i 4 8 SCIENCE PROGRESS 



of the complementary arc) was the cosine, O T (which cuts the 

 circle) the secant; and so forth. So that the well-known theorem 



sin 2 6 + cos ' 2 = I 



would be read by a mediaeval geometer as " The square of the 

 sine added to the square of the cosine gives the square of 

 the radius." 



It is further to be noted that in the computation of tables, 

 the values of the trigonometrical functions were expressed 

 in integers ; so that when additional accuracy was required tables 

 of sines, etc., were computed on the assumption that the 

 " radius " was proportionately large. Thus, in the sixteenth 

 century, Rheticus computed tables for every ten seconds of 

 the first quadrant, taking the radius as 1,000,000,000,000,000; 

 whilst Pitiscus added to this table a few of the first sines 

 computed to the radius 10,000,000,000,000,000,000,000. This 

 manner of presentation is, of course, the mediaeval equivalent 

 of our modern phrase, " correct to so many places of decimals." 



In more ancient times trigonometrical measurements of 

 angles were based upon the computation of the chord of double 

 the arc, that is to say, in the preceding figure the chord B B' 

 was put in relation to the arc A B. Thus in the first 

 century a.d. Menelaus defines the "nadir" of an arc to be 

 the right line subtending the double of the arc ; a table of 

 such arcs and chords was constructed and exhibited by Ptolemy 

 in the second century a.d., in which the chords of various 

 arcs are calculated at intervals of half a degree. The relation 

 of the half-chord to the arc — what we should call the sine of 

 the arc — was known to the Greeks but its familiar use in 

 trigonometry is due to the Hindu mathematicians, whose 

 knowledge, through their Arab pupils, slowly filtered through 

 to the West. 



The earliest trigonometric tables of note are those of 

 Johannes Muller, commonly called Regiomontanus (1436-76), 

 who completed an earlier table of sines by Peurbach, in which 

 the radius was taken as 600,000 and the sines computed for 

 every minute of the quadrant. Afterwards, leaving the relics 

 of the sexagesimal notation implied in the above radius, he 

 made a fresh computation of the sines to every minute of the 

 quadrant, taking the radius as 1,000,000. We have also from 

 Regiomontanus a table of tangents— a table which he called 



