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# Piece: https://leibnizlegible.com/api/records/60689/text
# Catalogue record: 60689 — [Aufz. zur Integration irrationaler Integranden]
# Incipit: (1) y = 10√[x+m] + 11√[x+n]
# Date: 1695
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60689&katnr=57676
# Work: https://leibnizlegible.com/work/00068197
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folios: Bl. 137–138 (from the shelfmark LH 35, 13, 2a Bl. 137-138): 4 pages, canvases 273–276, page ids 00068197:0273 to 00068197:0276
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=273
# Lines: 340 recognised on 4 of 4 pages, mean confidence 0.79
# 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.
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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 137r — 00068197:0273
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 138v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000273.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 111 lines, status machine
(1
y = 10 LX = m + 11 Ex+n
gy = 1010m + 1. LK = 10. 10 F 11. LIX +2 01S, X+ m. X+n.
sit v OX+m +2 EXEn, et VV = 202om + 20. 20 X + 2. 20. 21 Tx + m, X + n
31. 21n 21. 21.
Et  qL 30 sxam + 3) IX+n e ZZ 303o m 4 30. 30 X + 2. 30. 31 b, x+m, Xtu
31. 31 2,, +31. 31.
sit jam a+bgy Zv0 et etRyg = 22
ubi a, b, c, f sunt data
Ex aeq. 7 per 2 et 4, ut omnia fiant identica seu X maneat in determinata fier
a + 10bm + 11 bn 2 26m + 21n et  10b4 11b  20 +21 et f 10. 11b = 20. 21
Et ex oq. 8 per 2 et 6 similiter fiet:
(14)
(+ 16fm + 1fn = 36 m +31a et 10f+15F  30 +3 et 10. 1f = 30. 31
2
Ex 1 fit 21= 10. 1 b: 20 unde ex consit a FYSbm + I1bn= 2omet 10. 11B 20. Et ex 1opr 15
Et ex 10 peris  (2t: 10b + 1 b Z 2d + 1 10. 12P 20. Aequatio 17 multiplicata
e
per =n addatur ad aeq. 16 fiet: a + 18bm-n= 28 m-n osimiliter
aeg. 17 multiphcarta per - m addatur ad aeq. 1b. fiet a + 12 b n- m L 10Vb n- m
21
i a18 fit 20= a:mn + 10b et ex 19fit = a: man + n b = 10. 1 b. 20
(22)
Cergo per 720. 21. fit = a: m. n. 4 11b= 10. 11b: a :mon  10b d
(24)
sit  a iman (per conpend) = p et fiet - pp + pb. 112 - pb 10 (+ b 10. 11) = (10. 1 b
23
63
a est duarum
num
1d
Pari jure et H - 10 0 = C. f Sed pari jure 11. 11 = 10. 10 = C: fm- fn. Ergo fit a: b= c: f
eris
i sunt vianeta
quod est absurdum, et contra hypothesin, quia a, b, c, f sunt datae imo in easu talis proportionis evanesceret
indeterminata. Ergo haec via succedere non potest.
interim notandum hoc theorema: si  pro y sic assumi debeat
Si pro y sie atsumi debeat 10YX +m + 1 extn ut a fbyu aequetur quadrato
et
unde patet non posse ta aliam
a, ab(a 20X+m +2HXIn) fore _11. 11 - 10. 10 M - n = a : b
el aequari tali anto, velut C+ fyy aeq. cgto (a 30 Trtm +g1VXFN)
20. 20 =, g; m.n + 1010b et 21 fore 10. 11 b: 20
nam fieret a: b = c: f. Ergo si formula C+ fyy seu  af. b + fyy, coincidit priori a+ byus
m
talem formulam ad hui
martiplicatae tantum per  f: b
a +bgy aequari conto a 20 exam +21 sx+n,
et et Lun aequari onto a 30 sxtm +21 tran
indefinite,
potuissent absolute aequam itri vationati
a numeris rationalibus; quia X+m et X +n
simul possunt aequari quadratis indefinite.
Video igitur confugiendum esse
nanc nt
ad  compositam ex tribus irrationalibus
sed quae in se invicem ductae nunquam sub vinculo
contineant plus quam duas dimensiones. Ita enim
idem fiet in lateribus quadratorum 
quo facto ilti na et entractiones
si res processistet, seu si simul potuisset
quod sufficit ad scopum quadra turarum. Nam
formula summandam constabit ex Hc partibus,
varum quaelibet per se quasi sola esset revo-
eabilis est ad summationem  quantitatis.
IX = 1C L. _X F. 1
yy = 1eM 10X
possit explicari. Rem ergo tenta
Y I. X AR
- 210. 
Or
3x
Da Rorkum +D. Exin, ut pauso antet
_ a 30 l.. +3) f... ut dite
3c
 +byy = VD, 32C+ Z2 fX 1-2C
22c +28b10m (I1. n
= 2 0 I + 2CZ
re + 22(1d 1 + 32f10r2 1,
22b(0 + 2261 a 262
rationalis, etsi tota formula composita per nt r
rationale non aequetur indefinita non
bimus pagina sequent
s
12 p A It X
p 23
A
sed video 2
22 et 72
quae considerali
possunt t
22b. Z. 10. 11 fF. 20 2, Bluna
32 f. Y. 10. 11.= X- 30. 31
nhil
Adese
Ex 5 sit 2Y- L2b1c.1 e 20 CF tinc p7
AE
e
fit _ B10  b11 AEB + D 1Y b: 20
Ex 7 etst sit
3
C R + bloME L bi n Dom + 10= 1, VH. 20
multiglis etut _ 9. .  P  = 1. et pred iunstio
addetur ad aegq
+cetL 21. et, et
AD
n V10 + a +b10 M = - n
 20m Vt nubetur q es
ractio. 10 h per A et 9 habi tur
alter 20.
abc abd, acd bed

## Folio 137v — 00068197:0274
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 138r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000274.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 84 lines, status machine
SiQuarritur ( san bgus
C+ f yy
et earum ope porro ad ratioales.
dy
reducenda ad quadraturas conteas
sit y  1osxim, xen + 1 Sxtm, rp + 1 TXEnKIP.
as
12. 12
G 21.1. B D, 1. 12 nb
Eodem mods  fiat x et vV pro 10, 1, 12 porendo 20, 21, 22
5) Z et 23
30731,32
Debet esse a + B a L bgy = vV et 1C+ fyy 2Z
fiet a + 10. 10 mubt 11. 11 mpb f 12. 12 npb = 20 20 mn + 20u mp 4 2222 up
10. 10m tu b + 1. it o + p b. + 12. n nepb ( L 20. 20 men + 21. 22 m + + 22. 22 n+p
d
1. 10 bt ambF1. 2b= 20 20 +21. 2 +22. 2V 18. 11= 26 21 191 20. 22 112 lu
Habemus ergo ex aequationes resolvendas similiter prodibuntaliae sex si velum
efficere arq.. . pro deunt ergo aequationes resolvendae duodecim. literas vero
arbitrarias habemus quidem etiou 12 din spectem, sed vevera pauciores unt quia
ea
itaque remiser
redu ad pauciores reducuntur en enssimenter incraremur
sim Y, V, Z valortu, pro 10 scripsissemus 10+ 10) et in laeteris simedtm incraremur 
praestire non licebit. atbitiarius, et quia
edo copparatitiae, ita in sil
a unam et
t
litiae
abe
eodem redit aquenue quadrato firmulami atbusutcg? quod ea fiet si faciamus.
15 15 ut
 itaque potuis aquemus quadrato suoductum exe a + bgy in et fG9, vel quod
atdt coga
y= hn u s ip + _ 1B LXAM, eta + 1A Sxtm, XEP. + I LXF nxIP
y9= 1010 m 1040 X
13. 13 mn
inte
1312 m
1414 m
15aG 
 et quadratum sit a latere lujus quadatoquadhatum assurget
ad. X  rationalem per X) et X per IXam, Xt Et et smiles
Habemus ergo comparationes X, X, X, X, X; et X f quod sunt 5;
et _ ter X. Ju., XI.., X s. quae sunt 9 in summa 14
quadratum autem excitab v0 (= arbusrtcy) excilibimus a latere
D = _ 20 SXIn, XIn +21 Lxim, xip +2t Xtn, Xep
31X
sed ita prodeut
diterae coparatitiae tantum 12, cum nempe 10, 1, 12, 20, 21, 22; 30, 31, 32 m, n, p.
rus 
praestat ergo rem omnem reducere ad Muthplice
sinfimi gradus
aqualitates Et primum quidem quia duas jam habemus aequevalas
qua formulas simplices ut  veluli Xra et x1b, jam simul quadrate aequare
possumu, reducemus Triplices ad lales smplices
Ergo
e
sint X +  X + a y+b, _+c equanda catis. Nam
et ipsum athyu +cist aequare eto huc alibi reduxi.  sint ergo
quadinti vD, ZZ, VY, nempe y+ a = vv y +6 = ZZ et +C= 
sit y = 1aY ( id + AX +2O XX + 2 et eX Lrrm, Nin + 3IHE SXAm, Xrp + aD FX (Xtn, xIp
sit D = 12 tX I Y SIVSLS + C0 AI + UM + 120Y_m n + 13oCXrm Nip + 140 SxtR, X+p
vD =  1I. u2 F 2. 119. u+ X + uD MDXX +Z.  Bu Uxm em + 2 1. prAux Sxemxtp +E. II. 2. NX Exin, rp
2.Ggego Uorzex
210 n. 2 luo
2. 120 2. Margox
sut conparaies ter novem, id est 27. Arbitrariae vero in y, sunt 12, in ipsis V, Z, _, sunt ter quinque seu 15
summa etiam 27. Quod tamen vereor ut sufficiat, quico versih quantitatibus inrationes
ter quol novas lucrate sumus in yuo iolidem prodeunt novi termini
Et ita videtur res laudem futura in rolestate. Ino falso
mminuitur numerus literarum. sed si yfacias 14 +13X + 12X + uX + 10X F 2F ZX Y2OX Nxam pra
et ita porro, lucrabimur in y arbitrarias S,e V, Z, V, in qualibet 4 sunt in summa 17
novae arbitrariae. Sed aequaciones
fiet gy 210. 10 X  1010 in X A 1B. 10 mn + ZIBNIX + EIBAM SKEAN, KAP +E. LBI2 X +Z1AIE N Lnm ip tE Mt enre p Eem Eta
11. 11.
I.ne
s
qu revocando ad  ratiores
geneuliuest
12 p
e
Arb. Req
44 42

## Folio 138r — 00068197:0275
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 137v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000275.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 74 lines, status machine
ea 1B.)
y 10 spem + uxtn +XFp yy Z 16XA Iom 4, 10. A LXtm Xtn Y, 10 12 Sxra, arp 1. 11 12 Extmetn
L=30
31
Et. 1e p
aequationes comparatitia  decem, cum arbitrariae adsint c2
o +Nyta astbre a + bo, 10 nitrc
C
11. 13(41)
∞, 10r erio b = 20+21+22 10m + en + 12p + a = 2om +rin + up. 10.1 = 20. 21 101 2022 
sineslvamue Ex aeg.  11, 10, 97 habebitur 20, 21, 22 tollentur, ne 22 11. 12: 21= (ex 10) 10. 12:20
ergo 11. 20 = 10. 21, seu 21 11. 20: 10  = (per 9) 10. 11: 20. Ergo fet 10 = 20 idemque firet
in caeteris, quod est absundum et glan ae qu. 7 et 8. videam, an hoc sufficiat, ita enim
In caeleris quad est aljurdum.
sed videtur res successuro pro casu triplicis aequilitalis
pi nempe y + a, y+ b, ytc, debearit aequari quadratis vo Z p  respective
9+aLvop ytb et y tC =  p
Nempe sit y- 10 exem e, t x+n + V
a
nempe
Nempe sit g= ta ot ri 1oram + tXin
a
y = 10 SXIm. XIn + 11 S ptm. Xtp + 1 Sx +n. XIp + 13X + 14
4
v= 20 + xim + 21 Sxtn + 22 Extp
(5
Z=, 30 Ynem + 31 YX+n + 32 YX+ p
= 40 sram + 41 Exin + 4u extp
20.20  F 20.20 m
21. 211.
22.2.
Vr= 120. 21 Tumgnm + 2. 20. 2 tumxop t 2 de Sumnip. Et esdem modo se habet Z et 1
Hinc 2. 20. 21= 10 2. 20. 22 =11 2. 11. 22 =12. Et par est
10d
Et quia pari puie se habent 30, 31, 32. vel 40, 41, 42. ad 10, 11, 12. Hinc
15)
(14)
fiet 20. 21 = 30. 310 f(2 40. 41 et 20. 22 = 30. 32 = 40 42 et
12
21. 22 = 31. 32= 41. 42. Porro ex 49. 12, 14, 16, eoque et ex 13, 15,17,
(17
(16)
(20)
21
fit 20.20 =30 30 = 40. 40, et 21.21 = _ 31. 31=41. 41.
19
((Na 20. 2 = 35 31 Ergo 31 = Nam ex 16 fit 32 = 21, 22: 31. (Ex 14) f = 202
Ergo 21. 30 = 20. 31. seu 31 = 21. 30: 20 (sed ex 12) = 20. 21: 30 Ergo 32
Ergo 30.30 = 20. 20, idemque in caeteris; cum par sit ratio
22
Hiac possumus facere v = 20 X+m + 21 X+n - Z2 IX+ p
Z= 20 (xIm -Z1 YXIn  + 22SXFp, et pL2otmtm 
Hinc ut resol ita satis factum est aequationibus 12, 13, 14, 15, 1b, 17. supersunt
25)
equationes pro jam patis faciemus et aegqe 9, 10, 11. Est ergo exhii est  = 12:, 22t=
ex 10)  11:, 2.20 Ergo (ex 24 et 25) fil 11. 21 = 12. 20. seu
seu 212 12. 20 : 1
2
unde fit 20. 20
Ergo 21 = 2bc: b = a: 20,
t
sit d= 20 trim + 2 Sn tE S ta+byu=vo e+ syus = LZ. Ex 5 et 6 explicatis rascuntur
Ergo 20. 20 = ab:c
(ex 9) 2. 10. 20; unde fit 10 12 = 2. H. _ seu
210 et2 2. 10. 11 : 12 et 21 et pari jure
30)
21. 21 = l2. 10. 12.12 et 22 = 2. 11. 12: 10
20
X +c
Y v = a + Te sX + M. X LH.
21. 21 n
22. 22 
20 d =a 20. 22 = b 21. 52 =c
22 = C: 21 = b: 20. Ergo 21b.

## Folio 138v — 00068197:0276
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 137r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000276.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 71 lines, status machine
y = +10 (Xtm + u YXtn F 1 XEp + 1 (XEm. pta + 2X1 m. Xp +18 (arn. xep. CG
15X
F 1gX  +1g
261x
 e
x
 = 20 Sx+m + 21 SRt + a Yx 1 et 23 Sxe et 24. Sum Xoptas saen ip ra
fit vp = 2.21. 23 plrim t2. 20. BM Sxen +2. 2024m Spep +2. 20. 25 Lxem. xtn xrp cru sr 
2. 21. 14. 8
12a 20
2. 2225
2a 21
2. 21. 25 n
2a22
un
21224
2. 22. 23
s0d video hii ascendi ad factum ex tibus sub vincuto contentis
et liaet tolli possit prostat tamen hoc vilari. Sic ergo melius  sumplit
litera 5. V =
DZ = 30
2p = 40
16X
2. 25. 25M Spt mex ip
2. 2424 sxtn. xF
2a23 tampty
24. 24. lxrm. x+p
Ext. +p
+23+ 24X Etsliter  et Y
10 sxam, ran + 1 Sx+ m xep + B2 Xtn, Xtp + 13
ctX
20
f
er
zao
120. 27
9= aGXtm, en +D (Xim XAn AE SxIn, XIp + 1B+14X L15 XX
 y =
cit
22
ta
+ 14 XD liteuc q con prntiti)
+ 18 xr
23 + 24X. Et similiter Z, et 2 per 30
33
Literae in summa 4+ 3. 5+8
34Comparatiti
43 44
aequaliones 3. 9. utrobis
Aant 27.
B=22.  sed videter bienior fore calculus si xR a + byy +CH deteamus aquare v0
2. 20423 quarati
r
et in gy ex labit 1. 4. x 4. . X + a . X s. .. + a. + . X su  . 4+ n +. R S  + A.-. +. .X.....
et quadibet et est = - - d. K +.. X +. X 4+ . x + _+ X + ,x 4. x S. .. .. +. xF . pF .  ... 4. + uxr. pr. X ....
speu sit v=
yy .n + 
ComnaBmX L1CX
lmprrchiax i 
1 ersinx 10x
32X
pA +2IX D +Epn +2YuX 8
21b 12, +2. 1012X 
210. 111 =210. 1IX
Revera videtur catculus fieri
Paculo implicatior
Et ideo omnia ista quadrando assurgetur ad X, X X. X. 1
Et, postea pervenietur ad X, X. X. X. sed revem camen puict culculus
videtur commodior licet cretrioribus incognitis sit opus.
ieo 29 trqux.6
~H2 p, EII IX.