502 SCIENCE PROGRESS 



a painting of Napier given in Mark Napier's Memoirs of John Napier of 

 MercJiiston (1834), and contains a reproduction of one of the pages of tables 

 in the Descriptio. There is not much that is new in this lecture, but it is 

 pleasantly written and of some real scientific interest. Its object is to give an 

 account, as concise as may be, of the conception of a logarithm in the mind of 

 Napier, and of the methods by which he actually constructed his table of 

 logarithms (p. 6). " Napier's conception of a logarithm involved a perfectly 

 clear apprehension of the nature and consequences of a certain functional re- 

 lationship, at a time when no general conception of such a relationship had 

 been formulated, or existed in the minds of mathematicians, and before the 

 intuitional aspect of that relationship had been clarified by means of the great 

 invention of co-ordinate geometry made later in the century by Rene Descartes " 

 (p. 7). " On the theoretical side, Napier's representation [of numbers and their 

 logarithms] by continuously moving points involved the conception of a functional 

 relationship between two continuous variables, where Stifel and others had 

 merely considered the relationship between two discrete sets of numbers. This 

 was in itself a step of the greatest importance in the development of mathematical 

 analysis" (p. 45). 



A short biography of that very extraordinary man John Napier is given, 

 and his deep interest in theology as well as his interest in warlike inventions, 

 agriculture, and magical practices are referred to. Also there is a reference 

 (pp. 11-12) to his early mathematical work which was only published in 1839 

 by Mark Napier. Prof. Hobson supposes (p. 12) that Napier put aside his 

 early mathematical work in order to devote himself to the discovery of means 

 of diminishing the labour involved in numerical computations. Napier was 

 "led probably by the circumstances of the time," for the second half of the 

 sixteenth century was a time in which Continental mathematicians devoted a 

 great deal of attention to the calculation of trigonometrical tables. There is 

 strong evidence that Napier had fairly begun the great invention of logarithms 

 in 1594 (p. 13). About the time of the publication of the Descriptio in 1614, 

 Napier devised several mechanical aids for the performance of multiplications, 

 divisions, and the extraction of square and cube roots, "for the sake of those 

 who may prefer to work with the natural numbers." Napier published an 

 account of these inventions in 1617. The Descriptio did not contain an account 

 of the methods by which the Canon of logarithms was constructed ; this was 

 contained in the Constructio, published posthumously but written several years 

 before 1614 (p. 15), and a short account of the contents of the Descriptio and 

 Constructio is given on pp. 18-21. Coming closer to the nature of Napier's 

 invention, the definition of a logarithm by the help of points moving on straight 

 lines, as given in the Descriptio, is set forth on pp. 23-6. and Napier's method 

 of calculating logarithms is described on pp. 27-8. The steps in the construction 

 of the Canon are described, from the Constructio, on pp. 28-40. There are 

 also interesting sections on the introduction of the decimal point by Napier 

 (pp. 21-3), on the reception of the logarithms (pp. 16-18), on other tables of 

 logarithms (pp. 40-3), and on Napier's predecessors and his one rival in the 

 invention of logarithms, Jobst Biirgi (pp. 43-7). 



One point in all the expositions of Napier's work which the present reviewer 

 has seen seems rather remarkable. It is the prominence given to Napier's 

 kinematical definition of a logarithm. We know that Napier did not come 

 upon his invention by kinematical means, and we can only suppose that he 

 took to defining his logarithms in such a way for the sake of generality. It seems 



