For the time came when the instruments for measuring the positions of the stars and planets had been much improved, largely due to TYCHO BRAHE (1546-1601), a Dane of noble birth, who was the keenest and most careful observer that astronomy had yet produced. { 28} His observations enabled his friend and pupil, JOHANN KEPLER, (1571-1630), to subject the planetary movements to a far more searching examination than had yet been attempted, and he discovered that the Sun is in the plane of the orbit of each of the planets, and also in its +line of apsides+--that is to say, the line joining the two points of the orbit which are respectively nearest and furthest from the Sun. Copernicus had not been aware of either of these two relations, but their discovery greatly strengthened the Copernican theory.
Then for many years Kepler tried one expedient after another in order to find a combination of circular motions which would satisfy the problem before him, until at length he was led to discard the circle and try a different curve--the oval or ellipse. Now the property of a circle is that every point of it is situated at the same distance from the centre, but in an ellipse there are two points within it, the "foci," and the sum of the distances of any point on the circumference from these two foci is constant. If the two foci are at a great distance from each other, then the ellipse is very long and narrow; if the foci are close together, the ellipse differs very little from a circle; and if we imagine that the two foci actually coincide, the ellipse becomes a circle. When Kepler tried motion in an ellipse instead of motion in a circle, he found that it represented correctly the motions of all the planets without any need for epicycles, and that in each case the Sun occupied one of the foci. And though the planet did not move at a uniform speed in the ellipse, yet its motion was governed by a uniform law, for the straight line joining the planet to the Sun, the "+radius vector+," passed over equal areas of space in equal periods of time.
These two discoveries are known as Kepler's First and Second Laws. His Third Law connects all the planets together. It was known that the outer planets not only take longer to revolve round the Sun than the inner, but that their actual motion in space is slower, and Kepler found that this actual speed of motion is inversely as the square root of its distance from the Sun; or, if the square of the speed of a planet be multiplied by its distance from the Sun, we get the same result in each case. This is usually expressed by saying that the cube of the distance is proportional to the square of the time of revolution. Thus the varying rate of motion of each planet in its orbit is not only subject to a single law, but the very different speeds of the different planets are also all subject to a law that is the same for all.