The idea of

on a former occasion, we briefly referred. Newton was a singularly happy instance of the universal and exact applicability of his metaphor. When a function increases or decreases up to a certain point, and thence again decreases or increases, at that point it is a maximum or minimum. It is obvious that its velocity of increase or decrease varies up to that point where it is absolutely nothing; and then begins to increase again. Thus, by finding the value of the quantities involved which will make the fluxion become nothing, we find the maximum or minimum.

Another kind of maximum or minimum, abounding in problems of the highest interest, but far more difficult than the last mentioned, is that in which the function itself is required to be found which will be the greatest or least under certain conditions. One class of such problems we have before referred to, as known by the name of iso-perimetrical problems. But the investigation, in its more extended form, exercised the talents of the two Bernoullis, and has received from them, and from subsequent analysts, the most complete investigation.

In a sketch like the present, it is of course impossible to enter into any adequate account of the different branches into which the direct or inverse parts of the calculus extend themselves. It must suffice to say, that the latter, as it is the most difficult and important, so it is by far the most extensive. One very large and important class of problems which it has to consider, are those which involve two or more variables mixed up with their differentials in any manner of combination in an equation. The difficulty here is that of separating them, if possible, so that each variable shall stand combined with only its own differential; but the process is only capable of being effected in certain cases; and though extensive classes of such differential equations, as they are termed, have been solved, especially by the early labours of the two Bernoullis, and have subsequently occupied the researches of the most eminent

mathematicians, yet even at the present day the subject is invested with difficulties which have not been

overcome.

Progress of the Fluxional Methods.

Newton and the English geometers did very little towards perfecting this branch of analysis. The continental mathematicians, on the other hand, were zealously engaged in improving its methods and pushing forward its applications. Leibnitz published in the "Acta Eruditorum," and other journals, a number of papers full of original views and important hints, thrown out very briefly, and requiring the elucidations which his illustrious friends the Bernoullis and others were always so willing and able to supply. Their tracts, like his, were scattered in the different periodical works of that time; and several years elapsed before any complete treatise explained the general methods, and illustrated them by examples. The first book in which this was done, so far, at least, as concerned the differential or direct calculus, was the "Analyse des infiniment Petits" of the marquis de l'Hôpital, published in 1696, a work of great merit, and which did much to diffuse a knowledge of the calculus. The author was a man of considerable genius and indefatigable industry, and enjoyed the advantages of instructive intercourse with John Bernoulli. In the collection of the works of the latter (not published till 1724) is inserted a tract of some length on the integral calculus, written in 1691, as is expressly mentioned, for the use of M. de l'Hôpital, to whose work it would seem intended as a sequel.

Newton, besides his letters, inserted in the "Commercium Epistolicum," had written at an early period, as we before observed, the " Analysis per æquationes," &c. But this, together with another tract, was not published till 1711. The treatise on the quadrature of curves, though written in 1666, did not appear till 1704, when, together with the "Enumeratio linearum

tertii ordinis," it was appended to the "Optics." The treatise on fluxions, translated by Colson, was not published till after the author's death (in 1736); and another analytical tract remained unpublished, till bishop Horsley collected and edited Newton's works, in 1779. The "Arithmetica Universalis" was published by Whiston, in 1707.

Such were the principal productions of Newton on subjects of pure mathematics. And it is certainly a curious fact, and one of much interest, as illustrative of the genius of Newton's character, that not one of them was voluntarily published by himself. When his youthful composition on the " Quadrature of Curves" had been extolled by Barrow, and shown to, and eagerly copied by Collins, and when he had been urged by the former to publish it, he could not be induced to do so. And afterwards, referring to this circumstance, and the partial success in such researches which Mercator had attained, he says, "I suspected that Mercator must

have known the extraction of roots as well as the reduction of fractions into series by division; or at least that others, having learnt to employ division for this purpose, would discover the rest before I myself should be old enough to appear before the public; and, therefore, I began henceforward to look upon such researches with less interest." The tracts at the end of the first edition of the " Optics," he tells us, he was compelled to print on account of the plagiarisms from the MSS. of them lent to his friends.

The "Arithmetica Universalis," which contains the substance of his lectures as Lucasian professor, was obtained by Whiston, without the consent of the author ; perhaps taken down at the lecture, and published surreptitiously, and, as has been alleged, in a way involving an unjustifiable breach of confidence. All this singular reluctance to make known the valuable truths he had discovered, has been the subject of much observation and conjecture among his biographers; and a variety of suppositions have been made, which to us appear

Χ

quite uncalled for, since the whole singularity of the case seems easily resolvable into the natural consequences of the peculiar character of Newton's temperament.

From the first we may trace, in the very constitution of this great man, a morbid sensitiveness of mind, and an excessive shyness of disposition. This led him to retire from the public gaze, and to reserve his most valuable discoveries for the mere information of his friends. The same peculiarity we have already seen manifested in regard to his optical works; and he would appear actually to have viewed with regret the publication of those discoveries, when his repose had been disturbed by the controversies which resulted. He thought his celebrity dearly purchased at the price of his tranquillity; and, perhaps, had personal reputation been the only consideration, he reasoned wisely; but when mankind were to be benefited by the promulgation of his discoveries, the question surely assumed a different aspect. But it was probably more to the general reserve of his disposition than to any positive apprehensions of critical controversy, that we are to ascribe his backwardness to produce his mathematical inventions. One thing is certain; that controversy was not, in fact, at all prevented by such precaution, if it be regarded in that light; and indeed it must have been obvious, that could Newton have foreseen the questions which were soon to be agitated, he might, by the mere immediate publication of an explicit account of the method of fluxions, have entirely prevented the whole of those discussions which afterwards occasioned so much vexation to himself, and such unhappy and even disgraceful hostility among the mathematicians of the age. It may, however, be fairly said, that no circumstances at the time could have led any one to foresee such a dispute. From what has been already stated, nothing would appear more unlikely than that the two great rivals would be drawn into hostility; each candidly admitted the just claims of the other, and there seemed nothing to excite discord or jealousy.

One circumstance which tended to limit the applications, and impede the progress of the fluxional system, was the fondness which Newton evinced for the synthetical method in delivering his propositions. This is conspicuous throughout the quadrature of curves, and is upheld and defended in the treatise on fluxions. By synthetical methods we do not here mean the peculiar style of geometrical demonstrations, but the delivery of the results announced as propositions, each proved upon independent considerations. This method may be well adapted for the mere communication of elementary truths; but it has this capital defect, that it does not put the student in possession of those general principles by which the truths in question were discovered, or by which he can be led to the discovery of others. Newton seems, in fact, to have been guided in his choice rather by considerations of taste than of utility; for, in the 'Fluxions," he thus expresses himself on the subject:"After the area of a curve has been found and constructed, we should consider about the demonstration of the construction, that, laying aside all algebraical calculation as much as may be, the theorem may be adorned and made elegant, so as to become fit for public view." (§ 107.) He exemplified these principles very advantageously in that work; but whatever may be thought of this preference abstractedly, it certainly cannot be commended in point of utility: and unquestionably the spirit thus instilled into the English school of mathematicians, had, for a long time after, the worst effect in cramping their energies, and impeding the applications of the calculus, and the progress of mathematical discovery.

We must here, though very briefly, refer to that modification of the same principle of limiting ratios, which Newton adopted with a special reference to certain geometrical cases, under the name of "prime and ultimate ratios." This doctrine he has fully and elegantly expounded in the lemmas which form the introduction to the first book of the Principia; where, with