72 A SHORT HISTORY OF SCIENCE 



and P respectively. They are to be so revolved if possible that they 

 shall cross at right angles at T, and that S T and TR shall be respect- 

 ively equal to the lengths between which mean pro- 

 portionals are to be inserted. Then by similar triangles 



ST:PT = PT:QT = QT:RT 



PT and QT are the required mean proportionals. If 

 ST is taken equal to twice TR the special case of the 

 duplication of the cube is represented. 



To Plato is attributed a systematic method for finding numbers 

 which may be sides of right triangles, his method being essentially 

 an extension of the Pythagorean already described. Plato's 

 Timaeus dialogue is indeed an important source of our information 

 in regard to Pythagorean mathematics. Plato speaks with em- 

 phatic scorn of the shameful ignorance of mensuration on the 

 part of his countrymen. 



He is unworthy of the name of man who is ignorant of the fact 

 that the diagonal of a square is incommensurable with its side. 



While predominantly interested in geometry, Plato's arithmetical 

 attainments were considerable for his time. He made, for ex- 

 ample, a correct statement about the 59 divisions of 5040. 



Arithmetic has a very great and elevating effect, compelling the 

 soul to reason about abstract number, and if visible or tangible ob- 

 jects are obtruding upon the argument, refusing to be satisfied. 



Plato, Republic. 



... It would be proper then, Glaucon, to lay down laws for 

 this branch of science and persuade those about to engage in the 

 most important state-matters to apply themselves to computation, 

 and study it, not in the common vulgar fashion, but with the view 

 of arriving at the contemplation of the nature of numbers by the in- 

 tellect itself, not for the sake of buying and selling as anxious 

 merchants and retailers, but for war also, and that the soul may 

 acquire a facility in turning itself from what is in the course of gen- 

 eration to truth and real being. Plato, Republic. 



