130 H. S. CARSLAW. 



NAPIER'S LOGARITHMS: THE DEVELOPMENT OP 

 HIS THEORY. 



By H. S. Oarslaw, Sc. d. 



[Read before the Royal Society of N. S. Wales, August 2, 1916.'] 



Introductory. 

 § 1. This paper deals with Napier's idea of a logarithm. 1 

 In my view there are three distinct stages in the develop- 

 ment of this idea in his work. In the first he is concerned 

 with a one-one correspondence between the terms of a 

 Geometrical Progression and the terms of an Arithmetical 

 Progression. There are traces of this in the Constructio 2 

 in his use of the series 



10', 10' (l --L), 10' (l - -L J, etc., 



and in the word logarithm itself, derived from Aoyos aotfyios, 

 and generally taken to mean "the number of the ratios." 

 In the second he has passed from this correspondence, and 

 his logarithms are given by the well known kinematical 

 definition, which forms the foundation of the theory of the 

 Constructio. In the third, referred to in the Appendix to 

 the Constructio, he has reached the idea of a logarithm as 

 defined by the property : — 



1 In a previous paper in these Proceedings:— The Discovery of Loga- 

 rithms by Napier of Merchiston— Vol. 48, pp. 42 -72, 1914 — the question 

 of the construction of the logarithms of the Canon has been discussed. 



2 The Mirifici Logarithmorum Canonis Constructio was published in 1619, 

 two years after Napier's death, but had been written several years before 

 his Mirifici Logarithmorum Canonis Descriptio, published in 1614. I shall 

 refer to these works as the Constructio and the Descriptio. The Descriptio 

 was translated into English by Wright (1616), and Filipowski (1857), the 

 Constructio by Macdonald (1889). The former is a rare book, both, in the 

 original and in translation. Several of the more important pages are 

 reproduced in the Napier Tercentenary Memorial Volume, Plates I - VI, 

 (London, 1915). 



