OPTICS. 



rays would have diverged, after passing 

 through it, as if they had come from a ra- 

 diant point at double the distance of C 

 from the glass ; that is, as if the radiant 

 had been at the distance of a whole dia- 

 meter of the glass's convexity. 



If rays come more converging to such a 

 glass, than parallel rays diverge after 

 passing through it, they will continue to 

 converge after passing through it ; but 

 will not meet so soon as if no glass had 

 been in the way; and will incline towards 

 the same side to which they would have 

 diverged, if they had come parallel to the 

 glass. 



Of Refection. When a ray of light 

 falls upon any body, it is reflected, so 

 that the angle of incidence is equal to the 

 angle of reflection ; and this is the fun- 

 damental fact upon which all the pro- 

 perties of mirrors depend. This has 

 been attempted to be proved upon the 

 principle of the composition and resolu- 

 tion of forces or motion : let the motion 

 of the incident ray be expressed by A C 

 (fig. 2); then A D will express the parallel 

 motion, and A B the perpendicular mo- 

 tion. The perpendicular motion after re- 

 flection will be equal to that before re- 

 flection, and therefore may be express- 

 ed by D C = A D. The parallel motion, 

 not being affected by reflection, con- 

 tinues uniform, and will be expressed by 

 D M = A D ; therefore the course of the 

 ray will be C M, and by a well-known pro- 

 position in Euclid A C D = D C M. The 

 fact may, however, be proved by expe- 

 riment in various ways ; the following 

 method will be readily understood. 



Having described a semicircle on a 

 smooth board, and from the circumference 

 let fall a perpendicular bisecting the dia- 

 meter, on each side of the perpendicular 

 cut off equal parts of the circumference ; 

 draw lines from the points in which those 

 equal parts are cut off to the centre ; 

 place three pins perpendicular to the 

 board, one at each point of section in the 

 circumference, and one at the centre ; and 

 place the board perpendicular to a plane 

 mirror. Then look along one of the pins 

 in the circumference to that in the cen- 

 tre, and the other pin in the circumfer- 

 ence will appear in the same line produc- 

 ed with the first, which shews that the 

 ray which comes from the second pin, is 

 reflected from the mirror at the centre of 

 the eye, in the same angle in which it fell 

 on the mirror. 2. Let a ray of light, 

 passing through a small hole into a dark 

 room, be reflected from a plane mirror, 

 at equal distances from the point of re- 



flection, the incident, and the reflected 

 ray, will be at the same height from the 

 surface. 



Again, if from a centre, C, with the ra- 

 dius, C A, the circle, A M P, be describ- 

 ed, the arc, A O, will be found equal to 

 the arc, O M, therefore the angle of inci- 

 dence is equal to the angle of reflection. 

 The same is found to hold in all cases 

 when the rays are reflected at a curv- 

 ed surface, whether it be convex or con- 

 cave. 



With regard to plane specula, it is 

 found that the image and the object form- 

 ed by it are equally distant from the spe- 

 culum, at opposite sides : they are also 

 equal, and similarly situated. 



When parallel rays, as dfa, C m b, e Ic, 

 (fig. 9) fall upon a concave mirror, A B, 

 they will be reflected back from that mir- 

 ror, and meet in a point, m, at half the 

 distance of the surface of the mirror from 

 C, the centre of its concavity ; for they 

 will be reflected at as great an angle 

 from the perpendicular to the surface of 

 the mirror, as they fell upon it, with re- 

 gard to that perpendicular, but on the 

 other side thereof. Thus, let C be the 

 centre of the concavity of the mirror, A 

 b B, and let the parallel rays, dfa, C m b t 

 and e Ic, fall upon it at the points, a, b 

 and c. Draw the lines, C i a, C m If, and 

 C h c, from the centre, C, to these points ; 

 and all these lines will be perpendicular 

 to the surface of the mirror, because they 

 proceed thereto like so many radii from 

 its centre. Make the angle, C a h, equal 

 to the angle daC, and draw the line, am 

 h, which will be the direction of the ray, 

 dfa, after it is reflected from the point of 

 the mirror : so that the angle of inci- 

 dence, da C, is equal to the angle of re- 

 flection, C a // ; the rays making equal 

 angles with the perpendicular, C i a, on 

 its opposite sides. Draw also the per- 

 pendicular, C h c, to the point, c, where 

 the ray, e I c, touches the mirror ; and 

 having made the angle, C ci, equal to the 

 angle, Gee, draw the line, c m i, which 

 will be the course of the ray, el c, after it 

 is reflected from the mirror. The ray, 

 C m b, passes through the centre of con- 

 cavity of the mirror, and falls upon it at 

 b, perpendicular to it ; and is therefore 

 reflected back from it in the same line, 

 b m C. All these reflected rays meet in 

 the point, m ; and in that point the image 

 of the body which emits the parallel rays, 

 d a, C b, and e c, will be formed ; which 

 point is distant from the mirror equal 

 to half the radius, b m C, of its con- 

 cavity. 



