the pupil of the eye; and in order to produce distinct vision the rays of each such pencil must either be parallel or slightly divergent. Thus the sun, moon, and planets are seen distinctly, although so distant, by parallel rays; and the least distance from the eye at which an object can be seen distinctly varies in different persons, according to the power of the natural lenses which, in fact, form the eye. 1. When any object is brought nearer to the eye than this without the intervention of a lens, the vision becomes confused. If, however, the rays of light proceeding from the object were, by the interposition of a convex lens, rendered less divergent or parallel, the vision would be again distinct. 2. It is further necessary for distinct vision that the intensity of the light be not less than a certain intensity, as may easily be exemplified by gradually closing the shutters of a room, and thus diminishing the intensity of the light proceeding from the objects in the room, when they will grow more and more indistinct. 3. Lastly, it is also necessary, for distinct vision of any object, that the axes of the pencils proceeding from its extreme parts enter the eye under an angle not less than a certain angle, so that, however strong a light be thrown upon an object, if this object be very minute, or removed to a distance very great with respect to its magnitude, it will not be seen by the naked eye. If, however, by the assistance of a lens, or combination of lenses, a sufficient number of rays to produce the required intensity of light can be collected from each point of an object, and passed through the pupil of the eye, and if at the same time the axes of the extreme pencils are bent by this lens, or combination of lenses, so as to enter the eye under a sufficiently large angle, while the rays of each pencil are made parallel, or but slightly divergent, then vision will ensue, no matter how minute, how distant, or how imperfectly illuminated the object may be.

When a pencil of rays proceeding from a point of an object passes through a lens, the rays which pass through at different distances from its center will diverge from or converge to different points, so that the whole pencil will not any longer diverge from or converge to a single point; and from this cause the image of one point will overlap the image of another, and an indistinctness of the object will arise. This source of indistinctness is called the aberration. A combination of lenses may, however, be formed, so that the aberration of one shall be corrected by the aberration of the others Such a combination is said to be aplanatic

Since rays of light proceed in every direction from the points of visible objects, the pencils of light intercepted by a lens for the first time are all centrical, that is, their axes all pass through the center of the lens; but the pencils, upon emerging from this first lens are already determinate both in extent and direction, and consequently will fall, some of them at least, eccentrically upon a second lens, that is, their axes will meet this lens at different distances from its center. The axes of the most eccentrical pencils will then, after emergence, cross the axis of vision nearer to the lens than will those of the more nearly central pencils; and thus, while the center of the object is seen distinctly, the parts at a distance from the center will be distorted, or vice versa. This source of error is called the spherical confusion. The spherical confusion is diminished by dividing the desired deviations, or bendings of the axes, between two or more lenses; and it is found by opticians to be a good rule to divide the deviations equally amongst the lenses employed.

The most important source of indistinctness, however, is the dispersion of each ray into rays of different colours refracted at different angles (p. 68) which is called the chromatic dispersion. The effect of this dispersion upon a centrical pencil is partly analogous to the spherical aberration, causing the images of neighbouring points to be of finite extent and overlap one another; and it, moreover, fringes the image with colour. An eccentrical pencil is separated by this dispersion into pencils of rays of different colours, the axes of which are bent at different angles; and the imperfection arising from this cause is far greater than that from both the spherical aberration and spherical confusion.

Before stating the manner in which the imperfections arising from the chromatic dispersion are remedied, it will be expedient to explain what is understood by the focal length of a lens. Now a pencil of parallel rays, after passing through a lens, becomes either convergent or divergent, as the lens is convex or concave, and the distance from the point to which the pencil converges, or from which it diverges, to the surface of the lens, is called the focal length of the lens.

To find practically the Focal Length of a Convex Lens.— Place a lighted candle at one extremity of a scale of inches and parts, with which the lens has been connected in such a manner as to slide along, and always have its axis parallel to the scale. A flat piece of card is also to be made to slide along, so as to be always in a line with the light and the lens,

the lens being between the light and the card. The lens and card are then to be moved along, backwards and forwards, till the least distance between the card and light is discovered at which a clear image of the light is formed upon the card and this distance is four times the focal length.

The imperfection arising from the chromatic dispersion is remedied, for the centrical pencil, by making a compound lens of two or more lenses of different substances, as flint glass and crown glass, which are fitted close together, and are of such focal lengths that the chromatic dispersion of one is counteracted by the chromatic dispersion of the other. The effect of the chromatic dispersion upon an eccentrical pencil is remedied by setting two or more lenses at proper distances depending upon their focal lengths. Such a combination of lenses is called an achromatic eye-piece.

When an object is placed at a distance from a convex lens greater than its focal length, the divergent pencils of rays, proceeding from every point of the object, become, after passing through the lens, convergent, and at a certain distance from the lens, having converged nearly to points again, form there an inverted image of the object. The essential difference between any point in this image, and the corresponding point in the object itself, is, that the latter emits light in all directions, while the light from the former is limited to the pencil which has been transmitted through the lens, and is consequently determinate both in magnitude and direction. If, however, a screen be placed at the required * distance from the lens, a picture of the object in an inverted position will be formed upon this screen, and from each point of this picture light will be emitted in all directions in the same manner as from the points of the object itself. The single pencil of light from any point of the object, transmitted through the lens, supplies, however, in this case, the light for all the pencils emitted from the corresponding point of the image; and a very strong light must therefore be thrown upon the object to give a moderate brightness to the picture; more especially if the picture be of larger dimensions than the

1 1 1

*If u be the distance from the lens at which the object is placed, ƒ the focal length of the lens, then v, the distance from the lens at which the image is formed, is determined from the equation magnitude of the image is to that of the object as r to u.

v

The linear

น

E

object A portion of the light is also absorbed by the leus itself

REFLECTORS.

When a ray of light is reflected at a plane surface, the reflection takes place in a plane perpendicular to the reflecting surface, and the incident and reflected rays make equal angles with this surface. Thus, if QA represent a ray of light incident upon a plane reflector at the point A, and the plane of the paper represent the plane which contains QA, and is perpendicular to the re

A

R

flecting surface, intersecting it in the line RA R', then making the angle q'AR' in the plane QAR' equal to the angle Q AR, AQ′ will represent the course of the reflected ray.

Ε

R

The effect of a plane reflector upon the pencils of light which fall upon it is to change the direction of all the rays forming each pencil without altering the angles at which the several rays of the pencil are inclined to one another, so that the divergency or convergency of the pencils remains the same after reflection as before, and the objects from which they proceed appear to be at the same distances behind the mirror as they really are in front of it. Thus, a pencil of light diverging from a point of an object at P, after reflection at the point R of a plane mirror, appears to proceed from the point p' on the line PMP, perpendicular to the mirror, at the distance M P behind the mirror, equal to the distance MP. The point P', from which, after reflection, the pencil appears to have diverged, is called a virtual focus; and the apparent image of the object behind the mirror is called a virtual image.

The uses of a single plane reflector in mathematical instruments are nearly the same as the uses of a prism: viz., either to alter the apparent position of an object, so as to make its visual image coincide with the real image of some other object, as in the prismatic compass (described hereafter), or merely to change the direction of the pencils for the purposes of more convenient observation, as in the Newtonian telescope, (see page 84), the diagonal eye-piece (see page 83), &c.

When a ray of light, proceeding in a plane at right angles to each of two plane mirrors, which are inclined to each other at any angle whatever, is successively reflected at the plane sur

S

faces of each of the mirrors, the total deviation of the ray double the angle of inclination of the mirrors. For let 1i and Hh represent sections of the two mirrors made by the plane of incidence at right angles to each of them, and let si represent the course of the incident ray: then the ray sI is reflected at I into the direction IH, making with 1i the angle HIA, equal to the angle sii, and is again reflected at H into the direction H E, making, with Hh, the angle E HA equal to the angle IH h. Now the angle AHV, being equal to the exterior angle IH h, is equal to the two angles HIA and HAI; and because the vertical angles

h

H

is

A V H and IV E are equal, and that the three angles of every triangle are equal to two right angles, therefore the two angles VIE and SEH are, together, equal to the two angles A H V and HAI, and therefore to the angle HIA and twice the angle HAI (since AHV has been proved equal to HIA and HAI); but VIE, being equal to the vertical angle s Ii, is equal to the angle HIA: therefore, taking away these equals, the remainder, the angle SEH, is equal to the remainder, twice the angle HAI. Q.E.D.

This property of two plane reflectors enables us by their aid to measure the angle subtended at the eye by any two objects whatever, and is the foundation of the construction of Hadley's Quadrant, and the improvements upon it: viz., Hadley's Sextant, and Troughton's Reflecting Circle, hereafter to be described.

Note.-Plane reflectors are usually made of glass silvered at the back; and, as reflection takes place at each surface of the glass, there is formed a secondary image, which must not be mistaken for the primary and distinct image intended to be observed.

ON CURVILINEAR REFLECTORS.

Spherical reflectors, or specula, as they are called, produce upon pencils of rays results precisely similar, with one exception, to those produced by lenses. Thus, a concave reflector makes the rays of the pencils incident upon it more con vergent, and corresponds in its uses with a convex lens; while a convex reflector makes the rays of the incident pencils more divergent after reflection, and corresponds in its uses with a