The motion of the fourth satellite is less tractable; it does not so readily form an easy system with the others. After these great successes the two astronomers naturally proceeded to study the mutual perturbations of all other bodies in the solar system. And one very remarkable discovery they made concerning the earth and moon, an account of which will be interesting, though the details and processes of calculation are quite beyond us in a course like this. Astronomical theory had become so nearly perfect by this time, and observations so accurate, that it was possible to calculate many astronomical events forwards or backwards, over even a thousand years or more, with admirable precision. Now, Halley had studied some records of ancient eclipses, and had calculated back by means of the lunar theory to see whether the calculation of the time they ought to occur would agree with the record of the time they did occur. To his surprise he found a discrepancy, not a large one, but still one quite noticeable. To state it as we know it now:--An eclipse a century ago happened twelve seconds later than it ought to have happened by theory; two centuries back the error amounted to forty-eight seconds, in three centuries it would be 108 seconds, and so on; the lag depending on the square of the time. By research, and help from scholars, he succeeded in obtaining the records of some very ancient eclipses indeed. One in Egypt towards the end of the tenth century A.D.; another in 201 A.D.; another a little before Christ; and one, the oldest of all of which any authentic record has been preserved, observed by the Chaldæan astronomers in Babylon in the reign of Hezekiah. Calculating back to this splendid old record of a solar eclipse, over the intervening 2,400 years, the calculated and the observed times were found to disagree by nearly two hours. Pondering over an explanation of the discrepancy, Halley guessed that it must be because the moon's motion was not uniform, it must be going quicker and quicker, gaining twelve seconds each century on its previous gain--a discovery announced by him as "the acceleration of the moon's mean motion." The month was constantly getting shorter. What was the physical cause of this acceleration according to the theory of gravitation? Many attacked the question, but all failed. This was the problem Laplace set himself to work out. A singular and beautiful result rewarded his efforts. You know that the earth describes an elliptic orbit round the sun: and that an ellipse is a circle with a certain amount of flattening or "excentricity."[26] Well, Laplace found that the excentricity of the earth's orbit must be changing, getting slightly less; and that this change of excentricity would have an effect upon the length of the month. It would make the moon go quicker.