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 P D A to A, where it is least, and increases continually as it moves along the arc A Q P 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 C A 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 A Q P, 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 P D A Q, fig. 10.
Fig. 10.
Note 45, [p. 9]. Mean motion. Equable motion in a circle P E A 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.
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 E ♈ e ♎ 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° 24ʹ 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 E ♈ e ♎ is also everywhere ninety degrees distant from its poles P and p. The angle P C N, 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.