THE GENESIS OF LOGARITHMS 161 



in the continued proportion of 10,000,000 to 9,999,999. Thus 

 we have in these three tables a series of numbers in geometrical 

 progression, which numbers also coincide very nearly with 

 those in a table of natural sines from 90 to 30 . 



It remains to show how, to each of these " natural " numbers, 

 Napier appended the corresponding " artificial " number or 

 logarithm. 1 The portion of the Constructio (§§ XXII. -XXVI.) 

 immediately following the discussion of the formation of the 

 three tables given above is concerned with the definition of 

 logarithms which we have previously explained. Proposition 



XXVII. proves, as before mentioned, that nothing is the 

 logarithm of radius {i.e. of the sine of 90°). Proposition 



XXVIII. is of fundamental importance; as an illustration of 

 Napier's methods, we proceed to give his proof, as far as 

 possible in his own manner. 



The proposition states that, if r be radius and 5 any given 

 sine, then the logarithm of 5 is greater than r — s and less than 



T d S 



1 i _ . . _| 



g g 



b c 



a 



Let TS represent radius, and let a point g start from T with 

 a velocity proportional to TS, its velocity when at any point 

 d being proportional to d S, dS being taken to represent any 

 given sine 5. Simultaneously with the departure of g from T, 

 another point a moves from b with a uniform velocity equal 

 to the initial velocity of g; if, then, when g is at d, a is at c, 

 b c is called the logarithm of d S. 



In his proof of the proposition quoted above, Napier pro- 

 duces the line S T to 0, so that S is to 7^5 as TS is to dS. 



y 

 Hence it follows that T is equal to (r — s)— ; and since Td 



is equal to r — s, we have to prove that b c is greater than Td 

 and less than T. This Napier proves by assuming that the 

 moving point g starts from 0, its velocity decreasing according 

 to the geometrical law in such a manner that when g arrives 



1 See Appendix C. 



II 



