OF NEGATIVE QUANTITIES. 



247 



= 1 + (M — M') + x^ — y:2 — ^ ^^• 



B«M«^« 



(M-M')«x» 



.-. 1 + B w^ + -j^g- + &c. = 1 + (M - M') a? + 12 

 . . un — iM — m — ^^ ^ ^. ^^8 + ^^s ^ gjc./ 

 = 2|^ + y + -5- + &c.|; 



+ &c. 



Let n =ii then m = 



if_l 



1+ + 1 

 Let 1* =3/ ; then ?/fi — 1 = 0, 



an equation, one of whose roots is "^ ~ 

 .*. substituting this value for 1*, 

 1 + v'"^ 

 m = 



- 1 



_ _i + ^_3 _ (^3+ypr).A/-i 



1 + -/-S 



+ 1 



3+ \/—3 



3+ V-3 



(\^3 + ^/'—i).^/-l _ a/-1 



(^3"+ V~^)WS ~ V"3 ' 



...B.i = 2{^+i(^)' + i.(^)%.4, 



Now 12 "I — I — ^ (-7s) + 5 i'TTs) ~~ ^^' f ^s a series, which expresses 

 the value of the circumference of a circle, whose radius is unity ; 

 Let this series = c, then B = c ^— 1, 



.-. 1 = 1 + cxy- 1 - j-2 - ^273 V- 1 + 1727374 + ^^- ' 



.*. We have by means of mere algebraical operations, without the introduc- 

 tion of any geometrical considerations, expanded l' 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. 



