OPTICS 



313 



A n m s, &c. Thus the ray a, after being refracted, 

 will go on in the direction g h, and the d will go on in 

 the opposite direction r s, ami so of the rest. If all 

 these rays g k, r s, &c., were traced back to the point, 

 where, if they continued the same direction, they 

 would meet, that point would be found in x, which is 

 the centre of the circle which is of the same curva- 

 ture as the lens. This point is called the virtual. 

 or imaginary focus. The imaginary focus of a con- 

 cave lens is analogous in distance to the focus of a 

 convex lens, for when the lens is double concave, 

 then the focus is in the centre of the circle ; when it 

 is plano-concave it is double that distance, or in the 

 circumference when the rays are parallel. When 

 the rays diverge before they meet the lens, then 

 this will aild to the divergence after refraction, and 

 the imaginary focus will be nearer to the lens than 

 the centre of curvature ; on the other hand, if the 

 rays before refraction be convergent, then will the 

 divergence after refraction be so much less, and the 

 focus proportionally more distant. 



It is of importance to determine the image formed 

 by a lens, the radius of curvature being known, 

 and also the distance of the object whose rays are 

 refracted. Multiply the distance of the focus (for 

 parallel rays) by the distance of the object, and di- 

 vide by the difference of these two quantities, the 

 quotient is the distance of the image from the lens ; 

 thus, if the focal distance of the lens be six inches, 

 and the distance of the object twelve, then the distance 

 of the image will be twelve inches, if the focus be the 

 same, and the distance of the object eight, then the 

 distance of the image is twenty-four inches. 



Of Reflection. When light falls upon a body, a 

 portion of it is thrown back, or reflected from its 

 surface, according to a regular law, the explanation 

 of which constitutes that branch of optics called catop- 

 trics, a word derived from two Greek words, one of 

 which signifies from, or against, and the other to see, 

 because things are seen by light reflected from bodies. 

 When a ray of light falls upon any body, it is reflect- 

 ed so that the angle of incidence is equal to the angle 

 of reflection ; and this is the fundamental fact upon 

 which all the properties of mirrors depend. Let a 

 ray of light, passing through a small hole into a dark 

 room, be reflected from a plane mirror ; at equal dis- 

 tances from the point of reflection, the incident and 

 the reflected ray will be at the same height from the 

 surface. The same is found to hold in all cases, 

 when the rays are reflected at a curved surface, 

 whether it be convex or concave. The rays which 

 proceed from any remote terrestrial object, are nearly 

 parallel at the mirror ; not strictly so, but come 

 diverging to it in several pencils, or, as it were, 

 bundles of rays, from each point of the side of the 

 object next the mirror ; therefore they will not be 

 converged to a point at the distance of half the radius 

 of the mirror's concavity from its reflecting surface, 

 but in separate points, at a greater distance from the 

 mirror. And the nearer the object is to the mirror, 

 the farther these points will be from it ; and an in- 

 verted image of the object will be formed in it, which 

 will seem to hang pendent in the air, and will be seen 

 by an eye placed beyond it (with regard to the mir- 

 ror), in all respects like the object, and as distinct as 

 the object itself. If a man place himself before a large 

 concave mirror, but farther from it than the centre of 

 its concavity, he will see an inverted image of him- 

 self in the air, between him and the mirror, of a less 

 size than himself; and if he hold out his hand towards 

 the mirror, the hand of the image will come out 

 towards his hand, and coincide with it, of an equal 

 bulk, when his hand is in the centre of concavity, 

 and he will imagine that he may shake hands with his 

 image. If lie reach his hand farther, the hand of the 



image will pass by his hand, and come between it and 

 his body ; and if he move his hand towards either side, 

 the hand of the image will move towards the other; 

 so that, whatever way the object moves, the image 

 will move the contrary way. A bystander will see 

 nothing of the image, because none of the reflected 

 rays that form it enter his eyes. The images formed 

 by convex specula are in positions similar to those of 

 their objects ; and those also formed by concave spe- 

 cula, when the object is between the surface and the 

 principal focus : in these cases the image is only 

 imaginary, as the reflected rays never come to the 

 foci, from whence they seem to radiate. In all other 

 cases of reflection from concave specula, the images 

 are in positions contrary to those of their objects ; 

 and these images are real, for the rays, after reflec- 

 tion, do come to their respective foci. 



These observations will be more clearly understood 

 by reference to diagrams. That the angle of inci- 

 dence is equal to the angle of reflection is proved by 

 the fact, that in order to show the image of the body 

 reflected from a plane mirror, it is only necessary that 

 the mirror should be half the length of the body. 

 Let a b, fig. 9, be a plane mirror, and let A i>e a 

 spectator looking at the reflection of his own person. 

 The ray A a, from the eye, will be reflected back in 

 the same direction, for in this case it strikes the mir- 

 ror at right angles to its surface ; but the ray C i, 

 from the foot, impinges upon the mirror obliquely at 

 b, and is reflected in the direction b A, the line C b, 

 the line of incidence, forming an equal angle with the 

 line b x, that the line of reflection, b A, does. The 

 spectator will see his image as if it were at E D. 

 The fact of the angle of incidence being equal to the 

 angle of reflection, affords an easy method of measur- 

 ing the heights of objects. Lay a mirror upon the 

 ground, and looking at the mirror, walk back until 

 the image of the top of the object is seen reflected 

 by the mirror, then use this proportion: As the dis- 

 tance of the foot from the mirror is to the height of 

 the eye above the ground, so is the distance of the 

 mirror from the bottom of the object to the height of 

 the object. The law of reflection, we have been con- 

 sidering, holds equally true if the mirror be curved, 

 i. e., the angle of incidence is always equal to the 

 angle of reflection. Thus, in a concave mirror, A B, 

 fig. 10, let a b, C d e, f, be parallel rays from some 

 distant object, these rays will be reflected so as to 

 meet in the point C, which is the centre of curvature 

 of the mirror ; and this must of necessity be their 

 course, for if the mirror be supposed to be made up 

 of a great many little planes, it would be found, by 

 making the angles of incidence and reflection equal, 

 that all the rays would meet in the central point C, 

 which is called the focus of the mirror. An image 

 of the object will be formed at the point C, in the 

 same manner as would be done in the focus of a con- 

 vex lens. When the incident rays are not parallel, 

 the point of convergence will not be in the point C, 

 the centre of curvature ; for let rays from the arrow, 

 M E, fig. 11, fall upon the mirror A B, which is 

 concave like the former, the rays will flow from 

 every point of the arrow, as M, to every -point in the 

 surface of the mirror. Take the rays that flow from 

 M to the points A, C, and B. These rays diverge 

 before they come to the mirror, which divergence 

 will partly destroy the convergence after reflection, 

 and consequently the rays will not meet in C, the 

 centre of curvature, but at a point more remote ; 

 they will meet in points, and form an inverted 

 image of the arrow at e m. To find the distance of 

 the image, the distance of the object and radius of 

 curvature being given, multiply the object's distance 

 by the radius, and divide the product by thrice the 

 distance, minus the radius, the quotient is the distance 



