His second law (a far more difficult one to prove) states that a line drawn from a planet to the sun sweeps over equal areas in equal times. These two laws were published in his great work, _Astronomia Nova, sen. Physica Coelestis tradita commentariis de Motibus Stelloe; Martis_, Prague, 1609. It took him nine years more[3] to discover his third law, that the squares of the periodic times are proportional to the cubes of the mean distances from the sun. These three laws contain implicitly the law of universal gravitation. They are simply an alternative way of expressing that law in dealing with planets, not particles. Only, the power of the greatest human intellect is so utterly feeble that the meaning of the words in Kepler’s three laws could not be understood until expounded by the logic of Newton’s dynamics. The joy with which Kepler contemplated the final demonstration of these laws, the evolution of which had occupied twenty years, can hardly be imagined by us. He has given some idea of it in a passage in his work on _Harmonics_, which is not now quoted, only lest someone might say it was egotistical—a term which is simply grotesque when applied to such a man with such a life’s work accomplished. The whole book, _Astronomia Nova_, is a pleasure to read; the mass of observations that are used, and the ingenuity of the propositions, contrast strongly with the loose and imperfectly supported explanations of all his predecessors; and the indulgent reader will excuse the devotion of a few lines to an example of the ingenuity and beauty of his methods. It may seem a hopeless task to find out the true paths of Mars and the earth (at that time when their shape even was not known) from the observations giving only the relative direction from night to night. Now, Kepler had twenty years of observations of Mars to deal with. This enabled him to use a new method, to find the earth’s orbit. Observe the date at any time when Mars is in opposition. The earth’s position E at that date gives the longitude of Mars M. His period is 687 days. Now choose dates before and after the principal date at intervals of 687 days and its multiples. Mars is in each case in the same position. Now for any date when Mars is at M and the earth at E3 the date of the year gives the angle E3SM. And the observation of Tycho gives the direction of Mars compared with the sun, SE3M. So all the angles of the triangle SEM in any of these positions of E are known, and also the ratios of SE1, SE2, SE3, SE4 to SM and to each other. For the orbit of Mars observations were chosen at intervals of a year, when the earth was always in the same place.