TRACT 19. TRIGONOMETRICAL TABLES, &C. 291

in 1576, in the 60th year of his age. He Conceived, andexecuted, the great design of computing the triangularcanon for every 10 seconds of the quadrant, to the radiu31000000000000000, consisting of 1, followed by 15 ciphers.The series of sines which Rheticus computed to this radius,for every 10 seconds, and for every single second in the firstand last degree of the quadrant, was published in folio at Franc-fort, 1613, by Pitiscus , who himself added a few of the firstsines computed to the radius 10000000000000000000000.

But the large work, or whole trigonometrical canon com-puted by Rheticus , was published in 1596 by Valentine Otho,mathematician to the Electoral Prince Palatine. This vastwork contains all the three series for the whole canon ofright-angled triangles (being similar to the sines, tangentsand secants, by which names I shall call them), with all thedifferences of the numbers, to the radius 10000000000.

Prefixed to these tables, are several books on their con-struction and use, in plane and spherical trigonometry, &c.Of these, the first three are by Rheticus himself; namely,book the 1st, containing the demonstrations of 9 lemmas,concerning the properties of certain lines drawn in and aboutcircles: the 2d book contains 10propositions, relating to thesines and cosines of arcs, together with those of their sumsand differences, their halves and doubles, &c. The 3d bookteaches, in 13 propositions, the construction of the canon tothe radius 1000000000000000. By some of the common pro-perties of geometry, having determined the sines of a fewprincipal arcs, as 30°, 36°, &c, in the first proposition, bycontinual bisections, he finds the sines of various other arcs,down to 45 minutes. Then, in the 2d proposition, by thetheorems for the sums and differences of arcs, he finds all thesines and cosines, up to 90 degrees, in a series of arcs differ-ing by 1° 3o'. And, in the 3d proposition, by the continualaddition of 45', he obtains all the sines and cosines in the serieswhose common difference is 45'. In the 4th proposition, be-ginning with 45', and continually bisecting, he finds the sinesand cosines of the series of half arcs, till he arrives at the arc

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