and on Algebra as the Science of Pure Time. 359 
will tend to the given ratio of largeness denoted by ) ; results which we may express 
by the following written sentence, 
if Lc=sB, then L x=1 and Ly=d, (216.) 
prefixing the symbol 4, (namely the initial letter L of the Latin word Limes, distin- 
guished by a bar drawn under it,) to the respective marks of the variable moment c 
and variable ratios’ 4, y, in order to denote the respective limits to which those 
variables tend, while we vary the selection of one of them, and therefore also of the 
rest. Again, we may choose the moment c nearer and nearer to p, and then the 
ratio x will tend to the given ratio of largeness denoted by b, while the ratio y will 
tend to the ratio of equality ; that is, 
if Lc=p, then L w=), Ly=1; C217.) 
and if we conceive a continuous progression of moments c from-B to p, we shall also 
have a continuous progression of ratios x, determining higher and higher degrees of 
relative largeness (of the increasing step c—a as compared with the fixed step B—) 
from the ratio of equality 1 to the given ratio of largeness >, together with another 
continuous but opposite progression of ratios y, determining lower and lower degrees 
of relative largeness (of the fixed step D—A as compared with the increasing step 
c—A) from the same given ratio of largeness ) down to the ratio of equality 1; so 
that we cannot avoid conceiving the existence of some one determined state of the 
progression of the moment c, for which the two progressions of ratio meet, and for 
which they give . 
D—A _C—A 
(iy. i 
aaxb=y=a, that is : (218.) 
having given at first y > #, and giving afterwards y < #. And since, in general, 
D—A Cc—A D—A 
C—sA) B-—A B—A 
, that is, (a a x b).x x=), (219.) 
we can and must by (218.) and (214.), conceive the existence of a positive ratio a 
which shall satisfy the condition (209.), ax a=), if b > 1, that is, we must conceive 
the existence of a positive square-root of 6, if 6 denote any positive ratio of large- 
ness. A reasoning of an entirely similar kind would prove that we must conceive the 
existence of a positive square-root of 6, when ) denotes any positive ratio of small- 
ness, (b < 13) andif b denote the positive ratio of equality, (J=1,) then it evi- 
dently has that ratio of equality itself for a positive- square-root. We see then by 
