152 



SCIENCE PROGRESS 



continually halving we have, taking, say, 84° and 87 as our 

 starting-points : 



II 



It is clear that this process can be extended largely by 

 continually taking one of the complementary sines as the 

 starting-point for the halving process ; and thus a large number 

 of the sines may be computed and tabulated. 



Returning now to table I, we see that the sines of 22' 30" 

 and u' 15" are, to the accuracy needed, in the same ratio as 

 their arcs, and thus sine 1' is obtained by simply dividing 

 32, 724/8 by u£, giving 2,909 as a result, and this is, of course, 

 exactly T \ of sine 45'; so, multiplying 2,909 by 1, 2, 3, etc., we 

 have sine 1', sine 2', etc., up to sine 45'. 



Theorems for the sums and differences of two sines were 

 known and these, combined with the theorems already given for 

 halving, doubling, etc., enabled the calculators to compute any 

 required sine from the knowledge of those given in the pre- 

 liminary tables and the sines of small arcs. 



Usually the sines for the first 30° and last 30 in the quadrant 

 were computed in this way. The remaining gap from 30 to 6o° 

 was filled up by using the following theorem : 



" The difference between the sines of two arcs that are 

 equally distant from 6o c is equal to the sine of half the differences 

 of these arcs." That is, in modern notation 



sin (60 + 6) - sin (60 - 6) = sin \ (60 + 8 - 60 - 6) = sin 6 



And hence 



sin (60 - 6) = sin (60 + 6) - sin 6. 



Now if 6 be less than 30 , sin (60 + 6) and sin 6 are by 

 hypothesis known ; and hence sin (60 — 0), which lies between 

 30 and 60°, is obtained by a simple subtraction. 



The canon of sines (and also cosines) being thus completed 



