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into the general nature of those expansions, to justify the assumption of the series with which the proof

sets out.

2. The proof proceeds upon the supposition of a numerical equality between (a + x)*, (a + x)-m, (a + x), and their respective expansions continued without limit, which equality does not exist. There is not, necessarily, even an approximation in value between those quantities and their several expansions, when particular numbers are put in the places of a and x.

3. The sophism of shifting the hypothesis is next introduced; that is, results are obtained by making a supposition in one part of the demonstration, which, if it had been made in a preceding part, would have wholly stopt the process. The value of x is made equal to that of y in one equation, after building upon another equation, in which if that supposition had been made, the latter equation could not have been obtained.

4. The equality of the coefficients of the corresponding powers of r in two series of the form A+ Bx + Cx2 + &c. = a + bx + c x2 + &c. is asserted; and as this is asserted without proof, it is to be supposed that the common method of inferring that equality, by making x = 0, which proceeds upon a petitio principii, is deemed suf

ficient.

One, or more of these principal objections may

be made to most of the other proofs of the Binomial Theorem which are best known.

That published by LAGRANGE in his Theorie des Fonctions Analytiques, has been rendered much less objectionable by the learned author of the Principles of Analytical Calculation, in whose work it appears. Still, it hardly seems judicious to employ the symbol of equality, where no equality exists, although the reader is forewarned of it. This is, in reality, not merely an extension, but a change, of the meaning of the sign. There is, perhaps, some impropriety in denoting by it the relation subsisting between a variable quantity and its limit; still less, then, ought to be used where no such approximation necessarily takes place. There are degrees of inequality, but none of equality. An objection of such a nature, however, is comparatively light. But the proof is made to depend on the assumption, that if (a + x + x) *** be expanded as a binomial, first by considering a + x as one quantity, then by taking x + z as one quantity, the resulting series shall be identical, whatever the index of the expansion be, whether integral or fractional, positive or negative. That this is true when the index is a positive integer, will be readily granted; for it may be intuitively perceived. But can it fairly be assumed to obtain in any other case? How is it that the mind assents to any general proposition? It instantaneously verifies the included assertion, or

negation, by having recourse to definitions, or to some obvious instances. to which the proposition is applicable, and in which it is manifestly true, independently of any particularity belonging to any one of the instances considered. If there be not this perception of the agreement, or disagreement, of ideas, no genuine conviction can follow. The plausibility of the enunciation of a proposition may, indeed, be such as to win a hasty consent from the indolent; but, without that necessary verification, there can be no real knowledge. That" the same result must be had, when the same algebraical operation has been performed on the same quantity" is a proposition, at first sight, sufficiently plausible; but in the application made

LAGRANGE of that assertion, the quantity operated upon cannot, strictly speaking, be said to be the same quantity in all the cases; and when the proposition is more precisely enunciated, it becomes necessary to resort to the usual method, in order to judge whether it be true or false.

m

Now + is the index of several operations, and

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there is none of them which manifestly produces two identical series, when (a+x) +≈ and a +(x+%) have been subjected to it. In so simple a case as that in which the index is 1, the resulting series are far from being manifestly the same; and they are in no case necessarily so, excepting that in which the index is a whole positive number; for then only is there a numerical equality between M

(a+x+x)" and its whole expansion. The assertion is, undoubtedly, true; but it requires, and admits of, a proof, as much as the Binomial Theorem itself; and it does not appear that any advantage would be gained by previously establishing the truth of this assertion.

The faults, here imputed to LANDEN and LAGRANGE, seem to have arisen from the desire of being concise, where conciseness is not attainable, and where precision should have been chiefly aimed at ; and from the affectation of generalizing too hastily, where all the included particulars do not readily occur to the mind, and where they have scarcely enough in common, to furnish the basis of a demonstration equally applicable to each of them.

A proof very lately published, in the second part of the Philosophical Transactions, for the year 1816, which sets out from the assumption, that (a + x)m. (a + y)" = [(a + x). (a+y)]", whether m be positive or negative, integral or fractional, is evidently liable to some of the objections which have been urged against the methods of LANDEN and LAGRANGE. The equation thus assumed for the foundation of a general proof, is intended to comprehend some cases, in which, strictly speaking, no equality obtains. Or, if it be otherwise explained, in the form of the results of the operations, designated by the index m, an identity which is far from being self-evident.

it

supposes,

Considered with respect to exactness, Professor ROBERTSON'S proof of the Binomial Theorem, has great merit; and this is the place to acknowledge that his manner of inferring the form of the coefficients of the expanded binomial, when the index is a positive fraction, has been adopted in the following attempt. The author had, indeed, previously applied the same principle, in the case of a negative index; but, before he had perceived its application to the former case, it fell in his way to see the Professor's demonstration. It is remarkable that Mr. Robertson himself is not exact, when he investigates the expansion of (a + x)-m. He does not obtain it without the same shifting of the hypothesis, the same reduction of the value of a to 0, a supposition implicitly excluded in the previous reasoning, which has been noted in Landen's proof. He also places signs of equality between quantities which are not equal.

After what has been said, the reader will not look for conciseness in the following demonstration; but the Author has studiously endeavoured to avoid the imperfections which have here been pointed out. His original intention was merely to supply the defects, and to rectify the errors of other writers better known than himself. It appeared to him to be necessary, both to the correction of Landen's proof, and to the demonstration of the assumption made by Lagrange, that the ratio of any two successive coefficients, in,

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