A9a Agb
A94= A4 b ~
+Aquat eRges +bV
24 A. A4 a = _4 b 2. = a + b = 2kqab
2a 2 ab + a + b - 2 Rg a b = 3 a + 36 = 2Rqa b
2a + 2b - a - b + 2A9a6 =
iiorum infinit
Apex erdifastigumue Analyseos atque Aritunehaa
infinitorum, ex datis rationibus partium certas pro
2
11. 4. +. 1.5 + 1.
1t