But first it is desirable to explain briefly certain so-called “laws” which govern the motions of all the planets. These are known as Kepler's laws of planetary motion, and are three in number. The demonstration of their truth would carry us beyond the scope of this book, and consequently we shall merely state them as they are recognised by astronomers. 1st Law: The orbit of every planet is an ellipse, having the sun situated in one of the foci. 2d Law: The radius vector of a planet describes equal areas in equal times. By the radius vector is meant the straight line joining the planet to the sun, and the law declares that as the planet moves round the sun, the area of space swept over by this line in any given time, say one day, is equal to the area which it will sweep over in any other equal length of time. If the orbit were a circle it is evident at a glance that the law must be true, because then the sun would be situated in the centre of the circle, the length of the radius vector, no matter where the planet might be in the orbit, would never vary, and the area swept over by it in one day would be equal to the area swept over in any other day, because all these areas would be precisely similar and equal triangles. But Kepler discovered that the same thing is true when the orbit is an ellipse, and when, in consequence of the eccentricity of the orbit, the planet is sometimes farther from the sun than at other times. As the triangular area swept over in a given time increases in length with the planet's recession from the sun, it diminishes in breadth just enough to make up the difference which would otherwise exist between the different areas. This law grows out of the fact that the force of gravitation varies inversely with the square of the distance. 3d Law: The squares of the periods (_i. e._ times of revolution in their orbits) of the different planets are proportional to the cubes of their mean (average) distances from the sun. The meaning of this will be best explained by an example. Suppose one planet, whose distance we know, has a period only one-eighth as long as that of another planet, whose distance we do not know. Then Kepler's third law enables us to calculate the distance of the second planet. Call the period of the first planet 1, and that of the second 8, and also call the distance of the first 1, since all we really need to know is the _relative_ distance of the second, from which its distance in miles is readily deduced by comparison with the distance of the first. Then, by the law, 1^2 : 8^2 : 1^3 : x^3 (“x” representing the unknown quantity). Now, this is simply a problem in proportion where the product of the means is equal to the product of the extremes. But 1^2 = 1, and also 1^3= 1; therefore x^3 = 8^2, and x = ∛(8^2) (the cube root of the square of 8), which is 4. Thus we see that the distance of the second planet must be four times that of the first.