any other curved line. When the axis is perpendicular to the base, the solid is a right cone. If a right cone with a circular base be cut at right angles to the base by a plane passing through the apex, the section will be a triangle. If the cone be cut through both sides by a plane parallel to the base, the section will be a circle. If the cone be cut slanting quite through both sides, the section will be an ellipse, fig. 6. If the cone be cut parallel to one of the sloping sides as A B, the section will be a para. Fig. 7. Fig. 8. P B B bola, fig. 7. And if the plane cut only one side of the cone, and be not parallel to the other, the section will be a hyperbola, fig. 8. Thus there are five conic sections. NOTE 23, p. 5. Inverse square of distance. The attraction of one body for another at the distance of two miles is four times less than at the distance of one mile; at three miles, it is nine times less than at one; at four miles, it is sixteen times less, and so on. That is, the gravitating force decreases in intensity as the squares of the distance increase. NOTE 24, p. 5. Ellipse. One of the conic sections, fig. 6. An ellipse may be drawn by fixing the ends of a string to two points, S and F, in a sheet of paper, and then carrying the point of a pencil round in the loop of the string kept stretched, the length of the string being greater than the distance between the two points. The points S and F are called the foci, C the centre, SC or CF the excentricity, AP the major axis, QD the minor axis, and PS the focal distance. It is evident that, the less the excentricity C S, the nearer does the ellipse approach to a circle; and from the construction it is clear that the length of the string Sm F is equal to the major axis P A. If T t be a tangent to the ellipse at m, then the angle Tm S is equal to the angle tm F; and, as this is true for every point in the ellipse, it follows that, in an elliptical reflecting surface, rays of light or sound coming from one focus S will be reflected by the surface to the other focus F, since the angle of incidence is equal to the angle of reflection by the theories of light and sound. NOTE 25, p. 5. Periodic time. The time in which a planet or comet performs a revolution round the sun, or a satellite about its planet. U NOTE 26, p. 5. Kepler discovered three laws in the planetary motions by which the principle of gravitation is established:-1st law, That the Fig. 9. des radii vectores of the planets and comets describe areas proportional to the time.Let fig. 9 be the orbit of a planet; then, supposing the spaces or areas PSp, pSa, a Sb, &c., equal to one another, the radius vector SP, which is the line joining the centres of the sun and planet, passes over these equal spaces in equal times; that is, if the line SP passes to Sp in one day, it will come to Sa in two days, to Sb in three days, and so on. 2nd law, That the orbits or paths of the planets and comets are conic sections, having the sun in one of their foci. The orbits of the planets and satellites are curves like fig. 6 or 9, called ellipses, having the sun in the focus S. Several comets are known to move in ellipses; but the greater part seem to move in parabolas, fig. 7, having the sun in S, though it is probable that they really move in very long flat ellipses; others appear to move in hyperbolas, like fig. 8. The third law is, that the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun. The square of a number is that number multiplied by itself, and the cube of a number is that number twice multiplied by itself. example, the squares of the numbers 2, 3, 4, &c., are 4, 9, 16, &c., but their cubes are 8, 27, 64, &c. Then the squares of the numbers representing the periodic times of two planets are to one another as the cubes of the numbers representing their mean distances from the sun. So that, three of these quantities being known, the other may be found by the rule of three. The mean distances are measured in miles or terrestrial radii, and the periodic times are estimated in years, days, and parts of a day. Kepler's laws extend to the satellites. For NOTE 27, p. 5. Mass. The quantity of matter in a given bulk. It is proportional to the density and volume or bulk conjointly., In NOTE 28, p. 5. Gravitation proportional to mass. But for the resistance of the air, all bodies would fall to the ground in equal times. fact, a hundred equal particles of matter at equal distances from the surface of the earth would fall to the ground in parallel straight lines with equal rapidity, and no change whatever would take place in the circumstances of their descent, if 99 of them were united in one solid mass; for the solid mass and the single particle would touch the ground at the same instant, were it not for the resistance of the air. NOTE 29, p. 5. Primary signifies, in astronomy, the planet about which a satellite revolves. The earth is primary to the moon. NOTE 30, p. 6. Rotation. Motion round an axis, real or imaginary. NOTE 31, p. 7. Compression of a spheroid. The flattening at the poles. It is equal to the difference between the greatest and least diameters, divided by the greatest, these quantities being expressed in some standard measure, as miles. NOTE 32, p. 7. Satellites. Small bodies revolving about some of the planets. The moon is a satellite to the earth. NOTE 33, p. 7. Nutation. A nodding motion in the earth's axis while in rotation, similar to that observed in the spinning of a top. It is produced by the attraction of the sun and moon on the protuberant matter at the terrestrial equator. NOTE 34, p. 7. Axis of rotation. The line, real or imaginary, about which a body revolves. The axis of the earth's rotation is that diameter, or imaginary line, passing through the centre and both, poles. Fig. being the earth, N S is the axis of rotation. NOTE 35, p. 7. Nutation of lunar orbit. The action of the bulging matter at the earth's equator on the moon occasions a variation in the inclination of the lunar orbit to the plane of the ecliptic. Suppose the plane Np n, fig. 13, to be the orbit of the moon, and N m n the plane of the ecliptic, the earth's action on the moon causes the angle p N m to become less or greater than its mean state. The nutation in the lunar orbit is the reaction of the nutation in the earth's axis. NOTE 36, p. 7. Translated. Carried forward in space. The NOTE 37, p. 7. Force proportional to velocity. Since a force is measured by its effect, the motions of the bodies of the solar system among themselves would be the same whether the system be at rest or not. real motion of a person walking the deck of a ship at sea is compounded of his own motion and that of the ship, yet each takes place independently of the other. We walk about as if the earth were at rest, though it has the double motion of rotation on its axis and revolution round the sun. NOTE 38, p. 8. Tangent. A straight line which touches a curved line in one point without cutting it. In fig. 4, m T is tangent to the curve in the point m. In a circle the tangent is at right angles to the radius, C m. NOTE 39, p. 8. Motion in an elliptical orbit. A planet m, fig. 6, moves round the sun at S in an ellipse P D A Q, in consequence of two forces, one urging it in the direction of the tangent m T, and another pulling it towards the sun in the direction m S. Its velocity, which is greatest at P, decreases throughout the arc to PDA to A, where it is least, and increases continually as it moves along the arc AQP till it comes to P again. The whole force producing the elliptical motion varies inversely as the square of the distance. See note 23. NOTE 40, p. 8. Radii vectores. Imaginary lines adjoining the centre of the sun and the centre of a planet or comet, or the centres of a planet and its satellite. In the circle, the radii are all equal; but in an ellipse, fig. 6, the radius vector S A is greater, and S P less than all the others. The radii vectores S Q, S D, are equal to CA or C P, half the major axis P A, and consequently equal to the mean distance. A planet is at its mean distance from the sun when in the points Q and D. NOTE 41, p. 8. Equal areas in equal times. See Kepler's 1st law, in note 26, p. 5. NOTE 42, p. 8. Major axis. The line P A, fig. 6 or 10. NOTE 43, p. 8. If the planet described a circle, &c. The motion of a planet about the sun, in a circle A B P, fig. 10, whose radius C A is equal to the planet's mean distance from him, would be equable, that is, its velocity, or speed, would always be the same. Whereas, if it moved in the ellipse AQP, its speed would be continually varying, by note 39; but its motion is such, that the time elapsing between its departure from P and its return to that point again would be the same whether it moved in the circle or in the ellipse; for these curves coincide in the points P and A. NOTE 44, p. 8. True motion. The motion of a body in its real orbit NOTE 45, p. 9. Mean motion. Equable motion in a circle P EA B, fig. 10, at the mean distance C P or C m, in the time that the body would accomplish a revolution in its elliptical orbit P D A Q. Fig. 11. NOTE 46, p. 9. The equinox. Fig. 11 represents the celestial sphere, and C its centre, where the earth is supposed to be. q Q is the equinoctial or great circle, traced in the starry heavens by an imaginary extension of the plane of the terrestrial equator, and Eope is the ecliptic, or apparent path of the sun round the earth., the intersection of these two planes, is the line of the equinoxes; is the vernal equinox, and the autumnal. When the sun is in these points, the days and nights are equal. They are distant from one another by a semicircle, or two right angles. The points E and e are the solstices, where the sun is at his greatest distance from the equinoctial. The equinoctial is everywhere ninety degrees distant from its poles N and S, which are two points diametrically opposite to one another, where the axis of the earth's rotation, if prolonged, would meet the heavens. The northern celestial pole N is within 1° 24' of the pole star. As the latitude of any place on the surface of the earth is equal to the height of the pole above the horizon, it is easily determined by observation. The ecliptic Eye is also everywhere ninety degrees distant from its poles P and p. The angle PCN, between the poles P and N of the equinoctial and ecliptic, is equal to the angle e C Q, called the obliquity of the ecliptic. S NOTE 47, p. 9. Longitude. The vernal equinox, ∞, fig. 11, is the zero point in the heavens whence celestial longitudes, or the angular motions of the celestial bodies, are estimated from west to east, the direction in which they all revolve. The vernal equinox is generally called the first point of Aries, though these two points have not coincided since the early ages of astronomy, about 2233 years ago, on account of a motion in the equinoctial points, to be explained hereafter. If S op, fig. 10, be the line of the equinoxes, and ∞ the vernal equinox, the true longitude of a planet p is the angle o S p, and its mean longitude is the angle ∞ Cm, the sun being in S. Celestial longitude is the angular distance of a heavenly body from the vernal equinox; whereas terrestrial longitude is the angular distance of a place on the surface of the earth from a meridian arbitrarily chosen, as that of Greenwich. NOTE 48, pp. 9, 58. Equation of the centre. The difference between Cm and Sp, fig. 10; that is, the difference between the true and mean longitudes of a planet or satellite. The true and mean places only coincide in the points P and A; in every other point of the orbit, the true place is either before or behind the mean place. In moving from A through the arc A Q P, the true place p is behind the mean place m ; and through the arc PDA the true place is before the mean place. At its maximum, the equation of the centre measures C S, the excentricity of the orbit, since it is the difference between the motion of a body in an ellipse and in a circle whose diameter A P is the major axis of the ellipse. NOTE 49, p. 9. Apsides. The points P and A, fig. 10, at the extremities of the major axis of an orbit. P is commonly called the perihelion, a Greek term signifying round the sun; and the point A is called the aphelion, a Greek term signifying at a distance from the sun. NOTE 50, p. 9. Ninety degrees. A circle is divided into 360 equal parts, or degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. It is usual to write these quantities thus, 15° 16' 10", which means fifteen degrees, sixteen minutes, and ten seconds. It is clear that an arc m n, fig. 4, measures the angle m Cn; hence we may say, an arc of so many degrees, or an angle of so many degrees; for, if there be ten degrees in the angle m C n, there will be ten degrees in the arc m n. It is evident that there are 90° in a right angle, m C d, or quadrant, since it is the fourth part of 360°. NOTE 51, p. 9. Quadratures. A celestial body is said to be in quadrature when it is 90 degrees distant from the sun. For example, in fig. 14, if d be the sun, S the earth, and p the moon, then the moon is said to be in quadrature when she is in either of the points Q or D, because the angles QS d and D S d, which measure her apparent distance from the sun, are right angles. NOTE 52, p. 9. Excentricity. Deviation from circular form. In fig. 6, C S is the excentricity of the orbit PQA D. The less CS, the more nearly does the orbit or ellipse approach the circular form; and, when C S is zero, the ellipse becomes a circle. NOTE 53, p. 9. Inclination of an orbit. Let S, fig. 12, be the centre of the sun, PN An the orbit of a planet moving from west to east in the |