856 



SCIENCE 



[N. S. Vol. XL. No. 1041 



On the other hand, such a tahle is very useful 

 as a check in the development of theorems 

 relating to prime numbers. Mathematical 

 interest along this line has been greatly stim- 

 ulated in recent years by the publication of 

 the elegant work, in two volumes, entitled 

 " Handbuch der Lehre von der Verteilung der 

 Prinzahlen" by E. Landau, of Gottingen, Ger- 

 many. 



The prime numbers contained in the pres- 

 ent volume can be found by means of the given 

 factor table, but it is much easier to use the 

 present table in case the only question under 

 consideration is whether a given large num- 

 ber, within the limits of this table, is prime 

 or composite. Each page contains 100 rows 

 and SO columns of numbers, and hence there 

 are 5,000 different prime numbers on a page. 

 It is therefore very easy to determine, by 

 means of this table, the number of prime num- 

 bers lying between any two numbers within 

 the limits of the table. 



The Introduction covers fifteen pages and 

 deals with various questions relating to prime 

 numbers. It includes a table exhibiting the 

 actual numbers of prime numbers at intervals 

 of 50,000 up to 10,000,000, and comparing 

 them with the approximate numbers of these 

 primes according to ^ the formulas of Rie- 

 mann, TchebychefE (Cebysev) and Legendre. 

 It is somewhat surprising to find that the In- 

 troduction contains evidences of carelessness 

 while the body of the work seems to have been 

 prepared with the greatest care. 



In fact, at least three inaccuracies appear 

 on the first page of the Introduction. Line 

 twenty begins with the word " infinite " in- 

 stead of " finite." In line thirty-seven of the 

 first column it is stated that Eratosthenes was 

 a contemporary of Euclid. As a matter of 

 fact it is not known whether Euclid was still 

 living when Eratosthenes was born. We know 

 very little about the life of Euclid, and it is 

 distinctly stated in Giinther's " Gesehichte 

 der Mathematik," 1908, page 83, that we do 

 not know whether Euclid and Eratosthenes 

 were contemporaries. In line sixteen of the 

 second column of the first page the symbol 

 22» should be replaced by 2^n. 



In referring to these inaccuracies in the 

 Introduction it is not implied that they af- 

 fect seriously the value of the book. On the 

 contrary, we desire to emphasize the fact that 

 the table is not to be judged by its Introduc- 

 tion. Professor Lehmer realizes very keenly 

 the great importance of accuracy in listed 

 results, and he has made a careful study of 

 methods which tend to insure the greatest pos- 

 sible accuracy. In view of the enormous 

 amount of labor involved in testing the accu- 

 racy of such tables sufficiently to pass reliable 

 judgment, the reviewer bases his confidence in 

 the accuracy of the present table on the 

 methods used by the author, and not on his 

 own direct observations. 



In closing we may refer briefly to the fol- 

 lowing interesting sentence which appears on 

 page X of the Introduction : " It is hardly 

 likely, indeed, that any theorem of importance 

 in the Theory of Numbers was ever discovered 

 which was not found in the first place by ob- 

 servation of listed results." Professor Leh- 

 mer's comprehensive knowledge of the devel- 

 opments in Number Theory gives great weight 

 to this striking emphasis on the importance of 

 listed results. To the reviewer the quotation 

 appears to emphasize too much the usefulness 

 of the method under consideration, especially 

 as regards the developments in the theory of 

 algebraic numbers. G. A. Miller 



Univeesitt op Illinois 



Natural Sines to Every Second of Arc, and 

 Eight Places of Decimals. Computed by 

 E. GiFFORD from Eheticus. Manchester. 

 Printed by Abel Heywood & Son, 47 to 61 

 Lever Street. 1914. Pp. 543. 

 Among the extensive trigonometric tables 

 which were calculated during the sixteenth 

 century those of Eheticus occupy the most 

 prominent place. That an immense amount 

 of labor, devotion and perseverance was in- 

 volved in the preparation of such tables may 

 be seen from the fact that Eheticus employed 

 computers for twelve years at his own expense.^ 

 His " Opus Palatinum," published posthu- 

 1 BraunmuU, ' ' Vorlesungen liber Gesehichte der 

 Trigonometrie, " Vol. 1, 1900, p. 212. 



