and on Algebra as the Science of Pure Time. — 357 
example, it is impossible to insert a mean proportional between the two contra- 
directional (effective) steps a and © 9 a, that is, it is impossible to find any step b 
which shall satisfy the conditions of the continued analogy 
: (205.) 
or any number or ratio a which shall satisfy the equation 
axa=09: (206.) 
whereas it is possible to insert in two different ways a mean proportional b between 
the two co-directional (effective) steps a and 9 a, or to satisfy by two different steps 
b (namely, by the step 3 a, and also by the opposite step © 3 a) the conditions of the 
continued analogy 
ee (207.) 
and it is possible to satisfy by two different ratios a the equation 
axa=9, (208.) 
namely, either by the ratio 3 or by the opposite ratio 6 8. In general, we may 
agree to express the two opposite ratios a which satisfy the equation 
axa=b (> 0), (209.) 
by the two symbols 
/b (> 0) and 0 vb (<0), (210.) 
b and vb being positive ratios, but © vb being contra-positive ; for example, 
/9=3, OvV9=08. (211.) 
With this notation we may represent the two opposite steps of which each is a mean 
proportional between two given co-directional (effective) steps a and v, by the 
symbols 
Wena and O = x a3 (212.) 
and shall have for each the equation of a continued analogy, 
b! rw yp =; 
Wf Be Peak, Dey apa, (213.) 
