EXERCISE 17.--Drive two pins into a smooth board an inch apart and fasten to them the ends of a string a foot long. Take up the slack of the string with the point of a lead pencil and, keeping the string drawn taut, move the pencil point over the board into every possible position. The curve thus traced will be an ellipse having the pins at the two points which are called its foci. In the case of the planetary orbits one focus of the ellipse is vacant, and, in accordance with the first law, the center of the sun is at the other focus. In Fig. 17 the dot, inside the orbit of Mercury, which is marked _a_, shows the position of the vacant focus of the orbit of Mars, and the dot _b_ is the vacant focus of Mercury's orbit. The orbits of Venus and the earth are so nearly circular that their vacant foci lie very close to the sun and are not marked in the figure. The line drawn from the sun to any point of the orbit (the string from pin to pencil point) is a _radius vector_. The point midway between the pins is the _center_ of the ellipse, and the distance of either pin from the center measures the _eccentricity_ of the ellipse. Draw several ellipses with the same length of string, but with the pins at different distances apart, and note that the greater the eccentricity the flatter is the ellipse, but that all of them have the same length. If both pins were driven into the same hole, what kind of an ellipse would you get? The Second Law was worked out by Kepler as his answer to a problem suggested by the first law. In Fig. 17 it is apparent from a mere inspection of the orbit of Mercury that this planet travels much faster on one side of its orbit than on the other, the distance covered in ten days between the numbers 10 and 20 being more than fifty per cent greater than that between 50 and 60. The same difference is found, though usually in less degree, for every other planet, and Kepler's problem was to discover a means by which to mark upon the orbit the figures showing the positions of the planet at the end of equal intervals of time. His solution of this problem, contained in the second law, asserts that if we draw radii vectors from the sun to each of the marked points taken at equal time intervals around the orbit, then the area of the sector formed by two adjacent radii vectores and the arc included between them is equal to the area of each and every other such sector, the short radii vectores being spread apart so as to include a long arc between them while the long radii vectores have a short arc. In Kepler's form of stating the law the radius vector is supposed to travel with the planet and in each day to sweep over the same fractional part of the total area of the orbit. The spacing of the numbers in Fig. 17 was done by means of this law.