DEVELOPMENT OF MATHEMATICAL THOUGHT. 731 



diagonal ? Assume that we prolong the side of the square 

 indefinitely, we have a clear conception of the position 

 of the numbers 15, 20, 30, &c. ; but what is the exact 

 number corresponding to the length of the diagonal ? 

 This led to the invention of irrational numbers : it 

 became evident that by introducing the square root of 

 the number 2 we could accurately express the desired 

 number by an algebraical operation. But there are 

 other definite measurements in practical geometry which 

 do not present themselves in the form of straight lines, 

 such as the circumference of a circle with a given radius. 

 Can they, like irrational quantities, be expressed by 

 definite algebraical operations ? Practice had early in- 

 vented methods for finding such numbers by enclosing 

 them within narrower and narrower limits ; and an 

 arithmetical algorithm, the decimal fraction, was in- 

 vented which expressed the process in a compact and 

 easily intelligible form. Among these decimal fractions 

 there were those which were infinite — the first instances 

 of infinite series — progressing by a clearly defined rule 

 of succession of terms ; others there were which did not 

 show a rule of succession that could be easily grasped. 

 Much time was spent in devising methods for calculat- 

 ing and writing down, e.g., the decimals of the numbers 

 TT and c} 



It will be seen from this very cursory reference to 

 the practical elements of mathematical thought how 

 the ideas or mental factors which we deal with and 



' The transcendent nature of the 

 numbers e and it was first proved 

 by Hermite and Prof. Lindemann. 

 The proofs have been gradually 



simplified. A lucid statement will 

 be found in Klein's ' Famous 

 Problems,' p. 49 sqq. 



