NUMERICAL APERTURE. 15? 



equal to A C, and is called the cosine (written cos :) of the angle 



A. 



AC , 



. • . = cos A. 



A B 



Through X, draw X K perpendicular to O X, and meeting A K 

 in K ; then the line K X touches the circle T F X, but does not cut 

 it, and is called a tangent. Now, X K is the tangent of the angle 

 A to the radius A F, which is equal to A X. 



K X B C , . 

 •■• AX=XC -^"^^- 



The cotangent, written cot., is the tangent of the complement 

 of A. 



Therefore, if B A C is a right-angled triangle, having the right 

 angle at C, 



BC . . AC . B C ^ . 



-T— FT = sm A : = cos A ; — — = tan A. 



A B ' A B AC 



It IS evident also that tan A = -> and also that when A 



cos A 



is vtrrj' small the sine is almost equal to the arc subtending A. 



For example, the length of the sine 3'' in a circle of 10 inches 



radius is •523 inch, and tlie length of the arc is "524 inch. The 



difference therefore is of an inch. 



1000 



We can now proceed to the chief law of refraction. 



If a piece of string is made fast to a nail in the wall and held 

 loosely by the other end in the hand, and a slight up-and-down 

 motion given to the hand, the particles of the string are given 

 an up-and-down motion, and it is thrown into a series of elevations 

 and depressions, and the string apparetitly moves on, though, as a 

 matter of fact, it does not. This is called an undulatory, or wave 

 movement, from the familiar example of the waves of the sea. 

 Light is propagated in the same manner, but the movement is 

 believed to be a double one, at right angles to each other, as is 

 shown by experiments with polarised light. The propagation of 

 light takes place slower in a dense medium than in a rarer one. 



Now let ISI N (Fig, 2) be the limiting surface between two 

 media of different densities, and let A O B C be a parallel beam 

 of light striking it obliquely, and let A O meet M N in O, and 



