6 SCIENTIFIC METHOD 



the mathematical, the empirical, the explanatory, and the 

 verificatory~rnethod, and shows to what subjects each of 

 these four methods must be applied. We shall roughly 

 follow him, but endeavour also to show that his account 

 of deductive methods is imperfect. 



The first method is the mathematical, which consists 

 in the main of principles ancTdeductions about numbers 

 in arithmetic and algebra, and about magnitudes in 

 geometry. Mathematical principles are so obvious as 

 to seem a priori. It is so quickly perceived that one 

 and one are two, and a triangle is a three-sided recti- 

 lineal figure, that philosophers have been led to suppose 

 that the mind knows these principles by adding some- 

 thing to experience. Connected with this a priori 

 hypothesis is another a constructive hypothesis that 

 the mind beginning arithmetic with abstract units con- 

 structs numbers, and beginning geometry with points 

 constructs lines, surfaces and solids of three dimensions, 

 and might according to some have constructed mag- 

 nitudes with less or more dimensions. These views 

 have the history of mathematics against them. Man 

 began arithmetic with experience of the number of his 

 fingers and toes, and geometry with experience of the 

 magnitude of his hands, feet, and arms. He went on to 

 use these concrete bodies as standards to measure other 

 bodies. Geometry means the measurement of lands ; 

 and the most ancient Egyptian book of mathematics, 

 the papyrus of Ahmes, about 1 700 B.C. 1 , measures barns, 

 pyramids, and obelisks, and treats solid bodies before 

 proceeding to abstract surfaces. Mathematics, in short, 

 , began with concrete bodies, such as could only be reached 

 by means of experience, and only gradually receded 



1 See Gow's History of Greek Mathematics, pp. 16 seq., 126 seq. 



