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as the square of the distance increases. He then found, that the three great facts in astronomy, which form the laws of Kepler, gave the most complete evidence to the system of gravitation. The first of them, the proportionality of the areas described by the radius vector to the times in which they are described, is the peculiar character of the motions produced by an original impulse impressed on a body, combined with a centripetal force continually urging it to a given centre. The sscond law, that the planets describe ellipses, having the sun in one of the foci, common to them all, coincides with this proposition, that a body under the influence of a centripetal force, varying as the square of the distance inversely, and having any projectile force whatever originally impressed on it, must describe a conic section having one focus in the centre of force, which section, if the projectile force does not exceed a certain limit, will become an ellipse. The third law, that the squares of the periodic times are as the cubes of the distances, is a property which belongs to the bodies describing elliptic orbits under the conditions just stated.
But, says Professor Playfair, did the principle which appeared thus to unite the great bodies of the universe act only on those bodies? Did it reside merely in their centres, or was it a force common to all the particles of matter? Was it a fact that every particle of matter had a tendency to unite with every other? Or was that tendency directed only to particular centres ? It could hardly be doubted that the tendency was common to all the particles of matter. The centres of the great bodies had no properties as mathematical points, they had none but what they derived from the material particles distributed around them. But the question admitted of being brought to a better test than that of such general reasoning as the preceding. The bodies between which this tendency had been observed to take place were all round bodies, and either spherical or nearly so, but whether great or small, they seemed to gravitate toward one another according to the same law. This alleged principle has, however, been often controverted; and within this year (1821) Sir RICHARD PHILLIPS has published a series of Essays, in which he asserts that the phenomena of planetary aggregation are owing to the two-fold mo tions, which direct all their parts towards their centres : that the orbicular motions of the planets arise from the sun's action on the elastic medium of space, diffused inversely as the squares of the distances; that the rotatory motions arise from the unequal action of the same medium on the near and remote extremities of planetary axes; that heat is atomic motion; and, in fine, that all force, and all phenomena arise universally from matter affected by or multiplied into motion. Matter multiplied by motion, says Sir Richard, produces force, change, or phenomena, and not matter multiplied by attraction, or by repulsion, or by gravitation; and it is therefore the business of philosophers to trace the special motions which in every case unite with matter and produce particular phenomena.
NEWTON now perceived that, in the earth, another force was combined with gravity, and that the figure resulting from that combination could not be exactly spherical. The diurnal revolution of the earth, he knew, must produce a centrifugal force, which would act most powerfully on the parts most distant from the axis. The amount of this centrifugal force is greatest at the equator, and being measured by the momentary recess of any point from the tangent, which was known from the earth's rotation, it could be compared with the force of gravity at the same place, measured in like manner by the descent of a heavy body in its first moment of its fall.
The precession, that is, the retrogradation of the equinoctial points, had been long known to astronomers; its rate had been measured by a comparison of ancient and modern observations, and found to amount nearly to 50' annually, so as to complete an entire revolution of the heavens in 25,920 years. The honour of assigning the true cause of this phenomenon was reserved for the most cautions of philosophers. He was directed to this by a certain analogy observed between the precession of the equinoxes and the retrogradation of the moon's nodes, a phenomenon to which his calculus had been already successfully applied. The spheroidal shell or ring of matter which surrounds the earth, in the direction of the equator, being one half above the plane of the ecliptic and the other half below, is subjected to the action of the solar force, the tendency of which is to make thiş ring turn on the line of its intersection with the ecliptic, so as ultimately to coincide with the plane of that circle. This, accordingly, would have happened long since, if the earth had not revolved on its axis. The effect of the rotation of the spheroidal ring from west to east, at the same time that it is drawn down toward the plane of the ecliptic, is to preserve the inclination of these two planes unchanged, but to make their intersection move in a direction opposite to that of the diurnal rotation, that is, from east to west, or contrary to the order of the signs.
By an analysis also of the force of gravity, Newton explained those inequalities in the elevation of the waters of the ocean to which we give the name of tides. The motion of Comets yet remained to be discussed. Newton shewed that the orbit of the comet must be a conic section, having the sun in one of its foci, and might either be an eclipse, a parabola, or even a hyperbola, according to the relation between the force of projection and the force tending to the centre.
For more than thirty years after the publication of these discoveries, the system of vortices kept its ground, and a translation from the French into Latin of the Physics of Rohault, a work entirely Cartesian, continued at Cambridge to be the text for philosophical instruction. About the year 1718, a new and more elegant translation of the same book was published by Dr. Samuel Clarke, with the addition of notes, in which that profound and ingenious writer explained the views of Newton on the principal objects of discussion, so that the notes contained virtually a refutation of the text; they did so, however, only virtually, all appearance of argument and controversy being carefully avoided. Whether this escaped the notice of the learned Doctors or not is uncertain, but the new translation, from its better Latinity, and the name of the editor, was readily ada mitted to all the academical honours which the old one had enjoyed. Thus, the stratagem of Dr. Clarke completely succeeded, and the Newtonian philosophy first entered the university of Cambridge under the protection of the Cartesian.
The Aberration of the fixed stars is the discovery of Dr. BRADLEY ; he and his friend MOLYNEUUX, in the end of the year 1725, were occupied in searching for the parallax of the fixed stars by means of a zenith sector. The sector was erected at Kew ; it was of great radíus, and furnished with a telescope twenty-four feet in length, with which they proposed to observe the transits of stars near the zenith, according to a method that was first suggested by HOOKE, and pursued by him so far as to induce him to think that he had actually discovered the parallax of, Draconis, the bright star in the head of the dragon, on which he made his observations. They began their observations of the transits of the same star on the 3d of December, when the distance from the zenith at which it passed was carefully marked. By the observations of the subsequent days the star seemed to be moving to the south; and about the beginning of March, in the following year, it had 20' to the south, and was then nearly stationary. In the beginning of June it had come back to the same situation where it was first observed, and from thence it continued its motion northward till September, when it was about 20" north of the point where it was first seen, its whole change of declination having amounted to 40". This motion occasioned a deal of surprise to the two observers, as it lay the contrary way to what it would have done if it had proceeded from the parallax of the star. The repetition of the observations, however, confirmed their accuracy; and they were afterwards pursued by Dr. Bradley, with another sector, of a less radius, but still of one sufficiently great to measure a star's zenith distance to half a second. It embraced a larger arch, and admitted of the observations being extended to stars that passed at a more considerable distance from the zenith. It then occurred to Bradley that the appearances might arise from the progressive motion of the earth in its orbit. He saw that Römer's observation concerning the time that light takes to go from the sun to the earth gave a ready expression for the velocity of light compared with that of the earth. The proportion, however, which he assumed as best suited to his observations was somewhat different; it was that of 10313 to 1, which made the radius of the circle of aberration 20", and the transverse axis of the ellipse in every case, or the whole change of position, 40%. It was the shorter axis which Bradley had actually observed in the case of Draconis, that star being very near the solstitial colure, so that its changes of declination and of latitude are almost the same. In order to show the truth of his theory, he computed the aberration of different stars, and, on comparing the results with his observations, the coincidence appeared almost perfect, so that no doubt remained concerning the truth of the principle on which he had founded his calculations.
Though the integral calculus, as it was left by the first inventors and their contemporaries, was a very powerful instrument of investigation, it required many improvements to fit it for extending the philosophy of Newton to its utmost limits. A brief enumeration of the principal improvements which it has actually received in the last seventy or eighty years, will very much assist us in appreciating its great importance. Descartes is celebrated for having applied algebra to geometry; and Euler hardly deserves less credit for having applied the same science to trigonometry. Though we ascribe the invention of this calculus to Euler, we are aware that the first attempt toward it was made by a mathematician of far inferior note, MAYER, who, in the Petersburgh Commentaries for 1727, published a paper on analytical trigonometry. In that memoir, the geometrical theorems, which serve as the basis of this new species of arithmetic, are pointed out; but the extension of the method, the introduction of a convenient notation, and of a peculiar algorithm, are the work of Euler. By means of these, the sines and cosines of arches are mul tiplied into one another, and raised to any power, with a simplicity unknown in any other part of algebra, being expressed by the sines and cosines of multiple arches, of one dimension only, or of no higher power than the first. It is incredible of how great advantage this method has proved in all the parts of the higher geometry, but more especially in theresearches of physical astronomy. Besides, the facility which this calculus gives to all the reasonings and computations into which it is introduced, from the elementary problems of geometry to the finding of fluents and the summing of series, makes it one of the most valuable resources in mathematical science.
An improvement in the integral calculus, made by M. D'ALEM. BERT, has doubled its power, and added to it a territory not inferior in extent to all that it before possessed. This is the method of par. tial differences, or, as we must call it, of partial fuxions. It was diecovered by the geometer just named, when he was inquiring into the nature of the figures successively assumed by a musical string during the time of its vibrations. When a variable quantity is a function of other two variable quantities, as the ordinates belonging to the different abscissæ in these curves must necessarily be, (for they are functions both of the abscissæ and of the time counted from the beginning of the vibrations,) it becomes convenient to consider how that quantity varies, while each of the other two varies singly, the remaining one being supposed constant. Without this simplification, it would, in most cases, be quite impossible to subject such complicated functions to any rules of reasoning whatsoever.
The calculus of partial differences, therefore, is of great utility in all the more complicated problems both of pure and mixt mathe. matics: every thing relating to the motion of fluids that is not purely clementary, falls within its range; and in all the more difficult researches of physical astronomy, it has been introduced with great advantage. The first idea of this new method, and the first application of it, are due to D'Alembert; it is from Euler, however, that we derive the form and notation that have generally been adopted.
Another great addition made to the integral calculus, is the invention of La Grange, and is known by the name of the Calculus varialionum. The ordinary problems of determining the greatest and least states of a given function of one or more variable quantities, is easily reduced to the direct method of Auxions, or the differential calculus, and was indeed one of the first classes of questions to which those methods were applied. But when the function that is to be a marimum or a minimum, is not given in its form; or when the curve, expressing that function, is not known by any other property, but that, in certain circumstances, it is to be the greatest or least possible, the solution is infinitely more difficult; and science seems to have no hold of the question by which to reduce it to a mathematical investigation. The problem of the line of swiftest descent is of this nature; and though, from some facilities which this and other particular instances afforded, they were resolved, by the ingenuity of mathematicians, before any method generally applicable to them was known, yet such a method could not but be regarded as a great desideratum in mathematical science. The genius of Euler had gone far to supply it, when La Grange, taking a view entirely different, fell upon a method extremely convenient, and, considering the difficulty of the problem, the most simple that could be expected. The supposition it proceeds on is greatly more general than that of the Auxionary or differential calculus. In this last, the fluxions or changes of the variable quantities are restricted by certain laws. The fluxion abscissa of the ordinate, for example, has a relation to the fluxion of the that is determined by the nature of the curve to which they both belong. But in the method of variations, the change of the ordinate may be any whatever ; it may no longer be bounded by the original curve, but it may pass into another, having to the former no determinate relation. This is the calculus of La Grange; and, though it was invented expressly with a view to the problems just mentioned, it has been found of great use in many physical questions with which those problems are not immecliately connected.
Among the improvements of the higher geometry, besides those which, like the preceding, consisted of methods entirely new, the extension of the more ordinary methods to the integration of a vast number of formulas, the investigation of many new theorems concerning quadratures, and concerning the solution of fluxionary equations of all orders, had completely changed the appearance of the calculus; 80 that Newton or Leibnitz, had they returned to the world any time since the middle of the last century, would have been unable, without