September 22, 1916] 



SCIENCE 



429 



of logarithms to 28 figures of the numbers 

 from 1 to 10,000 and to 15 figures of the num- 

 bers from 100,000 to 200,000 will require a 

 volume of 1,100 pages. 



In the paper by Mr. H. S. Gay the final 

 formula? are unfortunately incorrectly printed 

 (p. 367). Corrected these should read, as 

 follows : 



i less than 45°. 



90 



.01147 + .006 cos , 



.01147 + .006 sin a' 



a greater than 45°. 



It is interesting to note that the author, a 

 practising engineer, arrived at his approxi- 

 mate determinations of the sine and cosine 

 by a consideration of first and second differ- 

 ences; similar considerations appear in the 

 earliest tables of sines, in the Hindu Surya 

 Siddhanta and in the work of Aryabhatta, a 

 Hindu astronomer of the sixth century a.d. 



The historical notes in connection with the 

 conception and development of logarithms are 

 of real interest. Professor David Eugene 

 Smith discusses admirably the treatment in 

 early works on algebra and arithmetic of the 

 law of exponents. Any careful study of the 

 evidence presented by Dr. Smith will show 

 that the way was being well prepared for the 

 invention of logarithms, so that no surprise 

 need be occasioned by the fact that other 

 claimants to the honor of the discovery have 

 their patriotic supporters. Professor Florian 

 Cajori meets in a definite and decisive manner 

 the arguments which have been advanced in 

 favor of the priority in the field of the Swiss 

 writer, Joost Biirgi, sometimes claimed as a 

 German. Cajori says : 



They compare Biirgi 's supposed date of inven- 

 tion with Napier's date of publication, and there- 

 from do not conclude, as they legitimately could, 

 that Biirgi was an independent inventor, but they 

 conclude, as they can not legitimately do, that 

 Biirgi 's invention was prior to Napier's, or that 

 Biirgi very probably lost priority simply because 

 of failure to publish his logarithms as soon as in- 

 vented by him. 



This memorial volume is marred by the mis- 

 taken efforts to ascribe to Napier the discovery 



of imaginaries, and the introduction of the 

 decimal point. Numerous writers, notably 

 Cardan and Bombelli, had a much more pro- 

 found grasp of imaginaries than is anywhere 

 exhibited by Napier. So far as the decimal 

 point is concerned Pitiscus in his " Trigonom- 

 etry " of 1612 preceded by four years Wright, 

 or Napier, in the use of the comma which ap- 

 pears in Wright's 1616 translation of the 

 " Descriptio " and in Napier's " Babdologiae " 

 of 1617; that Napier was familiar with the 

 work of Pitiscus is proved by the fact that in 

 both the " Descriptio " and the " Rabdologise " 

 Pitiscus is cited. The spread of the decimal 

 system was greatly facilitated by Napier's 

 adoption, but it is not warranted to ascribe to 

 him any " share in the improvement of decimal 

 arithmetic." 



The historical notes (pp. 159-161) to the 

 article on the " De Arte Logistica " are re- 

 plete with errors. In the dates on the progress 

 of arithmetical and algebraical printing Lucas 

 de Burgo comes first, followed by Cardan 

 with " the next known book." Arithmetics 

 printed before Cadan's work of 1539 occupy 192 

 pages of Smith's " Rara Arithmetica " while 

 in algebra the well-known works of Gram- 

 mateus and Ghaligai precede Cardan. Stifel 

 or Stifelius (not Stifellius) did not introduce 

 the ■+, — and V signs. Even the English 

 algebra by Eobert Recorde is cited as of date 

 1552, instead of 1557. The concluding remarks 

 to the effect that in Napier's day and for some 

 time afterwards arithmetic and algebra were 

 no part of the mathematical curriculum is 

 absurd. The solution of the cubic and the bi- 

 quadratic was effected nearly one hundred 

 years before the time of Napier's great pub- 

 lication; Vieta's introduction of literal coeffi- 

 cients preceded by more than twenty years; 

 the serious study of algebra and arithmetic 

 made in the time of Napier prepared the way 

 for the invention of the analytic geometry and 

 the calculus, introducing the era of modern 

 mathematics. 



To his contemporaries Napier's most cele- 

 brated work was "A Plaine Discovery of the 

 whole Eevelation of St. John," published in 



