Considering that the injurious effects of the secondary image arose from the images reflected from the two surfaces being formed near the same point, and at nearly the same focal distance, I formed a plan for destroying the secondary image, or at least counteracting its effects, by forming the concavity of the mirror next the eye of a portion of a sphere _different_ from that of the convex side which was silverised, and from which the principal image is formed. But, for a long time, I could find no opticians possessed of tools of a sufficient length of radii for accomplishing my design. At length a London working optician undertook to finish a glass speculum, according to my directions, which were, that the convex surface of the mirror should be ground on a tool which would produce a focal distance by reflection of about 4 feet; and that the concave surface should have its focal distance at about 3 feet 3 inches, so that the secondary image might be formed at about 9 inches, within the focal distance of the silverised side, and not interfere to disturb the principal image. But, either from ignorance or inattention, the artist mistook the radius for the half radius of concavity, and the speculum turned out to be only 23 inches focal distance by reflection. This mirror was fitted up as a telescope, on the aerial plan, and I found, as I expected, the secondary image completely destroyed. It produced a very beautiful and brilliant view of land objects, and even the brightest objects exhibited no double image. The mirror was nearly 5 inches in diameter, but the image was most accurately defined when the aperture was contracted to about 3 inches. It was fitted with a terrestrial eye-piece which produced a magnifying power of about 25 times. When directed to the moon, it gave a very distinct and luminous view of that orb, without the least appearance of a secondary image. But as the focal distance of the speculum was scarcely half the length I had prescribed, I did not apply to it any high astronomical powers; as I find, that these can only be applied with effect, in this construction, to a speculum of a considerable focal length. Happening to have at hand a convex lens 10 feet focal length, and 4 inches in diameter--the one side of which had been ground to a certain degree of concavity--I caused the convex side to be foliated, which produced a focus by reflection, at 13-1/2 inches distant. To this mirror I applied terrestrial powers of 15 and 24, with considerable distinctness. The power of 15 produced a very brilliant and distinct view of land objects. Had the mirror been at least 3 times the focal length, it would have formed an excellent telescope, with the same aperture. SECT. 6.--A REFLECTING TELESCOPE, WITH A SINGLE MIRROR AND NO EYE-PIECE. [Illustration: _figure 72._]