# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/27745/text
# Catalogue record: 27745 — Leibniz und Tschirnhaus: Rechnungen und Zeichnungen
# Incipit: x_3 [... qx + r]
# Date: 1679-10
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=27745&katnr=56782
# Work: https://leibnizlegible.com/work/00068206
# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 4
# Folios: Bl. 11–12 (from the shelfmark LH 35, 15, 4 Bl. 11-12): 4 pages, canvases 21–24, page ids 00068206:0021 to 00068206:0024
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068206&page=21
# Lines: 259 recognised on 3 of 4 pages, mean confidence 0.66
# 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 11r — 00068206:0021
# Source image: https://digitale-sammlungen.gwlb.de/content/00068206/jpgs/default/00000021.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 80 lines, status machine
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## Folio 11v — 00068206:0022
# The same scan as fol. 12r (00068206:0023), where its text is given.

## Folio 12r — 00068206:0023
# Scan: The same scan is also registered as fol. 11v; its text is given once, here.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068206/jpgs/default/00000023.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 130 lines, status machine
X xqX +r
XX ∞ qXa.
aX+btg3X
2
2
X X H qX +r
ax tozue 17X  Ab
14X +r
axt
ax +ZX
yg
b
axut bX + yX ∞ 4X + Y
ax + b Fy + bX + yx ∞ 9X +Y
ax - gX +
er axtoty
+ b+y
axpbx2
X3 x axp 467
XXX
yy x X9 ta
bX  cy+ d Z tC
Xxge tr
yx ∞ +r - Z
t
a
ra-Y
V7a) +r79 - rZ
raq Fraz
XA
Z
a - 9+ 7
a +9+Z- 2ag + 2a7-29
Sonse a atia a ra r
a-4+
=.
Y+c.
c t
2YZ
erzue
Ya -YZ a - 9tZ
+Z, a -9t
ra= raq +Z4 - rZ +Zat 39 +2
- 2Za9 +247-
12
t
XXL
a - r
X x
Xx a +B
X x a-
a
b
+6+c
e
c
a -6 -c
aa
tem
a- F.b- 1
xa
X
1 qX + Y
X X 
aX + Z
ax (+ ZX
axtaZe +ZX 1 qX + Y
s
Nit hinc
demonstratia quod non detud formili radicus
Cubicae imnginarias habentis, sine imaginaris
sit aliqua  a +F- b + Y a - (- b p c. posito 
Cesteatiam expressionem sine imaginariis, fiet.
Yar lb 1 C - 3 a - f- b B C - v
Ergo
Ergo 2a p, C - 2CV F 3CY. seu v - CD + C7 22
X H qX+ raysut XH bX + CX + dX + C
4
X
bxrxen dure
XX c
bx +CXFx  ex H qX  X sxt
b XAbXrbdxabe qbX qcX qdxtqC
XH qX + rx + SX + t
bx
Vxit β ext
at L. 6), C 2C J a- Lib + 3e a -b- 2ar. b ( (+ fb) atr. b
seu 2a 7 c - 3C + 3C V
a (t s- b) 1 C - 3ev + 3cv2 - v
= a(t s. 6)
X’H
X H bX + CX + dXFe
H bX + B.
X a - bB - x) X (seu X H
X p aX rx+ st
6+ c-9
X bX + CX + dx+ e
X5 bX + cx + dx +CX
X+p X   q X + YX + fX + t p  
- d C
B 6X Ap X L Bdx + p C
X - qX + r p d
CnX aH =
possibil.
seu v2- cB + C p E - C + 2a p = C + 2a. Ergo  + v +C Y S-C + 3a (Ego si 34 pE erit, v quantitas impossibilis aequalis ipsi 4C, - e42a
BC
12 3C
quod est absardim imo generalius adhuc ipsa C- 3CV + 3CV  2a cum requirat ipsam V solam ad constructionem eut possibilis C. determinuta per po 2C
di imporsibilem Adhuc talilius forre. ponatur a  DC. fiet C - 3CV + 3V  EB. Ergo C- ct GV 1 9V - 3V +2C B  v + 2C. Ergo C - 2Vp S. 2132G
sed omnium optine sic. quia jam =V + = B V. C +2 a et v est imaginam, s
Nilcil est
pt = CP+ CBH ZaV Madirembubicae affectae non posse exprimi sine imaginariis melius ita demonstratur, sit radius cubica non imarinarie expossa c. ejusvalor,
a - f- b
3C per Imaginarias Ya + (b + _ aequ. C. ubi una eademque imaginaria. compendio ea vocetur v. fiet: V +Y aequc. et Vv + Yaequ cV. e virvu
Sai. b
seu a, +S- b q + YV aequ CVY, pro vo valor ex aego 4. fiet a + 1 - b f YB aeqo ccp = cr. Ergo
 grecdinset tu prcolatias goen  se nas ri    p s sumtitasi n flliaribu
+
h = (bs- aequ pa +9S b, seu n. - 3bH
qu
q- 3Ch + b
eqt
ZH
sem an qx, SY
Concludendum quo
gaper
quoritur an X   senam X r
seu an X HAY. seu an qX. 19 4r
possibili. quod est as absurdum
X B aX +Z

## Folio 12v — 00068206:0024
# Source image: https://digitale-sammlungen.gwlb.de/content/00068206/jpgs/default/00000024.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 49 lines, status machine
X3 + y ∞a xyx b
X3 +y ∞ a Xy3 x b arod
 +y +2) xa xy + xt +y2x6,
N +9d +F t a
XXgy + XX22 +7988 ∞ 6b
Exryt +2yyxt + At x7
2xtry
xx +yy + 27. ∞ 8
Xyt xc
X +y 4 + 2F + Exrygue + 2e + 2y028 x bb
XLy +7 + +u xa
Xayy +Xx1V +y92t x 66- a
4,= +1  L yt 2+x
2x2yy1t + 2xxy et L. N yy 24
yy
2x + 2y9 + 27 26cc
X 62. 2abb + ea- ibca
ry + xt +yt +y2 +yt +2t x 6
x + yy +22 + et 26
X+ xy
a + b +C + dp
ab 5 X
X + xyt + xet 1ytt x
X + y3 +2 +C) x d
= 5XZ Y
ab  y
et
(ab. 1 y - 58 Zy
et
x a +a
n
ab C , Z
a AD
etC
ab
ryo
xxar Xx a +b +C X xacbeCI
X 17 ab y
ba
ac
a2
cC
ab2
e
et
yHtxd
a
V 1, X = 21
VY, X=V