2 Lamb, on Continuity. 



geometrical notion of magnitude is adopted. More pre- 

 cisely, it is assumed that every magnitude of the particular 

 kind under consideration can be represented by a length 

 O M, measured on an unlimited straight line X'X from 

 a fixed origin on it, to the right or left, according 

 as the magnitude is positive or negative. To every 

 magnitude of the kind in question there corresponds, 

 then, a definite point M, and when we say that a 

 magnitude admits of " continuous variation," it is implied 

 that the point M may occupy any position on the line 

 X'X , within (it may be) a certain range. 



Algebraically, any magnitude is represented by a 

 symbol, such as x, denoting the ratio which it bears to 

 some standard magnitude of its own kind. It is im- 

 material, for our present purpose, what basis we adopt 

 for a precise definition of the term " ratio." 



2. Suppose, now, that we have an endless sequence 

 of magnitudes of the same kind 



Xi, x 2 , x 3 , (i), 



each greater than the preceding, so that the differences 



are all positive. Suppose, further, that the magnitudes 

 (1) are all less than some finite quantity a. The 

 sequence will, in this case, have an "upper limit"; that 

 is to say, there will exist a certain quantity /ul, greater 

 than any one of the magnitudes (1), but such that, if 

 we proceed far enough in the sequence, the members 

 will ultimately exceed any assigned magnitude which is 

 less than /x. 



This has been justly characterised by Dedekind* as the 

 fundamental theorem of the Calculus. The proof, from 

 the geometrical point of view, is of course almost intuitive. 



* Stetigkeit und ivvationale Zahlen Brunswick, 1872. 



