i 9 6 SCIENCE PROGRESS 



of these progressions that Napier's mind grasped a great mathe- 

 matical principle, whose immediate application was invaluable 

 to him in carrying his calculations to a fitting conclusion. As 

 enunciated in the Descriptio this principle is that " the logarithms 

 of Proportionall numbers and quantities are equally differing." 

 This is the opening sentence of Chapter II of the Descriptio, 

 Chapter I having been taken up with his remarkable kinematical 

 definition of a logarithm in terms of the corresponding motions 

 of two points. By this ingenious definition Napier virtually 

 created the first transcendental function imagined by man. 

 This is now known as the exponential or logarithmic function 

 according as the one quantity or the other is taken as the inde- 

 pendent variable. The whole question is very clearly discussed 

 in the articles contributed to the Memorial Volume by Dr. J. 

 W. L. Glaisher and Prof. G. A. Gibson. 



Let us now return to the kinematical definition. 



o p 



a Q B 



Imagine two points P and Q to move in such a way that 

 while P, starting from O, describes its path with constant 

 velocity, Q, starting at a point A with the same velocity, moves 

 towards a goal B with a velocity at each point proportional 

 to the distance QB still to be described. The speed of Q 

 is thus constantly diminishing, and Q can never in a finite 

 time reach B. According to Napier's first system of loga- 

 rithms the distance OP travelled by P in the time taken by 

 Q to reach any position is the logarithm of the distance QB 

 still to be described. It is easy to see that in successive short 

 equal intervals of time the total distances travelled over by P 

 from the start form an arithmetical progression, while the 

 corresponding distances QB form a geometrical progression. 

 Napier's generalisation from a succession of numbers in geo- 

 metrical progression to the conception of a continuously varying 

 quantity whose rate of change is proportional to itself was a 

 remarkable mathematical achievement in an age which pos- 

 sessed no notation for such a conception. 



It is sometimes pointed out, somewhat critically, that what 

 are usually called Napierian Logarithms in modern books of 

 tables are not exactly those which Napier first tabulated in the 



