chord of double the arc. The Greeks and afterwards the Arabs divided the radius into equal parts, and each of these again
into smaller ones. The Hindoos expressed the length of the radius by parts of the circumference, saying that of the equal divisions of the latter, it took to measure the radius. Regiomontanus, to secure greater precision, constructed
one table of sines on a radius divided into parts, and another on a radius divided decimally into 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 and from to ;
and, later on, another with the radius , and proceeding from to . He began also the construction of tables of tangents and secants, to be carried to
the same degree of accuracy; but he died before finishing them. For twelve years he had had in continual employment several calculators. The work was completed by his pupil, Valentine Otho, in 1596. This was indeed a gigantic work,–-a monument
of German diligence and indefatigable perseverance. The tables were republished in 1613 by Pitiscus, who spared
no pains to free them of errors. Astronomical tables of so great a degree of accuracy had never been dreamed of by the Greeks, Hindoos, or Arabs. That Rhæticus was not a