indulge its adherents with the miserable delusion that it revealed to them the whole secret of the mechanism of the universe.

Borelli, in his theory of the Medicean stars, published in 1666, appears to have speculated more judiciously on the physical theory of the planets than any of his predecessors. He remarks that the motions of the planets round the sun, and those of the satellites round their respective primaries, must doubtless depend in each case on some virtue residing in the central body. He seems to have arrived at pretty accurate notions of the motion of a body in a circular orbit. He remarks that bodies so revolving have a tendency to recede from their centre of revolution, as in the case of a wheel revolving on its axle, or a stone whirled by a sling. When this force is equal to the tendency of the body to the centre, a compensation of effects takes place, and the body will neither approach nor recede from the centre of force, but will continually revolve round it.

Here, for the first time, an attempt is made to account for the motion of a body in a circular orbit, by means of a force directed continually to the centre of the circle. It must be admitted, however, that Borelli's explanation is at once imperfect and indistinct. He does not analyze the phenomenon of curvilinear motion into its constituent elements, but merely seeks to establish the necessity of a constant central force by an appeal to experiment. He rightly asserts that the body tends continually to recede from the centre, but he gives no account of the origin of this centrifugal force nor does he explain by what means the motion of the body in its orbit is continually kept up. His account of the last-mentioned part of the phenemenon is so obscure, that it is quite evident he had obtained only a very weak hold of the problem. After remarking that the compensatory effects of the two constant forces will maintain the body at a determinate distance from the centre, he then says, "therefore the planet will appear balanced and floating on the surface."*

Although Borelli's speculations possessed much merit, still they were not sufficiently clear to lead to any measurable results, and until a complete dynamical view of the problem of centripetal forces could be obtained, it was obviously hopeless to attempt its mathematical solution. Without stopping here to notice the partial researches of Hooke, Huygens, Wren, and Halley, we shall at once proceed to give some account of the immortal discoveries of NEWTON. This illustrious philosopher was born in the year 1642, at Woolsthorpe, in the county of Lincoln. Before attaining the years of maturity he made a multitude of beautiful discoveries in Analysis, and was even in possession of the method of Fluxions when he was only twenty-four years of age. He was now about to enter upon a field of speculation which was destined to offer a magnificent theatre for displaying the resources of that powerful instrument of investigation. Pemberton states that Newton, having quitted Cambridge, for Woolsthorpe, in 1665, to avoid the plague, was sitting one day in his garden, when he was led to reflect on the principle which causes all bodies to tend towards the centre of the earth. As this tendency did not appear to suffer any sensible diminution on the tops of the highest build

"Ideoque planeta libratus apparebit et supernatans." tarum in causis physicis deductæ. Florentiæ, 1666.

Theorica Mediceorum Plane

We shall have occasion to notice incidentally in the following pages the labours of these philosophers on the subject of centripetal forces. Newton commenced his researches at least as early as any of his contemporaries; nor does it appear, throughout all this career, that he was indebted to one or other of them for any of his ideas.

ings, or even on ascending the loftiest mountains, it occurred to him that it might possibly extend to the moon; and, if it did, might be the cause which retained that body in her orbit. Pursuing his meditations, he was led to imagine that a similar force directed continually towards the sun might retain the planets in their orbits. But a question naturally suggested by this generalization of his ideas was this-Did the solar force act with the same intensity on all the planets, or did it diminish with the distance from the centre, as the slower motion of the more remote planets seemed to indicate? His next step, therefore, was to determine, by a mathematical investigation, the magnitude of the force which retains a body in a circular orbit, the force being continually directed to the centre of the circle. The solution of this problem gave him an expression for the centripetal force in terms of the velocity of the body in its orbit and its distance from the centre, or, which amounts to the same thing, in terms of the periodic time and the distance. Hence, when the relation between these two elements was known, it was easy to express the force in terms of the distance alone, and by this means to ascertain the law according to which it varied. Now, Kepler had shown that the squares of the periodic times of the planets are proportional to the cubes of their distances from the sun; Newton hence inferred that the planets are retained in their orbits by a force directed towards the sun and varying inversely as the square of the distance from his centre.

The result of Newton's investigation relative to the law of attraction was strengthened by the analogy which other natural emanations. from centres offered: but it would manifestly have received a vast accession of support if it were found that the attraction exerted by the earth upon the moon, when compared with her attraction of objects at the surface, diminished also according to the same law of the distance. The solution of this question might, therefore, now be considered as the experimentum crucis which was to decide whether Newton had penetrated into the secret of the celestial motions, or whether he had been occupying his mind with speculations of a purely mathematical nature. Now, the force which determines the descent of a body at the surface of the earth is measured by the space through which it falls into a given small portion of time; and the force which retains the moon in her orbit is measured by the versed sine of the small arc de scribed by her in the same time; for, if no force had acted, the moon would have proceeded in the direction of a tangent to her orbit, and the versed sine being the measure of deflection from the tangent, indicates, therefore, the intensity of the deflecting force. It is obvious that, in order to compare these two small spaces, they must both be expressed in terms of the same unit, as a foot for example. Now, the versed sine of the lunar arc is readily expressed in terms of the radius of the orbit, and again the latter is derivable from the earth's radius by means of the lunar parallax. The question relative to the comparison of the two forces is, therefore, finally reduced to the determination of the distance in feet, between the centre of the earth and the surface. This object may be very readily effected when once the length of a given arc of the meridian is known; but, at the time we are considering, this point was by no means accurately ascertained. Newton employed in his calculation the rough estimate of 60 miles to a degree, which was in use among geographers and navigators; whereas the real length of a degree is about 694 miles. It may hence be readily inferred that the result obtained by him did not

satisfy his expectations. Having determined the earth's force upon the moon by diminishing the gravity of bodies at the surface in the ratio of the square of the distance from the centre, and then compared the result with the force indicated by the motion of the moon in her orbit, he found that, instead of the two quantities being exactly equal, the former exceeded the latter by about one-sixth. Deeming this discordance too great to justify his bold surmise, he laid the investigation aside, doubtless with the intention of reconsidering it at some future time.

Newton's attention was again called to the subject of centripetal forces, by a letter he received from Hooke, in 1679, relative to the path described by a projectile, taking into account the effect of the earth's diurnal motion. Hooke was unquestionably endowed with a genius of a very high order; but, partly from the desultory character of his researches, and partly from his deficiency in mathematical skill, he has not achieved results by any means commensurate with his great acuteness and originality. As early as the year 1666 he had illustrated, by means of a beautiful experiment, the motion of a body revolving in an ellipse under the influence of a force directed continually to the centre; and, in his letter to Newton on the occasion above referred to, he declared that, if gravity decreased according to the reciprocal of the square of the distance, the path of a projectile would be an ellipse, having the centre of the earth in the focus. Although this assertion was unaccompanied by any proof, and consequently did not possess any merit beyond that of a sagacious conjecture, still it excited a strong interest in the mind of Newton, who had already devoted much attention to the subject of central forces. His researches had hitherto been confined to bodies revolving in circular orbits: he now proposed to investigate the vastly more difficult question of a body revolving in an orbit of variable curvature.

Considering generally the motion of a body projected in free space, and exposed to the incessant action of a force tending towards a fixed centre, he arrived at the remarkable conclusion, that an imaginary line joining the centre of force and the body would constantly sweep over equal areas in equal times. Now Kepler had found that the planets revolve round the sun precisely according to this law; it followed, then, that all these bodies were retained in their orbits by a force directed continually to the centre of the sun.

It still remained for Newton to investigate the law of the force corresponding to the variation of the distance in the same orbit. According to Kepler's first law, the planets move in ellipses, having the sun in one of the foci. The question, therefore, was to determine the law of the force by which a body is compelled to revolve in an elliptic orbit, the force being continually directed to one of the foci of the ellipse. This problem is of a much more complex character than the similar one relative to a circular orbit. In order to form some idea of the difference between the two cases, we may remark generally, that when a body has once received an impulse in any direction, it would persevere with a uniform motion in the direction of the impulse, if it were not exposed to the influence of any extraneous force. Now, when a body revolves in a curvilinear orbit, it is continually changing the direction of its motion; this is, therefore, a clear proof that it is acted upon by some force which continually deflects it from the tangent to the orbit in the direction of which it is every instant naturally endeavouring to move. Now, the force required to retain a body in a curvilinear orbit at any given point depends partly on the curvature of the orbit and partly on the

velocity with which the body is moving; for, with the same velocity, but a greater amount of curvature, the body will require to be deflected in a given time through a greater space, and therefore the deflecting force must be more intense; and again, for the same amount of curvature, but a greater velocity, the body will be deflected in a less time through the same space, and therefore in this case also the force will be more intense. In order that the centripetal force may retain the body in its orbit without producing any other effect, it is necessary that it should constantly act at right angles to the tangent; for, if it act in an oblique direction, it will be partly expended in increasing or diminishing the tangential motion, according as the body is approaching to, or receding from, the centre of force. Now, when a body is compelled to revolve in a circular orbit by a force tending continually to the centre of the circle, the direction of the force is constantly perpendicular to the tangent; and therefore the force neither accelerates nor retards the body, but simply retains it in its orbit. The velocity of the body will, therefore, continue uniform, and, since the curvature of a circle is also uniform, it follows, from what we have already stated, that the centripetal force will have the same intensity for every point of the orbit.

But the question is much more complicated when we consider the motion of a body in an elliptic orbit. In this case, the force acts in an oblique direction with respect to the tangent at every point of the orbit except the two extremities of the major axis, and hence it is constantly expended, partly in deflecting the body into its orbit, and partly in accelerating or retarding the tangential motion. The velocity being therefore variable, and the same being true with respect to the curvature of the ellipse, it follows that the deflecting force which depends upon these two elements is also subject to continual variation. This force, however, which constantly acts at right angles to the tangent, can only be increased or diminished by means of a corresponding change in the intensity of the centripetal force, of which it forms one of the resolved parts. It follows, therefore, that the centripetal force varies not only from being more or less effectual in retaining the body in its orbit, but also because the elements upon which the effective part depends are also in a state of continual variation *.

The preceding remarks may serve in some degree to show the peculiar difficulties of the problem which now suggested itself to Newton. Enveloped as it was in complications and obscurities, his inventive genius devised the means of its solution, and he found that the centripetal force varied inversely as the square of the distance from the focus of the ellipse. This result accorded in a most satisfactory manner with the conclusion to which he was conducted by his previous researches, founded on the supposition of the planets revolving in circular orbits. Assuming the solar

• The resistance offered by a body to move in a curvilinear orbit has been termed its centrifugal force; it is therefore equal, and opposite to, the resolved part of the centripetal force, which acts perpendicularly to the tangent. Hence, when a body revolves in a circular orbit by means of a force directed to the centre of the circle, the centripetal and centrifugal forces will be equal; but, in every other case, the latter of these forces will exceed the former, and will tend not to the centre of force, but to the centre of the circle of curvature, corresponding to the infinitely small arc of the orbit in which the body is moving at the given instant. It is obvious that the centrifugal force has no positive existence; it merely arises from the resistance offered by the inertia of the body, in virtue of which the latter tends constantly to persevere in a straight line.

force to extend to the remotest planets, and to vary everywhere according to the inverse square of the distance from the sun, he demonstrated that the squares of the periodic times of the planets would be proportional to the cubes of their mean distances. This was the third of Kepler's famous laws of the planetary motions. It followed, therefore, that the law of the inverse square of the distance was true, not only when the distances related to the same orbit, but even when they were compared in different orbits. He had already arrived at this conclusion, by assuming the orbits to be circular, and now he found it to be demonstrable for the more rigorous case of elliptic orbits.

Notwithstanding the satisfactory nature of Newton's researches relative to the planets, the law of gravitation appeared to his cautious mind to be imperfectly established, so long as the serious discordance offered by the moon remained unexplained. A circumstance, however, had recently occurred, which induced him to suspect that the cause of this discordance lay in assuming an erroneous value for a degree of the meridian. We have mentioned that, in computing the earth's semi-diameter, he used the commonly received estimate of 60 miles to a degree. Picard, the French astronomer, however, having in the intermediate period measured an arc of the meridian with great care, and obtained a result considerably different, he resolved to repeat his previous calculation by means of it. To his unspeakable delight he now obtained a result which completely harmonized with his researches on the planets. Assuming that the semi-diameter of the lunar orbit was equal to 60 semi-diameters of the earth, he found that the space by which the moon is deflected from the tangent to her orbit in one minute is exactly equal to the space through which bodies at the earth's surface fall in one second. In order to appreciate the conclusiveness of this result, we may remark that, when a body is acted upon by a continuous force during a small portion of time, the space described by it in consequence varies in the direct ratio of the force and the square of the time. Hence if the force be supposed to vary in the inverse ratio of the square of the distance, the space will vary as the square of the time directly and the square of the distance inversely. It is clear, then, that when two bodies are placed at unequal distances from the centre of force, the minute spaces through which they are drawn by the force can only be equal, when the time, during which the more remote body is under the influence of the force, exceeds the corresponding time of the nearer body, in the same ratio in which its distance from the centre exceeds the corresponding distance of the other. Conversely, if two bodies fall through equal spaces in times which are to each other in the direct ratio of the distances from the centre of force, we may conclude that the force varies in the inverse ratio of the square of the distance *. Now, Newton assumed in his calculation that the moon is 60 times more distant from the centre of the earth than objects at the surface; and he found that the time occu

Let f f be the force of gravity at the earth's surface and at the moon, d d' the corresponding distances from the earth's centre, & s' the minute spaces through which bodies would fall at those distances in the times tt'; then, as mentioned in the text, we have s=a ft2, s=a ƒ ta being a constant quantity. Now, if we assume with Newton, 1 1

=

that ss, we have ƒ ft2; hence ƒ : ƒ' :: 72 : 72.

But Newton found that t: t'e

1

1

d: d'; therefore 7: 1 :: d2 : d2, and consequently, ƒ : ƒ' :: W2 : Ã2.