162 SCIENCE PROGRESS 



at T it has the velocity (proportional to TS), with which a 

 starts from b. In his own words : l 



" For in the same time that g is borne from o to T, g is 

 borne from T to d, because o T is such a part of o S as Td 

 is of TS and in the same time (by the definition of a logarithm) 

 is a borne from b to c ; so that o T, Td and be are distances 

 traversed in equal times ; but since g when moving between 

 T and o is swifter than at T, and between 7* and d slower but 

 at T equally swift with a ; it follows that o T the distance 

 traversed by g moving swiftly is greater and Td the dis- 

 tance traversed by g moving slowly is less than b c the 

 distance traversed by the point a with its medium motion, in 

 just the same moments of time ; the latter is, consequently, a 

 certain mean between the two former." 



The proposition can, of course, be demonstrated by means 

 of the relation already proved that 



r 

 logArf = r log, — ; 



from which we can easily show that 



(r - s) y >rlog,y> T - s, 



by writing the intermediate term in the form r log* ( i H — J 



and expanding. When r is very nearly equal to 5, the logarithm 



of 5 is, therefore, very nearly equal to the arithmetic mean of 



, .. .. . . , , (r + s) (r — s) 



the limits, i.e. is very nearly equal to — -. 



This proposition Napier uses at once to find the logarithm 



of the second term in the First Table, for, the first term being 



radius, its logarithm is, by definition, zero ; and the logarithm 



of the second term lying between the above limits lies therefore 



. . s r iooooooo 



between r — s = rooooooo and (r — s) - = = i ooooooi 



v 5 9999999 



and may therefore be taken with sufficient exactness as 1*00000005. 

 And this is also the common difference for the logarithms 

 of every number in the First Table ; hence, multiplying this 

 common difference by 2, 3, etc., the logarithms are readily 

 appended to the 100 terms constituting the First Table. 2 



To proceed from the First to the Second Table, another 



1 Construction Prop. XXVIII. 2 Ibid. Prop. XXIX.-XXXIII. 



