JOURNAL OF THE 



AC AB 



[im. 



A^B^ A^B^ 

 is what is called the incommensurable ratio, AB : A^ B^ 

 It is assumed, of course, that the successive units of meas- 

 ure all exactly divide A* B\ It may happen that one of 

 these units applied to AB will cause the point C to lie very 

 near the point B, but for a smaller unit the distance CB 

 will be greater than before, so that the variable CB is 

 sometimes decreasing and then again increasing, but as it 

 is always less than one of the parts into which A^ B' has 

 been divided, it can "become and remain" less than one 

 of the parts or less than any finite number that may be 

 assigned, however small; hence zero is its limit by the 

 definition. 



If in the ratio above we take A^ B^ as i (one foot say), we 

 have, limit i\C = AB from the last equation. AB and 

 AC can thus be regarded as incommensurable and com- 

 mensurable numbers respectively, and we see from the 

 above that an incommensurable number, as AB, is the 

 limit of a commensurable number as the number of parts 

 into which unity is divided is indefinitely increased. 



The student of algebra and geometry is familiar with 

 many applications of the theory of limits, such ^s: limit 

 of (i -f- yz -f y -f i^ 4- ...)--= 2, as the number of 

 terms of the series is increased indefinitely; the circle is 

 the limit of a regular inscribed or circumscribed polygon, 

 as the number of sides is indefinitely increased, etc., etc.; 

 so that no more illustrations need be given if these are care- 

 fully studied in connection with the first definition given 

 above to show that it is complete and meets fully every 

 case that arises. 



It may be observed, too, that although we can express 

 the length of a straight line or the perimeter of a polygon, 

 in terms of the length of a straight line, taken as a unit 



