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 Hanseatic League commanded the trade of the North. Close commercial relations existed between Germany and Italy. Italy, too, excelled in commercial activity and enterprise. We need only mention Venice, whose glory began with the crusades, and Florence, with her bankers and her manufacturers of silk and wool. These two cities became great intellectual centres. Thus, Italy, too, produced men in art, literature, and science, who shone forth in fullest splendour. In fact, Italy was the fatherland of what is termed the Renaissance.

For the first great contributions to the mathematical sciences we must, therefore, look to Italy and Germany. In Italy brilliant accessions were made to algebra, in Germany to astronomy and trigonometry.

On the threshold of this new era we meet in Germany with the figure of John Mueller, more generally called Regiomontanus (1436-1476). Chiefly to him we owe the revival of trigonometry. He studied astronomy and trigonometry at Vienna under the celebrated George Purbach. The latter perceived that the existing Latin translations of the Almagest were full of errors, and that Arabic authors had not remained true to the Greek original. Purbach therefore began to make a translation directly from the Greek. But he did not live to finish it. His work was continued by Regiomontanus, who went beyond his master. Regiomontanus learned the Greek language from Cardinal Bessarion, whom he followed to Italy, where he remained eight years collecting manuscripts from Greeks who had fled thither from the Turks. In addition to the translation of and the commentary on the Almagest, he prepared translations of the Conics of Apollonius, of Archimedes, and of the mechanical works of Heron. Regiomontanus and Purbach adopted the Hindoo sine in place of the Greek chord of double the arc. The Greeks and afterwards the Arabs divided the radius into 60 equal parts, and each of these again