THE GENESIS OF LOGARITHMS 157 



Now the initial velocity of C— Initial velocity of E — a 

 and the velocity of C is uniform. 

 Therefore 



dx 



-ji = a and x = at, 



the constant of integration vanishing, since x and t vanish 

 together. Hence 



or 



x , a 



x = \og N y = a\og e y 



a is the radius, which Napier took as io 7 units in length. 



So that we finally obtain 



io 7 

 logjvy = io 7 log, — . 



Now let us see how the actual tables constructed by Napier 

 were evolved. In the two rows of figures previously cited 

 the logarithms proceed in arithmetical progression, the numbers 

 in geometrical progression and such a geometrical progression 

 as we have cited shows increasingly large gaps. The problem 

 is to construct a series of numbers in geometrical progression 

 which shall yet be sufficiently close together to represent the 

 natural numbers or rather, in Napier's case, to represent the 

 sines of continually decreasing arcs, for, as has been said, 

 Napier's final object was the construction of a canon of 

 logarithms of sines. The manner in which this problem was 

 solved can best be demonstrated by a brief analysis of the 

 more important parts of Napier's posthumously published 

 work, the Construction which we now proceed to give. 



The full title of this work, which was, as has been noted, 

 published posthumously in 1619, is, literally translated — "The 

 Construction of the Wonderful Canon of Logarithms ; and their 

 relations to their own natural numbers ; with an Appendix 

 as to the making of another and better kind of Logarithms. 

 To which are added Propositions for the solution of Spherical 

 Triangles by an easier method : with Notes on them and on 

 the above-mentioned Appendix by the learned Henry Briggs. 



" By the Author and Inventor, John Napier, Baron of 

 Merchistoun, etc., in Scotland." 



Both this work and the Descriptio are, curiously enough, 

 the most neglected of Napier's works. This neglect is, of 



