and on Algebra as the Science of Pure Time. 369 
or a, is that since (by the 24th article) the square remains positive and unchanged 
when the root is changed from positive to contra-positive, in such a manner that 
Ola, xO "4% a> 256.) 
the square aa must be conceived as first constantly and continuously decreasing or 
retrograding towards 0, and afterwards constantly and continuously increasing or 
advancing from 0, if the root a be conceived as constantly and continuously increas- 
ing or advancing, in the general progression of ratio, from states indefinitely far from 
0 on the contra-positive side, to other states indefinitely far from 0, but on the posi- 
tive side in the progression. 
On Continued Analogies, or Series of Proportional Steps; and on Powers, and 
Roots, and Logarithms of Ratios. 
28. Four effective steps ab v’ b’ may be said to form a continued analogy or conti- 
nued proportion, a and b” being the extremes, and b and b’ the means, when they are 
connected by one common ratio in the following manner : 
as 
6S = (257.) 
and if we denote for abridgement this common ratio by a, we may write 
Be eX Bs Dt =e XG) X, Big Di) a MAL Xone (258.) 
Reciprocally, when » v b’ can be thus expressed, the four steps ab b’ b’ compose a 
continued analogy ; and it is clear that if the first extreme step 2 and the common 
ratio a be given, the other steps can be deduced by the multiplications (258.) It is 
easy also to perceive, that if the two extremes a and bv’ be given, the two means b and 
bv may be conceived to be determined (as necessarily connected with these) in one and 
in only one way ; and thus that the insertion of two mean proportionals between two 
given effective steps, is never impossible nor ambiguous, like the insertion of a single 
mean proportional, In fact, it follows from the theorems of multiplication that the 
product a x a x a, which may be called the cube of the number or ratio a, is not 
obliged (like the square a x a) to be always a positive ratio, but is positive or contra- 
positive according as a itself (which may be called the cube-root of this product 
