MATHEMATICS 43 



are not essential. They are, however, an entirely normal 

 and legitimate part of the subject, as I have explained above. 



Analytic geometry, the next branch of mathematics 

 usually taught, also has its origin in ancient ignorance. There 

 is no greater reason for its separation from elementary geo- 

 metry than that Euclid and his followers knew nothing about 

 it. The ideas themselves are now fairly widespread. Perhaps 

 measurement of position on the earth by latitude and longi- 

 tude is the best known instance of the general method. In- 

 deed, latitude and longitude give themselves an arithmetical 

 treatment of position that reduces geometry (the study of 

 position in space) to an arithmetical subject by enabling us 

 to substitue these numbers, latitude and longitude, for the 

 position in question. This scheme is also used commonly 

 by surveyors who locate objects accurately by measuring dis- 

 tances from any two lines on the earth's surface. In an 

 entirely different field a similar scheme is commonly used to 

 represent, for example, the fluctuation in the number of 

 students attending this University by representing this num- 

 ber by a curved line in a manner familiar to most of you. 

 The price of wheat, the temperature at different times of 

 day, and an endless number of other statistical figures are 

 commonly represented in the same way. 



If to arithmetic we add merely the shorthand of algebraic 

 symbolism, and denote numbers by letters, we arrive thorough- 

 ly in the sphere of analytical geometry, for we then have, 

 instead of arithmetic representation of position an algebraic 

 representation of position which is indeed the heart of analytic 

 geometry. Sufiice it to say here that this connection between 

 the two subjects, algebra and geometry, leads to easy methods 

 of proof of many geometric propositions, and also to easy 

 methods of geometric representation of algebraic facts. In 

 short, this is the whole of what we call analytics. 



It will seem curious to many of you that I should at 

 this point pass to what is known as the Function Theory, 



