March 18, 1880] 



NATURE 



463 



condensation obtained by giving the least prime divison 

 instead of all the divisors, cannot be put at less than 

 three to one. It must be observed also, that the pro- 

 cesses of division, which have to be performed when the 

 least divisors only are given, are definite divisions with a 

 known divisor, and do not involve the tentative process of 

 finding what the divisor is; it is just this tentative 

 process which it is the object of all such tables to avoid. 



Burckhardt's tables extend, as has already been stated, 

 to 3,036,000. They consequently tabulate no divisor 

 exceeding 1741, which is the prime number next below 

 the square root of 3,036,000, which lies between 1742 and 



'743- 



The celebrated German computer, Zacharias Dase, 

 began the task of extending this table to nine or ten 

 millions. There was then a prospect of the fourth, fifth, 

 and sixth millions being printed from a manuscript by 

 Crelle, so that Dase, instead of taking the fourth million, 

 began with the seventh. His task was interrupted by his 

 death, but was resumed by his friend Dr. Rosenberg. 

 The seventh, eighth, and ninth millions were published by 

 a Society called the Dase-V'erein, and printed in Hamburg, 

 and the tenth million exists in manuscript. These publi- 

 cations, however, were of little practical value to science, 

 so long as the gap between 3,036,000 (Burckhardt's final 

 limit), and 7,000,000 (Dase's initial limit) remained a 

 blank, and it was found that Crelle' s manuscript, which 

 had fallen into the possession of the Berlin Academy, was 

 not sufficiently reliable, in respect of accuracy, to supply 

 this gap. 



This blank Mr. Glaishcr has undertaken to fill up, and 

 the first instalment, the fourth million, is now before us. 

 With the assistance, towards the expenses of computation 

 and printing, of grants from the British Association, and 

 of the Government grants administered by the Royal 

 Society, but without any requital of his own toil, except 

 such as all good workers find in the satisfactory comple- 

 tion of their labour, he has secured for England a share 

 in the performance of this work. The fourth million, 

 added to Burckhardt's three millions, makes perfect work 

 as far as it goes. The fifth million is now going through 

 the press, and the manuscript of the sixth million is nearly 

 complete. When these are printed, the work of Dase and 

 Rosenberg will couple on, and we shall have, in a shape 

 available for immediate reference, a complete knowledge 

 of the divisors of all numbers up to nine millions. To 

 test a number nearly equal to nine millions might involve 

 our trying, as divisors, all the prime numbers from 7 to 

 2,999 inclusive. 



It would be premature to discuss the question of accu- 

 racy of performance until much more trial has been 

 made of the work than has been possible in the few days 

 which have elapsed since its appearance. Very good 

 guarantees, however, are afforded by the systematic 

 method in which the process of calculation has been 

 performed, as well as by the great experience which Mr. 

 Glaishcr has had in accurate computation, and again by 

 the numerical tests of comparing the number of primes 

 actually counted, within given limits, with the approxi- 

 mate numbers indicated by theory. 



It is well known that the frequency of the occurrence 

 of prime numbers in the neighbourhood of any large 

 number, x, is expressed by the reciprocal of the hyperbolic 



logarithm of x. Soldners' integral, / -= , should 



J a log x 



therefore express, with a high degree of approximation 



when x is large, the number of primes below a certain 



number. One difficulty of the application of this is, that 



the function integrated becomes infinite between the 



limits. Nevertheless a highly approximate formula for the 



number of primes below a high number x is given by the 



expression (due to Legendre) — 



log x — 1 "08366 

 A serious practical difficulty in attributing exactness to 

 any such formula, or in determining its constants to any 

 high approximation, lies in the irregular distribution of 

 the prime numbers. It not unfrequently happens that 

 two consecutive odd numbers are primes ;'that is so with 

 3,999,311 and 3,999,313. On the other hand there is no 

 prime number at all between 3,826,019 and 3,826,157, 

 which differ by 138. This variation of frequency effectu- 

 ally throws out any minute comparison between actual 

 counting, and analytical expressions for the number of 

 primes, founded on the assumption of regular continuity. 

 A discussion of this part of the subject is given in Mr. 

 Glaisher's introduction. 



For a full development of this and of the cognate 

 theorems, and of their limits, we must refer to the ex- 

 tremely valuable preface which Mr. Glaisher has prefixed 

 to his work. To that also we must refer for an account 

 of the ingenious methods used in abridging the enormous 

 labour of computation, and at the tame time of seizing 

 the advantages of the most systematic arrangement pos- 

 sible, in order to secure accurate work in the first 

 instance, and then the detection of error, if accidentally 

 committed. The amount of accuracy which it is possible 

 to obtain may be inferred from the fact, that after many 

 years' use, only two errors have yet been pointed out 

 in Burckhardt's extensive table, and that Chernac is 

 nearly as good. We have no reason to doubt that this 

 high standard of accuracy has been maintained by Mr. 

 Glaisher. 



Cur review would hardly be complete without some 

 remarks on the utility of this work. We have already- 

 pointed out the utter impossibility, as a practical question 

 to practical men, of ascertaining whether a given number 

 has divisors, and what they are, without the help of such 

 tables. One of the most obvious applications is to the 

 calculation of high logarithms. The larger logarithmic 

 tables, to a great number of figures, only extend from 1 to 

 20,000 in the case of common logarithms, and from I to 

 10,000 in the case of Napierian logarithms. When, there, 

 fore, such a logarithm is required for an incommensurable 

 number (as is commonly the case), it becomes necessary 

 to split it, either absolutely or approximately, into factors. 

 Now this series of tables, when complete, will give us at 

 sight the breaking up of the first seven figures of any 

 number, and by a little adjustment, turning upon t'le 

 formula a- — (J> ± c)-, suggested by Burckhardt, to a far 

 higher extent. For instance, Burckhardt himself gives as 

 an approximate value of ?r, 256 . 19 . 173 . 229 . 509 . 3203, 

 which (neglecting cyphers) is good for the first ten figures ; 

 and in the same way it has been found that 



log. 10 = 64. 5. 13. 13. 103. I09-54I -7oi +24844 

 up to the fourteenth decimal place. 



