of Edinburgh, Session 1869-70. 57 
adding the respective members of the preceding term to the doubles 
of those of the last, thus— 
1 1 3 7 17 41 99 239 » 
0’ 1’ 2’ 5’ 12’ 29’ 70’ 169’ C °' 
we form the well-known series converging to the ratio of the 
diagonal of a square to the side. 
Beginning with the terms 0, 1, if we add together the last two, 
thus— 
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, Ac., 
each term bears to the succeeding one a ratio approaching to that 
of the side of a regular pentagon to the diagonal thereof. 
If we assume the three terms 0, 0, 1, and continue the progres¬ 
sion by adding to the double of the last term, the difference of the 
two preceding ones, thus— 
0, 0, 1, 2, 5, 11, 25, 56, 126, 283, 636, 1429, &c., 
the ratio of each term to the following approaches to that of the 
side to the greater diagonal of a regular heptagon. 
Or again, beginning with the same three terms, if we form a 
progression by deducting the antepenult from the triple of the last 
term, thus— 
0, 0, 1, 3, 9, 26, 75, 216, 622, 1791, 5157, &c., 
we obtain an approximation to the ratio of the side to the longest 
diagonal of a regular enneagon. 
From these examples it would appear that important results may 
be expected from the study of this branch of Logistics. Now, the 
roots of quadratics were reached by the comparison of two magni¬ 
tudes, wherefore those of cubics may result from the comparison of 
three incommensurables; and analogously for equations of higher 
degrees. The comparison of several magnitudes thus forms the 
subject of the paper. 
Assuming three homogeneous quantities, A, B, C, arranged in 
the order of their magnitudes, we take the second B as often as 
possible from the greatest A, and obtain a remainder less than B; 
this remainder may or may not be greater than C. If it be greater, 
we take C as often as possible from it, and obtain a remainder D 
less than C, the least of the three quantities. B, C, D may now be 
