OF NEGATIVE QUANTITIES. 
247 
= 1 + (M — M') + + 
1 +B«®+ 5 ^ + &c. - 1 + (M - M’) x + < M ~ - + &c. 
,B» = M- M' = {“ + &,} 
^ r m 3 m h „ ~\ 
= 2 { m + T + T + &c -} ! 
Let n = £ ; then m = ; 
1+ + i 
Let 1 0 = y ; then y 6 — 1 =0, 
an equation, one of whose roots is 1 - + - ~ — - ; 
i 
substituting this value for IT, 
l + v' - — 3 
2 
m — 
— 1 
1 + V -3 
+ 1 
— 1 + 4/ -3 __ (\/3+ \/^T). V'-l 
3 + */~^3 3 + a/~^3 
_ (\/3 + V -l). V -1 _ 4 / — 1 _ 
( \/ 3 + — 1 ) • V" 3 4 / 3 
•• Bi = 2 {# +*(^) s+ m#) s + & 4 
B = 12 {vl — i (^3) + - &c.j. ■ ; 
Now 12 ^ (-^T;) 3 + 3- (^7§) 5 “ &c - } is a ser i es > which expresses 
the value of the circumference of a circle, whose radius is unity ; 
Let this series = c, then B = c y/~ 1, 
- r* rfi 
+ &c. ; 
••• 1* = > + c*J=i - &S- nrW=i + r 
2.3.4 
.-.We have by means of mere algebraical operations, without the introduc- 
tion of any geometrical considerations, expanded 1* in a series, which involves 
c, the circumference of a circle whose radius is unity. 
And this series was obtained by substituting for l” one of its impossible va- 
lues as they are called. 
