70 A SHORT HISTORY OF SCIENCE 



In the Laws he advises the study of music or the lyre to last 

 from the age of 13 years to 16, followed by mathematics, weights and 

 measures, and the astronomical calendar until 17. For a few picked 

 boys on the other hand in the Republic, he recommends before they 

 are 18, abstract and theoretical mathematics, theory of numbers, 

 plane and solid geometry, kinetics, and harmonics. Of arithmetic 

 he says, "Those who are born with a talent for it are quick at 

 learning, while even those who are slow at it have their general 

 intelligence much increased by studying it." "No branch of 

 education is so valuable a preparation for household management 

 and politics and all arts and crafts, sciences and professions, as 

 arithmetic; best of all by some divine art, it arouses the dull 

 and sleepy brain, and makes it studious, mindful, and sharp." 



The geometrical Greek view of numbers, exemplified in our 

 use of square and cube in algebra, is well illustrated by Thesetetus, 

 who says to Socrates that his teacher 



was giving us a lesson in roots, with diagrams, showing us that the 

 root of 3 and the root of 5 did not admit of linear measurement by the 

 foot (that is, were not rational). He took each root separately up to 

 17. There as it happened he stopped, so the other pupil and I de- 

 termined, since the roots were apparently infinite in number, to try 

 to find a single name which would embrace all these roots. We di- 

 vided all numbers into two parts. The number which has a square 

 root we likened to the geometrical square, and called 'square and 

 equilateral' (e.g. 4, 9, 16). The intermediate numbers, such as 3 and 

 5 and the rest which have no square root, but are made up of unequal 

 factors, we likened to the rectangle with unequal sides, and called 

 rectangular numbers. 



Under Plato's influence mathematics first acquired its unified 

 significance, as distinguished from geometry, computation, etc. 

 Accurate definitions were formulated, questions of possibility 

 considered, methods of proof criticized and systematized, logi- 

 cal rigor insisted upon. The philosophy of mathematics was 

 begun. The point is the boundary of the line; the line is the 

 boundary of the surface ; the surface is the boundary of the solid. 



