Note 105, [p. 36]. Square of time. If the times increase at the rate of 1, 2, 3, 4, &c., years or hundreds of years, the squares of the times will be 1, 4, 9, 16, &c., years or hundreds of years.
Note 106, [p. 37]. In all investigations hitherto made with regard to the acceleration, it was tacitly assumed that the areas described by the radius vector of the moon were not permanently altered; that is to say, that the tangential disturbing force produced no permanent effect. But Mr. Adams has discovered that, in consequence of the constant decrease in the excentricity of the earth’s orbit, there is a gradual change in the central disturbing force which affects the aërial velocity, and consequently it alters the amount of the acceleration by a very small quantity, as well as the variation and other periodical inequalities of the moon. On the latter, however, it has no permanent effect, because it affects them in opposite directions in very moderate intervals of time, whereas a very small error in the amount of the acceleration goes on increasing as long as the excentricity of the earth’s orbit diminishes, so that it would ultimately vitiate calculations of the moon’s place for distant periods of time. This shows how complicated the moon’s motions are, and what rigorous accuracy is required in their determination.
To give an idea of the labour requisite merely to perfect or correct the lunar tables, the moon’s place was determined by observation at the Greenwich Observatory in 6000 different points of her orbit, each of which was compared with the same points calculated from Baron Plana’s formulæ, and to do that sixteen computers were constantly employed for eight years. Since the longitude is determined by the motions of the moon, the lunar tables are of the greatest importance.
Note 107, [p. 37]. Mean anomaly. The mean anomaly of a planet is its angular distance from the perihelion, supposing it to move in a circle. The true anomaly is its angular distance from the perihelion in its elliptical orbit. For example, in fig. 10, the mean anomaly is P C m, and the true anomaly is P S p.
Note 108, pp. [38], [68]. Many circumferences. There are 360 degrees or 1,296,000 seconds in a circumference; and, as the acceleration of the moon only increases at the rate of eleven seconds in a century, it must be a prodigious number of ages before it accumulates to many circumferences.
Note 109, [p. 39]. Phases of the moon. The periodical changes in the enlightened part of her disc, from a crescent to a circle, depending upon her position with regard to the sun and earth.
Note 110, [p. 39]. Lunar eclipse. Let S, fig. 27, be the sun, E the earth, and m the moon. The space a A b is a section of the shadow, which has the form of a cone or sugar-loaf, and the spaces A a c, A b d, are the penumbra. The axis of the cone passes through A, and through E and S, the centres of the sun and earth, and n m nʹ is the path of the moon through the shadow.
Fig. 27.
Note 111, [p. 39]. Apparent diameter. The diameter of a celestial body as seen from the earth.