M NTMBERS, FIGURATE AND POLYGONAL. 



NUMBERS, THK.'KY OF. 



theological. It is not right that they should be utterly lost sight of. 

 for they form a dark background on which the merits of Sacrobosco, 

 Bradwardine, Kegiomontauus, and afterwards Tonstall and Recorde, 



XI MllKlts. lltil KATE and POLYGONAL. [NUMBERS, Amx- 



LATIi -XN OF.] 



M'MUKHS OF BERNOULLI. This name is given to certain 

 numben (we here see the mathematical use of the word, for they are 

 all fractions) first used by James Bernoulli, in his ' An Conjectandi.' 

 They are in fact (though not so considered by Bernoulli) the co-efficients 

 of the divided powers of x in 1 : (t * 1). We should hardly have given 

 them a place here, as our list of such ultimate references in mathe- 

 matics is by no means complete, if it were not that they only appear 

 to a sufficient extent in one English work that w know of (Peacock's 

 Examples). Let 



1 1 J_ B,* B^r> 



,ti _ J """ o * o 





2.3 



a form which it is shown to take. Then B,, B,, B 5 , Ac., are what are 

 called the numben of Bernoulli, and the following hist will show tw, 'in \ - 

 five of them, the first column being the index of B, the sec.. ml the 

 numerator of the fraction, and the thud its denominator. As for as 

 B W these are taken from Eider's Differential Calculus, all the rest 

 (and the logarithms) from Grunert's Supplement to Klugel, which 

 professes to take the additional numbers from a work of H. A. 

 Kothes, and the logarithms from Kytehvein's work on the higher 

 analysis : 



Numerator. 



No. 

 11 

 31 



51 



71 



95 



11 891 

 137 

 163617 

 17 43867 

 19174611 

 21 854513 

 23236364091 

 258553103 

 27 28749461029 

 298615841276005 

 31 7709321041217 

 33 2577687858367 

 8526315271553058477373 

 372929993913841559 

 89 261082718496449122051 

 41 1520097643918070802691 

 43 27883269579301024285023 

 45496451111593912163277961 



Denominator. 

 '6 

 30 

 42 

 80 

 66 



2730 

 6 



610 

 798 

 :::>.< 

 138 

 2780 

 6 

 870 



510 



6 



1919190 



6 



1 U80 



1806 



282 



47 5609403868997817fi8(i249127. r )47 4410 

 49 495057205241079648212477525 .66 



Thus the coefficient of a- 1 " : 1 . 2 . 8 . . . 15 in the development of 

 1 : ( - 1) is 3617 : 510, or n,.. The logarithms of the first eighteen 

 numben are as follows : 



No. Logarithm, 

 II 0-2218487496-1 



The higher numben may be approximately verified by the following 

 rule.' Let T be the ratio of the circumference of a circle to its 



1.2.3 ...... 2T-1.2* 



' ', 



We shall have some occasion to point, out the uses of the numben 

 of Bernoulli in the article SEIIIK*. The theory of these numben will 



s found in Peacocks . K,,],.^ J>itl, . 



CaJculu*,' Lacroix's ' Differential Calculus,' 3 vols., and in a very elabo- 



rate article (' BernouUischo Zahlen ') in the work of Grunert already 

 cited. 



'1 HERS, TH l'.< >l I V i ' V. The theory of numbers IK in fact the 

 science of whole or integer numben, and its most gem i.il ]<n>U 



D any equation whatsoever involving two or m.>ie unknown 

 quantities, or any number of equations between a greater nun i 

 unknown quantities, to determine every possible solution in which the 

 values of the unknown Ictten are whole numbers." It may also be 

 considered that the science extends to the determination of a'.: 

 tioua which contain nothing but rational or commensurable fi.. 

 all surd quantities or incommensurable!) .being excluded. 1 

 example, the equation x" + y*= 1000 were to be solved, x and y being 

 whole numbers or rational fractions, let the rational fractions reduced 

 to a common denominator bep : rand q : ; : thru th.- equation becomes 

 p' + o'=1000 z*; and if all possible whole values of j>, /, and ; be 

 found, all the fractional solutions of the former equation can be 

 exhibited. 



Connected with the science before us is a very large quan 

 properties of numben, of which it must be said that they can be 

 proved e.vtily enough, but cannot be expl i.-illy, in ret] 



the steps of an algebraical demonstration, we can . 

 result with common ainl self evident notions, which weni I 

 ju.-tify the coiicluiiion, to render it natuml. and destroy much oi the 

 curiosity, and even interest, with which it is looked at - 

 used to algebra, who hears of the conclusion for the first time. 1 n the 

 theory of numben it seems to us that the curious eh.-. 

 conclusions is not so much lessened by the demonstrations, and perhaps 

 this may be the reason why the science becomes a sort of passion, as 

 Legendre remarks, with most of those who take it up. The instances 

 given by the writer just cited, in his preface, will >-how tin' sort of 

 properties which we speak of. If c be any prime number, and N any 

 other number not divisible by r, then N r ~' 1 is always divisible by r. 

 Then 2 1, or 63, is divisible by 7. Again, if any prime n 

 divided by 4 leave a remainder 1, it is the sum of two square numben : 

 thus 13 is the sum of 9 and 4, 17 of 16 and 1, 29 of 25 and 4, &c. 



The theory of numbers is not of much immediate practical utility in 

 the applications of mathematics, which generally involve continuously 

 increasing magnitude, and in which therefore the introduction of whole 

 numben is matter of convenience, and not of nee. in, the 



data of such applications arc usually ouly approximate, so that an 

 answer in whole numbers, should such a thing occin . 

 possesses no particular interest. Hence this theory i.- Hit Ir ', u.h. <1 l.\ 

 .1 V.TV large class of mathematicians, among whom it a not unco; 

 to meet with a person deeply versed in the higher analy.-i.-. who does 

 not even know the principal results obtained by Gauss or Lc ^ 

 The subject is, in fact, an isolated part of mathematics, which : 

 taken up or not, at the choice of the student. It may possibly at some 

 future time be connected with ordinary analysis, that is to say, the 

 determination of the intiycr solutions of a set of equations may not be 

 so distinct a thiag from that of a mere solution, integer or not, as it is 

 at present. In fact, a hint given by M. Libri, in a tract presently to 

 be cited, docs give completely the means of assimilating the expression 

 of a problem in this theory to that of one in ordinary an 

 Suppose, for example, it is required to solve in whole munl>crs the 

 e.ju ition x- + if ! =x'. Let * represent two right angles; then it is well 

 known that sin ir x=0 when x is a whole number, and never else ; so 

 that " required a solution of .( J + y 1 =a 3 in whole number* " i- pn 

 the same problem as "required any solution of the three equ 

 :r" -t y- = ar, sin T x 0, sin ir y = 0." 



The earliest consideration of the theory of numben may have been 

 made in India [VlOA GA.MTA, in 1'ioc. l)iv.] ; but the 

 probably that of Diophaiitus, which consists of nothing .'.loins 



of this science, insomuch that the theory it .-el f 1 1 



the Diophantine analysis. The subject then rested. ...aking 



any progress, until the time of Bachet de Meziriac and i 

 editor and commentator of IMoph.-uitus. The subject rented 

 until the time of the man who literally left no part whatever of 

 mathematics iinaugment<-d, Enler. After him, Lagrangc, Liv 

 and Gauss applied themselves contemporaneously to < I 1.. 



works of the two latter are the separate treatises on this ]>dr: 



.in which the a. :th, matieal student r 



know it* present state. Various Memoirs of MM. C.iuchy aid 1 il.ii 

 may also be mentioned ; one in 



liein.itiqiic et de 1'hysiquc,' vol. , . 



in which the subject is made to have more resemblance than usual to 

 ordinary analysis. An elementary treatise <ui the tlic.ny of mn 



present state of the subject, is much wanted. The onlv 

 one, in English, that of IJarlow, was published in 1811 : it can 1 

 recommended for all that it includes. 



The |iis.|iii, -itioiies Arithi. MM d'.iun-wi.k. ISO] 



.- des Nombres' of Legendre (thud edition. n) has 



the advantage of coming later than that of Gauss (whi. -h 

 ion of Legendre'- ng method 



lions whi.li are more familiar to the ma'! works 



;n.ht\ : that of the Cennaii is condei ,ull of 



historical int'.nmat.ioii ; that of tin \v, but 



like most r'i. neh works, deficient in 



