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# Piece: https://leibnizlegible.com/api/records/499/text
# Catalogue record: 499 — [Aufz. zur Integration irrationaler Integranden]
# Incipit: a4 + x4, : x4 = (1) yy : aa, quaeritur ∫ydx.
# Date: 1676
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=499&katnr=58076
# Work: https://leibnizlegible.com/work/00068124
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folios: Bl. 102–103 (from the shelfmark LH 35, 8, 30 Bl. 102-103): 4 pages, canvases 203–206, page ids 00068124:0203 to 00068124:0206
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068124&page=203
# Lines: 208 recognised on 4 of 4 pages, mean confidence 0.77
# 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 102r — 00068124:0203
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 103v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000203.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 121 lines, status machine
absolutum non successit
e a +X: A -. gyiaa quaritur Sydx. fiei a FX= y a + aa X  = XEyy
en v X= a aa:, ry-aa igitur gy- aa debet aequari quadrato- quadrato
Sit y= 10 + 110 + 12VV,:, 20 + 21V + 22VV. et f= O: D siet gu -aa = OO - aa DD, : 9D
aequetur OD- a4. DD, quadrato quad lalo, ita enim postea etiam ayu-aa facile aequabitur
quaduloqu glo quadrato tantum enim superert, ut D aquetur  quadrato, quod facile
P = 30 + 31V+ VP
30 = 1. 10. 10 + 2. 10. 11V + 2. 1012YV
+1. 11. 1.. + 2. 11. 12B3 + 1. 12, 12V
a qu. 2. 2 1. t2. 21. 2 F 1. 2. 2.
aa W= a, 1. 20. 20 + 2. 20. 2 v + 2. 20 22 
yta =menn yea = nnp
y = m nn -a = nnp  a. se fel
nn- 20 m- p
n + pp = quado et nm. pp = quadr
4 10) 36 + 4. 30. 31V + 6303 i vV +4. 30. 31V + 31 p
pro v poratur ex v et Y fit 4.31: 30 = 10. 11 + 20. 21,, 10. 10 + 20. 30 Ex v et B fit 2. 30: 31 = 11. 12 + 21. 22, :, 12. n + 22 22
unde ex 11 et 2 fit 4. 10. 10+20. 20, 12. 12 + 22. 22 = 11. 12 + 21. 22, 10. 11+ 20. 2)
tex v v fit 383 - 10. 10 + 20 20, 1 12 +Z  quod et prodit multiplicando V e in v, et v in P
B (13
30. 314 = Ex v in V fit A30 3 = 10. 11 + 20. 21, 11. 12 + 2121, : 4 Et rursus
Et ex 14 et 15 fit
per V fit 30t 31 = (2) ( n.n + 21 21 + 10. 12 +, 20. 22 : 36
(2) 1. 1 + 21. 21 + 2. 1012 + 2. 20. 22
N (16)
Ces 31.31:, 30. 30 = 11. 12. + 21.2,:, 10. 11 + 20. 21 rividtdo v ser r
H), 11. 12 +21. 22, :, 10. 11 + 20. 21 per 17
en e
312. 30 12. 12 + 2222, :, 10.10+ 20. 205
Unde ex 19 per 13 fit seu 12. 12 +22. 22, (2 ) 10. 11 +20. 21 = 10. 10 +20. 20, (2l11. 12+21. 2
B (20
enexcteus  10. 10 + 20. 20, 12. 12. +22 22, (2(10. M2O. 2) =410. 10 + 20 287 (2 ( 11 12 + 21. 22
unde per 13 fil 11. 12 + 21. 22, 10. 11+20. 21 10. 11+ 20.2) = 4. 10. 10+ 20. 20, (. 1. 12 + 1. 22
30 1010 + 102 51 = n. 12+ 22. 22
seu R (3) 1041 + 20. 21 = 4. 1. 12. +21. 22, 10. 10. + 20. 10y
35= 10r0+ 2O1OFB). ex net 15 fit Ex n est 30 = A: 2. 30. 31 c (per 13)  C:, 6. 3131 fit 3. 31. A = 3dC
 2. 38. 3  A +6. 3030. 31. 31 = 
2. 30. 313=B
1 3A
D
seu 31:30=
Ex 12 est 3l = B:, 2. 30. 31 = C: 6, 30. 30 seu 3. 30B = 31C seu 30: 3=  C:, 3B
Ex 11 et 15 fit 2. 31: 30A: D Ex R et 16 fil 2. 30. 31 = B.E 
23 et 2+ fit 4 = BA: DE seu 4DE= AB quod etiontiiberi potera
multi plicanda 1 per 12, et 15 per 16 Ex 20 et 23 fa CD = 6AA, et ex 21, 2  CE= obB
BE
hine sit CC = GAD
sufficiunt 25, 26, 27 nam hinc sequitur 2
Cc = 36DE  AB = +DE) His tribus aequationibus adde aeq. 13.
et ex et 11 et 12, 31. 310, 30. 31 = A: B
Ex 13 et 28 habentur 30 et 31. Et ex aeq. 25, 26,27 quaerenda sunt
10, 11, 12, 20, 21 22
solutio horum in Epistatis mers ad du. Baronem odenhus
Kebretupetac
exa rua
tantu ejus ope, ineniendo quis tunc
vulor y, habeatur y+b, et ytc
qorm  cois divisor, rclinquens
itas quotientes dabit P, ipsi
quolientes dabunt m, et n
ad eoque et habebitur m:n seu li
seo modus efficiendi, utrabio sit
quacratus, adeoque et ut  se
intota sit quadratus
diseoraso, quod fitri pe
ex _bW. Wib- aa
9+b- n t- man: aa
fiet fyg. bb= mn et
m= y= m- b = mun + aal
et  mui A cando per 3
sit nftco+ infal- a, aa
fit Maa-aab =mnntaob
seu m- 2aab
tiat A~ mfK . b9
N infal a
aabn
non
gab utt e mna
yy - bb = mm: pB
y = ma ba
ga= sk I
titns aa cug inh u influca, 
hh. aa = nsl3 1
D th anfia + 2C+ a
adfancot aal
hih - nesll- 25
maeum
ergopoet aa
Efla+a
Instat Ergo mu
Restat austat 2gah = 2fsiK + faK
Staa: na e
rahe 2Fb4 at. son t=
fit m. 1bppn
Bsbraa, un
mmetrame
sed h.   autento 2htt aem 2s imeraea ser.
app-ann
y-bb
aequand
seu 28= - 61
qoquaad
i   5
appr an
ap- ad
-2uslm rin
anumm
e
32= 1. 7to
16 422-
uet
Aaua
sit a
ga + 2 ahb
99c0co +2900 ak + antiti- a
nan sit 10 = + si, fia
ad lt inth  ragir in fKk = in flg - 29 3= mna + bp
sit v- CB KV+ B
sfkKCOCo +ZB alteco n haat
naeram portesale
102. 103

## Folio 102v — 00068124:0204
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 103r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000204.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 1 line, status machine
40o lat D  sit B(2o) B- e+ ext

## Folio 103r — 00068124:0205
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 102v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000205.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 11 lines, status machine
d sar= ade
Eax
saruaneau sit nana d fa-
sret d SX +bX raex -
3X tbx +ac, dx 3XX + 2bX + avxabxXxt aex
(X+ bx + acX
X+ bXX + a CX
Ergo si sit d, C 1X... fiit  YX+ bX F acX, d0 L 3XX F 2bX + aCC
x3+ bX+ acX
pro b scimnmus nn, et m+ n et pro ac scubamus un, etque et sit de lxa en
d,O sp eu = LiX +  XX + max,, c sit de - + lxx+

## Folio 103v — 00068124:0206
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 102r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000206.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 75 lines, status machine
yy + ab
si possia tatid facere ratonalem habebo dimensionem curvae Hyperbolae. Nam si y-aaix
tiet Paxdx 1 dus ay = lat x dx. xx Habeo et meisionem ipsius TarX x seu dimensionem
curvae parabolae cubicae. Nam si y = x. zua fint dy = xX p: na et fiet  Suidus+ avax- var xiea
utraque enim summa iniri pt rationaliter, quod patet. Nam si sit S+X= y seu si fit
X. = aagy - ab seu x = Ilaagus - a b) tique pat unde si X habetur intionaliter datur summati
Mursus si sit Oaa b +x _: XF - y. seu yy X = aeb  aa X seu X N = a b:  yy-aa
qp
+rmm hpbens
ita rursus lantum opus est redd Dguaa fieri rationalem.
mo auplius videndum an hon hoc reperto omnis curva Conicasi
summam rationalem, Nempe sit  sitaq = Z2+ aC+ aecio et A aagus  adebet fieri
aequalis ipsi Z et fiel (Z + rac ZZ + 2 a 77. V + aa ee + 2ae: v + acc. vo  a2 (8
2aC 2 VV = 2ac82 + aa cevv  ab + 2ac0Dt at  seu-
+a sit ee ab et fiet 37 - 24 aceV, a cc 
tanec vo + 24e cev + acc
rahonalifatione ipsius  Saagus a b. Hinc etiam datur (gaZ. ergo datur Z2dq + aedque + (a c. V) dz
sit at.  co ergo dater codque
Eagos datur et Zduo
Z7 = 2ae + CCCOC:, 2aC + 2CC0 Ergo datur (d(raceCor cccoCO,, act 0)7
Nempe sit ay = 23 + av + ae firt argy - a b = (Z) +2)ZV + 2 aCZZ + ecvv + racev + an ée - ab = (2
22=ca0v +: 20+ 2ae
quia autem datur szclus ex posita
rationalisatione ipsius  aagus- aeb ergo dabitur et sydz. Ergo et S22dZ + Sevaque + Saedz. Ergo
videamus jam an omnis curva condea ad formam
mXX+ ad: nXX + a ,odx summari debet sit X7 = 50 et 2∞ dx = F du
Ergo et Sz d
catuur et sodZue
seu ax = fdb: 2YV, fiet  summand. ergo Sin f∞ +a;, nsv + a,  fdv: 21f0. se
ipsius  reduci possit
Jinfo +a 5fov + afV n fav: 2 si pro s0 scubatur aas.v pro do fiet: - acc: vV
et habebimus s infaa: v +a,: ns at v + af. B faa dv: vo. seu fiet
Generalius sit XX = asv Yaag,: v fiet 2Xdx = -agdv:p
seu dx = - gdv: v0): 2 NB:, 5ur a) ex et ex Y (mXX+ a) : (nXX + 2)) ax fit:
mo = max + in A3 + a  
anfv ratrusa matix
sit K
et ams
amabus
N = m in a XX + a = aagu dat _ a0 p  m  XX + n r a ausus a - m
seu XX +. 2 men a = s-
aneti
na hun
C+x B+ D.  = CC+ Vp + 2B)
s Bβ = .. . .. VVEr() -. nen
β= .
est autem is rationalis ex Hypolhesi = 8 tinc VV + 29γ - da V~ F 3+ 2 - du. 7
Ergo fit pV+2 Y, V = l
et et
yrb= mx gob = ep fiet sgs bb -mnpi aa et m pato = npaab seu p idela;
et fint yy- b0 =
mm - nn
agerentis istetqu
 mup- amno, hujus quaeditur sit amnp + axq quadratus
um np
nonel sala
Ergopo
ammpranmu ratun  aegum - aequn = oto sit am. n = W fint 2a  + aquoro- aq aus
a a o-aa eali mx
 7 +b,n P. ad ytc rp: aa Ygeb, stC= map: aa g = ump = gab, aa = n
sit p aub a 
un uocabo as pu heosbit Cmmanu). veit
yrbb minim r  i  
E ac v0 +2c
seu datur quiaantem datur  sytus - (d. gu. lb summat erd rationali  ex posita
2evn+ racv
 ah: co, fint 2Z- Ena race. co 4 adcl:, rea. co 1 cico, seu fic t
adeoue hutur et (a C: D) Azue
et. 23- et
dofall sinsvt avv: n + asv
0d
 ioast Z 6
sdaa. Erge
Vhrtae
eltis