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# Catalogue record: 28079 — Aufz. betr. Integration einer irrationalen Funktion
# Incipit: Si (a, x + 1, x + m, x + n) = ay quaeritur a [Integral]ydx quadratura
# Date: 1705
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=28079&katnr=56489
# Work: https://leibnizlegible.com/work/00068207
# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folios: Bl. 87 (from the shelfmark LH 35, 15, 5 Bl. 87.86v°): 2 pages, canvases 173–174, page ids 00068207:0173 to 00068207:0174
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068207&page=173
# Lines: 306 recognised on 2 of 2 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 87r — 00068207:0173
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000173.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 161 lines, status machine
(6.
2(= 31 -00, 21 + 32, :, 31 - 32
22= 32 00, 32+ 31, 1322 permatando + et .. 2 Hinc
22= 98, 31, 31 +32,: 31 32 Hinc per 4 fit 22 = 0036, 31132, : 131-32, vel mutatis signis
22 = 32 -00, 32+31, :, 32- 31, qua qui valox à valore 6 non disfert nisi permutatione notarum .. 1 et.. 2
duae coincidentiationes rationalium ita se habebunt:
O= + 2C + 22D +00)a
HE
fiet -D = (21 - 31) O + 00A, :, 22 - 32 = (2) -00) 31 0 4 00, 00V, : (22 - 60)3 resi dud equatio etandu Ot erit
310, - 32D
3=6033LO A 22700. 80Y. Unde duae concididentiationes
una constantium altera variantiur
Ex constantibus fit 22.32) = 000. 32) = 2 00. 22V - 00. 32Y. Et 220092 (. A), 32.
Et 22 = 60. 32 (R- Y) : (32A 0  00. 32 (V - 5) : (00Y - 32A) = (per 7) 100. 31 - 2. 32, :, 31- 32
200. 3]. 32Y -00. 00. (31+ 32)γ. = B1 + 32) 32.32
0. 315.
00. 32 -32.32
373 (00
s
Caeterum pari jare, si loco sublati primum  sub sustulissemus primum O, obaequs 10
fieret -0 = (er 32) D +00A, :, 21 - 31 = (22. 32 -00. 32)5) +00. 001, : (21. 31 - 00. 31
et ex constantium coincidentiatione fiet  (2 -00) 31X = (21 -31) 09 V seu 21. 31A - 2100Y = 0031 X - ~o3IV
Seu 2123) 00. 31 (V - 1): (00V - 31X)
it
Hinc sit ex 12 et 13 fit 22 = 5t: 100r- 32
A = 11V + 12Z
+ AB = H. 2bb + . 11. 2R vZ
+ I2. 2277
20000 a
12. 21
2
31:, 00r -317
B = 21 V + 227
-12.32 22
00. 00ag  )
+)
V(AC =) +1on + 1. + (VZ
at +1X32 22
+19. 31
+ 1
19. 31
e
ex aeq. 4
00 -32
31 - 0d
22121 = 1232,: 131 -11
22 = 11.32+12. 31 - 12. 21, : 11
22=4 g  12 11. 71EV,
22 = 11. 32+
C = 3PV +322 113]. Et = (1. 1. 2 = 1 3 1. 7 F tC. LC32
~ (+BC -)
an
IaBest
+21. 31BB F 21. 32VZ 22= 12. 21-31 3
+22322 + 22. 31
Etgi =0v. 00 an
32. 31 - 1. . Z tap. D
12, 31. 31
=12. 2. 31 (H2. 21. 11
I. 32. 3
12231 fit 21 = 411. 32 +12. 31 ( - 1)) 22 = 1 22 + 12 31 pA 32) 2 - 1  31 5
Ex coessici Ex FI. 22 = + 113= +21. 32
+1N. 21WV +, 12. 22
-11. 31.
H
quas duo seu 00,31. 32V = 00. 31. 32 AF00. 31. 321 00. 3232F 00 3100r - 31 32 00Y-
sit
12. 21
0000 abti
+ 21. 3IVV
= - 1. 31 -.
12. 31 = 11. 32
+22. 3222 
-12. 32-
00. 00 aY
possent jam poni 21 =22, salver generalitate
et fiet 31 1 - a 3
seu 1+12 = 31+32.
12. 37 - 11.32
XXI =1192
fit +11. 2IV (4, 2. 31 X et +12. 22Y = 122. 325
-11. 31Y == 11. 31X
=. 12. 32V = 12. 3291
21 = 11. 21 CX 22 = 12. 32 (V4 X)
tIr- 312
12V - 32X veles1. 3
et 11. 31 . 12V = 11.31. 32R = 12. 32. UY - 12. 32. 31X
ino nalva gempera titate pestumus faccut 31 11 =C 21= e0 = 2 ∞ L. 1Y (31 -11) = 31. 32X (11 - 31) seu -11. 12Y =, 31. 32X
32 2 11 42 -3) = 11. t2V . 31 K. fit na. 11 3, d Fer 311- 31 3K = 1. 12r
Posita 21F00 = 22 fit ut dixi N1 + 12 = 31+ 32
Ex 3 per 1 fil 21- 11 (1) 12. 31 - 11. 11 -11. 12 4 11. 31
00
1. 11 + 11. 12 -11. 31 = 12. 31
(23- H310 = (1. 7 (31) X 11) Et 22C n Ho W Hn H, s pre
tH. 0X
00. 12. 31 - 12. 31. 31f II. 12 31  I. 3 L∞  r. 00 Fo2 3 1- HC B AecrtB2. 413 13 4(.3
11 12V2 31322 31 11l1231F u. 3.
11. 32 A c2. 31
ex et 2 sit u tr. 31 = 1. pn. gA out A -. AV
ex 2r 3 fit 31 - 11(8) 11. 31. 31X = 11. 11. F2 12. 3.d 3yeA = 11. 12Y
ex 6 fit 1 11. 12. 31V - 1. 12 A 12. 3P2 = 11. 12. 31 31 = 00. 11. 31. 3) 00 L1. 11. 121 11. 4. C2Y + 12. 3131
11 + 12 = 31+ 32
H1.  21, - 31
00
X 11. 12. 11. 12Y = 11, 2. 11. 31X - 11. 12. 12. 3IX + 11. 12. 8. 31 X
100. 11 ... - 00. 11 . . . - 00. 11.... + 00. 11. - . .
abe abt aet ide
00: (F A) = U. 31 :, ur - 3)X seu 00. UY - 00. 3N = 11.31 (V + X) seu 31 - 00. uY Q vadtur ud
pari re ZL 00. T2V = 0a 32X = 12, 2(V+ Nse 3, = va 12Y. v0X + ar + 12d
Ponamus jam esse 21 = 00 = 22, quod sala generalitate licet, (H 00. HH = au 2A = 11.3
le
12 - 32 = 31 - 11 sen 11412 = 31+32 rursus 11. 3C = 1IY = 31 X, 5 en 11. 31. 12Y - 11, 312, 2A = 11. 12. 32Y - 12. 31. 32A
12.32 12V - 32X
eq
Hine 32 = 1. 4 12 31 = 11. 3, (12Y-324)., 12 (Hr 31N) ande Mu In m G m ESCN H tur -  313C
1I. 12. HHC+ 12 R 11V (31. 12. H1Y) - H. 12. 3CX - 12. 12. 318 + 31. 12 31d (3 3, CrV 1. 3F. 3EX
siet 31 = 11 H1 12 +, 11. 12 (1 + 12)4Y: (x 1. D2. G1V b1. CEfra 32 ∞ 111 12- 31
et fet 31. 31 = 2. H. 12. V + 1. 12X 1 12 12X - 1. 32 32 = 1. 31 (1 2Y - 3V): 12(∞H31 X
H1- 12. TTY + 1C. t. HY 3t. 2 HY - 1. 12 31 d 12, 12. 31X + 31, 12. 31 X = 1, 51. 12V - e. 35
27 = 11 + 12 = 31 = H5 11. 12. 3IV :, 11. 12V - 12. 31+ 11. 31 X, et multiplicando per cucem
n1. 11. 12Y (11. 12. 3,X)F11. 11. 31X + 12. 11. 12Y - 12. 12. 314 (+ 12. 11. 31X (31. 11. 121Z 31.X -31 41. 31d -(r. 12. 3IV
+11
13112.517
et fit 4 11731. 31X +2. 11. 1213944. 11. 125,+117 = 0 +11 75L. 31. 31 F2. 11. 12Y
L+ 12)
12. 1212
= 12. 1252
1 - X = l - n = r
11. 113B 00. 11Y - 00. 317
312 11. 12G + 12
 0d. UY00. 31N = 11. 3. X∞ fit 31 = 00. 11Y:, 00X L11F sit
super sunt ad huc binae aequationes
31+32 = 11+ 12
fit 32 = 00. 12V:, 00A + (LN
0on +B. 125
00 (1 + 12) d M
400.
60. 12Y =00. 32 X = 1B. 32H
= 11+ 12 et fiet v0. 00. 1IYA +, 30. 1. 12 _VE +00. 0012 aV = 90 . t1 2E   + 00 qu llin)
00. IIY, 00. 12V
sem seu 00. 00 fut AV d 10VO.  N YH. 1R NI 400 (11 + 12) a H 00. 0071) (11 + 12) = 2. 00. 11. ILV FC
+
Ergo
superest adhum una aequatio, 0oiust = HC. S
vo. 11. 3.
00x + 13I
313h -1
DcA tUK
En qua aequatione 11 et 12 se habent eodem modo.
fit 32 = 0a3+ 12. 31 (00 + 11 -31 7: 0 11 (00 + 31 - 11) = 1142 - 31 unde 9P. 12. 71 - 1. 31 51 =, ox 1- 4 0031 L 1 2(3d] - 31 31 = 2. 1. 3
Ii
(1231 (00 +1 - 31) = vo(iEK. sit 31. 00. 00 N + 31. 11.02A - 31. 31.00 + 3100. 13pY +31. 11. 12I - 31. 31. 12H = 1, 00, v0Y - 11. L1 00Y4 11 31. 00Y
v0A + 12R
fit 31. 31 (2+00) = d 00(1V = o0R - 11d - 12H   (12: 01) (C) = 11 (09 - 11) V
Ex 5.s. 2 fit 2 n 2B fit (B1 - 12) a 37 3, = - (2. 11. 12F  NX Cu) 31 + n. 12 (1) = me ( un L oorA Z) 31A 411 (er s  ta m 1 X
11 (00 = 11+31)
Ergo fit 31 =
Do Cnti  n Id a, ∞0 (irtiZ (E. IN. 1) X = 2. 1. 1V quo valire substituto in valore
resin (32 = 1 + 12. 31 debst Juodise valor ipsus 3. differens a vatoe ipsius 3 Ipla permutatine ipsorum 01 et 12
t
al

## Folio 87v — 00068207:0174
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000174.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 145 lines, status machine
sit Lia, XFL, XEm , XIn _) = ay quaeritur asy ax quadratura rationali
Quaeritur modus efficiendi ut factum ex X+L, X+m, X+n sit quadratice extrahibile
ex prae
 si AB, AC, BC, ducantur invicem produci Quadratum AABBCC
Manifestum est
fiat ergo. a,X HI = AB aixtm = AC, A _ aXtn = BC X = AB - at = AC - am = BC - a
Videndum quomodo assumi possint A, B, C
B = 21 SaO + 221a
Tentemus facere A = 11 s( + 12 fa D
Erit FAB = +11. 21 aC. + 11. 22 a SOD +AC - +11. 31aC + 11. 32a SOD
=ab +12. 32 a D + 12. 31 11
C = 31YaO + 32 VaO
+BC = +21. 31 a C + 21. 32a16)
=an +22. 32 a D +22. 31 .- 
-00. 00 an
00.00 am
+12. 22 a5 + 12. 21 4a0 in
Coincidentietur primus valor cum secundo, et tertius etiam secundo, irrationales interse, et rationales inter se
2 Ex secunda +21. 32 ) + 11. 32. 
+22. 31 + 12. 31
+12. 21 +n. 31
(2) 2. 11. 32t 2. 1231- 21 (12+ 52)
= al
00.00 al
Ex aequ. 1 et 2 fit 11. 22 = 21. 32 seu 22 =21 (12 - 32, :, 31 - 11)
Rursus a)q. et 2 eas in unum addendo fit +11. 22 + 2132 = 2. 11. 32 ande 22
12. 31
12. 21 7 22.3
+12. 21 4 22, 31
Et aequando duos valores ipsius 22 fit 21, 12 -32, 11+ 31 = 2. 1. 32 (131 7 - 215+1275+317
3252 - 111
D.
11. 32
+12ZC+ar, s+n. s + 31
fiet 21 = t12 31S1- 115 =, - 32t13i5 +27325- 1
11
2fF11. 325+317
- 12. 31)
tn. 3F2- 31.
Ex coincidentiatione prima
in aeq. 3
= 12. 32..
Secunda +21. 31 aO + 22. 32 a D 40000a C
I1. 31
Et et aequio per 2., et ita aequentur inter se
=HI. 31
- 12. 32
 2125 +217O + 00.00 h- 1t
eequentur Termini absoluti fit 22 B. 12
ol. 31 52 = 12
fieri 21() 1l3Cl- n). si teata
coinsedentiatio
 at 1  231912. 32 (- 17 ) l. n)
si tentaremus 2A AY 3+d aY feret 22 8
32s - 12Y
 et debet pari jure
12. 32
l. p. 72 et nr. - 11. 12. 31
sen 32= 2t HV FAV, . H. ande (ir. B. 31 3 13. 31. 3dCIB. (. H_L 11. 513rL
31 A = IIV
271EIaD
et evanescente  fiet aequatio nova
132s ~ +21(+22212310 400005+2712V
- 12.32
11 + 31
3  g1. 32 -u. 32
Lu32ZCM2L. SFII.32
et erit 21  +11. 31, 4VII. P + 11.32] quem Valorem subsitundo 22 = 41. 31 3l- 325
fiet
ex coincidentiationibus rationalium ponatur m- 1 A et pL n = V
0
2-12. 31
12.71 - 11 12FII31
(12. 1 = W I)3: H + 1 +2 124- EL. I1. 12V
Ne qui valores 21, e 22 sunt prioribus aequandi, nec puto permissum ut aliquid pro arbitrio
seu 2. H B. 3t _ = 31. 31 (12+ a) + (12 11) r (FF1)SIB + ( c V2
asumamus, quia ex coefficientibus 1, 12, 21, 22, 3, est duae aliquae tnutiliter quo ad b determinationem assumuntur
cum absorberi possent ab  et di
a0. V. 31 + 3. s1 .
8.32V Fo 71
Sic ergo (alculum resimamus A = 00 da  + 00 laD
ax = AB - al AC = am BC - an m l=
Lam est
391.II 21+32 = 31 +32 = ov. 315 + 00. 32X
0∞uf. 31  00V-321
B= 21 5ao + 2 la D C = 31 la O+ 32 +a
00 s unitas A V v - 5 = X
+99+106. 24 +00. 22  9D 40051 a F00. 32aSOD ~ + 21. 314O + 21. 32 a L0D
es
= +00, 22 aD +00. 21 a 4) = +00. 32 45) +00 31 a Y() D = +2. 32 a5) +22. 31 a YC)
22= 00 72 (31- 00), 3132
31200). 31-34
Ex prima identificatione irrationalium, fiet  21 + 22 = 31 + 32 Ex secunda fit 21. 32 + 22. 31200 (31 + 32)
- v0. 00 a R
-00100 ab
-00.0 am
Ex 1 et 2 fit 21. 32 +22 31 = 00 (21 +22). Ergo 22: 21 ~00 - 32, :, 31 - 00 Et ex 1rt 4 fit 3t +32 31. 3 fit
21+ 32 - 21, : 21 = 00 - 32; 31 -00. seu 31. 31 + 224 - 21. 21 _ - 0c 31 -ogg + ee 21 = 30 21 s 3221
seu 2t seu 31+32 -1 = 00- seu 31432 = 0s-3 + 31 - 00 ac denu 21 2 31 - 00, 31432:, 31 13231 32
-2,32+ 31,:, 32831. Ita enim
0
31-00
21 et 22, differcent solum permutando notas 1 et 2 dextras, ut opoatet
Ex prima identificatione rationalium, fit 21O + 22D, A 231O + 32D Exsecunda 21. 31 D, r2. 32D  o 00 310+00. nD
21. 31] +6000Y,: 100.32 quos vatores
=n. 21
Hos valores coincidentiando fit, ex constrantibus A: Y = ∞ E XC   00. 32 A - 21. 32 X = 00 321 - 00. 22V 
et  per 9; D 1270 + 00 A, :, 32. 22.  (per 10)- 00:
(13)  10.  00. 32 FR:, 22 = 00. 32 (r - 1):, or 32A seu 23 00. 32H:, ooV - 32 X
tiq
3 - 21. 22.32 421. 31. 32 00- 31. 327
EEd
Siut 31 et caleg
60. 31. 31 (+22. 31. 32 = 21. 22. 31 (+ ∞
eu 1021. 72 4 ∞3231 00. 31, 32 00. 22. 3, 15 21. 22 (32 - 31) fer 21 00. 32, 2t :, 00.  21. 31 f 21. 32
Ex C pro C variante ( tollenda fit +00]5 +21]32 =-
abe filfev. 31i3 2t. 3
13 e ad 2 f 209 213 =60 (31 +32 + 21. 22 (33-31, et
Bu Etpo C variante tullenda fit in quae qua aequatione ut et n2 servanter inter se legem justitiae, seu
d-. a(e - 2 31=C
0) f (22. 00) 325) + 0000. 00V
=(31 - 21) 00V, seu  21 = 0031 (V∞A) :, 00Y - 31X
duos t sulore tq 15, aequando sit do
M2- 00. dor 4003 = 22 (32
seu 2 00. 31T: ,100V - 312. sid idem debet prodire ex 14 et 15. Nan ebe 15 es Videamu  examen
unde si
accut
30 21. 31.92D
e
Ex 9 fiet d2 (22 - 32) D + 00X1 :, 31 -21 
Et ex rationalibus sit mdentificasis, fit (t ) 31X
equa
60.00. 21. 3cV = 00. 21. 32 32 = 00. 5t. 3 ∞. 90.5  H (100, 21 31 5) Etutle
frei (00 -3)2 (00+21) 3FN
Ex 15est 22= 00. 21. 32:, 00.3] 71. 7. 31 t ( 32 = ( 1) 31 + 32f2t, et fie
+
re
1- 21.3.
00. 21 32 = 00 3131, 0931. 32 - 00. 31. 21 ( 1 31  51. 31. 37 f rl. 31 3 ((21. 32. 3( 1 31. 32
60. 31 (32 - 1
fiet (00 3c)Y t
00. 2IV = 32. IY + Z1. 3IH = 0031 T. fil 21 = ( 31H. turt 31V
(X =
22- Fd, 1
32.91