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 startling fact that spherical trigonometry came to exist in a developed state earlier than plane trigonometry.

The remaining books of the Almagest are on astronomy. Ptolemy has written other works which have little or no bearing on mathematics, except one on geometry. Extracts from this book, made by Proclus, indicate that Ptolemy did not regard the parallel axiom of Euclid as self-evident, and that Ptolemy was the first of the long line of geometers from ancient time down to our own who toiled in the vain attempt to prove it.

Two prominent mathematicians of this time were Nicomachus and Theon of Smyrna. Their favourite study was theory of numbers. The investigations in this science culminated later in the algebra of Diophantus. But no important geometer appeared after Ptolemy for 150 years. The only occupant of this long gap was Sextus Julius Africanus, who wrote an unimportant work on geometry applied to the art of war, entitled Cestes.

Pappus, probably born about 340 A.D., in Alexandria, was the last great mathematician of the Alexandrian school. His genius was inferior to that of Archimedes, Apollonius, and Euclid, who flourished over 500 years earlier. But living, as he did, at a period when interest in geometry was declining, he towered above his contemporaries "like the peak of Teneriffa above the Atlantic." He is the author of a Commentary on the Almagest, a Commentary on Euclid's Elements, a Commentary on the Analemma of Diodorus,—a writer of whom nothing is known. All these works are lost. Proclus, probably quoting from the Commentary on Euclid, says that Pappus objected to the statement that an angle equal to a right angle is always itself a right angle.

The only work of Pappus still extant is his Mathematical Collections. This was originally in eight books, but the first