view must have communicated to the philosopher who first beheld it, may be conceived more easily than expressed. To the immediate impression which they made upon the sense, to the wonder they excited in all who saw them, was added the proof, which, on reflection, they afforded, of the close resemblance between the earth and the celestial bodies, whose divine nature had been so long and so erroneously contrasted with the ponderous and opaque substance of our globe. The earth and the planets were now proved to be bodies of the same kind, and views were entertained of the universe, more suitable to the simplicity and the magnificence of nature.

The author of such discoveries could not be forgiven. Galileo, accordingly, was twice brought before the Inquisition. The first time a council of seven cardinals pronounced a sentence which, for the sake of those disposed to believe that power can subdue truth, ought never to be forgotten. "That to maintain the sun to be immoveable, and without local motion, in the centre of the world, is an absurd proposition, false in philosophy, heretical in religion, and contrary to the testimony of Scripture. That it is equally absurd and false in philosophy to assert that the earth is not immoveable in the centre of the world, and, considered theologically, equally erroneous and heretical."

GASSENDI, the contemporary and countryman of Descartes, possessed great learning, with a very clear and sound understanding. He was a good observer, and an enlightened advocate of the Copernican system. He explained, in a very satisfactory manner, the connection between the laws of motion and the motion of the earth, and made experiments to show, that a body carried along by another acquires a motion which remains after it has ceased to be so carried.

Though the pendulum afforded a measure of time, in itself of the greatest exactness, the means of continuing its motion, without disturbing the time of its vibrations, was yet required to be found, and this by means of the clock, Huygens contrived most ingeniously to effect. The telescope had not yet served astronomy in all he capacities in which it could be useful. Huygens applied it to the measurement of small angles, forming it into the instrument which has since been called a micrometer. The best cye, when not aided by glasses, is not able to perceive an object which subtends an angle less than half a minute, or thirty seconds. But when we direct the axis of a telescope, which magnifies thirty times, to the same object, we are sure that it is within the thirtieth part of half a minute, that is, within one second of the point aimed at.

It should have been observed that the French astronomer, Picard, was the first who employed instruments furnished with telescopic sights, about the year 1665. The planes of the orbits, their inclinations to the orbit of Jupiter, and the lines in which they intersect that orbit, were all to be determined, as well as the times of revolution, and the distances of each from its primary. Add to this,

that it is only in a few points of their orbits that they can be observed with advantage. The best are at the times of immersion into the shadow of Jupiter, and emersion from it.

It would be gratifying to be able to observe, that the universities of Europe had contributed to the renovation of science. The fact is otherwise ;-they were often the fastnesses from which prejudice and error were latest of being expelled. They joined in persecuting the reformers of science. It has been seen, that the masters of the university of Paris were angry with Galileo for the experiments on the descent of bodies. Even the university of Oxford brought on itself the indelible disgrace of persecuting, in Friar Bacon, the first man who appears to have had a distinct view of the means by which the knowledge of the laws of nature must be acquired.

In the arts connected with OPTICS, the ancients had made some progress. They were sufficiently acquainted with the laws of reflection to construct mirrors both plane and spherical; they made them also conical.

ALHAZEN, a celebrated Arabian writer on optics, lived in the eleventh century; the anatomical structure of the eye was known to him; concerning the uses of the different parts he had only conjectures to offer; but on seeing single with two eyes, he made this very important remark, that, when corresponding parts of the retina are effected, we perceive but one image.

ROGER BACON appears to have made a near approach to the knowledge of lenses, and their use in assisting vision. He knew how to trace the progress of the rays of light through a spherical transparent body, and understood, what was the thing least obvious, how to determine the place of the image. It is probable, that the knowledge of the true properties of these glasses, whether theoretical or practical, may have had a share in introducing the use of lenses, and in the invention of spectacles, which took place not long

after.

The lapse of more than two hundred years brings us down to Maurolycus, who was distinguished for his skill in optics. He was acquainted with the crystalline lens, and conceived that its office is to transmit to the optic nerve the species of external objects; and in this process he does not consider the retina as any way concerned. This theory, though so imperfect, led him nevertheless to form a right judgment of the defects of short-sighted and long-sighted eyes. The same author appears also to have observed the caustic curve formed by reflection from a concave speculum.

A considerable step in optical discovery was made at this time by Baptista Porta, a Neapolitan, who invented the Camera Obscura, about the year 1560. The light was admitted through a small hole in the window-shutter of a dark room, and gave an inverted picture of the objects from which it proceeded, on the opposite wall. A lens was not employed in the first construction of this apparatus,

but was afterwards used; and Porta went so far as to consider how the effect might be produced without inversion. But the complete discovery of the truth relative to the constitution of the eye, and the functions of the different parts of which it consists, was left to Kepler, who, to the glory of finding out the true laws of the planetary system, added that of first analyzing the whole scheme of nature in the structure of the eye. He perceived the exact resemblance of this organ to the dark chamber, the rays entering the pupil being collected by the crystalline lens, and the other humours of the eye, into foci, which paint on the retina the inverted images of external objects. By another step of the process, the mind perceives the images thus formed, and refers them at the same time to things without.

ANTONIO DE DOMINIS, Archbishop of Spalatro, had the good fortune to fall upon the true explanation of the phenomenon of the Rainbow. Having placed a bottle of water opposite to the sun, and a little above his eye, he saw a beam of light issue from the under side of the bottle, which acquired different colours, in the same order, and with the same brilliancy as in the rainbow, when the bottle was a little raised or depressed. From comparing all the circumstances, he perceived that the rays had entered the bottle, and that, after two refractions from the convex part, and a reflection from the concave, they were returned to the eye tinged with different colours, occording to the angle at which the ray had entered. The rays that gave the same colour made the same angle with the surface, and hence all the drops that gave the same colour must be arranged in a circle, the centre of which was the point in the cloud opposite to the sun. This, though not a complete theory of the rainbow, and though it left a great deal to occupy the attention, first of Descartes, and afterwards of Newton, was perfectly just, and carried the explanation as far as the principles then understood allowed it to go.

The notion of Infinite Quantity had, as we have already seen, been for some time introduced into Geometry, and having become a subject of reasoning and calculation, had, in many instances, after facilitating the process of both, led to conclusions from which, as if by magic, the idea of infinity had entirely disappeared, and left the geometer or the algebraist in possession of valuable propositions, in which were involved no magnitudes but such as could be readily exhibited.

It was in this state of the sciences, that NEWTON began his mathematical studies, and, after a very short interval, his mathematical discoveries. The book, next to the elements, which was put into his hands, was Wallis's Arithmetic of Infinites, a work well fitted for suggesting new views in geometry, and calling into activity the powers of mathematical invention. Wallis had effected the quadrature of all those curves in which the value of one of the co-ordinates can be expressed in terms of the other, without involving either

fractional or negative exponents. Beyond this point neither his researches, nor those of any other geometer, had yet reached, and from this point the discoveries of Newton began. Proceeding on the same general principle with Wallis, as he himself tells us, the simple view which he took of the areas already computed, and of the terms of which each consisted, enabled him to discover the law which was common to them all, and under which the expression for the area of the circle, as well as of innumerable other curves, must needs be comprehended. In the case of the circle, as in all those where a fractional exponent appeared, the area was exhibited in the form of an infinite series.

The problem of the quadrature of the circle, and of so many other curves, being thus resolved, Newton immediately remarked, that the law of these series was, with a small alteration, the law for the series of terms which expresses the root of any binomial quantity whatsoever. Thus he was put in possession of another valuable discovery, the Binomial Theorem, and at the same time perceived that this last was in reality, in the order of things, placed before the other, and afforded a much easier access to such quadratures than the me thod of interpolation, which, though the first road, appeared now neither to be the easiest nor the most direct.

The first work in which Newton communicated any thing to the world on the subject of fluxions, was in the first edition of the Principia, in 1687, in the second Lemma of the second book. The principle of the fluxionary calculus was there pointed out, but nothing appeared that indicated the peculiar algorithm, or the new notation, which is so essential to that calculus. About this Newton had yet given no information; and it was only from the second volume of Wallis's Works, in 1693, that it became known to the world. It was no less than ten years after this, in 1704, that Newton himself first published a work on the new calculus, his Quadrature of Curves, more than twenty-eight years after it was written.

Of the new or infinitesimal analysis, we are to consider Newton as the first inventor; Leibnitz, a German mathematician, as the second; his discovery, though posterior in time, having been made independently of the other, and having no less claim to originality. It had the advantage also of being first made known to the world; an account of it, and of its peculiar algorithm, having been inserted in the first volume of the Acta Eruditorum, in 1684. Thus, while Newton's discovery remained a secret, communicated only to a few friends, the geometry of Leibnitz was spreading with great rapidity over the Continent. Two most able coadjutors, the brothers James and John 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. The reserve of Newton still kept his countrymen ignorant of his geometrical discoveries, and the first book that appeared in England on the new geometry,

was that of Craig, who professedly derived his knowledge from the writings of Leibnitz and his friends.

In order to form a correct estimate of the value of this discovery, it may be useful, says Professor Playfair, to look back at the steps by which the mathematical sciences arrived at it. When we attempt to trace those steps to their origin, we find the principle of the infinitesimal analysis making its first appearance in the method of Exhaustions, as exemplified in the writings of Euclid and Archimedes. These geometers observed, and, for what we know, were the first to observe, that the approach which a rectilineal figure may make to one that is curvilineal, by the increase of the number of its sides, the diminution of their magnitude, and a certain enlargement of the angles they contain, may be such that the properties of the former shall coincide so nearly with those of the latter, that no real difference can be supposed between them without involving a contradiction; and it was in ascertaining the conditions of this approach, and in showing the contradiction to be unavoidable, that the method of Exhaustions consisted. The demonstrations were strictly geometrical, but they were often complicated, always indirect, and of course synthetical, so that they did not explain the means by which they had been discovered.

At the distance of more than two thousand years, Cavalleri advanced a step farther, and, by the sacrifice of some apparent, though of no real accuracy, explained, in the method of indivisibles, a principle which could easily be made to assume the more rigid form of Exhaustions. This was a very important discovery-though the process was not analytical, the demonstrations were direct, and, when applied to the same subjects, led to the same conclusions which the ancient geometers had deduced; by an indirect proof also, such as those geometers had adopted, it could always be shewn that an absurdity followed from supposing the results deduced from the method of indivisibles to be other than rigorously true.

The method of Cavalleri was improved and extended by a number of geometers of great genius who followed him; Torricelli, Roberval, Fermat, Huygens, and Barrow, all observed the great advantage that arose from applying the general theorems concerning variable quantity to the cases where the quantities approached to one another infinitely near, that is, nearer than within any assigned difference. There was, however, as yet, no calculus adapted to these researches, that is, no general method of reasoning by help of arbitrary symbols.

But we must go back a step, in point of time, if we would trace accurately the history of this last improvement. Descartes, as has been shown in the former part of this outline, made a great revolution in the mathematical sciences, by applying algebra to the geometry of curves; or, more generally, by applying it to express the relations of variable quantity. This added infinitely to the value of the algebraic analysis, and to the extent of its investigations. The