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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 3
# Folio 42r, page id 00068000:0083
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axty =ll ax + aZ = mm
ax ay = pp ax - a Z = 99
2aX = U +pp = mm +94
ay + aZ = nn
ay - a Z = p F
Zay = Il - pp =nn +YY 2aZ = mm - 99 = nn = rY
2g = Ex tribus his aequationibus quaeramus l, p, n,r per m et  in medio positas
Ex aeq. 3 esto n+Y = S m + Fg n - Y = a m + = 9, fiet 2n = (S + aym 4 (S - 2) 9
et 2Y = ( - 3 ) +(Z + 2)9. vel compendio 2n = =m + 8G et 2t = d m q
a
Ex aequ. 1 fit l - nm = 99. pp, sit l + m = à G(4) D et l- m = a g (4) ap, unde fiet
2l = (a + 4) 4(4) CL - 2) p et 2m = ( - 4) 9(+) (+ + a) P. seu compendio al - h9() kp 2m - h a h
at 4, seu 2l = 2 2Km + hhr. KK
superest aequatio 2d a praestanda, quae est l- pp = nn +YY, in qua substituen substituendi inventi
valores ipsatum , a, P, Y fiat - Al + app + 4nn + Arr= ohinc prodibit
4Hkum  4 Khhr. - 4+ R mq h- 2hihkK + k gq
atit
+16 aa mm -16 ak mq
X
hk
Afm
aa
aq
dd mm
aa
ht
f2 fg mq
aa
+ 2fg mg
aa
aaht
+ TkKqq
ht
= 1899
a
+ ff
aa 7
2
Hic videamus an liceat destruere mm, faciendo - 4KK + 16 aa + fF + d
aa
tihi
tit
=0; ubi videanas an
aa
sufficiat explicare f et g, quae timplicius exhibentur zerat = S + a et 99 = L - a fiet ff + d = 2s + 2aa
h = 4 4 K= t -  t hh = Ht +.
aa
ad
KK
5
= H
et KK + 16aa = utlina +B Aaa: H 4F + f a aH aa
hh
hh
. An aliter si itidem faciamus
aa
a
1 230 +316 + h
Numerus eonux 10 + ub + 12bb + 13b3+ 14b+ 156 + etc
quadialum 10. +B + 2. 10. 11b + 2. 10. 12bb = 2. 10. 1363 + 2. 10. 14 b + 2. 10. 15b5+ etc
+ 1. 11. 11 bb + 2. 1P. 126342. 1. 1B6Y +2. 11. 14b3+ etc
50. 30 + 2. 50. 51b. + etc
2020 + 2. 20. 21 A etc
E0. 60 + 2. 60.61b tet
aa iaa
2 + 2, et -aHK laa =- AG 
a F16aaH=
+4 + 2aa H + at
+1. 12. 126+2. 12. 136342F
Aliter sunt hes aequalines Upp = mmtgu
25442a - - 3 - 99
aass
ll- pp = nn + VY mm - g9 = nm +rY nn -YI
sit p = l5, et fint l - pp = I= tf + ff N - 1+ 2lf - ff = mm t4q nn +rY, sen l- nn err+ H
rursus 2I - 2lf +ff = mm + 99, fiet l- u nntrr+ss,. 2f 
ubi substitu endo am rursus l = mn + gg- ff, : 2l - f = nn + rr + ff3: 2f, et fint.
2frum +2f99 -(273 = 2l nn + 2l rr + ilff - fnn- frr (f3), fiet
El = 2fmm + 2599 + fnn + frr,:, Ann + frr+S = nn + rr + ff,: F
fit 255mm + 2f5gg- f+ Snntffer = que nn + rr + H)
Rursus ob rumq nm - gq = nn -rr, fiat r = n - g, unde fint mn-99 nn - nd + 2ng-g
et n = min - 99 + gg,: 29 et r = mm -49F99 - (299,: 29
Ergo nn +rr = m + gt + gt - rmm gg (t 2mmgq - 29999) + m . + g2 + g - 2mm49 (2mmag. + 2 9999); +gq
b
seu  nntrr= f(m + 9+g)-mmgqe : gq = 00  Etsum + 2f49- 5 4 (500) = d(0ff f
sit vg m + 8g199. 89
+4f
fin O0) + OOff+ Zs= 2H (mm 199) - 254
et 13O + f = fi  (2(nm +49) - Z H)
et habemus m + q+ g - 2mmgus   _ggH = 2fGg ((2 (mm + 44) = Z fs)
quae estullima aequatio, affungens ad octaon gradu in in, g,e, et ad atum in F, quae efficiura
est ud pationglibus, et ext his tu quatuor vel potrus in effectu 3 liters,
bra sunt aritrarae
es
t
= 4nn
ad na
M = 20 + 21b etc
42 60 +61b + etc
r= 70 +71b + et
l = 10 + 11b A+ 12 bb + 13 U. + et p. = 50 + 5ib + et
-2+ 4