Fig. 68.—Diagram for Kepler's Laws.
After having thus allotted to each straight line its approximate length, let us join their extremities by a curve. What do we see before us? A geometrical figure widely different from a circle, for the diameters (i.e., the straight lines passing through the centre) are far from being equal. The figure is an ellipse.
If now we pass from the appearance to the reality, o will be the sun, and a a´ a´´, m m´ will indicate the terrestrial orbit, or the points of the curve successively occupied by the earth in movement. The moveable straight lines, free at one extremity, and at the other attached to the centre of the sun, are called the Vector heliocentric radii. By the help of this construction, you see that the point occupied by the sun is beyond or without the centre; this eccentric point is the focus of the ellipse, and the distance from this focus to the centre, its eccentricity. The extremity of the major axis, the nearest to the focus, is the perihelion, and its farthest extremity the aphelion. The difference of the angles formed by the vector radii indicate the inequality of the movements: to the greatest angle, the perihelion, corresponds the maximum of velocity (a a´ a´´), just as to the smallest, or aphelion, corresponds the minimum (m m´); the other angles mark the velocities intermediary between these two extremes. We have thus before us a series of triangles with their apices at the focus of the ellipse, and their bases on the contour of the curve.
But these latter are not sufficient for the mind, whose principal function lies in seeking unity among the variety of phenomena.
In what way are the variations of distance connected with the variations of velocity? What is the simplest expression of their relationship? These are questions which naturally presented themselves to Kepler's inquiring intellect. By dint of immeasurable patience, and recommencing more than once the same toil, this great astronomer discovered that the variable arc traversed by the earth (or, in appearance, the sun), in four-and-twenty hours, multiplied by one half the corresponding vector radius, is a constant quantity: is the product which, as elementary geometry teaches, gives the surface of a triangle. And, in fact, look at the matter carefully: the vector radii form triangles whose base is the arc traversed in the same interval of time, and whose apices rest upon the centre of the sun (or, in appearance, the observer, or the centre of the earth).
To fix these ideas thoroughly in our minds,—and a superficial knowledge is worse than useless,—let us imagine to ourselves a man holding horizontally extended a tube of a certain length, capable, like a telescope, of being lengthened or shortened at pleasure; and let us fancy him pivoting upon himself, in such a manner that he sweeps, every minute, exactly the same area or same quantity of surface, while varying perpetually the swiftness of movement and the length of the tube; this "ideal man" will have solved the problem whose solution is inscribed, in ineffaceable letters, on the machinery of our globe; he will describe around him an ellipse, of which he himself occupies one of the foci.
By this method of investigation and deduction, Kepler succeeded in breaking up the traditionary authority of the circle and of uniform movement. He broke it up for ever by two of his celebrated laws, which may be rendered in the following terms:—
1st, The orbit of the earth, as well as the curves described by the other planets, are ellipses, one of whose foci is represented by the sun;
2d, The heliocentric vector radius of a planet describes around the sun areas equal with the times; or, in other words, the surfaces described by the vector radii, in equal times, are also equal.