By the first law of Kepler, then, we learn that Saturn's orbit s s's's'" is an ellipse, and that the sun is situated at s, one of the

indicated by the circles mm', vv', мм', and JJ', the points m, v, м, and л being the perihelia of those orbits. The line & & in each orbit is the line of nodes, & being the rising node. In the case of Jupiter the greatest departures from the plane of the ecliptic are indicated by the lines i and jj'; in the other orbits the corresponding departures are too small to be thus represented. The angles of inclination of the orbits of Mars, Venus, and Mercury, to the ecliptic, are, respectively, 1° 51′ 5′′·5, 3° 23′ 33′′-2, and 7° 0′ 25′′-0. The corresponding angle in the case of Jupiter is 1° 18′ 36′′-7. It will be observed that the orbits of Mars and Mercury are more eccentric than those of the other members of the system. The dotted ring A A'A" marks the probable extent of the zone of asteroids, the orbits of four of which-Harmonia, Nemausa, Polyhymnia, and Nysa―are indicated respectively by the curves hh', a a', pp', and n n', the perihelia of these orbits being at h, a, p, and n. The two first are the least eccentric of the asteroidal orbits, and differ little from the circular form. The orbits of Nysa and Polyhymnia are remarkably eccentric. Professor Nichol remarks that 'Nysa recedes farther from the sun than any of the others, and, with the exception of Hæstia, approaches him the nearest.' If, however, the elements of the asteroidal orbits are correctly given by him in his 'Cyclopædia of the Physical Sciences' (article Asteroids), Hæstia is by no means remarkable for its near approach to the sun either as respects mean or perihelion distance, while the perihelion distance of Nysa is less than the mean distance of Mars. As will be seen from the figure, part of the orbit of Nysa absolutely falls within the orbit of Mars, a circumstance that will seem still more remarkable when it is considered that the centre of the ellipse in which Nysa moves lies outside the orbit of the earth-falling, in fact, very near the orbit of Mars. The orbit of Polyhymnia is not so eccentric as that of Nysa; yet the centre falls only just within the earth's orbit. To avoid confusion, the nodal lines of the four asteroidal orbits are not drawn in the figure; the following table indicates their positions, and the angles at which the planes of the four orbits are inclined to the ecliptic:—

It will be seen from this table that the path of Nysa does not actually intersect that of Mars.

The asteroid Melpomene is also remarkable for the close proximity of a part of its orbit to the aphelion of Mars.

It has been noticed by Mr. Cooper, of Markree Castle, that in the positions of the asteroidal orbits a speciality is observable which can hardly be the result of accident: --the perihelia and the ascending nodes are not distributed indifferently, but are found chiefly in the semicircle from 0° to 180°. The observation may be extended to the larger planets; all of those introduced in the figure have their perihelia and rising nodes within the semicircle from 330° to 150°, which- -more nearly than the semicircle just indicated-corresponds to the region in which the asteroidal perihelia and rising nodes are most remarkably crowded. The planets Uranus and Neptune do not

foci of this ellipse. The second law of Kepler indicates the law of Saturn's motion in this orbit, which may be illustrated as follows: - Suppose that PP', QQ, and RR are arcs over which Saturn passes in equal intervals of time; then Kepler's second law asserts that if straight lines s P, S P', s Q, s Q', S R, and S R′ be drawn (to avoid confusion, these lines are omitted in the figure), the areas SPP', SQQ and S R R', are equal. Since the sector SPP is plainly shorter than the sector s Q Q', and sqq' than S R R', it follows from the equality of these areas that the arc P P' is longer than the arc Qo', and Qo' than R R-increase in the breadth of the sectorial area compensating deficiency in length. In other words Saturn's velocity in his orbit increases as he approaches perihelion, and diminishes as he approaches aphelion. Thus, when he is near perihelion, he appears to be describing an orbit smaller than his actual orbit, with a velocity greater than his mean velocity; when he is near aphelion, these relations are reversed. His period, therefore, would appear too small, if determined when he is near perihelion, and too great if determined when he is near aphelion.*

deviate from the same law: the longitudes of their rising nodes are respectively 73° 14′ 38′′; and 130° 10′ 12′′-3, the longitudes of their perihelia 168° 27′ 24′′, and 47° 17′ 58′′. The speciality as regards the perihelia is certainly remarkable, and its physical interpretation worth seeking. The congregation of the rising nodes in the region indicated is obviously due to the choice of the ecliptic as the plane to which we refer the positions of the other orbital planes. Convenient as this selection is in many respects, it has its disadvantages; in fact, with the single exception of Mercury, no planet could be selected the plane of whose orbit is less suitable as a plane of reference in viewing the grander relations of the planetary scheme.

The orbits of Uranus and Neptune have not been introduced into the figure on account of their dimensions. The mean distance of Uranus from the sun is about twice, the mean distance of Neptune more than three times, that of Saturn. The eccentricities of the orbits are respectively 0466 and 0087, their inclinations to the plane of the ecliptic 0° 46′ 29′′-9 and 1° 46′ 59′′.

* The absolute velocity of a planet at any point of its orbit varies inversely as the length of the perpendicular on the tangent at that point: the angular velocity of the planet about the sun's centre varies inversely as the square of the planet's distance from the sun. There is a slight error in Nichol's statement that 'by an appropriate choice of an eccentric circular orbit the sun's motion relative to the earth or to any planet,' (or, which is the same thing, any planet's motion relatively to the sun), 'may be very closely approximated to,' on the supposition of uniform velocities. See article 'Eccentric' in Nichol's 'Cyclopædia of the Physical Sciences.' On such a supposition the angular velocity of a planet about the sun's centre would appear to vary inversely as the distance, instead of as the square of the distance of the planet.

The absolute dimensions of the ellipse in which Saturn moves are as follows: his mean distance from the sun (or half the greater axis of his orbit) is no less than 874,321,000 miles, his least distance (or ss) is 825,404,000 miles, and his greatest distance (or ss") is 923,238,000 miles. The eccentricity of the orbit is very nearly ⚫056. In this vast orbit he moves with a mean velocity of 21,160 miles an hour, sweeping out a mean hourly angle of 5'025 about the sun. He occupies 10759-2197106 days in moving once round his orbit, or in completing a sidereal revolution.*

Kepler next inquired whether there existed any relation between the periods of the planets and the dimensions of the planetary orbits. He selected the mean distances (or the semi-major axes of the orbits) for the comparison, considering that some relation might probably be found between the powers of these distances and of the periodic times. It was, however, only after many years' inquiry, that he arrived at the conclusion that it was here, and thus, that some new harmony in the planetary scheme was to be sought. One would have thought the rest of the work was simple; yet even when the very law he was seeking had occurred to him, two months and a half elapsed before he was able to verify it. Let us consider how the law might have been determined from the orbits and periods of Saturn and the earth. Calling the mean distance of the earth 1, Saturn's mean distance is 9.53885; again, calling the earth's period 1, Saturn's period is 29.4566:-now what relation (if any) exists between these numbers, 9.53885 and 29-4566, or their powers? The first is less than the second, but the square of the first is plainly greater than the square of the second; we must therefore try higher powers of the second number. Trying the next power, that is, the square of the second number, we immediately find the relation we are seeking; thus :-The square of the first number is less than the square of the second; but the next power, or the cube, of the first number is almost exactly equal to the square of the second.†

* All the elements of Saturn's orbit are undergoing slow processes of change; the natures and causes of some of these are examined further on; the tables of Appendix II. indicate the amount of the annual variation of each element.

+ The cube of 9-53885 is 867·9369; and the square of 29.4566 is 867-691, differing from the first by less than 0.246.

Here then is the required law, if, only, it shall appear that the relation is confirmed when we try it upon other pairs of planetary orbits. On trial it appears to be true for every such pair, and thus the third law of Kepler is established; viz., that,

3. The squares of the periodic times of the planets vary as the cubes of their mean distances.*

Such are the laws of Kepler-laws purely empirical as presented by him, but destined to prepare the way towards, if they did not directly lead up to, the grandest law of nature yet discovered by man-the law of universal gravitation. Strictly speaking, none of Kepler's laws are correct: the planets being of appreciable mass and exercising attractions upon each other and upon the sun, their motions deviate from the orbits they would follow if these conditions did not exist-orbits which would be strictly in accordance with the laws propounded by Kepler. The accuracy of the laws, however, corresponded with, if it did not surpass, the accuracy of instrumental observation in Kepler's time, and for many years following the announcement of his important discoveries.

In the latter half of the seventeenth century, Newton commenced the investigation of Kepler's laws. Kepler had sought to learn what are the paths of the planets, and what the laws they obey in pursuing those paths: Newton devoted the powers of his piercing intellect to inquire why the planets follow such paths and obey such laws. He sought, in fact, the physical interpretation of the observed phenomena.

Newton first proved that a body moving in such a manner with respect to any point that its radius vector describes equal areas about the point in equal times, is moving under the influence of forces constantly directed towards or from that point. According as the orbit thus described is concave or convex towards the point, the force acts towards or from the point. Since, then, each planet describes equal areas in equal times about the sun, and moves in an orbit whose convexity is towards him, the sun exerts an attractive force on each member of the system.

*The law may also be expressed as follows:-Fixed units of time and space being chosen, the square of the number expressing the periodic time of a planet bears a constant ratio to the cube of the number expressing the mean distance of the planet.

Secondly, Newton demonstrated that if a body revolves in an elliptical orbit (or in an orbit whose form is any of the conic sections) under a central attracting force residing in one of the foci, that force varies as the inverse square of the distance of the attracted body. He further showed that Kepler's third law was a necessary consequence of attraction so varying.

In obtaining these results, Newton may be considered to have empirically demonstrated the existence of an attractive force exerted by the sun's mass, and to have established the law under which that force acts. The reader must be careful, however, to distinguish such a result from the establishment of the great law of gravitation. The mere determination of the law of attraction exerted by the sun on the planets and by these on their satellites, however interesting, would have been neither particularly valuable nor-except in being demonstrated-novel. The idea of attractions so exerted, and the very law of such attractions, had occurred to many astronomers long before Newton's day; nor does it appear that Newton himself attached any great value to the result, thus far, of his inquiries into the planetary laws of Kepler.

The history of the process by which Newton arrived at the great discovery which has rendered his name famous has been repeated so often that it would be idle to give it here at length. The idea that the moon was retained in its orbit about the earth by the same attractive energy that causes unsupported bodies to fall to the earth,* appears to have occurred to Newton about the year

* The story of the apple, whose fall suggested the first idea of his great discovery to Newton, is probably apocryphal. Whether it is true or not, the manner in which it is usually related in works on popular science is calculated to lead to altogether erroneous ideas of the nature of Newton's discovery. It would not have been the question, 'Why does the apple fall?'-that Newton would have asked himself: the attraction of gravity had been known for many ages; the laws of its action on falling bodies had been discussed, however erroneously, by Aristotle, and had been correctly established by Galileo. The inquiry might have been suggested, 'What if this attraction of gravity, so familiar to philosophers, of whose operation I have just witnessed an effect, has a wider range of action? what if an attraction whose influence appears to be exerted alike on bodies of the most varying natures, and to be unaffected by differences of elementary conformation, of form, or of physical condition, in the bodies acted upon, is itself exerted equally by bodies so differing ; is a property depending not upon the quality but simply on the quantity of matter;—is, in fact, a "primitive power of nature," exerted by every atom in immeasurable space, with a range altogether