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 into 60 smaller ones. The Hindoos expressed the length of the radius by parts of the circumference, saying that of the 21,600 equal divisions of the latter, it took 3438 to measure the radius. Regiomontanus, to secure greater precision, constructed one table of sines on a radius divided into 600,000 parts, and another on a radius divided decimally into 10,000,000 divisions. He emphasised the use of the tangent in trigonometry. Following out some ideas of his master, he calculated a table of tangents. German mathematicians were not the first Europeans to use this function. In England it was known a century earlier to Bradwardine, who speaks of tangent (umbra recta) and cotangent (umbra versa), and to John Maudith. Regiomontanus was the author of an arithmetic and also of a complete treatise on trigonometry, containing solutions of both plane and spherical triangles. The form which he gave to trigonometry has been retained, in its main features, to the present day.

Regiomontanus ranks among the greatest men that Germany has ever produced. His complete mastery of astronomy and mathematics, and his enthusiasm for them, were of far-reaching influence throughout Germany. So great was his reputation, that Pope Sixtus IV, called him to Italy to improve the calendar. Regiomontanus left his beloved city of Nürnberg for Rome, where he died in the following year.

After the time of Purbach and Regiomontanus, trigonometry and especially the calculation of tables continued to occupy German scholars. More refined astronomical instruments were made, which gave observations of greater precision; but these would have been useless without trigonometrical tables of corresponding accuracy. Of the several tables calculated, that by Georg Joachim of Feldkirch in Tyrol, generally called Rhæticus, deserves special mention. He calculated a table of sines with the radius = 10,000,000,000 and from 10" to 10";