and on Algebra as the Science of Pure Time. 355 
And all the theorems of this article, respecting pairs of ratios or of steps, may easily 
be extended to the comparison and combination of more ratios or steps than two. 
In particular, when any number of ratios are to be added or multiplied together, we 
may arrange them in any arbitrary order; and in any multiplication of ratios, we 
may treat any one factor as the algebraic sum of any number of other ratios, or 
partial factors, and substitute each of these separately and successively for it, and the 
sum of the partial products thus obtained will be the total product sought. As ar 
example of the multiplication of ratios, considered thus as sums, it is plain from the 
foregoing principles that 
(d+ce)x(b+a)= fdx(b+a)} + {ex(b+a}} 
= (dx b) + (dxa) + (cx b) +(exa) 
=db+da+cb+ea, (199.) 
and that 
(b+a)x(b+a)=(bxb) + (2xbxa) + (axa) 
—bb+2ba+aa, (200.) 
whatever positive or contra-positive ratios may be denoted by a bc d. 
And though we have only considered effective steps, and positive or contra- 
positive ratios, (or algebraic numbers,) in the few last articles of this Essay, yet the 
results extend to null steps, and to null ratios, also; provided that for the reasons 
given in the 20th article we treat all such null steps as consequents only and not as 
antecedents of ratios, admitting null ratios themselves but not their reciprocals into our 
formule, or employing null numbers as multipliers only but not as divisors, in order 
to avoid the introduction of symbols which suggest either impossible or indeterminate 
operations. 
‘On the insertion of a Mean Proportional between two steps ; and on Impossible, 
Ambiguous, and Incommensurable Square-Roots of Ratios. 
24. Three effective steps ab b’ may be said to form a continued analogy or con- 
tinued proportion, when the ratio of v to b is the same as that of b to a, that is, 
when 
b 9g 
= =; (201.) 
VOL, XVII, 2 Beh Agy 
