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# Piece: https://leibnizlegible.com/api/records/980/text
# Catalogue record: 980 — [Aufz. zu Wurzelausdrücken]
# Incipit: Sit formula irrationalis, y = (1) ²√[ax³+abxx+accx+ad³] reducenda est
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=980&katnr=58561
# Work: https://leibnizlegible.com/work/00068051
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 6
# Folios: Bl. 16–17 (from the shelfmark LH 35, 6, 6 Bl. 16-17): 4 pages, canvases 31–34, page ids 00068051:0031 to 00068051:0034
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068051&page=31
# Lines: 244 recognised on 4 of 4 pages, mean confidence 0.80
# 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 16r — 00068051:0031
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 17v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068051/jpgs/default/00000031.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 77 lines, status machine
Vinit adn
sit formula irrationalisus = fax tabxx + ux + reducendues
ad talem, ubi sub vinculo indeterminata non ascendat ultra gradum rdm
explicando X rationaliter; ita nempe at ex non sit irrationalis. fiat
pu si posset tolli temminus unxi maximus res esset confecta. Sed tunc oportet
valorem ipsius X habere inde terminatam in nominatire de, quo max. pitum
ergo sentemus collere duos terminos infimes sibque X = inZ tnn: a
et fiet ax = nZ3 + 3mmnnzZ + 3 mnt Z + n6: aa
yy B2h
accx =
ad3
+ab mnZZ + 2 ab mmZ + abrt: aa
+aacc inz + aacemn; ac
+ad: aa
Verum hinc patet posse quidem tolli terminum infinium, per n
sed hoc sublato non est in pte ut tollatur penultimus per m,
quia m in eo evanescit
faciamus X = mZ + nn: y+ p habemus nominatorem 1Z+ p
in balore ipsius y. et ita non habemus intenta.
si la
quod si mulliplicemus numeralirem adhuc per Z+ p
ut videamus, an licet enim sic assurgatur ad Z, vider tamen
potent an liceat tollere duos terminos summos
sit ergo _. X = mZ + nn: Z+ p = 0: d, fiel
auyu = a O + ab O0D + acc OD) +ad 3
= a CD + ab 00D) + acc O V + ad 3: 3)
quaeramus teminus summos in a0D est am Z + aBmp
in
+3 amumBZ.
ab edD est, abmmiz tabmmp
+rabmun 33
ac cOD est facemZ Haacemp 23
in ac ad s est ad
R
2
+aaccnn 13)
93
+4ad B Z et X d = Z - nn. d,:, Z-
Ar est constans Ergo hoc casu res nonceder
Ut ergo autios terminos simul tollamus fiet m bmm + cc m + d =0
et p L am + rab mun 1 aaccn: et - p = Sammun + 2 ab mtn pa ce 
e
regulanter,
Hinc sequitur fimmutam quamvis irrationalem simili Methado
Es 19
1
deprimi posse ad inferiorem, et hanc rursus ad inferiorem, atque ita hinc sequibe
formutam quamvis posse ubbi extrahibilem. Hinc sequitur et compositas
reodi posse itales sit enim exempli causa
YEX + Px+q
V  YX pX∞ + qXar + Y aX + 1Xx + ex +d, reddatur primum
una ex partialibus extrahibilis, deinde et altera fiet denique non formula ratio
ncet  lactu suceta i qua arqur petato aotrum tinti
icet.
:aa
O: aa
O. aa
d. O= numZZ L2munZ+n
 2
D.
-Q13piZ3php
ID = 22 + 2pZ + B p
D-
2
2
D- 21 a p ebppi
+p
si estet b4c, -0, finet
u
y= Xa ida
et m = dH et  mm= dd
et M = d3 et fiet
p
=p= 3add Lad + 4d3 
seu p = = pn. d
et x = -dZ + nn :  - nn: d
am + 2 ab mm + zascemtad

## Folio 16v — 00068051:0032
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 17r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068051/jpgs/default/00000032.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 55 lines, status machine
X + bx +gCCKX + d X + e, non fuisset
opus multiplicatione per D. nam nominator per se fuissel
extrahibilis, ex his ,olet ejusdem difficultatis esse
reddere extrahibilem
 YX +eu et (x + et
Haec aitur Methadas est universalis pro extrahenta
ergo, XX + OpX + cc = 8: a. Ergo res praestarpt per
extractionem radicis Atici ex B.
sat si milubia cxtrahenda ex X meluodo praecedenti
etiam posset reduci ad X simplicem
posset etim sic procedi in YaXX + ap X + acC = y
ut tollamus tennum ultuum fiatus qasi err A pX_ = y = ZX,
sed missit nunc ubicis primum de quadraticis
fit in exemplo simpliciore ubi si res
succedit, poterit etiam in caeteris praestari
  X + B 0
sit X = in Z + nng: B Z + p
43 m+Z + amnnz + 6mmn ZZ + 4mu z + n,:, Zrap Z+ opph a p 
et (ext per 1 et 3. fiet y - 2sm Z+ + 4 m H Z + 6 mm n Z7 + amp Z + nd
Sin Ver b +4Vp. +6b pp + ab p + bpt ilpp
as
seu m = b Y-1
ut ergo bollantur duo terminis summi fiet  m =  - b+
jam seu p = - mnn, mb (= B -nn, in b: b. Y. b seu fier
p = nn: Y- b seu p = un: m et pp = n: mm su et fet
y = 6m mn y
omnt y
ita video evanesere omnia
F1 stet
et 4 nino y et n m
od esto
destruamus ergo potius terminos ultimos, fiet
A PP  = nn V. 1 : b et m = _ (nE : (1
6b pp = 6bn : bb
et Amnn = +b=i nn a 4bp = 4b nnF1: b n = - b
vel 6mmr = 6bbn 1 -1
ergo fit y- bnneY 30
 6b pp = 6b r -1
3:, Z2 + 2 nn , Z + nt  -1
posito X = - b  1Z + un,:, Z + quitt1. sodta csue aefiriostans.
na did. pr
26bn, U 2+ 4b n  Z+ B 13 . Z2+ 2pE + Pp 
id est 0
m
N5 nec quicquam hac via obtineri.
tiet 6mm nt= 6bb
et
radia (Obeca, sed radice Qvadratica. sed quaeritur po
a X + a p X + a CF  = y
ta cursus habebitur vator rationa
videamus.
C.I
1e
um
u

## Folio 17r — 00068051:0033
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 16v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068051/jpgs/default/00000033.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 67 lines, status machine
Metuebam ne x fieret constans,
non evenire, nam N
fit   Z- vn Y.1
b81
re sumamus securitatis causa.
V = VX + b sit X = mZ + nn: Z+ p
Q+ nn Y1
43 44
X = m Z + 4 mnZ + 6 mini ZZ+amn Z + n
L tap E + 6pp Z+ qpEF pF
+bt. +4b p. +6 b pp. +4(Ep. +b: 77 + 2 pZ + pp
ex plmo
In numertore aq. 4. tollamus duos terminos infimos, et fiet n + b p = 0 seu p = nn 3 -1: b
pursus ex perulbium +nin + 4bp =0 seu m = - b p: n. Est autem B3 = pt: p = - p : br: p
2
catens riguex 1et 3 fit = 2nd Z +grinu Z +6 mm Z Et 4mn Z+ n
Ergo p = = n: (39 = - n Y1: b s1 = - n 7 - Y -1  - b3 = Ex
n:B Y de  in -1 sed Y de Y-1 in Y est
 de, S-. in J-1 = 
2-1 et fit B = n V-,: b = - n: b ,
Ergo m = + b: Y-,  vel = b , D sed malo retinere priorem
quippe immediatius obtentam et tutiorem. ino certissimam.
 Nempe mn +  6p =0 Ergo nn + b p: p = 0 Ergo mn  N: p=o
ergo mn = n: nnd - 1: b Ergo m = b: -1
M + b = 0 nam n = - (+
mnn = mnn:m =  - b nn X1 : b = - b nn V. 2 Ergo nnn + b+ p = 0
b2p = (3 nn -1
Gmran =  n bb = -1
14D
2a
bN= bp. p n p) Ergo nmn + b pp
bpp = st n Y1: bb = bb n 11
sed 101 +  = c
-
Nam f-1 in 1- 1
nun fit vel 2.
= -1r vel +
aq.
bb nt in Y., + F
quod sa a t in Y- 1 ponatur esse -1 frest
-1 +  =
8 - 1 + 1 = 0 Videtur autem hoc praeferendum
quia 12t Y., c seu L idc Y.1
fecimus )-1 subtato vinculo Quid si feci Ergo et haerursus tantum vinculu
-
-1
tollemus an (2 1 quaeritur. sed si velim, quadrando ducere in se invice
2
em
c
q
4 414
fiel: tEs
-, sunamus esse Y1 e fiet mmn = n b: Va et b pp = bb  Y
et  et fiet minn + bB bbn in sE HL =
+1
 
et fieret y = 2b nn Z . 3+ un d :b et X= , Z +nn, ; q+ na . 1. b
Et
 X = Z + nn  -1  + nn d--: b. Ergo nnhil actum
es esti
pan p fecet ostri codta desderata
sed tamen id vedeo
V+1 +
+ 
=0 vel
7+1

## Folio 17v — 00068051:0034
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 16r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068051/jpgs/default/00000034.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 45 lines, status machine
sed nihilominus puto in casibus non nimis
simplicibus adeoque in hoc ipso puius reducto ad X
rem po ut in quadrando feri posse onstat rem posse
succevere. Id ergo reapse adhuc exploranum.
sane si consideretur esse X = inZ + nn Z+ p
et  in cubica gradu esse m+ bmm + cem + d=0
et p = un, +3mm + 2b m + cc,:, m + 2bmn + 3CC m + 4d
utique si x sit constans obortet, ut si ope aeqq 1, 2, et 3
tollamus m et p. simul evanescat X 3 alioqui X
non feret constans itaque calculum hunc generaliter instituendo
scremus quibus casibus reu succedere possibile sit
sit
X nd+ 2bmmZ + 3cCmZ + 4d
sit nn = h qp fiet q-
X est constans oportet, ut sit un: m = p
et fet  (m3 (2bmf + DCc,  + 4d = (3m  2b mm) f CCm)
seu fiet m = (CCm) + (3d = bm cem (d) seu nm- 4 d: b
et um =d: b quo substituto in aeg. 2 oritur deferminatio, seu
relatio inter datas
X9: V + ab ay: infiat vV = dx et vdo = L adx seu
di = 2 a dx sax fiet Pfaan XX + ab,  ax ac
seu fa du =  ax + a x hinc _ b = 0 et d = 0
in form  ax + ab XX + acCX + ad.
et cc = ab et fit
m = = ccm = - abm seu m = Y- ab
et - p = - haabnn  + aab nn et
= aab s- ab + 3aab -ab
seu p- un: Af- ab
quod vitare volebamus
sed ita in illud ipsum inciditur,
si esset x-cot h. foret
faco + 3ahcoco + dahtco + ak3, fiilt man m-  e  ab. H
+a b co + abh
fil (b)= h et cc = 3tk + ab cd = h + abh
et . p =
3a, - ab = 2hS- ab + hh + 2ah, nn, = h + ( ab
Cas, 3rab, a - ab - 2h f ab + th, f- ab -h
aha 
MMl
si sit vo (∞+ ab = vo (∞+ 2orZ + Z2. fit fit vr = ab- 22
ut (v + a610 - ur +Z do
v (ab) (a6) 2V ab + v7ZZ. an ac vv-2 sib iti vi
u
seu v v - 2atab Z: aa -Z
v mle H Vi +bV = 1+h