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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 158r, page id 00068196:0313
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CXX. XV, y CB, a xy = aa dus = - aa dx: XX
faxdx rdydu = dx SCa +tX) xx videamus an X + a aliqua
arte sic explicari possit, ut quadratice extrchibilis reddatur, et generalioris calculi
causa scribatur 1(X + C) = av
Fiat X = Jam Nan erit XX = am +2 a Imn et X = aamm + aam Ima
an
et  + aC= aamm +bnamn + 4 aam Imn
(
un
Assumatur av = ap + fV sint Xtac = aapp + 2asp ima. Hunc vala
es
un ac
rem priori identificando, fiet  iaequatis irrationalibus, fp = 2a m
Htnm
et aequatis rationalibus aappfsmu = aamm +buamn + aec.
Multiplicetur haec aequatio per ff, fiet aaspp + n=frum +6 af mntaffc
nu
et proffpp ponendo valorem daamm +8aamn, fint 4a um +8amn+ f mn=
ausfrum +6 aaffmnt affc vel (4a- aas5) (mm + nn) + (8  6aas ) mu-asp
nn
84+5 6aa mn = a affC. Et procedet Extractio
4a- aaff
a parte sinistra, si sit 842 + ft -6aaff, :, 4a -aaff = + 2, quod
seu mntun +
44 - aaf
aggredi possumus, quia f est arbitraria fiet 8a + f - 6aaf = + 8a   2aasf
quod si faciamus + ~ + fiet ff - baa =- 2aa et ff-4aa, et
fiet (8a +164- 244) nn = affC, quod non licet. nam
evanescet mn Ergo fiet + ∞ = et prodibit 8a + f - 6aass - - 8a + 2aaf
seu f + 16a 8aaff. seu ffe4aa =0 a1
et reciditur in absurdum prius incongruum
Sit x = Um + In ert X taC= Umm blm +Au mn
un +aC
siat av = _ ell + q Imn et fint x t BC = ce (4, ggmn + 2e llgimn, et
erit q= 2 al. e 2tarn 22a,mtn,: e, et prodibit aequatio:
Umm +CIImn = ee (4+ aaa (mm +2mu) nn
i nn
+aC
Fingatur
et ob mn-
aa
ee in
e
eel, fint ee (= alleell + aC, sed hoc non licet fieret
e da
4aa
aa
aua
enim c=0. Atiter et ouum taec
aa
2- aamm tbnamn rquam nm
Xtac= in raC
(psam + qan) ppam tepgarmu
Y= tam1 tan
quam
taa (mmrnutimn) ma =aulma fiat am +n) = C
a
ee
ta
ac aa  ( (mn) + uc = ee  10) (mta) non liaet
aa
aV = ar + qsin auvn = aatraguema + 2argtmu
c
fiet
ee 
Au(mn): 2ar Y =  2ll (m +n): ag
fit
qusmins +ggmn = lln
=El (m+n)= ga ac
aC
et 4 ( qu (mon) = U (
ac