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 His notation was totally different from ours. Thomas Bradwardine, archbishop of Canterbury, studied star-polygons,—a subject which has recently received renewed attention. The first appearance of such polygons was with Pythagoras and his school. We next meet with such polygons in the geometry of Boethius and also in the translation of Euclid from the Arabic by Athelard of Bath. Bradwardine's philosophic writings contain discussions on the infinite and the infinitesimal—subjects never since lost sight of. To England falls the honour of having produced the earliest European writers on trigonometry. The writings of Bradwardine, of Richard of Wallingford, and John Maudith, both professors at Oxford, and of Simon Bredon of Winchecombe, contain trigonometry drawn from Arabic sources.

The works of the Greek monk Maximus Planudes, who lived in the first half of the fourteenth century, are of interest only as showing that the Hindoo numerals were then known in Greece. A writer belonging, like Planudes, to the Byzantine school, was Moschopulus, who lived in Constantinople in the early part of the fifteenth century. To him appears to be due the introduction into Europe of magic squares. He wrote a treatise on this subject. Magic squares were known to the Arabs, and perhaps to the Hindoos. Mediaeval astrologers and physicians believed them to possess mystical properties and to be a charm against plague, when engraved on silver plate.

In 1494 was printed the Summa de Arithmetica, Geometria, Proportione et Proportionalita, written by the Tuscan monk Lucas Pacioli, who, as we remarked, first introduced symbols in algebra. This contains all the knowledge of his day on arithmetic, algebra, and trigonometry, and is the first comprehensive work which appeared after the Liber Abaci of Fibonacci. It contains little of importance which cannot be