and on Algebra as the Science of Pure Time. 337 
multiplying b as a multiplicand by the fractional number 5 as a multiplier, or finally 
by the complex act of first submultipling b by the denominator », and then multi- 
pling the result 2 by the numerator v. Under the same conditions, it is evident that 
we may return from c¢ to b by an inverse or reciprocal act of fractioning, namely, by 
that new complex act which is composed of submultipling instead of multipling by », 
and then multipling instead of submultipling by »; so that 
b="x ec, whene=~xbd: (123.) 
v B 
on which account we may write 
=!@x(2x»b), ande=~x ("x ce), (124.) 
v liad B v 
whatever (effective) steps may be denoted by » and c, and whatever (positive or con- 
tra-positive) whole numbers may be denoted by » and v. The two acts of fractioning, 
marked by the two fractional numbers ~ and =. are therefore opposite or reciprocal 
acts, of which each destroys or undoes the effect of the other; and the fractional 
numbers themselves may be called reciprocal fractional numbers, or, for shortness, 
reciprocal fractions : to mark which reciprocity we may use a new symbol 4, (namely, 
the initial letter of the word Reciprocatio, distinguished from the other uses of the 
same letter by being written in an inverted position,) that is, we may write 
=u 
Lu 
; (125.) 
v 
Be? 
whatever positive or contra-positive whole numbers may be marked by » andy. In 
this notation, 
v v wh v 
cf hile = tt) tS ae 126. 
Ysu(ut) sab at, (126.) 
or, to express the same thing in words, the reciprocal of the reciprocal of any frac- 
tional number is that fractional number itself. (Compare equation (57.) ). 
It is evident also, that 
1 ‘ 
ecto andb=4 xo, ifb=p xa; (127.) 
