150 



NUMERICAL APERTUEE. 



Let FAX (Fig. i) be any angle, and let us call it A. From 



Fig. I. 



the point A, with any radius, describe a circle B H O, cutting A F 

 in B. From B let fall a per^jendicular B C, on the other leg of the 

 angle A, and produce B C to meet the circle in H. Then B C is 

 called the sine of the angle A to the radius A B, and is (Euclid 

 III. 3) half the length of the chord B H. With the centre A, and 

 another radius A F, describe any other circle ; through F draw F G 

 parallel to B H, and meeting A X in G ; then F G is also the sine 

 A to radius A F. But by Euclid VI., 4 :— 



B C : F G : : A B : A F, 

 B C 



therefore 



A B 



is a constant quantity for the same angle. Now 



this fraction 



B C 

 A B 



turned into decimals is what is written down 



in mathematical tables as the natural sine of the angle A, which 

 is itself given in degrees. 



The angle A in the figure is less than a right angle, but it 

 might be greater. By a similar construction it will be seen that 

 sin A = sin (iSo"^ — A.) This is, however, unimportant for us, as in 

 objectives the semi-angle does not exceed 90''. 



If through A, A Y is drawn perpendicular to A X, the angle 

 Y A F = 90*^ A, and is called the complement of the angle A 

 and is equal to the angle ABC (Euclid I., 23). Its sine, B D, is 



