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._ The ancients had looked for equality in the movements of planets traversing the circumferences of circles: they were mistaken. It is true this equality exists; only, not where they supposed. If they had sought it in the surfaces described by the vector radii, they would have anticipated Kepler's discovery of the laws which govern our world. But their astronomical dogmas prevented them from seeing the path which led to this great discovery. Hence we may conclude that Dogma is an evil thing. [Illustration] FOOTNOTES: [Footnote 73: Ptolemæus, "Syntaxis Mathematicalis," iii. 3.] [Footnote 74: Certain astronomical calculations, so called because begun under the patronage of Rudolf II., Emperor of Germany (1576-1612).]