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# Piece: https://leibnizlegible.com/api/records/59887/text
# Catalogue record: 59887 — [Aufz. zur Integration irrationaler Integranden]
# Incipit: Quaeritur quadratura Rationalis, pro summanda Formula √(xx+ab,xx+ac)dx
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59887&katnr=57590
# Work: https://leibnizlegible.com/work/00068197
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folios: Bl. 9 (from the shelfmark LH 35, 13, 2a Bl. 9.70. 10): 2 pages, canvases 17–18, page ids 00068197:0017 to 00068197:0018
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=17
# Lines: 192 recognised on 2 of 2 pages, mean confidence 0.76
# 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 9r — 00068197:0017
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 70v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000017.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 119 lines, status machine
st Xea=   +D, Q sit XX = 0 + D + 210D, _, et natab  sit  = O+D et e = b +C
 abeolatit + aaa C + 4aatue 10D -9 gy Ubi explicare oportet item assumtas, ut aldi ipsius
taeqt
+2ae  16D.
00.00
QZyy fiat extrahibilis: Assumatut OD= at
 et 8= 10Z +ua, : 00 
gus aat  raa sat q +aat
1y firt 4Qgy = 00. 00aa tt 4. 00. 00 aatat t +4. 00.00  compen
quaeritur quadratura Rationalis prosummanda formula XX+ ab, XX + ac) x = aydr
 Orellaaeat.  ml- t  sist XX + ab Xx + ac = at +2a 0) +b8, at + 2al OD + CZ.: 48 = yet multiplicatione facta
Et ex p
10. 20 bc hh. +210. 1 aBcK t + 1. 11. 11. aabc
00. 10a et + 1. 00. 11 aae Y
compendio = mtir + nh + paa
+2. 00. 10 aerat R +2. 00. 11aae Vat
itaque erit 4, 00. ∞ aa +4. 00. 00 aet = qus t4. 00. 00aa tat
10. 10 bc. F1. I1. 11 aabc
+2. 10. I abC
00. 10 a 42, 00. 11. aae Vat
+1. 00. 11 aae
12. 00. 10. aetat
00. 117 H
et multiplicando actu ipso
quod ut aequetur quadhatet, fiet
a, Gt+na t +  aap
adeoque Amhtur natitaap)= m, Et
1400.00 a 400. 00 D+ 00. od a 11. 1 ab c B0. 00a aelat u 4600 iac
10. 10 bC. 4. 00. 00 aa t( 10. 1 bC 11. 11 abc + 10. 10bc. 2.90. 1ae Vat) 4. 10. 1 bbccae 
(00. 10,90. 400. 00 aat+ 00. 10 ae 11. 11 abc (4 00. 10a8. 2.00. 11aelat
+1. 0d 11 aaee + 160d a b0. 10
+400. 10geuia
Unde destructis destruendis fit (11. 11. at + 4 18 aC) (abc + ee) 
Be abc = b1(C + 2bc Abe = bb + CC 2bC = qu b. C) =  ttr fie et set ua ato quiuui
Ittaq+ 1 ( a 16na es alet 0 
11.N + 2. 8bc Vat 11 + 64bbbccat- 64 bbccat = 4at atat
10. 10
ahh. 10
aaha
atgat
et liet 11  qtat,  abc  V(16bbcc - h+). jam y =
purro y-, tr + un a, 1ot + na. Compendii causa ut m fiat valde simplex
R  409. 00 a araL = 10. 11 abe-2(0bge sa
d tac
e bes 2iaavat, lin (ubc. eo) - eeal; hdibe posto q re tat. al. L B2 = aaq
Zmatba Intum cosreiaRii be
et y = 4(00. a GnGi, 10t + na et n= 400 a v et 1 = 2
pollumus, facere (o. 10
n22.B uat, itoac
6bb ce +.) at
oss.
videamus an tat ab sic assumere liceat, ut pa lor ipsius n contaatur
et n = -4. d à dat 2:-
qbe.
iitaque si laceremus 24 ate
Del (+Nl6bbcc tor 44bc). + et
Dn  betch 1 ee (+ Hisblic) thh + +bC)
ecifinet
caiat ) 
orpliando u sitar dobali oucertei cet nadt uu meatremu,  i derea atvet
( +
Pqoiat abt . od. ae ra
et ( (tn =  abc taslla a  elibicet  1. ab.
e 
n = 2Vat 0 = 2 Re(G ebbbCc ht + abc +hh)
(16 - 
va n = 2aa fat, - 2 - ee (4.  Fet
seu n = 2aa fat, -2 -ee Y (4L F F
p = 2aa tat, - 2 hh b
Atibe
i
d +4 + 
boce 
(1 = Y) +.
ee (_ F, (16bbc C -h) + abc +hh)
bbbe bE 2bbcC) = (DbbbC - (a ccC (8 bbec = b - ct ( 2bbcC
 -(bb +CC) bC, lb bib qu (b+C) +bc
 qu (bb +CC) = 5DC (bbF cC) =
Nouc bd (cc b C AbC + 4bC (6 bb c)
gu (ob- qu (bb cc) +bc u (b+ c) = qu (b +C)
16bbcc h 4bc + tih, 4bc = hh = qu (b +C), 4bc = qu eb-c). Ergo M1bbba h)-(b+C) + bC6. 
edes
A(ebC = sgit Cb CD) sit ium limilatu vt erid dy = 4 b dc 44C db = ZCb+G) (2b - 4),. - (abc- qu baG
en in hac curate o eailionem ab: dem vis b limituu. Si altentra costansse voines cit db
et fiet nnaa aap seum
at, 10. 1b cL. 9. pl bre s 1 a
at 00. 1 40. v. 1abe voloP
lat
snlatae
bcotat
00. 00. a Vat. 10. 1 (ee- 9b0)
unef haieit
2.1 qu e dat
4 aaeeat al
unbe  eeat aallte
hbbec
itbCe)  00h1a ces. bC
cest carja 
sun
 i id e per differ
tiatem 
siet q
ie, dec
sn
rotet ti
(B N sed cum mul  dicuc
methodus
vinimus
entes cessat
manimi
differentialis, et
recurrendum est ad liq otr
ensa gua bb ec
he R 5,  et.
i diverri cadi
Aatene, in paginabias iat
: nobc
itaque si dapes relatio inter b et C, uli es
 in hac cuto ahe   ab de  eca blimitatu s allerta cpusse   et

## Folio 9v — 00068197:0018
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 70r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000018.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 73 lines, status machine
Hbc = qu (b.5) ( aequat sit C =  b -c et de K = n ( (b-c), fiet
et ca mb. at  c: b = m
p
ate
4bt 4b  b -c (2  qu (b. c) =  qu (b. C) ut st acb ML9-
aa
104
2
 at  lebo - b (er p , at 1a)
et 4b m =aa ( C) = bt (b-C) seu Atalm -aab talit: prei - nub nub m
seu 4 ab + 4 ab pie L aaba (= aab m + a abM) = elib a + tb m - l8 0t. 
et exa dividendo per ba, fit 4a + +um = da L  . seu atrm = Boat nn)
2aa CF  = - 2p + pp fiet la = pp - 2aa ip et a + 2p = +  p)
seu m = +  + pp + 2aa - 2pp - rap, : 2p, = 4 (f.. 2 p p 4 2a -2ap: 9
nun
fiet  b: C = 2a + (41 - E) pp + raa (rap,: rpa = atm,: m
et K =  (b -c) = d ac - Wmb = 1db. = pp -raa, b
a m
pat
pp- 2aa, b vel PP- 2aa,b
-pp-2aa,
=3pp + 2aa
at
(2a) + (+1-2) pp 4 raa (2ap)
4bc = quC b.c), posito b manere
et csieri  2apb., (+1-2) pp 4 2aa, seu C = -
est 
 vel
= pp- 2aa
rap b
31+2aa
si p sit negativa eligendum prius,  tC diversi signi a b, sxeundum
priorem expressiionem gir p sit affirmativa, et c diversi signispotest
utraque inservire ex pressio.
sit e sit data = Y fiet H 1- 2) pp + raa
2ap
seu apr = (+1 -2) pp = 2aa, et pp - rpr + rr = rr + 2aa
(+1-2)
544 154
et pr sen pp - 2rr (+ B rt traa (+1 22) si sit
5
4
et p = 2r+ Y(rY + 2a a (+ 1- 2)): + 1 - 2
Hinc si Y: a fit  D. C sit
si
F12
alioqui in imaginarias incidimus. Hac mefhodo dum reditur
quaesitum adrationalia, etiam habetur determinatio
sed recta 4bb 4bc = bb - 2bc + cc seu bbo  - b. bc + cC =0
b+ bb = 2. 3c b + 9cc = 9cc - cc = 9c, fit  b: c = 212:1
Cum vero patio C est minor, quam 1 ad 212, res ex impossibilis.
reditr ad imaginrias et ((4bc qu (b. C)) est imaginaria
dixx + lx tam X + (XX + ab, XXtaC) = AViXX t abi) + V A(XEtaC)
fit A qu (Zax +lx tam)  (X + ab xtac)
2X 2LXx. .)  (XX tab, xta C) fir 1
hPXX F Ex tan Y CX + ab XX + 40) = ZtCxC +
e
Rh XXF. . = 2V
 sit Au (hxY....) + (XX + ab, extaC)
aqu Z (XxA ab) + WV (Xx +ac)
An sic 62 + 10D = 210 + 1D si 669∞ + 0D F 2621 0D = 22 O F DrS + 2  10D
fil qu =89 et Z= R 28 et v - a
mt
Uibi solum C et D) sunt datae
. et  18640 + 0D = m66o + ac 44
ac
mmY
 sunt _ C et  quaestae variantes,, M quaesita non varians. 8 et Q dant
quatuor quaesitas non vartantes, et m est quinta, de er quae adhibendae pro
qunque asqrationibus identificatoris
si esset et + si B + D essen simelices foret _ d = a et Z= _ m teil t dG +BD rmnaC + au 9y D
 et f ubi non  non habet locum successs, sed sidendum an non in retento t
posito B, tD