These are two of the three celebrated theorems called Kepler's laws: the first is, that the planets move in ellipses round the sun, placed in the focus; the second, that the time of describing any arc is proportional in the same orbit to the area included between the arc and the two bounding distances from the sun. The third will be mentioned on another occasion, as it was not discovered till twelve years later. On the establishment of these two theorems, it became important to discover a method of measuring such elliptic areas, but this is a problem which cannot be accurately solved. Kepler, in offering it to the attention of geometricians, stated his belief that its solution was unattainable by direct processes, on account of the incommensurability of the arc and sine, on which the measurement of the two parts AQ_m_, SQ_m_ depends. "This," says he in conclusion, "this is my belief, and whoever shall shew my mistake, and point out the true solution, _Is erit mihi magnus Apollonius._" FOOTNOTES: [189] It is not very easy to carry the understanding aright among these Aristotelian ideas. Many at the present day might think they understood better what is meant, if for "form" had been written "nature." [190] De mundo nostro sublunari, Philosóphia Nova. Amstelodami, 1651. [191] Theoricæ novæ planetarum. G. Purbachii, Parisiis, 1553. CHAPTER VI. _Kepler appointed Professor at Linz—His second marriage—Publishes his new Method of Gauging—Refuses a Professorship at Bologna._