1 68 SCIENCE PROGRESS 



" Therefore, with 2 as the given root, and 10,000,000,000 

 as the index, seek for the number of places in the multiple, 

 and not for the multiple itself; and by our rule you will find 

 301,029,995, etc., to be the number of places sought, and the 

 Logarithm of the number 2." 



A method used by Briggs for finding the logarithms of 

 small prime numbers, which depended upon the formation of 

 a large number of geometric means between unity and the 

 given prime number, is fully outlined in the article " Logarithms " 

 above-mentioned and needs no further discussion here. 



By these methods Briggs computed the logarithms of all 



integers from 1 to 20,000 and from 90,000 to 100,000 to 14 



places of decimals. The gap from 20,000 to 90,000 was filled 



by the calculations of Adrian Vlacq, who computed his 



ogarithms to 10 places of decimals. 



It is interesting to note that an abusive mention of Vlacq 

 by Milton in his Defensio secundo pro populo Anglicano led 

 Vlacq to state simply and clearly the story of his life from 

 the age of 26. Any faithful account of one to whom mathe- 

 maticians are so much indebted — for the tables of Briggs and 

 Vlacq are the parents of all the logarithmic tables which 

 have succeeded them, no re-computation on such an extensive 

 scale having been made since — must necessarily possess great 

 interest, and " one is almost inclined to pardon Milton his 

 abuse, seeing that thereby we are made acquainted with what 

 would otherwise probably have always remained a mystery." : 



Here an account of the genesis of logarithms may fitly 

 close. Several points of minor interest remain — a considera- 

 tion, for example, of Kepler's logarithmic tables, which differ 

 from Napier's in one point only ; in Napier's Table the arc of 

 the quadrant is divided into a definite equal number of parts, 

 so that the sines corresponding to these angles are, in general, 

 irrational numbers. In Kepler's table the radius is divided 

 into a definite number of equal parts, so that the sines are 

 rational numbers, the corresponding angles or arcs being 

 irrational. 



Something might be said also of the very doubtful claim of 

 Joost Burgi (15 52-1632) to be an independent discoverer of 

 logarithms, a discussion of which may be found in several 

 of the standard histories of mathematics. 



1 Glaisher, Phil. Mag. October 1872. 



