Again, if from a centre, C, with the ra dius, C A, the circle, A M l', be describ ed, the arc, A 0, will be found equal to the arc, 0 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 df a, C m b, e l c, (fig. 9) IA 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 13, and let the parallel rays, d f a, C m b, and e lc, fall upon it at the points, a, b and c. Draw the lines, C i a, C m b, 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 da C, and draw the line, a m h, which will be the direction of the ray, df a, 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 h ; 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, C c e, 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, Cm 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.
The rays which proceed from any ce lestial object, may be esteemed parallel at the earth ; and, therefore, the images of that object will be formed at m, when the reflecting surface of the concave mirror is turned directly towards the ob ject. Hence the focus of the parallel rays is not in the centre of the mirror's concavity, but half way between the mir ror, and that centre. The rays which pro ceed from any remote terrestrial object are nearly parallel at the mirror ; not strictly so, but come diverging to it in separate 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 mir ror's concavity from its reflecting surface, but in separate points at a little greater distance from the mirror. And the near
er the object is to the mirror, the further these points will be from it ; and an in verted image of the object will be form ed in them, which will seem to hang pendent in the air ; and will be seen by an eye placed beyond it (with regard to the mirror), in all respects like the object, and as distinct as the object it self.
Let A c B (fig. 10), be the reflecting surface of a mirror, whose centre of con cavity is at C ; and let the upright ob. ject, D E, be placed beyond the centre, C, and send out a conical pencil of di verging rays from its upper extremity, D, to every point of the concave sur face of the mirror, A c B. But, to avoid confusion, we only draw three rays of that pencil ; as D A, D c, D B. From the centre of concavity, C, draw the three right lines, C A, C c, C B, touching the mirror in the same points where the aforesaid touch it, and all these lines will be perpendicular to the surface of the mirror. Make the angle, C A d equal to the angle, I) A C, and draw the right line A d, for the course of the reflected ray, D A : make the angle, C c d, equal to the angle, D c C, and draw the right line, c d, for the course of the reflected ray, D c ; make also the angle, C B equal to the angle, D B C, and draw the right light line, B d, for the course of the reflected ray, D B. All these re flected rays will meet in point d, where they will form the extremity, d, of the in verted image, e d, similar to the extremity, If, of the upright object, I) E. If the pencil of rays, Ef, Eg, E h, be also con tinued to the mirror, and their angles of reflection from it he made equal to their angles of incidence upon it, as in the for [Der pencil from I), they will meet at the point, e, by reflection, and form the extremity, e, of the image, e d, similar to the extremity, E, of the object, D E. As each intermediate point of the object be tween D and E, sends out a pencil of rays in like manner to every part of the mir ror, the rays of each pencil will be re flected back from it, and meet in all the intermediate points between the ex tremities, e and d, of the image ; and so. the whole image will be formed, not at i, half the distance of the mirror from its centre of concavity, C ; but at a greater distance between i and the ob ject, D E ; and the image will be invert ed with respect to the object. This be ing well understood, the reader will easily see how the image is formed by the large concave mirror of the reflecting telescope, when he comes to the de scription of that instrument. See TELE SCOPE.