conception, but which yet (as far as the most essential point of the result is concerned), has been confirmed as an inherent peculiarity in the nature of light, by the whole extent of modern research. The "Optics" is prefaced by a declaration, that its design is to explain the properties of light by means of experiment alone, without any admixture of hypothesis. In this respect alone it was quite a novelty in the age when it appeared: and when we compare the true philosophic spirit thus exhibited, with the extent, importance, and precision of the actual results, we shall be constrained to assign it a place in the first rank of experimental investigations, and to say that in that rank it stands alone.

Newton's Mathematical Discoveries.

In tracing the progress of those inventions which were directed to the object of overcoming the difficulties which elementary methods were incompetent to surmount, we have noticed the improvements successively made by some of the greatest mathematicians, which were chiefly called forth in the attempts to solve the problem of the quadrature of curvilinear areas, or the determination of an area equal to that of the curve expressed by some algebraical value in simple terms, derived from its equation. We have noticed the several principles which were suggested, but which received very limited applications; and the ingenious idea of Wallis, of interpolating in the series of known areas those which ought to hold an intermediate place.

Invention of the Methods of Series and Fluxions.

Newton, on taking up the subject, which, we have reason to believe, he did as early as 1665, soon extended the method of Wallis by the invention of general serieses for this purpose; and, at the same time, connected the process of such serieses with a highly refined method, which was, in like manner, an improvement upon

the hints supplied in the inventions of his prede

cessors.

The serieses which he deduced, as applying generally to the quadrature of curves, were, as he tells us, founded on Wallis's principle; and, from comparing the forms of those already obtained, he discovered a law which was common to them all. In such of these forms as involved fractional exponents, he found the series infinite; though, wherever he could render it convergent, it would of course give a value to any requisite degree of approximation. This, however, was closely connected with an object of more extensive importance and utility, a theorem for expressing any root of a quantity composed of two terms in a series involving certain combinations of those terms, according to a given law, and which would be convergent. To such a theorem he was almost immediately led; and, as he instantly perceived that roots were comprised under the same algebraic formulas as powers, by the mere adoption of the idea of fractional indices, he saw that this theorem was no other than the most general form of that for raising a binomial quantity to any given power, in a series of terms involving the index and the whole powers of the two terms in a certain regular progression. It hence acquired the name of the binomial theorem.

This theorem, besides its other numberless applications, was directly employed in the quadrature of curves, as supplying a more direct method than that originally adopted, and out of which it had in fact originated. The whole course of the discovery is related in detail, in a letter to Oldenburgh; one of a valuable series to which we shall often have occasion to refer, collected under the name of "Commercium Epistolicum." The application, then, consisted mainly in this; that he reduced the value of the ordinate of a curve into an infinite series of the integer powers of the abscissa by the binomial theorem, or by any simpler algebraical process, where the case admitted it. After this, other considerations, like those employed in the method of in

divisibles, would assign the small area corresponding to each term, and the sum of these would give the whole area required.

But such methods as those of indivisibles were operose, tedious, and unconnected by any very general principle. The fertility of Newton's genius soon supplied this part of the process also, and devised a method which, by the application of certain easy but highly general rules, gave the ready solution of all problems of this nature. A letter of Dr. Barrow, in the collection above referred to, in 1669, mentions, that some years before Newton had shown him a MS. treatise, but would not publish it, entitled "Analysis per æquationes numero terminorum infinitas," in which not only were such serieses employed, but the general connecting principle and method clearly pointed out, though certainly not given in precise formal rules, with a peculiar notation.

Newton first gave to the public an account, though a very brief one, of his method, in the second lemma of the second book of his Principia, in 1687; and the precise rules and notations were made known by Dr. Wallis, in the second volume of his works, in 1693, where he extracts an account of them from two letters of Newton, written in 1692. This method, then, which its inventor had called fluxions, was known for years only to his friends; but those friends soon included, by means of their correspondence, some of the first mathematicians of Europe. In the earlier stage of the investigation, however, the different serieses employed for quadratures were the principal objects of attention, and these were soon communicated to some of the most eminent geometers.

Leibnitz visited England in 1673; and, having formed an acquaintance with Oldenburgh, secretary to the Royal Society, continued a correspondence with him on his return to the Continent. At the time of his visit, he was but slightly conversant with mathematics ; but his powerful mind grasped every subject; and, after his return, he was soon able to enter into those

topics which were beginning to excite so much interest, and even in a condition to take a part in the discussion; and, in 1674, communicated to Oldenburgh some serieses of his own. In 1676, Newton, at the request of Oldenburgh, wrote an account of his method of quadratures, containing also his method of fluxions, concealed under an anagram. This was sent to Leibnitz; who, in 1677, replied to Oldenburgh by sending a short account of an equally general method which he had invented, which he called the differential calculus, and of which he gave the principal rules and notation. An account of it was first published in the "Acta Eruditorum," in 1684.

Thus, while Newton's method remained known only among his friends and correspondents, that of Leibnitz was publicly announced, and spreading rapidly on the Continent. Two most able coadjutors, the brothers John and James Bernoulli, joined their talents to those of the original inventor, and illustrated the new methods by the solution of a great variety of difficult and interesting problems. Such was the reserve of Newton, and so little were his methods known or followed up among his countrymen, that the first book which appeared in England on the new geometry (as it was called), was a treatise by Craig, professedly derived from the writings of Leibnitz and his disciples.

Thus far, then, the history of these important inventions is clear. Two valuable and general methods of analysis, applicable to the solution of the same classes of problems, though differing in their notation, and in the mode in which the first principle was conceived, were discovered, separately and independently, by Newton and Leibnitz, within a very short time of each other. Newton's invention was first in order of time, but not published to the world till long after. Leibnitz could not have derived his idea from that of Newton, that idea being concealed in cypher: he discovered it independently, though rather later in time, but was the first to publish it. Newton, in his Principia, gave testimony

to the independence of Leibnitz's discovery, and expressed a favourable opinion on its merits. Leibnitz, moreover, seemed equally willing to admit the claim of Newton. Leibnitz, indeed, enjoyed the advantage of seeing his calculus rapidly improving, and extending its applications in the hands of his friends and disciples, whilst that of Newton remained in comparative obscurity. But still nothing like rivalry or hostility appeared between the parties. A few years later, we shall find this tranquillity strangely disturbed: for the present we must briefly recur to some general view of the nature of the discoveries in question, and then proceed to other subjects of discussion, of which so many, and all of such high interest, crowd upon us at this most eventful period of scientific history.

General Idea of the Fluxional Calculus.

What we have before said of the methods of infinitesimals, and the attempts at the problems of tangents and quadratures, will have sufficed to show how near an approach had been made to general principles. The method of serieses is in itself, perhaps, at once the most satisfactory and the most easily applicable, in cases where an exact solution is impossible: the essential idea is one which is practically familiar to the mind of every one who has even gone so far as to calculate by decimal fractions. The notion of a value carried on to any number of places of decimals, but which, strictly speaking, never terminates, though each successive figure we obtain brings us, in a tenfold degree, nearer the truth, and though we may approach nearer to the exact value than to be able to assign or even conceive the difference, this notion, we repeat, is practically familiar to the mind of every computor. There is, then, nothing more than this in the principle of those applications of serieses to which we have referred. The object and nature of the series is simply to afford the means of expressing a succession of terms, each formed

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