66

completed. A portrait of Kepler is engraved in the seventh part of Boissard's Bibliotheca Chalcographica. It is not known whence this was taken, but it may, perhaps, be a copy of that which was engraved by desire of Bernegger in 1620. The likeness is said not to have been well preserved. "His heart and genius," says Kästner, are faithfully depicted in his writings; and that may console us, if we cannot entirely trust his portrait." In the preceding pages, it has been endeavoured to select such passages from his writings as might throw the greatest light on his character, with a subordinate reference only to the importance of the subjects treated. In conclusion, it may be well to support the opinion which has been ventured on the real nature of his triumphs, and on the danger of attempting to follow his method in the pursuit of truth, by the judgment pronounced by Delambre, as well on his failures as on his success.

Ein Calender

"Con

sidering these matters in another point of view, it is not impossible to convince ourselves that Kepler may have been always the same. Ardent, restless, burning to distinguish himself by his discoveries, he attempted everything; and having once obtained a glimpse of one, no labour was too hard for him in following or verifying it. All his attempts had not the same success, and, in fact, that was impossible. Those which have failed seem to us only fanciful; those which have been more fortunate appear sublime. When in search of that which really existed, he has sometimes found it; when he devoted himself to the pursuit of a chimera, he could not but fail; but even there he unfolded the same qualities, and that obstinate perseverance that must triumph over all difficulties but those which are insurmountable*.*

Histoire de l'Astronomie Moderne, Paris, 1821.

Dissertatio cum Nuncio Sidereo

Frankfurt, 1610, 4to. Francofurti, 1610, 4to.

Frankfurt, 1611, 4to.

Francofurti, 1611, 4to.

Vom Geburts Jahre des Heylandes

Strasburg, 1613, 4to.

Respons. ad epist S. Calvisiii

Ecloga Chronicæ

Francofurti, 1614, 4to.
Frankfurt, 1615, 4to.
Linci, 1615, 4to.
Lincii, 1616, 4to.
Lentis, 1618, 8vo.

Aug. Vindelic. 1619, 4to.

Linci, 1619, fol. Ulma, 1620 Lentus, 1622, 8vo.

Francofurti, 1622, 8vo.

Linz. 1623, 4to.

Marpurgi, 1624, fol.
Lentiis, 1625, 4to.

Francofurti, 1625, 8vo.

Ulmæ, 1627, fol.

Somnium

Tabulæ manuales

ISAAC NEWTON was born at Woolsthorpe, in Lincolnshire, on the 25th December, 1642 (O. S.) the year in which Galileo died. At his birth he was so small and weak that his life was despaired of. At the death of his father, which took place while he was yet an infant, the manor of Woolsthorpe, of which his family had been in possession several years, became his heritage. In a short time his mother married again; but this new alliance did not interfere with the performance of her duties towards her son. She sent him, at an early age, to the school of his native village, and afterwards, on attaining his twelfth year, to the neighbouring town of Grantham, that he might be instructed in the classics. Her intention, however, was not to make her son a mere scholar, but to give him those first principles of education which were considered necessary for every gentleman, and to render him able to manage his own estate. After a short period, therefore, she recalled him to Woolsthorpe, and began to employ him in domestic occupations. For these he soon showed himself neither fitted nor inclined. Already, during his residence at Grantham, Newton, though still a child, had made himself remarkable by a decided taste for various philosophical and mechanical inventions. He was boarded in the house of an apothecary, named Clarke, where, caring but little for the society of other children, he provided himself with a collection of saws, hammers, and other instruments, adapted to his size; these he employed with such skill and intelligence, that he was able to construct models of many kinds of machinery; he also made hour-glasses, acting by the descent of water, which marked the

time with extraordinary accuracy. A new windmill, of peculiar construction, having been erected in the vicinity of Grantham, Newton manifested a strong desire to discover the secret of its mechanism; and he accordingly went so often to watch the workmen employed in erecting it, that he was at length able to construct a model, which also turned with the wind, and worked as well as the mill itself; but with this difference, that he had added a mouse in the interior, which he called the miller, because it directed the mill, and ate up the flour, as a real miller might do. A certain acquaintance with drawing was necessary in these operations; to this art, though without a master, he successfully applied himself. The walls of his closet were soon covered with designs of all sorts, either copied from others, or taken from nature. These mechanical pursuits, which already implied considerable powers of invention and observation, occupied his attention to such a degree, that for them he neglected his studies in language; and, unless excited by particular circumstances, he ordinarily allowed himself to be surpassed by children of very inferior mental capacity. Having however, on some occasion, been surpassed by one of his class fellows, he determined to prevent the recurrence of such a mortification, and very shortly succeeded in placing himself at the head of them all.

It was after Newton had for several years cherished and, in part, unfolded so marked a disposition of mind, that his mother, having taken him home, wished to employ him in the affairs of her farm and household. The reader may easily judge that he had little inclination for such pursuits. More than once

B

he was sent by his mother on market-days to Grantham, to sell corn and other articles of farming produce, and desired to purchase the provisions required for the family; but as he was still very young, a confidential servant was sent with him to teach him how to market. On these occasions, however, Newton, immediately after riding into the town, allowed his attendant to perform the business for which he was sent, while he himself retired to the house of the apothecary where he had formerly lodged, and employed his time in reading some old book, till the hour of return arrived. At other times he did not even proceed so far as the town, but stopping on the road, occupied himself in study, under the shelter of a hedge, till the servant came back. With such ardent desire for mental improvement, we may easily conceive that his repugnance to rural occupations must have been extreme; as soon as he could escape from them, his happiness consisted in sitting under some tree, either reading, or modelling in wood, with his knife, various machines that he had seen. To this day is shewn, at Woolsthorpe, a sun-dial, constructed by him on the wall of the house in which he lived. It fronts the garden, and is at the height to which a child can reach. This irresistible passion, which urged young Newton to the study of science, at last overcame the obstacles which the habits or the prudence of his mother had thrown in his way. One of his uncles having one day found him under a hedge, with a book in his hand, entirely absorbed in meditation, took it from him, and discovered that he was working a mathematical problem. Struck with finding so serious and decided a disposition in so young a person, he urged Newton's mother no longer to thwart him, but to send him once more to pursue his studies at Grantham.*

There he remained till he reached his eighteenth year, when he removed to Cambridge, and was entered at TrinityCollege, in 1660. Since the beginning of the seventeenth century, a taste for the cultivation of mathematical knowledge had shown itself among the members of that University. The elements of algebra and geometry generally

These details of the infancy of Newton are taken chiefly from "Collections for the History of the Town and Soke of Grantham, containing authentie Memoirs of Sir Isaac Newton, &c. by Edmund Turner, (London, 1806.)" And from the Eloge on Newton, written by Fontenelle.

formed a part of the system of education, and Newton had the good fortune to find Dr. Barrow, professor; a man who, in addition to the merit of being one of the greatest mathematicians of his age, joined that of being the kindest instructor as well as the most zealous protector of the young genius growing up under his care.

Newton, in order to prepare himself for the public lessons, privately read the text books in advance, the better to follow the commentaries of the lecturer. These books were, Bishop Sanderson's Logic,* and Kepler's Treatise on Optics, from which it is evident the young learner must have made considerable progress in the elements of geometry when studying at Grantham. After Newton went to Cambridge, the process of the unfolding of his intellect, a subject so interesting in the study of the human mind, fortunately remains to us either described by himself or established in literary monuments, by which we are enabled accurately to trace its progress.

At this epoch, Descartes bore sway both in speculative and in natural philosophy. The authority of the metaphysical systems of his daring and fertile mind having succeeded to the empire which those of Aristotle had previously exercised, caused his method and his works to be adopted also in mathematics. Hence the geometry of Descartes was one of the first books that Newton read at Cambridge.

After Newton's persevering efforts, when reading alone, to make himself master of the elements of this science, explained so unconnectedly and imperfectly by other authors, he must have felt a lively pleasure on entering on the wide career that the French analyst was the first to open, and in which, having shown the connexion between algebraical equations and geometry, he discovers to us the use of that relation in solving, almost at sight, problems which, up to that time, had foiled the efforts of all the ancient and modern mathematicians. It is singular, however, that Newton, in his writings, has never mentioned Descartes favourably; and, on more than one occasion, has treated him with injustice.† He next proceeded, when

The title is Logica artis Compendium, auctore Robert Sanderson. Oxon. 8vo.

Particularly in his Optics, where he attributes the discovery of the true theory of the rainbow to Antonius de Dominis, Archbishop of Spalatro, leaving to Descartes only the merit of having "mended the ex

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 (1-2); 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 impartial 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. It is this 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, without 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 analytical 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 cf 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, unlimited 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 foundation; 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 "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 froin 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