152 



NATURE 



\_Dec. 15, 1887 



A rarer form is as follows : — 



^ 1 \-A _| 1 ^ 1 : \-\ 1 I 



I have noticed that this latter form seems more difficult for the 

 little musicians, one of whom in particular used to provoke me 

 by singing'the B most outrageously flat. I have been accustomed 

 to imitate these birds by whistling, and they very readily answer 

 my whistle. In this way the different forms of their theme 

 have become fixed in my memory. W. L. G'>odwin. 



Queen's University, Kingston, Canada, November 11, 



Who was Mr. Charles King ? 



Among the ingenious in many considerable parts of the 

 world, of whose undertakings, studies, and labours the Philo- 

 sophical Transactions of the years 1700 sqq. gave some account, 

 an able microscopist suddenly appears, of whose life and 

 work one would like to have more accurate information than 

 seems to be current. Perhaps a member of the Royal, or the 

 Royal Microscopical, Society may be able to supply some particu- 

 lars about this " Anglois anonyme," as Trembley calls him, and 

 willing to assist in rescuing his name from an undeserved oblivion. 

 His first contribution to the Philosophical Transactions — of very 

 little importance indeed — is to be found in No, 266, for 

 September and October 1700, pp. 672-673, under the title, 

 "A Letter from Mr. Charles King to Mr. Sam. Doudy, 

 F.R. S., concerning Crabs Eyes;" it is dated, " Little Wirley, 

 Decemb. 14," and subscribed, " Ch. King." In the copy of 

 the Transactions I have before me, a contemporary, who seems 

 to have been tolerably well informed, has inserted divers MS. 

 notes, remarks, and corrections ; he added here the words, 

 "Staffordsh«." to the locality, and "Student of Ch.Ch. Oxon." 

 to the subscription, which, as far as I know, does not recur in 

 any of the subsequent Transactions. But under the title, " Two 

 Letters from a Gentleman in the Country, relating to Mr. 

 Leuwenhoeck's Letter in Transaction, No. 283, Communicated 

 by Mr. C." (in No. 288, for November and December 1703, 

 pp. 1494-1501, with eight figures, text and illustrations being 

 both equally remarkable for the period), the same hand has again 

 inscribed the name of " Mr. Charles King," and filled up the 

 blanks left on pages 1494 and 1495 by the initials " W." and 

 " W. Ch. Esq." with the additions of " irley par. Com. 

 Stafford." and "Walter Chetw... of Ingestry Staffords*." (the 

 rest has been cut off by the binder of the volume), so that there 

 remains no reasonable doubt as to the truth of the identification. 

 Now we read in the second of these letters from the country, 

 • dated "July 5, I703,"p. 1501, "But of those " (viz. animalcula) 

 "(among other things) I last year gave an account to Sir Ch. 

 Holt, which I hear will shortly be publish'd in the Transactions." 

 I don't think it is bold to conjecture that the account here 

 alluded to had already been published, and is, in fact, the 

 article printed in No. 284, for March and April 1703, pp. 

 i357(/^w)-i372 (with excellent figures on the plate accompanying 

 that number), under the title of " An Extract of some Letters 

 sent to Sir C. H. relating to some Microspocal " {sic) "Obser- 

 vations. Communicated by Sir C. H. to the Publisher" (H. 

 Sloane) ; and no doutit these epistles may also be ascribed to 

 the same anonymous gentleman. 



In all the above-mentioned letters we have some early and 

 fiist-rate contributions to microscopical science, the importance 

 of which had been shortly before so evidently demonstrated 

 by the wonderful discoveries made by the improved magnifying- 

 glasses. 



Quceritur : Who was Mr. Charles King ? S. 



The Hague, November 27. 



NOTE ON A PROPOSED ADDITION TO THE 

 VOCABULAR V OF ORDINAR Y ARITHMETICS 



'T'HE total number of distinct primes which divide a 

 -*■ given number I call its Manifoldness or Multi- 

 plicity. 



* Perhaps I may without immodesty lay claim lo the appellation of the 

 Mathematical Adam, a* I believe that I have given more names (passed into 

 general circulation) to the creatures of the mathematical reason tnan all the 

 other mathematicians of the age combined. 



A number whose Manifoldness is ;z I call an «-foId 

 number. It may also be called an 7/-ary number, and 

 for « = I, 2, 3, 4, . . . . a unitary (or primary), a binary, 

 a ternary, a quaternary, .... number. Its prime divi- 

 sors I call the elements of a number ; the highest powers 

 of these elements which divide a number its components ; 

 the degrees of these powers its indices; so that the 

 indices of a number are the totality of the indices of its 

 several components. Thus, we may say, a prime is a 

 one-fold number whose index is unity. 



So, too, we may say that all the components but one of 

 an odd perfect number must have even indices, and that 

 the excepted one must have its base and index each of 

 them congruous to r to modulus 4. 



Again, a remarkable theorem of Euler, contained in a 

 memoir relating to the Divisors of Numbers (" Opuscula 

 Minora," vol. ii. p. 514), may be expressed by saying that 

 every even perfect number is a two-fold number, one of 

 whose components is a priine^ and such that when aug- 

 mented by unity it becomes a power of 2, and double the 

 other components 



Euler's function (^(«), which means the number of 

 numbers not exceeding n and prime to it, I call the totient 

 of n ; and in the new nomenclature we may enunciate 

 that the totient of a nuinber is equal to the product of 

 that number multiplied by the several excesses of unity 

 above the reciprocals of its elements. The numbers prime 

 to a number and less than it, I call its totitives. 



Thus we may express Wilson's generalized theorem by 

 saying that any number is contained as a factor in the 

 product of its totitives increased by unity if it is the 

 number 4, or a prime, or the double of a prime, and 

 diminished by unity in every other case. 



I am in the habit of representing the totient of « by the 

 symbol m, t (taken from the initial of the word it denotes) 



' It may be well to recall that a perfect number is one which is the 

 half of the sum of its divisors. The converse of the theorem in the text, viz. 

 that 2"(2" ' ' — 1), when 2" "*" ' — 1 is a prime, is a perfect number, is 

 enunciated and proved by Euclid in the 36th (the last proposition) of the 9th 

 Book of the " Elements," the second factor being expressed by him in the sum 

 of a geometric series whose first term is unity and the common ratio 2. In 

 Isaac Barrow's English translation, published in 1660, the enunciation is as 

 follows : — " If from a unite be taken how many numbers soever 1, A, B, C, D, 

 in double proportion continually, untill the whole added together E be a 

 prime number ; and if this whole E multiplying the last produce a number F, 

 that which is produced F shall be a perfect number." 



The direct theorem that every even perfect number is of the above form 

 could probably only have been proved with extreme difficulty, if at all, 

 by the resources of Greek Arithmetic. Euler's proof is not very easy to 

 follow in his own words, but is substantially as follows : 



Suppose P (an even perfect number) =: 2"A. Then, using in general 



X to denole the sum of the divisors of X, 



/> 



_/p /."/i _j.+.-, /i 



2"A 



Hence 



A 



2«+i _i 



Hence A = ^Q, and /a = 1 -|- yu. -t- Q + mQ -f . . ■ (if m be supposed > 1). 

 Hence unless ju = 1 and at the same time Q is a prime 



y^ 



yx > fx(Q 4- 1), 



is greater than itself. 

 A 

 Hence an even number P cannot be a perfect number if it is not of the form 

 2"(2" ' ' — 1), where 2" "'" ' — 1 is a prime, which of course implies that « -(- 1 

 must itself be a f rime. 



It is remarkable that Euler makes no reference to Euclid in proving his 

 own theorem. It must always stand to the credit of the Greek geometers 

 that they succeeded in discovering a class of perfect numbers which in all 

 probability are the only numbers which are perfect. Reference is made to 

 so-called, perfect numbers in Plato's " Republic," H, 546 B, and also by 

 Aristotle, Probl. I E 3 and " Metaph." A 5, which he attributes to Pytha- 

 goras, but which are purely fanciful and entitled to ni more serious con- 

 sideration than the late Dr. Cummings's ingenious speculations on the 

 number of the Beast. Mr. Margoliouth has pointed out to me that Muhamad 

 Al-Sharastani, in his " Book of Relig.ous and Philosophical S=cts." Careton, 

 1856, p. 267 of the Arabic text, assigns reasons for reg.ardingaU the numbers 

 up to 10 inclusive as perfect numbers. My particular attention was called to 

 perfect numbers by a letter from Mr. Christie, dated from "Carlton, Selby," 

 , containing some inquiries relative to the subject. 



