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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 94v, page id 00068200:0190
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btot. 1707
Quaeritur S dx t Cx  + ab XX + c) Lsydx sit x = tal + san
fiet XX = al. + 2atla  x = aa l + 2 aa ln + 4 aa l SIn LGy =Cqu (+ fY In)
fiet X tabXX + ac = aal + 6aa ln + aabl + ac + 4aa L2 An = yy = qu (ap + f L lu)
yy
in
= aapp +ffln -
+ 2aa b.
+ 2a fp sIn ubi s constans
fit BC 2al. + ab = fP et p = r + 2a l + ab
8
et aapp + Bl = a aa ll + & Ca In + 4 aa b l + aa bl =  H + 6 ln + bL + ac
seu (1 4 84g eua seu  (44a- f) H + (8aa 6) In + (4 aa - s5) bl ++ aa bb - ac ff = 0
Nn
nn +A aa
Yn
compendo
d + p In + bl + aq.
 In +   bl + aabb -ac f= 0= nn 
seu l + 8aa - 6f1(4
HK
4aa -ff
4aa - 
Examinavi an l et n possent reddi simplices, ope constantis arbitria
no
hujes nm
l= 10V + H et n = 20V +2+. quod F sed res non procissit, quod etiam praevideri poterat, quia
non procedit, si l’exponas per 30n +31, cum tamen prior hypothesis ad hanc simpliciorem utique reduci
poffit Efficiendum necups ut l, et n sunt simplices integrot ex eadem  variabili. Itaque nec
speranda aequationis depressio aut radicis extractio.
Cogitavi eanda an non fingi possit
aliquod imaginarium. Notav scilicet aequationem hujusmodi biliteram abmptotam posse
prodninem homoeoptotam, produci posse ex ravice non homeveoptota
fX.9X  +ah ducatur in-gx + fy + ah, popuodit - fg(xx + f+ Xy + atis 25+
rgy
seu Xx - 5t +9g xy +  abg - abif ( C - natih = 0
g
quae aequatio comparetur propositae simili ex +mxy +n GE + ap =
y
Ad
ponames h= a, nam nihil prodest; porn 5 +d = -   f E
n
12
a
=ahig 52)
5
e
-+
a
taahh
Ex ret aeg. 1 et 3 fint B + gg = aam: B. Ergo ex 4 et 3 firt f+g9 + 2fg = aa(m: p) -2(a. p)
Ergo f +g = (: p) s(mp - ap) similiter ex 4 et 3 fit H+99 - 259 = aa (m: p) + (a: p) adeoque
f-g = (:  Y (mptap). Ergo 2f = (: p), impFap)  SEmp ap) ut similiter erit
29= (. P 3d emp ap) - Emp Fap). Ergo ob aeq. 2 quae nondum usi sumus, seu 25 29 raa fiet
+
mpFa p) + Y(mp hap) Ampiap)- temp dap
n
2a
p, mp Fap) = emp ap) + semp rap t mp dap)
mp F ap = mp + ap
ubi reducendo ad communem
denominatorem fit
adeoque 2p . (mp vap)
aap
n
2a
n seu n = a  (mp. + rap)
a
Adeoque tandem un = mp + 2ap, quae est determinatio reperienda, in aequatione proposita
ubi p est terminus ultimus, n penultimus, n antepenultimus. Itaque jam pro X et y redeundo
ad priores literas b, n, firt  erit pultunus terminus aabb - acss penultimus erit b
+aa - ff,
antepenultumus erit daa -bHE et fiet bb = 8 B, aabb - a C st + 2 (aabb - ac 55) f aa - B)
Ana- H
et
W  = ud qu (499 - f5)3
seu (16abb =, e au sob) ans abld) = (8 a bb) BaC s6abb  + 6uc 5  8 a bb - 2a bitsp. 8 ses t. d )
dH
Aabb +E = aC s sit
1n
i eraneseente  a bb, potest aequatio feliciter dividi per s et fun multis destructis fiet
 (6 C f - 8aaf + 16a = seu f = 4aa 20.
sad eum valorem
ad ribere non licet, nam per eum imogniae indeterminatae in aequ. O Qvanescunt
quis si fariamus l = a fav +214, p0V + 11 1 n = 220a V + aa, :, 10v+ 1)
fiet Aaavb + 2. 21 aav + h1. 21 aa.
2030aa WV + 2. 20av +a
20 apWV + 1. 20. 2 apV +21aap
+ 1. aaV
1Ba bvV + 1. 11 ab V
+ 1. 10. 21 abV + 11. 21 ab
10. 20 ab WV + 1. 11. 20abv + H. ab
L
+1. 10alv
ita endu nihilo
minus procede
reductio qua
diaturae
 10V + 11
av +21a
10a vv + 11 aV
+ c0. 21 av + 11. 21 a
10v + 1
2oav + aa
10. 10 aquV +2. 10. 1 agu + H1. Haq
10.20 avv + n. 21
110ag0.
Ex primo termino habebitur 20 ex ultimo 21,
controhendi causa fiat divellatur terminus prinus
et
et
ex medio, 10
a + 20. 20a + 20 p = 0
ab + 20ab+ joaque = 0. quanquam nec
et fiat 
it opus, possumus enm 10 dissimulare e
cerminum primum ut
solum 20 quaerere, Ex
Cujus valorem subssituendo in . termino ultimo habemus 1
30 aV + 31  et quae rere solum 21, et
caeterae se at 20 et 31 proderunt harm dutae obserrandae utiliad vitandes cactum
e
errores.
prostat taare
n= 30 a + g: Av+2
ime proftatfmece
= 10 a B + H1,: V + 21
nnod.  2o ess fprturm f. F