FACTORIALS. 



FACTORIALS. 



conceivable, whether a be integer or fractional, under the usual and 

 easy extension to fractions of the idea of multiplication ; but it is not 

 or ought not to be, BO intelligible when n is a fraction. What are four 

 and a half multiplications by 36 ? The beginner will say, four multi- 

 plications by 36, followed by a multiplication by 18 ; but this mode oi 

 defining breaks down immediately, for the two half operations would 

 make more than the whole : two successive multiplications by 18 are 

 equivalent to more than a multiplication by 36. It is multiplication 

 by 6 which is the half operation to multiplication by 36. It is true 

 that we do not apply the phrase fraction of an operation in our descrip- 

 tive language ; but we apply the symbol in our symbols. For just 



as every a* in a" denotes one multiplication by a, every a'denotes 

 that multiplication which twice repeated is equivalent to one multipli- 

 cation by a. In like manner a 1 is the multiplier which being used 7 

 times, gives the same result as a used 3 times. We are not going to 

 give the theory of simple powers, but only to put it in connection with 

 what follows ; and the reader will do well to observe, that in the very 

 first ideas of ratios [ADDITION OP RATIOS] the notion of numerical 

 quantity entering as a multiplier in repeated operations was so much in 

 the minds of those who framed Euclid's language, that they spoke of 

 what were really multiplications as if they had been additions. The 

 same thing may be traced in calling 100 to 1 the duplicate ratio of 

 that of 10 to 1 [RATIO], and 10 to 1 the subduplicate ratio of that of 

 100 to 1 : duplicate means double, and subduplicate means half. The 

 beginner must learn to understand numbers with reference to their 

 force as indices of operation, and even the advanced student may 

 'I more study of this part of the subject than he suspects himself 

 wanting. 



Again, to establish the equation 0(x,7n) = .r" when m is an integer, 

 is not the same thing as establishing 



in fact the symbol x* in algebra is well known [Rooi.] to be in its 

 complete meaning 



-y'.r- ( cog -.2ii + V - 1 sin -.2fcir ) 

 ( n n j 



where k is any integer. 



Next after the operations of powers and roots, nothing occurs more 

 frequently in mathematical formuUe than successions of multiplica- 

 tions in which the multiplier is not always the same, as 1 . 2 . 3 .... n, 

 a(a + b) (a + 24) . . . . (a + n 16). The various hints which had been 

 given of the interpolation of fractional meanings, such as that of 

 Wallis, and others of Leibnitz, the Bernouillis, Stirling, to., have been 

 extended with great power by the French and German mathematicians 

 of the last eighty years. Two different lines were taken in the two 

 countries. The Germans first began to consider how the ordinary 

 notation might be extended. Vandermonde proposed to denote 

 m(m 1) (m - 2) ---- to factors by [m] ; the bracket* distinguish- 

 ing it from m* in the usual sense. Hindenburg, followed by Kramp 

 and most of the Germans, proposed a much better notation. Con- 

 rig .< as denoting m unaltered factors, they made room in the 

 nyinlxjl f'r a part expressive of the permanence of the factor, and wrote 

 it *" I . Thus it became a particular case of x* I ", which was made to 

 stand for m factors, the first of which is x, and which alter by a at 

 every step ; giving 



1 = x(x-a) (x-Za) ..... (x-iifTl 

 (x + ^lo)l- = x*\* 



:nid so on. This notation certainly opens the road to convenient 

 expression of a large number of striking formula; : take its binomial 

 theorem for instance, 



which is perfectly analogous to the ordinary theorem. 

 Also the following : 



+ ( 



^ + (MA)"' ~- 



which is true for all values of m, and gives the binomial theorem if 

 '= !, and the exponential theorem if A* = l. a = 0. The 

 analogous theorem to Taylor's is 



i i is well known. 



We think it is to be regretted that this notation has not been more 

 I in England : we do not remember at this moment any writer 

 who ha made much use of it, except Mr. Peter Nicholson, in his 

 works on Involution and on Increment*. 



A name was to be found for this extension. The notion of calling 

 x 1 , x 3 , 3*, &c. the powers of .r, was an extension of the term as used by 

 Euclid, which applied to the square on a line only. Not that the 

 square on a line was originally called its power, but that the power of 

 a line was measured by the capacity of its square. The object of the 

 old geometers was to reduce every area to a sqxiare, which enabled 

 them to describe it by one line ; and hence a line seems to have been 

 considered as having more or less power (of inclosing space) according 

 as its square was greater or less : the power being measured by the 

 magnitude of the square. The phraseology seems to have reached 

 those who were not geometers : thus Diogenes Laertius tells us that 

 Pythagoras discovered that * the ' subtending side of a right-angled 

 triangle is as powerful as the two containing sides together.' But 

 those who will smile at the idea of the power of a line residing in its 

 square, will laugh outright at the notion of Kramp, who proposed, 

 seeing x* represents the pmvers of x, that the symbol .r 1 " should 

 represent its numerical faculties (facultds numeriques). From the 

 powers and the faculties we might have reached, possibly, the feelings 

 and opinions, had it not been for Arbogast, who proposed to call the 

 different cases of .r*' by the name of factorials, a term which has now 

 gained considerable currency among the German writers, and was 

 approved by Kramp himself. 



The French, on the other hand, follow Euler and Legendre in con- 

 necting the factorials from the outset with definite integrals, and the 

 latter in adopting a specific notation, not derived from that of powers. 

 Legendre signifies 1 .2 . 3 . . . . n by F (n + 1), and hence the name <5f 

 gamma-functions has been applied to them : they are best called, 

 factorial fuiu < 



We shall give a slight account of the subject so far as it is in the 

 way to be speedily reckoned among the elementary parts of mathe- 

 matics. 



A series or a product of terms is only distinctly conceivable when 

 ri is integer, but if it can be represented by a function in which n 

 enters as a usual symbol of magnitude, and not as a number of terms 

 or operations, then the function is intelligible, though not the 

 representative of the series, when n is a fraction. To take a very 

 simple case : it would be absurd to demand the value of 1 + 2 + 3 + 



+ (n l) + n, when n is a fraction; but this series is 4(ra+l), 



which is always intelligible. 



The equation 



1 + 2 + 3 + + = JH (i + l) 



is absurd except when n is a positive integer. In the times when the 

 phrase ' less than nothing ' was invented, it would have been said 

 boldly that 2J terms of this series are | x 24 x 3J, or 35-=-8, and that 

 7 terms of it are J (7) (6) or 21. All that we should now say is 



that the function which, when n is integer, is equal to 1 + + n 



becomes 35-=-8, and 21, when is 2J and 7. Whether we are likely 

 to be the gainers by refusing extensions of language which naturally 

 present themselves, remains to be seen : it seems to us that ' 7 terms 

 of the series l + 2 + ... + w'is a very innocent abbreviation of ' the 

 value, when n = 7, of the function which, when re is a positive integer, 

 is always equal to 1 + 2 + ... + .' But at any rate, mathematicians are 

 now in the habit of passing from expressions in which n is an index of 

 number of operations, to the equivalents in which n is only an index 

 of magnitude, and of using the latter in the most general sense. 



But there is an infinite number of ways of representing, for example, 

 a function which is 1 + 2 + .... + when n is an integer. If $n be a 

 function which is unity whenever u is an integer, such as cos27rn, 

 l + sin2-)i, &c., then Jre (" + !) * <t>n answers the condition as well as 

 4" ( + 1). It is usual however to start with a radical function which 

 is free from periodic multipliers, and there is generally no difficulty 

 in deciding upon the selection. In all the cases which are most 

 useful, the radical function is the one which is clear of all sines and 

 cosines. 



But it is to be remembered that in this branch of the subject we 

 have not advanced so far as to make it coextensive with the theory of 

 powers : it is in fact precisely in the condition of the theory of powers 

 before the discovery of the multiplicity of values in x* when n is 

 fractional. We are thus limited to an arithmetical view of the subject. 



Some writers have censured Legendre for employing a new symbol r 

 + l), when 1"" was already in use : if, which may be doubtful, he 

 had heard of the latter before he invented the former, he would, in 

 our opinion, still have acted judiciously in inventing the additional 

 symbol. He might have argued that it would not be wise to associate 

 the second symbol with notions which are only true of the arithmetical 

 case of it. As soon as the complete theory of the expression shall be 

 given, 1" is ready for it: in the mean while F (ra + 1) expresses the 



M 



arithmetical case of it, just as V x m expresses that of .r". 



We translate quite literally, to show that I.aortius was not geometer 

 enough to know that the subtending and containing was said of the right angle, 

 not of the triangle. His words are ;,., Si,T which Kraus, who in his turn 

 was not geometer enough to venture the rendering of jpfc^ M into Latin, 

 :ranslates laatundrm valrre. \Ve take I,acrtius as meaning that the hypothenuse 

 was M powerful as the two sides together : whether he understood hia own 

 ihrase, or only caught it from the geometers, is another question. Tor othe 

 uses of the same phrase, sec IRRATIONAL 



