one woul, have imagined that gratitude for benefits received would have formed a part of their doctrine. And here I cannot refrain from stating how completely we are the jest of the whole continent, for being betrayed by men on whom we are still lavishing imniense sums of money, and for barbouring in the bosom of our own country those who are sending intelligence of importance to our enemies.' This work, which is gratefully dedicated to Dr. Jenner, is occasionally varied by the author's poetical talents, and con. tains on the whole, considering its size, no small portion of information. , Art. III. A Reply to a Monthly Reviewer, in which is inserted Euler's Demonstration of the Binomial Theorem. By Abram Robertson, D.D. FR.S, Savilian Professor of Geometry. 8vo. ' is. 6d. Oxford, Cooke, &c. ; London, Payne and M•Inlay, &c. 1808. In our Number for February 1807, we reviewed a DemonstraItion of the Binomial Theorem inserted by Profrssor Robert. son in the Transactions of the Royal Society for 1806, part 2d; and we said that the demonstration was not new, since it was the same with that of Euler, in Novi Comm. Petrop. 1774. Again, in our Review for September 1807, we took notice of a paper from the same author, on the Precession of the Equinoxes, inserted in the Transactions for 1807, Part I., and we remarked that the chief part of the Paper was taken from Thomas Simpson's Miscellaneous Tracts, 1757. As it might have been expected, Mr. (now Dr.) Robertson was dissatished with our strictures; and with almost mare than the usual irascibility of an author, he has published a Reply, in which he calls us by many hard names, says that our criticism is weak and ignorant, and asserts that his De. monstration of the Binomial Theorem is not like that of Euler, a'nd that he did not borrow from Thomas Simpson. Now in the fate of our remarks there is something singular. Dr. R. contends that they are imbecile and insignificant, and yet he deems it necessary to make a continued effort to crush them during forty pages :-he perused them, he says, with perfect composure, yet never did a reply exhibit plainer symptoms of irritation ;--and he is determined to prove them to be unjust, yet, by a most sing alar act of exculpation, he furnishes the very matter which establishes their justness beyond a doubt. This is no party representation, no smartness of retort com. mon and allowable in argum¢ntation, and said in the extravagant spirit and language of controversy, but the statement of a plain fact. We observed that the Professor's demonstration was similar similar to that of Euler :-o rebut this charge, he prints parts of his own and of Euler's proof; and what do these shew?--that the proofs are dissimilar?-or that the resemblance is slight and imperfect? no such thing; they incontrovertibly establish the similarity.-It will be difficuli to find a like act of infatuation : an instance of candour so ludicrous in its excess, or a conduct so unavailingly bold. We also shall now proceed to give the essential parts of Euler's and of Dr. R.'s proof; and we shall beg leave, for the use of our readers and of Dr. R., to explain Euler's symbols, according to the meaning which he himself affixes to them. This being done, we are satisfied that every person who is in. the least imbued with Mathematical Science will recognize the similarity, or rather the identity, of the two demonstrations. (4) In page 107, Euler, after having reduced (a + b)a to the form (1 + x)", says that " When n is an integer positive number, (1 + x)* is known to be equal to the series - it ext t . . - . *? + &c. but if n be not an integer positive number, we may regard the value of the series as unknown, and use for it the sign [n]; so that, generally, we may put [n] = Itnx + n. *? + &c. of which we know only at present that, in the case in which n is an integer positive number, [n] = (1+x)*: but in oiher cases the values that belong to this sign [n] may be investigated by the me thod which follows: whence it will appear that generally [n] = (1+x)", whatever numbers are put for the exponent the “(5) To conduct this investigation, let us multiply two series of this kind, or two like signs [n] and [m] together, that we may obtain a series equal to the product [m] [n], which it is evident will be expressed by a form of this kind, 1 + 8x+BX+ &c. ; and that it may appear in what manner the coefficients are to be determined by m and n, let us begin the multiplication 1 [m]=1+ .*+ *. " ! **+ 8c. 2 [m].[n] = 1 + (m + n) x + - tmn + ,202 + &c. “ If now we compare this product with the assumed form it Ax + Bx. + &c. it appears that . ?-9 m + m + I A=m+n, B=" * tmn + K + 2 « (6) In “ (6) In the same manner as we have been enabled to determine the first two coefficients A and B, by m and n, we may determine, if the multiplication be continued, the coefficients C and D &c. by the same letters m and n, alihough the calculation would soon become troublesome and operose. In the mean time, we may hence safely conclude that all the coefficients A, B, C, &c. ought to be formed after a certain determinate manner by the two letters m and n, though we are ignorant of the law or principle of their formation: but here it behoves us principally to observe, that this law of formation does not depend on the nature of the letters m and n, but will be the same whether m and n denote whole numbers, or any other letters whatsoever. Let this reasoning, somewhat refined, be well noted, since on it the whole force of our demonstration depends. " (7) Hence an easy method is opened to us, of finding the true values of all the coefficients A, B, C, &c. while we regard the letters m and n as integers, since the same determinations arise as if they denoted any other numbers whatsoever : but the letters mand n being considered whole numbers, we shall certainly have [m] = (1+x)" ,and [n] =(1+x)", whence the product of these formulæ will be [m] : [n] = (1+x)m+1: but this power is evolved into a series mtn mtn min-1.2 1+**** + == *2 + &c. & I "TIT* « Now, then,' if we consider m and n to be general, this series ought to be denoted by the sign [m+]; whence we obtain this remarkable property, that [m] . [n] = [m + n], whatever numbers are substituted for those letters *. - (8) Since, then, two forms [m] and [n] of this kind, multiplied into each other, produce a simple form of the same nature, so also, if several like forms be multiplied into each other, they may be reduced into a simple form ; for we have the following reductions: [m] . [n] = [m + n] [m]. [n]. [P]. [9] = [m+n+p+g] &c. Hence, if all the numbers m, n, p, &c: are taken equal to one of them, m, for instance, we shall have the followiug reductions: [m]? = [2m]; [m] = [3m], [m]* = [4m] &c. and generally [m]a = [am]. * It is to be recollected that [m] stands for i+mx + m. *** &c. &c. but [m], Euler says, = (1+x}m (m an integer): in other words, (i+x)m evolved is represented by the above series, and (1+x)*(n an integer) is represented by a like series,.'. (1+mx+&c.) (1+nx + &c ), or [m] . [n] = (1 +0:1m (1+x)* = (1+x)*+* (m and n integers) (1+x)-411 = 1+m+nx + &c. : but, since the law of the formation of the coefficients has been shewn to be the same whatever mand n are, .-: [m] [n] = 1+ m +1x &c. whatever m and n are, and therefore == [m+n]. Rev. ." (9) These * (9) These things being premised, let i denote any integer positive number, and let us first put 2m=i, so that m= , and the first of the last forms gives = [i]: but, because i is an integer number, [i] will = (1+x)"; and so, [ ] = (1+x)*: whence, by extracting the root, [] =(1+x) • ; and thus much therefore we have obtained, that the Newtonian theorem is true also in those cases in which the exponent n is a fraction of this form . " (10) In like manner, if we put 3 m =i, so that m=?, the second of the preceding forms gives 6 ]=[i]=(1+x); hence by extracting the root we obtain - =(1+x) * -; and thus our theorem is also true if the exponent ’n shall be a fraction of the forn - Generally, also, it is clear that l __] =(1+x)ė: so that it is now demonstrated that our theorem is true if for the exponent n any fraction, as -, be taken, whence its truth is established for all positive numbers that can be substituted for the ex. ponent n. " (u) It only remains, then, to establish its truth in those cases in which the exponeni n is a negative number. For this purpose, let us call to our aid the reduction first found; that is, [im]. [n] = [m+ n], where m denotes a positive number whether it be whole or broken ; so that, as it has been shewn, [m] =(1+x)". Put now n=-m, and m + 11 = 0, and therefore [0] =(1+*;° =1; which being substituted, the preceding formula gives (1+x)". [-m] = 1, whence we get [~m] = (1+x)m ~!" m =(1+x)"; and thus also the Newtonian theorem is shewn to be true, when the exponent n is a Acgative number." Such is Euler's demonstration : that which follows, was given by Dr. R. in the Philusophical Transactions for 1806, . Part 2. “ (15) By the general principles of involution a+b=a"xit 5 jo za"x1+x", by putting x = By article 130.3+al"=1+nx and by the same article 1+x1" i+mx + m. .--*4 + &c. But by the general prin. ciples of involution, and article 13, 17x)" x 1+xm=i+xa+= 1+ x + mx+n+m: *++-+*° +n+m.*** .***-?x' + ntm-1 atm-2 nim-3 3 x4 + &c. when n and m are whole numbers, • Hence it is evident that if the series equal to 17x7" be multiplied by the series equal to 1+xm, the product must be equal to the series which is equal to itx}"*". Now the two first mentioned series being multiplied into one another, and the parts being arranged according to the powers of x, the several products will stand as in the following representation. 403 + m . . .n ** + &c. For the sake of reference hereafter, let this be called multiplica. tion A. • Now with respect to the coefficients prefixed to the several powers of x, in the foregoing multiplication, two observations are to be made, by means of which the demonstration of the theorem may be extended to fractional exponents. In the first place, supposing n and m to be whole numbers, the sum of the coefficients prefixed to any individual power of x, in multiplication A, must be equal to the coefficient prefixed to the bame power of x in the binomial series 1+n+ mx+n+ m u s • *? - ttm- tm-2 . -- 2 n m -3 m. ann. *?+n+m. + &c. The certainty of this circumstance rests partly on the 13th article, and partly on a plain axiorn, viz. that cquals being multiplied by equals the products are cqual. tim---! tntm. |