1. There is a great resemblance between the properties of _convex_ lenses and _concave_ mirrors. They both form an inverted focal image of any remote object, by the convergence of the pencil of rays. In those instruments whose performances are the effects of reflection, as reflecting telescopes, the concave mirror is substituted in the place of the convex lens. The whole effect of these instruments, in bringing to view remote objects in heaven and on earth, entirely depends on the property of a concave mirror in forming _images_ of objects in its focus. 2. The image of an object placed beyond the centre, is less than the object; if the object be placed between the principal focus and the centre, the image is greater than the object. In both cases the image is inverted. 3. When the object is placed between the focus and the mirror, the image situated _behind_ the mirror is greater than the object, and it has the same direction: in proportion as the object approaches the focus, the image becomes larger and more distant. These and similar results are proved by placing a lighted candle at different distances from a concave mirror. 4. An eye cannot see an image in the air except it be placed in the diverging rays; but if the image be received on a piece of white paper, it may be seen in any position of the eye, as the rays are then reflected in every direction. 5. If a picture drawn according to the rules of perspective, be placed before a large concave speculum, a little nearer than its principal focus, the image of the picture will appear extremely natural, and very nearly like the real objects whence it was taken. Not only are the objects considerably magnified, so as to approach to their natural size, but they have also different apparent distances, as in nature, so that the view of the inside of a church appears very like what it is in reality, and representations of landscapes appear very nearly, as they do from the spot whence they were taken. In this respect a large concave speculum may be made to serve nearly the same purpose, as the Optical Diagonal Machine, in viewing perspective prints. 6. The concave speculum is that alone which is used as the great mirror which forms the first image in reflecting telescopes; and it is likewise the only kind of speculum used as the small mirror, in that construction of the instrument called the _Gregorian Reflector_. _Quantity of light reflected by polished surfaces._