about twenty-one years old, to read the works of Wallis, and appears to have taken peculiar delight in studying the remarkable treatise of this analyst, entitled Arithmetica infinitorum. It was his custom, when reading, to note down what appeared to him capable of being improved; and, by following up the ideas of Wallis, he was led to many important discoveries: for instance, Wallis had given the quadrature of curves, whose ordinates are expressed by any integral and positive power of (1-x); and had observed, that if, between the areas so calculated, we could interpolate the areas of other curves, the ordinates of which constituted, with the former ordinates, a geometrical progression, the area of the curve, whose ordinate was a mean proportional between 1 and (1-x) would express a circular surface, in terms of the square of its radius. In order to effect this interpolation, Newton began to seek, empirically, the arithmetical law of the co-efficients of the series already obtained.* Having found it, he rendered it more general, by expressing it algebraically. He then perceived that this interpolation gave him the expression in series of radical quantities, composed of several terms; but, not blindly trusting to the induction that had conducted him to this important result, he directly verified it by multiplying each series by itself the number of times required by the index of the root, and he found, in fact, that this multiplication re-produced exactly the quantity from which it had been deduced. When he had thus ascertained that this form of series really gave the development of radical quantities, he was obviously led to consider that they might be obtained still more directly, by applying to the proposed quantities the process used in arithmetic for extracting

plication of the exterior bow;" and yet every im partial reader, who refers to the original works, will see that the theory of Descartes is exact and complete, either as to the cause of the bow, its forma tion, or its size, and that he was only unacquainted notwithstanding his ignorance relative to this part of the phenomenon, Descartes, with great sagacity, refers it to another experimental fact, by assimilating it to the colours formed by prisms. formation of colours that Newton has so completely explained by the unequal refrangibility of the rays of light; but all the rest of the explanation is due. to Descartes. The book of Dominis contains absolutely nothing but explications entirely vague, with out any calculation or real result.

with the cause of the different colours; and even,

These details are mentioned by Newton himself, in a letter sent through Oldenburg to Leibnitz, dated October 24, 1676. It is No. LV. in the Com mercium Epistolicum, published by order of the Royal Society of London.

roots. This attempt perfectly succeeded, and again gave the same series, which he had previously discovered by indirect means; but it made them depend on a much more general method, since it permitted him to express, analytically, any powers whatever of polynomials, their quotients, and their roots; by operating upon and considering these quantities as the developments of powers corresponding to integral, negative, or fractional exponents. It is, in fact, in the generality and in the uniformity given to these developments in which the discovery of Newton really consists: for Wallis had remarked before him, with regard to monomial quantities, the analogy of quotients and roots, with integral powers, expressed according to the notation of Descartes; nay, more, Pascal had given a rule for forming, directly, any term of an expanded power of a binomial, the exponent being an integer. But whatever might be the merit of these observations, they were incomplete, and wanted generality, from not being expressed in an algebraical form. In fact, this step made by Newton was indispensable for discovering the development of functions into infinite series. Thus was found out the celebrated formula of such constant use in modern analysis, known by the name of the Binomial Theorem of Newton; and not only did he discover it, but he further perceived that there is scarcely any analytical research in which the use of it is not necessary, or at least possible. He immediately made a great number of the most important of these applications, solving, in this way, by series, with unexampled facility and exactness, questions which, up to that time, had not even been attempted, or of which solutions had been obtained only when the real difficulties of the case were removed by particular limitations. It was thus that he obtained the quadrature of the hyperbola 'and of many other curves, the numerical values of which he amused himself in computing to as many decimal places nearly as had previously been employed in the case of the circle alone: such pleasure did he take in observing the singular effect of these new analyti cal expressions, which, when capable of being determined exactly, stopped after a certain number of terms; and, in the opposite case, extended themselves indefinitely, while approximating more and more to the truth. Nor did he confine his application of these formula to the

areas of curves and their rectification, but extended it, to the surfaces of solids, to the determination of their contents, and the situation of their centres of gravity. To understand how this method of reducing into series could conduct him to such results, we must recollect that, in 1665, Wallis, in his Arithmetica infinitorum, had shown that the area of all curves may be found whose ordinate is expressed by any integral power of the abscissa; and he had given the expression for this area in terms of the ordinate. Now, by reducing into series the more complicated functions of the abscissa which represent the ordinates, Newton changed them into a series of monomial terms, to each of which he was able to apply the rule of Wallis. He thus obtained as many portions of the whole area as there were terms, and by their addition obtained the total. But the far more extensive, and, in some respects, un-: limited applications that Newton made of this rule, depended on a general principle which he had made out, and which consisted in the determining, from the manner in which quantities gradually increase, what are the values to which they ultimately arrive. To effect this, Newton regards them not as the aggregates of small homogeneous parts, but as the results of continued motion; so that, according to this mode of conception, lines are described by the movement of points, surfaces by that of lines, solids by that of surfaces, and angles by the rotation of their sides. Again-considering that the quantities so formed are greater or smaller in equal times, according as the velocity with which they are developed is more or less rapid, he endeavours to determine their ultimate values from the expression for these velocities, which he calls Fluxions, naming the quantities themselves Fluents. In fact, when any given curve, surface, or solid is generated in this manner, the different elements which either compose or belong to it, such as the ordinates, the abscissæ, the lengths of the arcs, the solid contents, the inclinations of the tangent planes, and of the tangents, all vary differently and unequally, but nevertheless according to a regular law depending on the equation of the curve, surface, or solid under consideration.

Hence Newton was able to deduce from this equation the fluxions of all these elements, in terms of any one of

the variables, and of the fluxion of this variable, considered as indeterminate; then, by expanding into series, he transformed the expression, so obtained, into finite, or infinite series of monomial terms, to which Wallis's rule became applicable: thus, by applying it successively to each, and taking the sum of the results, he obtained the ultimate value, i. e. the fluent of the element he had been considering. It is in this that the method of fluxions consists, of which Newton from that time laid the founda-. tion; and which, eleven years later, Leibnitz again discovered, and presented to the world in a different form, that, namely, of the modern Differential calculus. It were impossible to enumerate the various discoveries in mathematical analysis, and in natural philosophy, that this calculus has given rise to; it is sufficient to remark, that there is scarcely a question of the least difficulty in pure or mixed mathematics that does not depend on it, or which could be solved without its aid. Newton made all these analytical discoveries before the year 1665, that is, before completing his twenty-third year. He collected and arranged them in a manuscript, entitled

66

Analysis per æquationes numero terminorum infinitas." He did not, however, publish, or even communicate it to any one, partly, perhaps, from a backwardness to attain sudden notoriety, though more probably from his having already conceived the idea of applying this calculus to the determination of the laws of natural phenomena, anticipating that the analytical methods which he had discovered would be to him instruments for working out the most important results. It is at least certain, that, satisfied with the possession of this treasure, he kept it in reserve, and turned his attention more closely towards objects of natural philosophy. At this time (1665), he quitted Cambridge to avoid the plague, and retired to Woolsthorpe. In this retreat he was able to abandon himself, without interruption, to that philosophical meditation which appears to have been essential to his happiness.

The following anecdote is related by Pemberton, the contemporary and friend of Newton.-Voltaire, in his 'Elements of Philosophy,' says that Mrs. Conduit, Newton's niece, attested the fact.

One day, as he was sitting under an apple-tree, (which is still shown) an apple fell before him; and this incident

abandon his leading notion, but, in conformity with the character of his contemplative mind, he resolved not yet to divulge it, but to wait until study and reflection should reveal to him the unknown cause which modified a law indicated by such strong analogies. This took place in 1665-6. During the latter year, the danger of the plague having ceased, he returned to Cambridge, but he did not disclose his secret to any one, not even to his instructor, Dr. Barrow. It was not till two years afterwards, 1668, that Newton communicated to the latter, who was then engaged in publishing his lectures on Optics, certain theorems relating to the optical properties of curved surfaces, of which Barrow makes very honourable mention in his preface. Newton had now become a colleague of his former tutor, having been admitted master of arts the preceding year. At length in the same year (1668) an occurrence in the scientific world compelled him to declare himself. Mercator* printed and published, towards the end of this year, a book called Logarithmotechnia, in which he had succeeded in obtaining the area of the hyperbola referred to its asymptotes, by expanding its ordinate into an infinite series; this he did by means of common division, as Wallis had done in the case of fractions of the form

1

1

-x

then, considering each term of this series separately, as representing a particular ordinate, he applied to it Wallis's method for curves, whose ordinates are expressed by a single term, and the sum of the partial areas so obtained, gave him the value of the whole area. This was the first example given to the world of obtaining the quadrature of a curve by expanding its ordinate into an infinite series. And it was also the main secret in the general method which Newton had invented for all problems of this nature. The novelty of the invention caused it to be received with general applause. Collins, a gentleman well known to science and philosophy at that time, hastened to send Mercator's book to his friend Barrow, who communicated it to Newton. The latter had no sooner glanced over it, than recognizing his own fundamental idea, he immediately went home, to find the manuscript; in which he had explained his own method, and

Born in Holstein: he passed the greater part of his life in England.

presented it to Barrow; this was the treatise Analysis per æquationes numero terminorum infinitas. Barrow was struck with astonishment at seeing so rich a collection of analytical discoveries of far greater importance than the par ticular one which then excited such general admiration. Perhaps, too, he must have been still more surprised at their young author having been able to keep them so profoundly secret. He immediately wrote about them to Col. lins, who, in return, entreated Barrow to procure for him the sight of so precious a manuscript. Collins obtained his request, and happily, before returning the work, took a copy of it, which being found after his death, among his papers, and published in 1711, has determined beyond dispute, by the date which it bore, at what period Newton made the memorable discovery of expansion by series, and of the method of fluxions. It would have been natural to suppose that an interference with his own discoveries would at last have induced Newton to publish his methods; but he preferred still to keep them secret. “ I suspected," says he, "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 ap pear before the public, and, therefore, I began henceforward to look upon such researches with less interest."*

It were difficult to explain this reserve and indifference by the feelings of extreme modesty alone; but we may come near the truth by considering what were the habits of Newton, and by figuring to ourselves the new and extraordinary allurements of another discovery which he had just made, and which he already enjoyed in secret; for in general, the effort of thinking was with him so strong, that it entirely abstracted his attention from other mat ters, and confined him exclusively to one object. Thus we know that he never was occupied at the same time with two different scientific investigations. And we find, even in the most beautiful of his works, the simple, yet expressive avowal of the disgust with which his most curious researches had always finally inspired him, from his ideas being

Com. Epist. LVI. At the end of the Optics.

continually, and for a long time, directed to the same object. This might, perhaps, also have in part been caused by a discouraging conviction, that he would seldom be understood and followed in the chain of his reasoning; since others, in order to do so, must be as deeply immersed in the subject and as abstracted from other matters as himself. Be this as it may, when Mercator's work appeared, a new series of discoveries of a totally different nature had taken hold of Newton's thoughts.

In the course of 1666, he had accidentally been led to make some observations on the refraction of light through prisms. These experiments, which he had at first tried merely from amusement, or curiosity, soon offered to him most important results. They led him to conclude that light, as it emanates from radiating bodies, such as the sun, for instance, is not a simple and homogeneous substance, but that it is composed of a number of rays endowed with unequal refrangibility, and possessing different colouring properties. The inequality of the refraction undergone by these rays in the same body, when they enter at the same angle of incidence, enabled him to separate them; and thus, having them unmixed and pure, he was able to study their individual properties. But the breaking out of the plague, which in this year compelled him to take refuge in the country, having separated him from his instruments, and deprived him of the means of making experiments, turned his attention to other objects. More than two years elapsed before he returned to these researches, on finding himself about to be appointed lecturer on optics in room of Dr. Barrow, who in 1669 generously retired in order to make way for him. He then endeavoured to mature his first results, and was led to a multitude of observations no less admirable from their novelty and importance, than for the sagacity, address, and method, with which he perfected and connected them. He composed a complete treatise, in which the fundamental properties of light were unfolded, established, and arranged, by means of experiment alone, without any admixture of hypothesis, a novelty at that time almost as surprising as these properties themselves. This formed the text of the lectures he began in Cambridge 1669, when scarcely twentyseven years old, and thus we see,

from what we have related concerning the succession of his ideas, that the method of Fluxions, the theory of universal gravitation, and the decomposition of light, i. e. the three grand discoveries which form the glory of his life, were conceived in his mind before the completion of his twenty-fourth year.

Although the lectures of Newton on optics must inevitably in the end have given publicity to his labours on light, he still refrained from publishing, wishing probably to reserve to himself the opportunity of adding a complete analysis of certain curious properties, of which, as yet, he had had but a slight glimpse. We refer to the intermittences of reflection and refraction which take place in thin plates, and perhaps in the ultimate particles of all bodies. It was not till two years later, that he made known some of his researches, and soon afterwards he was induced to give them full publicity. In 1671 he had been proposed as a Fellow of the Royal Society of London, and was elected on the 11th of January, 1672. In order that he might be qualified to receive this distinction, the rules of the society required that he should declare himself desirous of becoming a Fellow, and he could not do so in a more honourable manner than by offering some scientific communication. He forwarded to them a description of a new arrangement for reflecting telescopes, which rendered them more commodious in use by diminishing their length without weakening their magnifying powers. With regard to this invention, in which Newton had been preceded, probably without knowing it, by Gregory the Scotch mathematician, and by a Frenchman of the name of Cassegrain, it is merely necessary to observe that the construction offers in practice some inconveniences, which cause it to be little used. Nevertheless, when he presented a model of it,* of his own construction, it made a great impression in his favour among the members of the society, to whom probably the construction of Gregory's telescope was not yet well known. The letter which Newton wrote to the society on this occasion, ends with the following characteristic expression:-" I am very sensible of the honour done me by the Bishop of Sarum, in proposing me Candidate, and which I hope will be

This model, made by Newton himself, is still preserved in the Library of the Royal Society.