Feb. 28, 1889] 



NATURE 



413 



Xew South Wales, November 30, 1887). The New South 

 Wales species is, I think, identical with that found in Queens- 

 land, and I should be inclined to doubt the distinctness of the 

 \ictorian species recorded by Mr. Dendy in Nature (p. 366), 

 ind previously by Mr. Fletcher. 



Mr. Dendy appears to lay some stress on the differences of 



lour as between his specimen and the specimens of P. 

 ■karti hitherto described, but it must be remembered that in 

 line species of Peripatus — e.g. capensis and novm-zealandiis — 

 the range of individual colour-variation is very considerable. 



All the species that I have seen are very beautiful when 

 alive ; but the beauty, which is partly due to the texture of the 

 skin, is very hard to reproduce in a drawing. 



It is a remarkable fact that a creature which lives so entirely 

 in the dark as does Peripatus should present such rich 

 coloration and such complicated markings. 



The egg of Peripatus leuckarti is heavily yolked and of a fair 

 size, but smaller apparently than that of the New Zealand 

 species. Its development cannot fail to be of the greatest in- 

 terest, and it is sincerely to be hoped that the Australian 

 zoologists will lose no time in working it out. 



A. Sedgwick. 



Trinity College, Cambridge, February 18. 



Anthelia. 



I HAVE been following with much interest your notices of 

 anthelia, and was about to add my mite to the information 

 given, when, by the mail just in, I have your issue of October 25 

 last, wherein is a notice of the phenomenon as observed in 

 Ceylon. I have witnessed it there scores and scores of times in 

 my early tramps bird collecting, and I have also seen it at the 

 Cape, in Brazil, on the Amazon, in Fiji, and in this island. On 

 turning up my dear old friend Sir E. Tennant's book on Ceylon, 

 I find that at p. 73, vol. i., he gives a very fair figure of the 

 effect produced. It may be, as he says, that the Buddhists took 

 from it the idea of a "halo" or "flame" for the head of 

 Buddha, but there is one peculiarity about these flames that 

 always struck me. In whatever position you find the Buddha, 

 the flame is invariably in a straight line with the body even if 

 the figure is recumbent. In form it always resembles the 

 "tongues of fire" depicted by old painters as falling on the 

 apostles on the Day of Pentecost. 



I have seen many instances of what I suppose may be called 

 "anthelia "in calm water, but the appearance is usually more 

 rayed. I have an exquisite engraving in my print collection of 

 the " Madonna and Dead Christ " by Aldegrever (1502-58). It has 

 often occurred to me, in looking at it, that the artist has taken 

 his idea of the halo round the Virgin's head from the appearance 

 presented by the "anthelia" in water. There is the same 

 luminous centre, and then the divergent rays. The halo round 

 the head of the dead Christ in her lap is a four-cornered luminous 

 star, issuing rays, of which three points only are visible — like 

 nothing in nature with which I am acquainted. 



E. L. Layard. 



British Consulate, Noumea, January 3. 



i 



Mass and Inertia. 



I AM pleased to see that Dr. Lodge has adopted my suggestion 

 made in the Engineer about four years ago of using the term 

 inertia for the quantity mass-acceleration. In making the sug- 

 gestion I considered that I merely asked a return to the meaning 

 implied by Newton in the phrase " vis inertia." 



Unless this is the meaning of the term, the reason why 'S.vtr^ 

 is called moment of inertia is almost incomprehensible. With 

 it the connection is obvious ; fo", if;// is the angular acceleration 

 of a body about an axis, and r the distance of any particle, its 

 linear acceleration is ^\/r, its inertia m^r. and its moment of 

 inertia rm\^r, or mrpr". As the angular acceleration is the same 

 for all particles of the body, the moment of inertia of the body 

 is ^"Zmr^. 



As Dr. Lodge mentions that he is bringing the matter before 

 the British Association Committee on Units and Nomenclature, 

 might I suggest that in future Swr^ should be called the moment 

 of inertia constant, thereby implying the existence of the variable 

 Jfactor \\/, the angular acceleration, in the expression for moment 

 of inertia. E. Lousley. 



Royal College of Science, Dublin, February 16. 



To find the Factors of any Proposed Number. 



It has long been a desideratum of mathematicians to discover 

 a formula or method for ascertaining the factors of any proposed 

 number, and also determining whether it be a prime or not. 

 Their endeavours during the twenty centuries that have elapsed 

 since Eratosthenes (B.C. 276-196) made the first recorded 

 attempt to produce a practical rule for the purpose have not 

 been attended with success. 



As it may interest many readers of Nature, and others, I 

 propose, with a few preliminary remarks, to make known a 

 simple arithmetical method by which this desideratum can now 

 be attained. 



Factors of an even number can readily be found, as 2 is always 

 one of them, but it is not always so easy to find the factors of 

 an odd number, especially if it be a high one, and, if the 

 number be the product of two primes, the difficulty in this 

 respect is still greater, because they are its only factors. 

 Hitherto they could be ascertained only by trying in succession, 

 as divisors, the prime numbers of less magnitude than its square 

 root. 



To find by such process the factors of 8616460799 (the square 

 root of which is between 92824 and 92825), it might, possibly, be 

 necessary to try 8967 prime numbers as divisors (out of the 8969 

 that there are) before they could be ascertained. By my process, 

 division sums are altogether avoided. This high number occurs 

 in a chapter on " Induction as an Inverse Operation," in 

 "Principles of Science," by Stanley Jevons, second edition. 

 His emphatic remarks as to the difficulties attending on inverse 

 operations in general, and particularly those with reference to 

 finding the factors of this number, were the incentive to my 

 endeavouring to discover some process for ascertaining them which 

 might possibly have escaped being previously tried. He states : — 

 " The inverse process in mathematics is far more difficult than 

 the direct process. ... In an infinite majority of cases it 

 surpasses the resources of mathematicians. . . . There are no 

 infallible rules for its accomplishment. ... It must be done by 

 trial, ... by guess-work. . . . This difficulty occurs in many 

 scientific processes. . . . Can any reader say what two numbers 

 multiplied together will produce 8616460799? I think it un- 

 likely that anyone but myself will ever know. They are two 

 prime numbers, and can only be discovered by trying in succes- 

 sion a long series of prime divisors, until the right one be fallen 

 upon. The work would probably occupy a good computer 

 many weeks. It occupied only a few minutes to multiply them 

 together." 



Mr. Jevons adds: "There is no direct process known for 

 discovering whether any number be a prime or not, except by 

 the process known as the 'sieve of Eratosthenes,' the results 

 being registered in tables of prime numbers." 



In the article on prime numbers in " Rees's Cyclopaedia " (ed. 

 1819), the writer states: "It is in fact demonstrable that no 

 such formula " (for discovering whether a number be a prime or 

 not) " can be found, though some formulae of this kind are 

 remarkable for the number of primes included in them." 



The difficulty of finding the factors of numbers is also 

 referred to by the eminent writer (at that time President of the 

 Mathematical Society) — under the initials C. W. M. — of an in- 

 teresting review of " Glaisher's Factor Tables," in Nature, vol. 

 xxi. p. 462. In course of his remarks he mentions the number 

 3979769, and respecting it says : "It would require hundreds 

 of division sums to ascertain by trial that it had 1979 for a 

 divisor, and that consequently it was the product of 1979 x 201 1 ;" 

 and he adds, ". . . . 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 practical limits." 



These extracts affiard conclusive evidence that no direct rule 

 or method has hitherto been known, by which the factors of a 

 number could be ascertained, and also that it is considered it 

 would be a task of almost insuperable difficulty to devise one. 

 Yet it seemed to me not unreasonable to think that, as two 

 factors multiplied together formed a product, it ought to be pos- 

 sible to unmultiply or split up (as " C. W. M." expresses it) that 

 product into its factors again, "without the enormous labour of 

 trying for its divisors." 



Strongly impressed with this idea, I attempted to realize it, 

 and before long succeeded in discovering a simple arithmetical 

 process for the purpose, and diffisrent from any previously tried. 

 When applied to find the factors of 8616460799, instead of 

 "many weeks being occupied " in the task, it showed, within a 

 very reasonable time, that they were 96079 x 89681. When 



