MATHEMATICS 47 



increases. This is a case of a limit of a sum ; it is typical 

 of a large group of practical cases which arise in geometry 

 and in science, in which we wish to find the total limit of a 

 variable sum. \\'e call the study of such limits the Integral 

 Calculus. Indeed, the integral calculus includes also a most 

 ingenious device for finding the limits of such sums, but that 

 is neither here nor there; the subject-matter is just such limits 

 as I have mentioned. To illustrate their occurrence in science, 

 I will say that the work done in walking up a hill can be 

 figured out step by step, even though the inclination of the 

 hill change from step to step. To find the work done on a body 

 which slides up a hill without taking any steps we should 

 proceed to a limit in which the number of steps increases with- 

 out any bound. This is typical of the problems which occur 

 in the integral calculus. 



Finally, another type of limit which gives rise to a chap- 

 ter in mathematics is that of I ti finite Series, of which the 

 best known examples are the geometric progressions, like 1 



plus ^ plus %. plus ys plus etc., forever. Suffice it to 



say that such infinite series occur in mathematics and that 

 we cannot avoid them. That they depend upon limits will 

 be clear if I point out that we simply cannot add all of the 

 terms together, that what we mean is therefore the limit 

 approached as we add more and more and more terms. Such 

 series occur in great profusion in all sciences which have 

 advanced to the standpoint of mathematical expression ; in 

 physics, in engineering, in astronomy, their occurence is con- 

 stant. These form the subject matter of one great branch 

 of mathematics — the study of infinite series, or, more broadly, 

 the study of infinite processes. 



Perhaps I have reached a limit past which I should not 

 go in an attempt to explain roughly the various fields into 

 which mathematics goes in its process of drawing necessary 

 conclusions. Perhaps no reason exists why mathematics 

 should not proceed to other fields and extend its methods to 



