DEVELOPMENT OF MATHEMATICAL THOUGHT. 697 



The attraction of the heavenly bodies varies with the 

 distance, the velocity of a falling stone or the cooling 

 of a hot body varies with the interval of time which 

 has lapsed or flown. We are now so much accus- 

 tomed to represent such dependence by curves drawn on 

 paper, that we hardly realise the great step in advance 

 towards definiteness and intelligibility that this device 

 marks in all natural sciences and in many practical 

 pursuits. But the representation of the natural con- 

 nections of varying quantities by curves also forms the 

 connecting link with the other class of researches just 

 mentioned. Descartes had shown how to represent 

 algebraical formula? by curves in the plane and in space ; 

 and at the beginning of the nineteenth century this 

 method was modified by Gauss and Cauchy so as to 

 deal also with the extended conception of number 

 which embraced the imaginary unit. Two questions 

 arise, Is it possible to represent every arbitrary de- 

 pendence such as we meet with in the graphical descrip- 

 tion of natural phenomena by a mathematical formula — 

 i.e., by a formula denoting several specified mathematical 

 operations in well-defined connections ? and the inverse 

 question, Is it possible to represent every well-defined 

 arrangement of symbols denoting special mathematical 

 operations graphically by curves in the plane or in 

 space ? The former question is one of vital importance 

 in the progress of astronomy, physics, chemistry, and 

 many other sciences, and has accordingly occupied many 

 eminent analysts ever since Fourier gave the first ap- 

 proximative answer in his well-known series : the latter 

 question can only be answered by much stricter defini- 



