PROGRESS OF MATHEMATICS AND MECHANICS 243 



Rheticus, Kepler, and others in connection with the development of 

 the new astronomy, Napier made a vastly more important inven- 

 tion. His definition of a logarithm rests on the following kinetic 

 basis : 



Q 



_s 



7V_ _i _Si 



T S is a straight line of definite length ; 7\ <Si extends to the right 

 indefinitely. Moving points P and Pi start from T and T\ with 

 equal initial speeds; the latter continues at the same rate, the 

 former is retarded so that its speed is always proportional to its 

 distance from S. If equal intervals are taken on T\ Si the cor- 

 responding intervals in TS will grow smaller to the right. When P 

 is at any position Q the logarithm of QS is represented by 

 the corresponding length 7\Qi on the other line. It may be 

 shown in fact that if in our notation PS = x, T\P\ = y, TS = I, 



fi 1* t* 



: = : . This conception involving a functional relation be- 

 dy I 



tween two variables went much deeper than the comparison 

 of discrete numbers by Stifel. 



Napier's conception of a logarithm involved a perfectly clear 

 apprehension of the nature and consequences of a certain functional 

 relationship, at a time when no general conception of such a relation- 

 ship had been formulated, or existed in the minds of mathematicians, 

 and before the intuitional aspect of that relationship had been clarified 

 by means of the great invention of coordinate geometry made later 

 in the century by Rene* Descartes. A modern mathematician re- 

 gards the logarithmic function as the inverse of an exponential func- 

 tion; and it may seem to us, familiar as we all are with the use of 

 operations involving indices, that the conception of a logarithm 

 would present itself in that connection as a fairly obvious one. We 

 must however remember that, at the time of Napier, the notion of an 

 index, in its generality, was no part of the stock of ideas of a mathe- 

 matician, and that the exponential notation was not yet in use. 



Hobson. 



