i 5 o SCIENCE PROGRESS 



and cosecant, when r is taken to represent the base and per- 

 pendicular respectively of the right-angled triangle. 



Such expressions are clearly rational and by giving p, in 

 succession, different values the canon we are discussing was 

 computed. 



It should be mentioned that this work, which is of great 

 rarity, was published anonymously. Both Hutton and Montucla 

 agree in ascribing it to Vieta from internal evidence and from 

 the fact that Vieta repeatedly mentions it in his other works. 



But perhaps the most massive of all such tables is that 

 computed by George Joachim Rheticus (1514-76), a pupil of 

 Copernicus and professor of mathematics at Wittenburg. The 

 work, which was published in 1596 by Valentine Otho, gives 

 tables of sines, tangents, secants, etc., computed to the radius 

 io 10 and for every 10" in the quadrant, together with their 

 differences. Theorems and explanations are given for the 

 construction of the canon to the radius io 15 and, as in Vieta's 

 work before-mentioned, the trigonometrical ratios are con- 

 sidered as being represented by the sides of right-angled 

 triangles. The computations to the radius io 15 , which were 

 made proceeding by steps of 10" and for every separate second 

 in the first and last degrees of the quadrant, were published in 

 161 3 by Pitiscus, who added to the canon a few sines calculated 

 to the radius io 22 . 



With the mention of Lansberg's tables (1591) and Pitiscus's 

 trigonometry (1599), our enumeration of the principal tables in 

 vogue at the beginning of the seventeenth century may be 

 considered to be fairly complete ; and now, remembering the 

 absence of all logarithmic aids to computation and considering the 

 large number of significant figures to which the calculations were 

 carried, one can well imagine what a slow, tedious and laborious 

 process was the construction of such tables. Rheticus, indeed, 

 in the compilation of his canon incurred an expense of thousands 

 of gulden, having a large staff of computers continuously 

 employed for a space of twelve years. 



The methods used for computation were, of course, very 

 varied : I give here a brief analysis of one process, which will 

 serve as an example. 



By the theorems of elementary geometry, the lengths of the 

 sides of a few of the regular figures inscribed in a circle of 

 given radius (io 8 , io 10 or whatever figure may be chosen) can 



