34 ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 



ised, often after the lapse of centuries. The history of mathe- 

 matics shows numerous instances of mathematical results which 

 were originally the outcome of a mere desire to extend the science, 

 suddenly receiving in astronomy, mechanics, or in physics practical 

 applications which their originators could scarce have dreamt of. 

 Thus Apollonius erected in ancient times the stately edifice of the 

 properties of conic sections, without having any idea that the plan- 

 ets moved about the sun in conic sections, and that a Kepler and a 

 Newton were one day to come who should apply these properties to 

 explaining and calculating the motions of the planets about the 

 sun. The question of the practical availability of its results in other 

 fields has at no period exercised more than a subordinate influence 

 on mathematical inquiry. Particularly is this true of modern math- 

 ematical research, whether the same consist in the extended de- 

 velopment of isolated theories or in uniting under a higher point of 

 view theories heretofore regarded as different.* 



This independence of its character has rendered the results of 

 pure mathematics independent also of the accidental direction which 

 the development of civilisation has taken on our planet ; so that 

 the remark is not altogether without justification, that if beings en- 

 dowed with intelligence existed on other planets, the truths of math- 

 ematics would afford the only basis of an understanding with them. 

 Uninterruptedly and wholly from its own resources mathematics has 

 built itself up. It is scarcely credible to a person not versed in the 

 science, that mathematicians can derive satisfaction from the com- 

 fortless and wearisome operation of heaping up demonstration on 

 demonstration, of rivetting truth on truth, and of tormenting them- 

 selves with self-imposed problems, whose solution stands no one in 

 stead, and affords satisfaction to no one but the solver himself. Yet 

 this self-sufficiency of mathematicians becomes a little more intel- 

 ligible when we reflect that the progress which has been made, par- 

 ticularly in the last few decades, and which is uninfluenced from 

 without, does not consist solely in the accumulation of new truths 



*Cf. Felix Klein, "Remarks Given at the Opening of the Mathematical and 

 Astronomical Congress at Chicago." The Monist (Vol. IV. No. i, October, 1893). 



