added towards the end of the Middle Ages to account for the supposed changes in the obliquity of the ecliptic. A few writers invented a larger number. Outside these spheres mediaeval thought usually placed the Empyrean or Heaven. The accompanying diagram illustrates the whole arrangement. [Illustration: FIG. 36.—The celestial spheres. From Apian’s _Cosmographia_.] These spheres, which were almost entirely fanciful and in no serious way even professed to account for the details of the celestial motions, are of course quite different from the circles known as deferents and epicycles, which Hipparchus and Ptolemy used. These were mere geometrical abstractions, which enabled the planetary motions to be represented with tolerable accuracy. Each planet moved freely in space, its motion being represented or described (not _controlled_) by a particular geometrical arrangement of circles. Purbach suggested a compromise by hollowing out Aristotle’s crystal spheres till there was room for Ptolemy’s epicycles inside! From the new Nürnberg press were issued also a succession of almanacks which, like those of to-day, gave the public useful information about moveable feasts, the phases of the moon, eclipses, etc.; and, in addition, a volume of less popular _Ephemerides_, with astronomical information of a fuller and more exact character for a period of about 30 years. This contained, among other things, astronomical data for finding latitude and longitude at sea, for which Regiomontanus had invented a new method.[43] The superiority of these tables over any others available was such that they were used on several of the great voyages of discovery of this period, probably by Columbus himself on his first voyage to America. In 1475 Regiomontanus was invited to Rome by the Pope to assist in a reform of the calendar, but died there the next year at the early age of forty. Walther carried on his friend’s work and took a number of good observations; he was the first to make any successful attempt to allow for the atmospheric refraction of which Ptolemy had probably had some knowledge (chapter II., § 46); to him is due also the practice of obtaining the position of the sun by comparison with Venus instead of with the moon (chapter II., § 39), the much slower motion of the planet rendering greater accuracy possible. After Walther’s death other observers of less merit carried on the work, and a Nürnberg astronomical school of some kind lasted into the 17th century. 69. A few minor discoveries in astronomy belong to this or to a slightly later period and may conveniently be dealt with here.