476 Prof. H. S. Carslaw on Napier's Logarithms : 



With a limiting tilt of the aeroplane of 90°, the vertical 

 component of the earth's magnetic field produces its maximum 

 effect, the magnetic intensity being 0' 18 tan 67° = 0*18 x 2*3. 

 For = 90°, the deflecting couple should equal this 

 (0'18x2*3) ; but the field due to the compensating magnets 

 is then 0'18 X tan (deflexion), therefore for proper compen- 

 sation tan (deflexion) = 2*3. 



This condition is seen from the curves (fig. 9, p. 471) to 

 be very nearly fulfilled with the 4-inch magnets at a distance 

 d = 18 cm. The sine curve having maximum value 2"5 is 

 plotted as a line of dots, and is shown to be very nearly 

 coincident with the corrector curve ; showing that the com- 

 pensation for vertical oomponent would be very nearly perfect 

 with 4-inch magnets at 18 cm. Any discrepancies are due to 

 the fact that the field produced by the compensating card is 

 not uniform, and the card magnets are of considerable size. 



L. Napier's Logarithms : the Development of his Theory. 

 By Professor H. S. Carslaw, Sydney, N.S. W* 



§ 1. Introductory. 



^j^HIS paper deals with Napier's idea of a logarithm. 

 A In my view there are three distinct stages in the 

 development 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 Arith- 

 metical Progression. There are traces of this in the 

 Constructio f in his use of the series 



and in the word logarithm itself, derived from X070? apiOfios, 

 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 



* Communicated by the Author. Read to the Roy. Soc. New South 

 Wales, Aug. 2, 1916. 



t The Mirijici Logarithmorum Canonis Constructio was published in 

 1619, two years after Napier's death, but had been written several years 

 before his Mirijici 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 of the latter are reproduced in the ' Napier Tercen- 

 tenary Memorial Volume,' Plates I.-Vl. (London, 1915). 



