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# Piece: https://leibnizlegible.com/api/records/962/text
# Catalogue record: 962 — [Aufz. zu irrationalen Integranden]
# Incipit: Cogitaveram olim ad irrationalium quadraturas pervenire reducendo ad
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=962&katnr=58543
# Work: https://leibnizlegible.com/work/00068049
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 4
# Folios: Bl. 86–87 (from the shelfmark LH 35, 6, 4 Bl. 86-87): 4 pages, canvases 173–176, page ids 00068049:0173 to 00068049:0176
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068049&page=173
# Lines: 173 recognised on 4 of 4 pages, mean confidence 0.83
# 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 86r — 00068049:0173
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 87v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068049/jpgs/default/00000173.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 33 lines, status machine
son sucedul csta CX C epta ultima paguua
logilaveram olim ad irrationalium quadraturas pervenire reducendo
ad aequationes rationales seu extrahibilitatem per explicationem
obtinemt. Sed adverti posten id fore difficillimum, et haud scro
an semper possibile. Sed via anagogica sese melior obtulit
primum ergo si  adsit n si plures sint partes valoris
unumquodque membrum,
Summandi, redurantur ad communem un quaeque pars summetur
separatim quasi solanstaret. Nata
Si Membrum aliquod
habeal nominatorem irrationalem is fial rationalis multiplicando
quod semperfieri potest, ita si plura oriantur membra ob Numeratorem
plura membra habentem, unum quodque membrum rursus se paratim tractetur
habens nominatorem rationalem. Si membri numerator sit dx
irrationalis, erit vel radi communis, vel radix universalis. Si sit
particularis
radix Universalis, ifficiatur ejus extrahibilitas. si fieri potest
verlo g. sit mnt nX + XX +  ptX V99+ rX +XY
ubi m, n, P, q,V, datae;
b+∞+Dx + C+ X
fiat
Vm +nX +XX + Sg+ DEtSX+tXAX
b + cx + dX +  C+ fx + gX.
quidem cex, terminos vero comparando octo, quos tamen
s
gonnihul augere licet, explicatione dat ipsius X
ed misis hunc radicious a universalibus, consideremus
particulares, ul 4+ pF + sx + tx+ 0. Hic non vidlo
Hic si duae essent radices aequales divisor aliquis esset extrahibilis, adeoque daretur
depressio. Verum sola Auctione radicum non possunt, ex radicibus aqb
in aequalibus fier aequales sint enim X + a est Xtb radices fiat X= y+ C
pet y+a +C et y+ b+C. sed ita non possunt fieri a+ C  et b+C aequile
sut suppono a et b sint inaequales. An ergo sic licebit procedere

## Folio 86v — 00068049:0174
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 87r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068049/jpgs/default/00000174.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 45 lines, status machine
in Pacto ex X+N. X+b. X+C. X+ C
X +a et X +b radices fiat X =  y + nn et fiet
+mG + RN et my+ nn
+ ay+ pa
y + p
b3+ pb seu radices erunt y + untpa et 94 HNF pb
inta
fiet nntpa, :mta = un+ pb,: m+ b, adeoque fiet
num, + nnb + pam (+ pab) = (numt una + pbm (F pba
et sit nn = Pm, b-a,: b= a et fit nn = pm sed
tunc fit X = mytum seu x = m. Videamus an sic
y + p
liceat X = myy + ny+ p,: yyt qqy+r2
fit + myus + ny + p in
+ agus + ady + art
my + ny + p3
by + bgG + br3
sed tollere licet
yt quae pro-
veniunt quia
lollere oportetet in radicibus vel summm vel ulluum terminum
ande
quoram utrumque supponeret esse a = b. saltem vicleamus
an licatulium. Videamus an saltem efficere
liceat ambas radices extrahibiles, fiet
N F2nnag + uqqu = mph +mar + ap + aarr
en+2nmbq +bbqqu = mp + mbr ZBp + bbrr
n-mp= ua Mr+p Lunga + rr- q9 aa
N-mp-
Mr + p - 2nnq b + rr- 99 bb
Id qua ineo quod video obtineri posse neque enim inde sequitur
ut neas ex quont quo quod a et b sint aequales sed solum
quod sunt radices ejusdeu aequationis. ope harum duarum
aequativum tollamus m, fil m= n + 2nnaq + auque = ap - aarr,: p +atY
p
fit (B7
Znnaqp+ aaqquep- ap= aarrp n brf 2nnagbr aaqybr
apbrr aarrber = n p + 2nnbq B + bb9q p - bp  p3- bbrr p3
Sed quia restant
ntar At2nubus art boggarL bparr- bbrrarr
duae radices necesse est tertiam quoque esse extrahibilem, et item
habebitu novus valor ipsius M. ubi tantum t pro a vel Doccurrit C
Et iamvementus duns ex  eris ra hests qualua literis n, P, qra 2taque
et m
m+ b

## Folio 87r — 00068049:0175
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 86v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068049/jpgs/default/00000175.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 44 lines, status machine
itaque videfui saccedere haec Methodus ad reddendam explicationis
pro parte
ope exharibilem
V X + . X erc. faciendo X = inyy t n y + p, + tqu
idem videtar succedere pro _ Vax + bX G quae cota faciendu
extra ribilis
m areme nominatore restat in extraribi
et, ob imparem numes hdimensionum
latas ergo non debet et in nunda numeratore restare.
Ut ergo molius calculum subducamus, patet ad ex trahibilita
tem formula X +.. XCrc. opos esse tantum btans
aquatiomau comparatitiis quatuor pro X.. nibus pro X una
quamvo Et ideo pro X ascendi debere ad y
perfecta extractione super omnia opus est pro radicibus
hiniversalibus. Et hauc puto unicam esse viam
oc  perfecte resolvendo
sit ergo formula quae eanque ex qua extrahenda sit radio
quadratica. Ea sit dimensionum e
assumatur valor
ipsis X, in quo y sit dimenssonum n habebimus arbitrarias 2n +)
opus autem habemus  literis e aequationibus
seu literis C-1 si e sit par, vel e, si sit impar. sed si
ergodebec
omnimodam velimus extractionem ut pro radicibus universalibus
oportet es in casus e imparis  facere denominatorem ipsius
Naloris Y extratibilem. seu non ero itaque in numeralore
sunt arbiliaris 1 = I in nanipato nominatore qui par erunt
nF n: 2 =I et fient arbitrariae n - 1+ n. 2 - 1 seu
Haec alia quoque Mechodo iniri possunt sit 
C
beune
fiet 3n = bG:2 =
.. etc. formula, ubi e par aequunda quadrato, op est aquation  radice
cujus  duensionet sunt l. 2; opus est ergo et opus est explicatione ut sit X= 
e
v
12..
apu et aequationibus saritibis e  haba vero s Ascenditur ergo ad Himentiones ne
2 Ca
et aequandum est quod prodit, quadrato a radice olihart dimensionum ne–
et prodeunt aequationas comnparatitiae ne-1, habemus arbitrarias ne: E (=1) + 2n( 1
sit 2nC -2 = ne + an, seu ne = An + 2 siu e= 4+ 2 iaqu num
habemus salis apbitamum Et. male suppossin in radicibus rex n reddendis exra
subilitus. Ino quans radice sufticere undi huberariam

## Folio 87v — 00068049:0176
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 86r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068049/jpgs/default/00000176.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 51 lines, status machine
tm ficas
X + bX +C sit X
In sic dxo X + C fiat X = (F 1:  M + B
 d = aXV dx = die 3 dy yy: (2m+ nist
dx xtC
1
+ nV -7dy yy
simc
m + ny 3
12)m+ny
vel sic dy Sic siat V = LX + e: XXX + ma
fit LV - = XX + mad 2hX  = 2X ∞ hX +  dx
et
.7+C=
datu Ergo
summa
AXF ma
+h XX + l3.: XX + ma,
XXFma 2EX = hx (32X (hXX + 1.
+ C XY 4 CiMa)
dx
il Xx+ma
ema
Xx + ma
haes reduci pt ad hoc unum ut solum detur sum
poces patet et hanc dari
hil llXx+ma
a
X+9. X-q
e
X-C
D: Aqq
 O
2
sit ma = -9q
7
X+9. X-
4
X+9. X-9 X+G. X-9
X+4. XC
14
q
4-0:44-
a
∞14.
XAg. X- G Xtq. X. 9 X+G
2
et 9. XG.
t
oi
C-