convex surface that is polished, it will form a convex spherical mirror; if it be the concave surface, it will be a concave mirror.

We must here first observe, that when a ray of light falls on any curved surface whatever, it will be reflected in the same manner as from a plane touching the point of that surface where it falls. Thus, if a tangent be drawn at the point of reflection to the surface of a spherical mirror, in the plane of the incident ray and of the centre, the ray will be reflected, making with that tangent an angle of reflection equal to the angle of incidence.

PROBLEM XXXIV.

The place of an object, and that of the eye being given ; to determine in a spherical mirror, the point of reflection, and the place of the image.

THE solution of these two problems is not so easy in regard to spherical as to plane mirrors; for when the eye and the object are at unequal distances from the mirror, the determination of the point of reflection necessarily depends on principles which require the assistance of the higher geometry; and this point cannot be assigned in the circumference of the circle without employing one of the conic sections. For this reason, we shall omit the construction, and only observe that there is one extremely simple, in which two hyperbolas between their asymptotes are employed: one of these determines the point of reflection on the convex surface, and the second the point of reflection on the concave surface.

It will be sufficient for us here to take, notice of

one property belonging to this point. Let B be the object (fig. 30 pl. 9), A the place of the eye, E the point of reflection from the convex surface of the spherical mirror DE L, the centre of which is C; also let FG be a tangent to the point E, in the plane of the lines BC and A C, which it meets in I and i; and let the reflected ray A E, when produced, intersect the line BC in H: the points H and I will be so situated, that we shall have the following proportion as BC is to CH, so is BI to IH.

In like manner, if BE be produced till it meet AC in h, we shall have, AC: Ch:: Ai: ih; proportions which will be equally true in the case of reflection from a concave surface.

In regard to the place of the image, opticians have long admitted it as a principle that it is in the point H, where the reflected ray meets the perpendicular drawn from the object to the mirror. But this supposition, though it serves pretty well to shew how the images of objects are less in convex, and larger in concave, than they are in plane mirrors, has no foundation in physics, and at present is considered as absolutely false.

A more philosophical principle advanced by Dr. Barrow is, that the eye perceives the image of the object in that point where the rays forming the small divergent bundle, which enters the pupil of the eye, meet together. It is indeed natural to think that this divergency, as it is greater when the object is near and less when it is distant, ought to enable the eye to judge of distance.

By this principle also we are enabled to assign a pretty plausible reason for the diminution of objects in convex, and their enlargement in concave mirrors for the convexity of the former renders the

rays, which compose each bundle that enters the eye, more divergent than if they fell on a plane mirror; consequently the point where they meet in the central ray produced, is much nearer. It may. even be demonstrated, that in convex mirrors it is much nearer, and in concave much farther distant than the point H, considered by the ancients, and the greater part of the moderns, as the place of the image. In a word, it is concluded that in convex mirrors this image will be still more contracted, and in concave ones more extended, than the ancients supposed; which will account for the apparent enlargement of objects in the latter, and their diminu tion in the former.

We must however allow that even this principle is attended with difficulties, which Dr. Barrow, the author of it, does not conceal, and to which he confesses he never saw a satisfactory answer. This induced Dr. Smith, in his Treatise on Optics, to propose another; but we shall not here enter into a discussion on this subject, as it would be too dry and abstruse for the generality of readers.

PROBLEM XXXV.

The principal properties of spherical mirrors, both

convex and concave.

Ist. THE first and principal property of convex mirrors is, that they represent objects less than they would be if seen in a plane mirror at the same distance. This may be demonstrated independently of the place of the image: for it can be shewn that the extreme rays of an object, however placed, which enter the eye after being reflected by a convex mir.

ror, form a less angle, and consequently paint a less image on the retina, than if they had been reflected by a plane mirror, which never changes that angle. But, the judgment which the eye in general forms respecting the magnitude of objects, depends on the magnitude of that angle, and that image, unless modified by some particular cause.

On the other hand, in concave mirrors it may be easily demonstrated, that the extreme rays of an object, in whatever manner situated, make a greater angle on arriving at the eye than they would do if reflected from a plane mirror; consequently the appearance of the object, for the above reason, must be much greater.

2. In a convex mirror, however great be the distance of the object, its image is never farther from the surface than half the radius; so that a straight line perpendicular to the mirror, were it even infinite, would not appear to extend farther within the mirror, than the fourth part of the diameter of the circle of which it is a segment.

But in a concave mirror, the image of a line perpendicular to the mirror is always longer than the line itself; and if this line be equal to half the radius, its image will appear to be infinitely produced.

3. In convex mirrors, the appearance of a curved line, concentric to the mirror, is a circular line also concentric to the mirror; but the appearance of a straight line, or plane surface, presented to the mirror, is always convex on the outside, or towards the eye.

In a concave mirror, the contrary is the case: the image of a rectilineal or plane object appears concave towards the eye.

4th. A convex mirror disperses the rays; that is

to say, if they fall on its surface parallel, it reflects them divergent; if they fall divergent, it reflects them still more divergent, according to circum

stances.

On this property, of concave spherical mirrors, is founded the use made of them for collecting the sun's rays into a small space, where their heat, multiplied in the ratio of their condensation, produces astonishing effects. But this subject deserves to be treated of separately.

PROBLEM XXXVI.

Of Burning Mirrors.

THE properties of burning mirrors may be deduced from the following proposition :

If a ray of light fall very near the axis of a concave spherical surface, and parallel to that axis, it will be reflected in such a manner, as to meet it at a distance from the mirror nearly equal to half the radius.

FOR let ABC (fig. 31 pl. 9) be the concave surface of a well polished spherical mirror, of which D is the centre, and DB the semi-diameter in the direction of the axis; if E F be a ray of light parallel to BD, it will be reflected in the direction of FG, which will intersect the diameter B D in a certain point G. But the point G will always be nearer to the surface of the mirror than to the centre. For if the radius D F be drawn, we shall have the angles DFE and DFG equal, consequently the angles DFE and GDF will also be equal, since the latter, on account of the parallel