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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 184r, page id 00068195:0367
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Sed cum successus videatur mirus, et pene, incredibilis; operae pretium, est exemplum
ipsum Fractare speciali seu proprio calculi, ut facilius appareat si quis adsit lapsus
 ergo aequatio - aayy + mn XX +ac (mn) X + aa cc = 0. Hanc ut praestemus prationalter
ad lustitutum nostrum, sumamus _ X: a = 100V + 10Ia,: 200V+ 201a. et y. a = 300Y + 7019) :, 200V + 201 a. Fiat
+ 201. 201..
1. 101. 200..
aequ. 0.
Ex termino si
200cC + C (mtn) 100. 200 + _ 100 mn = 300.  a=v
ubisi faciamus 100 mn -300 aa =v. fiet
300: 100 = Vmn : a et 200 = - 100, m+n,
Ex Termino summo destructo fiet
201CC - + 101 (mtn) 201 C + 10emn - 301aa = 0
fiet 201C = C - = 101 (m +n) + L Soi qu m An)- toinm + 4. 3ota)
aequ. 0 ubi ut destruamus quod est in vinculo radicali fiet ,onm - 2toi mitaedoia
2. 200.
quae aequatio est imaginaria, sed hoc nos non moramur, et fiet
. 301 : 101 = m - n, Y-1, : 2 et fiet 201: 10, = t1): 2C
Rursus ex Termino medius destruendo est 201 = (t300. 30l aa - 2. 190. 101 mn - 101. 200 (n + n) C, 1 2 20
seu  100. 101 gu (m+n)  = 100 (mn)  (m = n) D 101 = 2. 100. 101 mn + 101. 100 (mAn: C) om + r) C
opparet divisis omnibus per  100. 105 evanescere
sitasrrationalem, aut ullam novam ad hibes
restrictionem;  sed maneat 201 c = = 101 (mtr) + L 1 (101101 qu ( m - n) + 4. 301.301 a4)
Eno non licet in valere ipsius 201. Destinete
sed ex Termino medio fil 201 = 2. 300. 301 aa - 2100. 101mn - 101. 200(M + n)C ; 2. 200 ∞C + 100 (n + n) C
seu 201C =  2. 100 mn) 301 - 2. 100. 101 mn + 101. 19qu (m 12), :,  2 man) Y + 1e (u i)
seu 201C = 2. 100. 301 armn + q (mm. + nn) 100. 101, : - f. 100 (m + n) 
= ∞= 101 (mFn) + L (1oi qu (Mn) + 4. 301 aa)
(1430
seu 2. 100. 301 a tmn = 2. 100 101 mn (4 100. 101 qusm tn) = (10B. 101 qu ( +n) = 100 (m + 1) L(io qu mg
4. 100. 30i aa mn) + 4 G00dioi mn _ 100 iqu emn. n) +(30) vitum + nn)  4 100i 30i m aa
_. tot. et amn Amn 4. 30 mntnn) aa + 8. 101. 30amnumn + 101 qu (mm. nn) = 0
seu 
= 8. (1003101. 30lamnima
ioimmnn
et fiet 2. 301 a =- 2. N. masma +)  m id - (qu (mm. na) - 4 mmnn) nm + rn)). n
F 200. 200 Cc + 200C. W n) 1009 + 105. 100 a f - 300. 300 aa = 0
00 : 100. = sf: a) - taf: a
201. 201CC + 201C 1019 8 a(, 101. 101 af = 0. Ergo 201C - - 5 1619 + 5 110199.  as
Ergo 200C + 100g = 0 seu 200: 100 = = g: c
301. 301 aa
2 200, 201cc f201 c. 1009 + 101. 200c9 + 2. 100. 101 af - 2. 300. 301 aa =
Hs ex deniue ex permino medio est
X = - = 11 + Y (i- 4. 12)) q pro 300 et pro 700 ponendo valores per 100, et 300 a = 100 Yaf 200C - 
fis -(2. 1009201C+201C. 1009) 101. 10099 + 2. 100. 1014f - 2. 100 1a f 301a = 0
et divisis omnibus per 100 fit 201C9 = - 1099 + 2. 101af - 2. 301 a 1a f
Ergo ex priori valore ipsius 201C fit (- 101 99 + (  (10i gg - 4. 10r a5 + 4. 301 a a)-
sit 9g aaf = ah, et fiet
9 (101 ah + 4. 301 aa)   101 ah - 4. 3oratas, et quadrando
=(2, 10199 + 4 101af - 4. 301 a ra f
utrobique siet roj atigu + 4. 50i aague - (10i aatt) + 16 20i a3f + 8 101. 30+ aa h lat
10X = - 4 11 4 3 (1. 4.10. 13)
aequ.0
100. 101 (m +n) C (- 1m 4 1). 20)
10. 10XX + G. 10X. 11 +10.12
200. 200.. +2. 200201.
AX + IX + Z 1 = L, 11 - 12)
19XX FIIX + 12 =0
rqu 411 = 10. 12
seu 62
XX + IPX + 1220
10XX + 10X. 11 + L. 11
Aayy = 24, 300. 300VP +2. 300301 aV + 301. 301 aa1
+mnXX = +mn, 100. 100.. + 2. 100. 1011. +101. 101..
raesnm) X = csmn), 100. 200 .- + 1. 100. 201 .. 4 101. 201..
taacC = cc,
estructo
Te
44 2elaa 9g
ccuce
seu (1019 = 4. 10i afgg + 4. 301 aagge = (10 q -(8) 101 a599 + 8 agg Saf. 10130) + rt s 3eur. B 
4. Ze aavt
1o1 aatihreroias
B F16. 10C a t = 32. 101. 301a f (af + 16301 af bi 
Sit jam Aeg ag - 4 af = ah fiet 4. 301  ah 8, 4 4. 10 afah  8. 101 30r a atratat Z0
301 aR - 101 sa f =0 videamus au res actu succedat
Is vulo Ergo 201C = L 101G +  10, Cah + a(af vat)) sei 2ot  toc (:2C) g+ Hatita aalsi
seu
200 =  100g: C  300 300 = 100 raf ia 301 =10Naf: a
Em XB: a = 100B + 101a, :, = 100(9: C)D + 1O1 C2C) 9+(ah + 4 a a f : h)1)
seu 201 = 101 (: 2c),94 Hah + 4 aa (f: h))
sit formula af xx + BC9X + CC, ubi pro X: a substituatur vator 100∞ + 11a, : -100G: d)v+ 201
afxX = aB p0o po V + 2. 100af101 aV = 101a s1o1 aa
G 109.10)
F10099..
1. 10059201 _.. + 101. 201 aa C9
1. 10099101().
= (2.) 10029. 201 . . + 201 201 aa cc 
100. hocas
n 100. 101 39
ubi 20+ postea etam expli
Sed hinc ortur absurdim
nam si 301: 101 = taf: a
fiet 301: 101 = 300: 100
quod non licet. Itaque
res non procedit. 
ta y esset in ratione data
ad X, contra hypothesin
C9X=
cc =