462 



NATURE 



[March 18, 1880 



The only explanation which can at present be advanced 

 to account for the difference between the observations of 

 Deville and Troost on the one hand, and Meyer and 

 Crafts on the other, is that in the experiments of the 

 former the iodine was gradually converted into vapour, 

 whereas the method adopted by the latter involves the 

 almost instantaneous volatilisation of the iodine ; in the 

 case of some organic compounds a difference of this kind 

 in the mode of heating is known to exercise a considerable 

 and in many respects similar influence on the result, so 

 that this explanation is not unsupported by analogy. 



Great difficulty was experienced in determining the 

 density of free bromine in consequence of the explosive 

 rapidity with which it is converted into gas when intro- 

 duced into the intensely-heated bulb of the density appa- 

 ratus. The results obtained are not accordant, but all lie 

 between the number corresponding to the molecular 

 formula Br 2 and that required on the assumption that 

 dissociation takes place to the same extent as in the case 

 of iodine. Using platinic bromide, PtBr 4 , however, 

 instead of free bromine, Meyer and Ziiblin find that a 

 reduction in density takes place precisely of the character 

 of that observed for chlorine from platinous chloride and 

 for iodine. Thus at a temperature of about 1,570° the 

 observed density in two experiments was 378 and yb\, 

 3'64 being exactly two-thirds of the density corresponding 

 to the molecular formula Br 2 . 



As yet Meyer has told us nothing of the nature of the 

 dissociation products of the three halogens ; their deter- 

 mination and separation will probably be attended with 

 great experimental difficulties, but the problem could not 

 well be placed in abler hands, and we trust that ere long 

 we may be able to congratulate him on the accomplish- 

 ment of this the crowning triumph of his labours. 



Henry E. Armstrong 



GLAISHER'S FACTOR TABLES 

 Factor Table for the Fourth Million. By James 

 Glaisher, F.R.S. (London : Taylor and Francis, 

 1880.) 



THERE is no general method of ascertaining whether 

 one number is divisible, without remainder, by 

 another specified number (less than its half) except by 

 actual trial, or by the knowledge, otherwise acquired, of 

 all the divisors of the first number. If then the second 

 is not among these, it is also known that it is not 

 a divisor of the first number. The knowledge of 

 whether a specified number has any divisors at all, and if 

 so what they are, is only to be obtained in general by 

 trying it with all possible divisors less than its square 

 root. The process can be shortened, but only to a limited 

 extent, and, speaking generally, it would require hundreds 

 of division sums, to ascertain by trial that 3,979,769 had 

 1979 for a divisor, and was consequently the product of 

 1979 and 201 r. 



It is, however, frequently important to mathematicians 

 to know how to split up any given number into its 

 divisors or factors, and this without the enormous labour 

 which may be involved in actually trying for its divisors, 

 especially as there is no general mathematical principle 

 which enables us to dispense with the trial, or even to 

 shorten it so as to bring it within practicable limits. The 



alternative is to tabulate numbers up to a given limit, 

 and to indicate, for each, whether it has divisors, and 

 what they are. It is not necessary, or usual, to include 

 in such tables every number without exception ; for an 

 inspection of the last figure of any number tells us 

 whether it is divisible by two or by five; and the old rule 

 of "casting out the nines" tells us whether it is divisible 

 by three. These considerations greatly reduce the number 

 which it is necessary to tabulate ; for, among the first 

 300 numbers, 150 are even, that is to say, divisible by 2 ; 

 and of the remaining 150, 50 are divisible by 3; while of 

 the 100 left after that, 20 are divisible by 5. The exclu- 

 sion of the numbers divisible by 2, 3, or 5 thus reduces 

 the number of tabular entries required, from 300 to 80, 

 and this proportion holds all through the table, as well as 

 for the first 300 numbers. It will be observed that the 

 last two figures of these 80 numbers remain the same for 

 every batch of 300. This facilitates the tabulation, and 

 advantage has been taken of this facility in printing the 

 Tables. 



The first extensive tables of this kind were those 

 published by the Austrian General, Baron von Vega, at 

 the close of the last century. These extended from 1 to 

 108,000, and thus give all the divisors of the numbers not 

 divisible by 2, 3, and 5 within those limits. The next 

 table was that of Chernac, a Polish Professor of Mathe- 

 matics at Deventer, in Over-yssel, which was published in 

 1S11. It contained all the divisors of all numbers, not 

 divisible by 2, 3, and 5 up to 1,012,000. It forms a very 

 thick quarto volume of over 1,000 pages. 



The next extension was made by Burckhardt (1814-17), 

 who published a series of three volumes, giving, not all 

 the divisors, but the least prime divisor, of all numbers 

 (except those divisible by 2, 3, and 5) up to 3,036,000. 

 This is not quite so convenient, as a matter of immediate 

 reference, as giving all the divisors ; but it answers every 

 necessary purpose. For example, when we know that 

 3,999,589 has 11 for its least divisor, we can find by 

 actual division that the quotient is 363.599^ We "look 

 out" this number in the earlier part of the table, being 

 sure of finding it there, seeing that 11 was the least 

 divisor of its multiple ; we find its least divisor to be 31. 

 Performing the division by 31, we obtain the quotient 

 11,729. We " look out " this again in the earlier part of 

 the table, and we find that 37 is the least divisor. Per- 

 forming this division, we obtain 317 as the quotient. 

 Since this is less than 37 X 37, we know that it can have 

 no divisors except unity and itself, or that it is prime. 

 If, instead of the least prime divisor, all the divisors had 

 been given, we should at once have found from the 

 table 



3,999>5 8 9 = n X 3> X 37 X 3'7- 

 There is an obvious advantage in the more complete 

 table. Unfortunately it is balanced by the practical 

 inconvenience of size, and "a great book is a great evil." 

 What this practically comes to may be judged of from 

 the remark that Chernac's table, which gives all the prime 

 factors from 1 to 1,019,000, takes 1020 quarto pages; 

 while Burckhardt' s, which gives only the least prime divisor, 

 contains the numbers from 1 to 3,036,000 in 336 quarto 

 pages. It is true that Burckhardt's table is more closely 

 printed than Chernac's, with somewhat smaller type, and 

 a slightly larger form; but, making] all allowances, the 



