432 



SCIENCE 



[N. S. Vol. XXXII. No. 822 



to present the whole series of experiments 

 from which he drew his conclusions, while in 

 the repeated experiments of Basils and Pun- 

 nett all results were tabulated. This unsci- 

 entific attitude seems to pervade the whole of 

 Eusso's work, and so long as his methods are 

 thus at fault, it is not worth while to consider 

 the bearing of his results, particularly in the 

 face of direct contradiction by other investi- 

 gators going over the same ground. 



Finally the enquiry concerning the possi- 

 bility of reversing the operation of Mendelian 

 dominance in cross breedings may be consid- 

 ered. Here again we have to do with faulty 

 experimental methods. Russo claims to be 

 able to make white dominant over black in the 

 first generation of hybrids by treating the 

 white mother with lecithin injections before 

 breeding, but practically no attempt is made 

 to analyze the racial composition of the ani- 

 mals used in breeding. Such experiments as 

 he presents in support of this contention would 

 have no standing whatever with experienced 

 breeders and it may be said without any exag- 

 geration that in such a presentation of his 

 case he has forfeited entirely the serious con- 

 sideration of his work. A detailed analysis of 

 this part of his studies has been recently given 

 by Castle ° and will not be repeated here. 

 It is much to be regretted that an extended 

 investigation like this of Russo's should be 

 vitiated by untrustworthy methods, for such 

 lines of work need following out and are ex- 

 tremely valuable in furthering an analysis of 

 the relations existing between the germ cells 

 and the parental bodies. That the author will 

 present his work purged of the serious errors 

 it now contains must be the hope of all his 

 fellow workers. C. E. McClung 



Factor Tables for the First Ten Millions, con- 

 taining the smallest factor of every number 

 not divisible by 2, 3, 5 or 7 between the 

 limits and 10,017,000. By Derrick Nor- 

 man Lehmer. Washington, D. C, Carnegie 

 Institution of Washington, Publication No. 

 105. 1909. Pp. xvi -1-476. Price $20. 

 The publication of the best and most exten- 

 sive work in any language, on an old and 

 'Loo. ait. 



important subject, is eminently worthy of 

 recognition, especially when the preparation of 

 such a work demanded the most painstaking 

 care and unselfish devotion to the interests of 

 science. Prime numbers and factors of com- 

 posite numbers are among the oldest as well 

 as among the newest objects of study in math- 

 ematics. The perennial interest in these sub- 

 jects bears testimony to their importance and 

 helpfulness in our eiiorts towards stronger in- 

 struments of thought and towards a more ra- 

 tional intellectual penetration into the phys- 

 ical laws which we encounter on all sides. 



While it may be true that integral numbers 

 do not occupy comparatively as large a place 

 in our present mathematical thinking as they 

 once did, they still constitute, according to 

 Minkowski, " the fountain-head of all mathe- 

 matics " and they enter prominently into 

 many of our mental processes. We are not 

 infrequently brought to questions whose solu- 

 tions are expedited by a knowledge of the 

 existence of primes or of the factors of large 

 composite numbers. Under such circum- 

 stances one will naturally turn hereafter to 

 the tables before us with an unusual confidence 

 in their correctness and a high appreciation 

 of their great extent. 



The pages of the present table are very large 

 — about sixteen inches long and twelve inches 

 wide. " Each horizontal line of the table 

 covers 210 numbers. The multiples of 2, 3, 

 5 and 7 are not listed. As there are 100 lines 

 on each page it follows that each page will 

 serve to find the smallest factor of 21,000 con- 

 secutive numbers. The largest and smallest 

 of these are given at the top of the page. 

 These numbers then indicate at a glance the 

 page that contains the smallest divisor of the 

 given number." To find the smallest factor 

 of a given number without the aid of an 

 auxiliary table, it is necessary to divide the 

 number by 210 and to locate the quotient and 

 the remainder in the table. By means of these 

 two numbers it is very easy to find the smallest 

 factor of the number in question, if it is com- 

 posite but not divisible by 2, 3, 5 or 7. The 

 division by 210 may be avoided by means of 

 an auxiliary table. 



In his preface the author states that " The 



