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# Catalogue record: 168 — De Radicibus Aequationum inveniendis
# Incipit: Suppono quod in x radice aequationis generalis rationale esse potest
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=168&katnr=59136
# Work: https://leibnizlegible.com/work/00068231
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 18
# Folios: Bl. 12–13 (from the shelfmark LH 35, 3 A 18 Bl. 12-13): 4 pages, canvases 23–26, page ids 00068231:0023 to 00068231:0026
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068231&page=23
# Lines: 104 recognised on 2 of 4 pages, mean confidence 0.86
# 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 12r — 00068231:0023
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 13v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068231/jpgs/default/00000023.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 55 lines, status machine
luppono
De Radicibus Aequationum inveniendis
quod
Certum est  in radice aequationis generalis rationale esse potest
id esse terminum secundum, numero radicum divisum ut si sit
tunc
aequatio X + p X + gX + YX + 5x + + = 0 erit pars rationalis
radicis generalis = p: 5. Oportet enim esse X = p + etc. quia
erit
sola litera p ipsi X est homogenea
itaue quod in Xx rationale, est pp- 29 dices divis pers
p= 3p9 + 3Y
divisper
p2- 4p 9+ 4pr +. 299- 45
p. 5P9 + 5pr + 5p qps-59r +S
pono autem sublato termino 2d0 p, esse X = Va +  +Yc + Id et ment
= h + l + m + n = sh  = RC
constat esse XE - 5yX +5Z
eehh
sit hl = y et hlm = Z et hemn= co
+ syH X 5yZ. = _ h
=sld XX - 2y
X3 = 3yX + 3Z
X - 4yx + +ZX L 2yy-400
M
C
Amn
X- 5 yX + 5ZX +syyx  - 5yZ
5CoX
Quoniam autem h seu 1 + P+ m +n seu a + b + C +d sunt rationalis quantitas
ex hypothesi quod sunt radices aequationis quadrato-quadrntriae ravalis:)
Et eren si sintequanttas hinc pateb etiam quod X - syx etc. sit quantitas
Nobis id agendum est ut inveniamus h absolute. Quod utique inveniemus, si
rationalis
sciamus quid sit rationale in X et in 5yX et in 5ZX et in SyyX et in 3coX et
in 57Z. Deinde quaerere etiam debetimus quid sit SCE. His enim rationalibus
in unum alditis debet conflari valor totus, quippe rationalis ipsius h5, et oportet
destrui quid quid irrationale intercurrit.
ex Tabula alibi coepta
Eodem modo quaeremus  h l absolute tantur enim aequationes
pro hl (seu y) XC, h. m tim similiter inveniemus hVM
a
12
et denique s lmn horum autem valoribus habitis, utique habebitur aequatio
quadrato- quadratia pro h vel l vel m vel n.
Verum quia prolixum foret nimis investigare quid sit racale in yxd in Z
altinibe
in yyx in cox in yZ; et ita magis adhuc in aequationibus pro h3l, vel hFm et
Rideo reperi insigne compendium, cujus ope sufficial solum nosse quid sit ravate
co in aeq. pro h,
in X et ejus potestatibus. Nompe tollantur tliterae Y, Z
vel alia; ope equationum paulo ante positarum, et Hh h h  vocando O, D, 4
supererunt in valore ipsius h, solae
X, O, D, E et quidem hae nihil continent rarale
sed an nt facta ex ipsis, et estum potentiae

## Folio 12v — 00068231:0024
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 13r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068231/jpgs/default/00000024.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 49 lines, status machine
Sed res omnis proponenda est distinctius.
er X+ pX gX X+ SX + =0
sit aequatio, cujus radices quin  HE, L, M, N, O quarum quae libet vocatur X.
radicis unius partes irrationales quatuor h+l+ m +n, compendio h = X
H + L + M + N + O = p
H vel L vel M vel N vel O = X
HLMN vel dc = 2
h + l + m + n = X
HL + HM etc seu HL = 9 HLM = Y HLMN = 5 HLMNA. F
HL vel HMvel Adc = Y
ALMNO = TI
hl + hon et = hl = y
HLm vel HLNecc = Z. HLM NoelSC = 2 HLMN- 
hilm = 3
Elium = co
hlumco in veniri potest quid sit rationale in qualibet potentia ipsius 
ut in X, X,  etc.
Inveniri etiam pt quid sit rationale in  vel in qualibet potentia
ipsius Y, ut in V V etC. et ita quoque in Z, 2, et
. Sed non aeque vid etur facile definire quid sit ravate in ipsis
qu
ex h, l,mn copositis seu in y, Z
ie invicem et cum X.
Xx = Ah + 2hl
co, et et horum poteneris atque factis in
h, C, in n = sa sb sc B
ergo quod in X est rationale erit et in 2ab
X3=  + 3hl + etm
erit ravale in 3ab r abc 3h C+ 6hlur
illud manifestum videtur hi, h, tc. nihil continere ravale
jam htrx eliam nihil continet rationaleSa + sb + SC +Sd inta tbd
dat sa + sab
videndum an in generalo calculo unquam aliquid ravale prodeat ex formula
je
tale, eut forma maequaliter se habente in exemplo, ut Sao vel hat
Nota. Quae farimus pro X, e ex p, qr; eadem facere
possumus pro _Y ex 9, seu HL ex HL. HM (cujus valot invenori potest)
ex HL. HM. HN (cujus valor etm inveniri pb quae et eunt HL. HL. HA MC. Ma. aV
ad tactand habendum y crsque Y,
Y1
e
is
loco
Vecum illud in quaestione est, an
liceat inde transires ad y.  seu an possit haberi ontor ipsi,
y et
eigue quadrulorum potcabiar, itat ut coincidant y er Y. vel saltem, ut
quoad rationalitates ut se X et ejus potentiae habeot ad P, q,r, ec. ita se
et et ejus potenliae habeant ad qseu HL. HC: HM, eC.

## Folio 13r — 00068231:0025
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 12v); this page is its right half.
# No recognised text on this page (recognized).

## Folio 13v — 00068231:0026
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 12r); this page is its left half.
# No recognised text on this page (recognized).