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# Piece: https://leibnizlegible.com/api/records/26989/text
# Catalogue record: 26989 — Math. Aufzeichnung
# Incipit: [...]a + b _z aqv
# Date: 1680
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=26989&katnr=56959
# Work: https://leibnizlegible.com/work/00068124
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folios: Bl. 125–126 (from the shelfmark LH 35, 8, 30 Bl. 125-126): 4 pages, canvases 251–254, page ids 00068124:0251 to 00068124:0254
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068124&page=251
# Lines: 389 recognised on 3 of 4 pages, mean confidence 0.76
# 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.
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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 125r — 00068124:0251
# Scan: The same scan is also registered as fol. 126v; its text is given once, here.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000251.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 195 lines, status machine
B3. 1+ f.2 aequ 2 +3, 8, sat 43, 1 S.L + P.F tat egu = 14 3F. 1 +B2 s-1 aequ 5 + L- 2 Ergo 3s+ sur aequ - 1 + s. 2 aqu  1 + -
4( X - l aequ. 0. fier: X aequ. 1. item X + X + 1 aequ. 0 et X _u + X L  N + - et X +  B Y. = - = 1 1+ s- 3.
Nam . - 1  3, f.3 4d, =1, -3 - 3 f-3 aqu. 8. et 2S, (3) etiam aequ. 8.
Hinicratio eroris patet.
 25
115
S1 seu - 1+ s- 3 aequ 2 31
Nimitum omnis aequatio  hugmodi,  qua in qua radix cubica extrahitur, habet
vel +9
haec X aequ. 1. vel X - 1 aequ. v. habet tres radices 1. - 1 + s- 3. et - 1 - L- 3
itaque haec quoque aequatio in qua quaeritur. 5+ s-2,  +5 - 1-2 habet tres radices. Equae in hoc quidem casu omnes
tres radices. exempli gratia
tres sunt  reales. Et quidem calculo nostro (paginae versae) rperietur radix vera seu affimativa, non vero falsa seu 
negativa quae est =2. Et alia exhibenda esset series, pro inveni endis caeteris, aequatio erit.  iaego 4. rgo 9110. et -2  t g -  B 5. P B.2
et fier: X aequ gX + 10. cujus una radix est - 2. Nam - 8 aequ B2 g in -2 + 10 aqu
Ergo B, B 27. Ergo P H 9.
X X - 9X = 10 dividendo per X+2 fiet: X aequ 2X + 5 cujus radiees quaeramus. X 1X + A aqu  4 +5. sit X 1 aequ _ Y9 aeqo +3 seu Xp +) vel AB
E= 2X - 5
Ergo aequatio hoc dip habet radices falsa X d2 et -2 unamque veram +4
X - 2X + 1 aequ 1+5 aequ. 6. seu X  1 aequ 16. seu X aequ 1+ 16. vel 1 - 16.
bin decimalibus est.
L
9x pB
AA)
4 A
 60000000000 est 2.449 erc. Cui si addas, 1. fit 3. 44 9
Idem prodibit si quaeras 5+ +.2 = 15- 1.2 per seriem infinitam.
aequ 3 (seu X aequ 2)
XHaX + 4
X. Haxr aX
Nimirum dividatur per S1. faet 15+ s. 2 + Y 5 - s.2 aequ 1 + Y. 2 + 31 - s- 25
ex 3s quod et verum esse comperietur
15
14 308 (6 13090 (8
243
X 4 AX + 4
etc
- 0031668614 (8 = 00 233 80039610 erc
1 = - - - 2 Y) 2 oury erg v0 8 1) 04 51, (+ vo  g qubus s5
X +4di_ u X
=4 1x = 16k = 10
X VX - 12X - 
et .... . . .... .... in 15 aequ 3. 449
782782 110
6561 413343 334807836
Nam Si+ L. 2+ S - (.E, H D2 - = b - = b -
in decimalibus
Et quia b in deci seu  in decinaitus est - 808p
posito b aequ 1-22. unde b aequ
= 100vevIEg,  + 0000000076611711795E
ab+2ic
sit X pX pX A. qaequi ponatur una ejus
Rudix inventa esse R. fiet etiam  px.
Subtrahantur haeduae aequationes a se invicema pq
ec:
y3- R3∞ pX + pR(= 4+4 aaqu.0. haec aequatio
dividatur per X = R. fe proomet XRx + R. paequ. 0.
et R X aequ Sp-ZR = ZR
400o vors 153b0s et
hini fier deniq . .   . . . . . . p + 2 + 0o777 7172. 12005 2bp. bi1728c 1
ubi sufficit sallem duos priores termins 2. et 0017 in unum uddifiet 2017. jam. I5 in decimalibus est 1709. et 2017 pe 1709 dabit 3449053. Hubemus ergo 3. 449. quod quaerobatu
decer
Itaque mirfica illam difficultatem resolvimus. Nam : videbatur debere prodire - 2. et pti quia produxeramus 5+s-2 aut 25 - s- 2
end
eb
de
. Sa + s-2
be
-E cubice mulliplicata ergo erat 15+ f- 2 + 5 - sir aequ. = 2. quia - 1 + 2, +,  1 - s-2 aequ 2. quomodo ergo
qu.
15 = s.2
et 16
prodt quomodo ergo prodit 3. 449. seu 3+0449 sed responsio estftis nunc patet nimirum
ires habebit valores, unum verum 3. 449. duos falsus, unum -2. alterum 1 - 56. seu 1 - 2449. seu - 1. 449.
XBh. sei Xn Jh et x= XX + X Th + Ihh ro XX + i + iti 7   - 4 stit pi th
XH est  vel aequ. + 2sR. 14 s.3 et X + Th p + h S. 3,  seu X
E
At ost 3s
 tus vgo. 0. Ergo Vcompretendit utramus radicm. Vvidendum ergo an ductum ex radicibsumma ratealis aequationis: B6 10v + 25 2sA FY -1 sa
firoveuiet v. + 13 - 10 aequo
et verum
ve
13+ s.2 plures habet adces
 rego B
Cve 5 + s et V - bd aqu. 2 seu N- 1 (V) 
5 = S
17
e
2 3
125
49
32.75
16375
900ef2gg
et  citem
6.76
20. 38
e
∞1
2.687 8. 78 4, 8. 856
2726000
2 2b
799000+2
545
 Re
loco
348100
2 gC
3888
46656
2038
6114
4).
Thp - 1+ Y. 2L
noterimu
damplices
fres:
10
2000 vit aequi4so , - 2ibs
=3s.1) 0L,  3 L.1 a3 S.1.
Jamensus per 20n 3B,, 3 seu per 17. 38 continuetur
divisio nempe primum sumatur 8 ++ quod
didatur ricuo invento quotiente, idque.
345S, pliqdandium sipocet continuando, inde quatens
ad simplia
A3D =B26, 26, -1 aequ 18, 18+2626
+  = f. 3- +  aqu. 1000. 1000 Ququ aequ 10.
2H1. (13 sat 47S- + = 16. ergo
+8 +
A2 ihc pono 20Z 17 3 portubice
1t.
uuns
fiet Xvcl x (3 + V L Tubice dat: 27 + 27 5. 1 - - s. a G
aeque. 3 aequ. 18426f-j
6 fot 144 1Yat 4 tal 3-is-2 + 256
8 jam + contra aeque
2, 6. 9 9) V1 s. (in 1S. onli  6 l. 1 .. -
6. S gforte lamensatis cufrendo a
est habin solum composita ex
numeris D. et 9f-) Quaeerit
bis jnus terminomm quia
mago
dibident
iin
12
1090
a +
a  l2b
187 261.1 1d
3b 107
(+ 6 1 (b.76.  
7 1000 + 10n, 1, 1b t. 1
76
1000
18F26S.1 investigetur communi extatend
more hoc modo: primum extanhatur radix 18
quantur ibet exacte vel ex 19000D eut 2. 6
Ove
hujus quadrtam est 6. 76. quo dicitums 26 mece cubica otrauandi
quod et havius ficcipt hc modo extruhatur audix ex 19
quotiens jam habitus est 2. 6.  cujus quadinto. seu ex 44 qua est d.5 differentia inter 319 et
Rriplicato pono instituatur divisiolus estantem 6)t(6) .t et 216 est 09 ergo Y18+ 2b tat est cinciter
142V a1
quadratur AZ
Hujus liplum est
Eexponedta est et dicula cuborum
composilorum sita ut numer
199+ 999141 sint latera
tour tubit summus sterotia
 l
dit nien eo ducit ens 2
in vel 
s 26 + 09 fat  imo error operatio
6723sis.
 Trbica continuanda est sic:
eor An et ab initio sic:
sout s
. .Li l ar. 6 51
+18 V0C
1) significat 1S.1.
(C2) sinilicat 2f.I
(1)) significat = 1
A 167
E.d onfinta des bis n
per V3+3
Bi tao:p +5. - 1083410 t7
extrahibilis fiat
K - yX-
X - 9 X = 1
vlnranturs productum digid
6
.1 Nt
524
eit
32 e

## Folio 125v — 00068124:0252
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 126r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000252.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 90 lines, status machine
a+ b aqu a + Z a b + 2. 5-1 a b + 22 2 a b + 2F12. 723,
sit Laequ L. fiet.
Si Zaequ  fiet:
in decimalibus si addas
duplum orom esti
1 + i
3141 + 3, - 1
4-
12
1, 2,3
X 1672
1,23, 4
asy 5
4 4
+ 2567 2
Caelerum ex his etiam apparebit quomodo nomina numeratores procedere debeant
- 1
1 + 008 aequ 14 0001 A = 0000001.
81
4 0000009001
10
24373
 ese
d082304537a
aequ 1 +000051D8DB. - 0000 0BD12500 +00000000000b250V erc summa
+ 280
seu radix cubica ex 13+ 3le + e + 3C12. id est quaeritur l. e. Series proveniet hujus modi:
3+1 + e
er. 25.
80  se
B) Pt 3e ( 3el
mtn aequ. 1. 01
V + 21 laequ. 1 + ot aequ. 1 + e2 - 00 441
10601
10001000
avs100100001
100100001
10001
100500001
l
Y 1 + ou aequ. 1 + 001  00 001001 + 00000011011 cu equ
(10 equ 10 00 t +
2)raequ Y10. aequ
10001
. 100 1000110001
6
dobodd
0011b00101 f+H
Y 10. 01 aequ (, 4101 aqu , + 0101
Nhabeatus una Radix ejus _ R - p R - q. hunc
subtuhendo a priore fieri X R3 - p X + pR aequ. 0.
+ 10010001
- 0000000100100001
= 10010000
00000vVOVVOIIe
dividendo per X -R. fiet: X + R X + R aequ. 0.
et X aego +RX + p p - Z N et x + = R B Y p - 2 t seu Xp Sp. 2 f - L R
At aequ. = 1 + s.3 Exgo E si jame . multipiretur per 3 quia h. 1 idem quod R.
recte multiplicari potest
ero et Yh, idem quod ith in c. ergo quia S1 nes habet valores unum 1. alteropndios - 1 4+S.3
etiam 3str duos habebit natores, unum h. alterum Th in  alterum YE in =1 + sa
Hinc quique 35 + s- 2 aequ. - 1 +. s - 2. multiplicando per - 1 4 f.3 fict: + L 4, sa3 -+ s. 14 1 s6
5+ f- 2 aequ. L + V-  - s.  4 (3 Eodemmodo S5+ s. 2 aequ- 1 + Su2 nulti per - 1 + f.3
5 = S.2 aequ. L = s. 3 + Y-= 
+13
Ergo 25 5. Y- 2 + Y5 - S. 2 aequ. 1 f 16. quod
em, nam extidendo radice eu his duobu binonous X
i
x - g X - pX - quequ. 0.
ve
 2440, xqu. 16,
+
unde puto nominalores semper non alios forequam progressionis Geometricae duplae
+ 3
 aesi atb 3r an BZa
fiet:
10001
000
seriem infinitam prodiit: 3. +49 ex) id est 1+ 16. est
curtium binomium multiplicandum per V =1 Sad
 rarilibus. Regula vulgaris.. ne
72913
00
At ∞
00
+ 1
101 c
10
0001
101

## Folio 126r — 00068124:0253
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 125v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000253.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 104 lines, status machine
-
1, 273, 4, 5,6
25t162412
Z ) X2,27 2 4. 2326 5
1, 2, 34, 5,657
204372
similiter pro extroctione radicis cubicae erens
12. 8
 erit
-4 - 12 -24 - 3 
ECa -01  - 112, - 241 3t 
ut per
8 + 10
 it 123 177+ 1145
fei + , - 1
115 +. 135, d
72  1 558 1724. 6 1333
Eti 3
-8
+2, 5- 2 5, 8
1oi, 9 17. 27
12 Si
.- 51
1 - 14
2,58,11 3
235, 5, 293
1237 5, 8
9023350039410 10
FC L 21 312 a-
2-51
a b + 2722  a b+ 2a
n nominatores progressionis Geometricae triplae, et ita porro. Quare utia est in numeris adhiberi algorithmum dyadicum pro extractione
radicis quadratae, ternarium pro extractione radicis cubicae. Et nihil ominus semper apparebit ratio transferendi ex una in alteram,
 1 +1
1d C)) 8)50bC
sit l+ e. cujus cubus l + 31 e + 3le + e. dividatur in duas partes l + 3le, et C + 3e l. ex quibus prior fiat aequal. a. posterior aequal. b. et quaeritur (7) a + b
21 0500008
et 02aequ. 010
m +n. aequ m + 2mn+ n. sit maequ. 1. et n aequ fiet: mn aequ, 1 + or + 001 aequ 1. jam, 2 aequ 10 fiet 1 + 010 + 001. aequ 1. 011 quod est quadr. de 1. 01
1+01 sit seu + + et 1 + 0110 + oi ( 1. +  1 + 0b1 aequ AB. t 1001
00 1 1
potus sic. Maequ. 1. n aequ 000c. 100
1100 ( +qu. V + 0010001 aequ. l _ + 00010001  = 000 30091001000D1+ 0000000VVV1001000110001
3n-q
100 100
10 1000 1 +001+ 60d 00e1
42 - 4 - 5- 7- 
13
410000001250000
888
0062500000
P
a Z aequ
poihrru
0 fr 1.
0. 0 1
ErDI. Q aequ ool 31 000
1s  0000l
128 1 00000b
811
004 Xi.
672
600e
∞d00
2,3-3, 3- 43.5 36,6
6561737
00+69 1403 d
+10000d00000000000000
+0000500000
0000000 125000
525
 H
6000001
1010001
1. 000000000000
00004SIIOII
0000100
.0b10b1 101
tt 2.7
19683, 3
1. 00l 0101001 1 lberet esse 1, 01000000300
set a aqul et baqu =
sint (1 = L - - +
- 5
128
aequ - 1
1100101 0 L0 V  l
4000b+81 10
7
256
l
qu0.
256
0000000011
01011000001
000000
M
a
00000014∞ 11
0000000101
00r101 1
6
3001010001
1000001100

## Folio 126v — 00068124:0254
# The same scan as fol. 125r (00068124:0251), where its text is given.