or Reflecting Sector, an instrument invented by Mr William Garrard, for the purpose of measuring angles, particularly small ones, with a greater degree of accuracy than can be done by Hadley's quadrant or by the sextant.

The frame of this instrument is similar to that of Hadley's quadrant, having two radii, a limb, and braces; but with this difference, that the further radius is produced upwards of four inches beyond the centre of motion of the index; and the great speculum, or what is called the index-glass in Hadley's quadrant, being placed there, is called the upper centre. In this instrument there is no provision for the back observation. Antimeter. The horizon-glass is like that in Hadley's quadrant; there are two sight vanes, to suit two different situations of the large speculum or object glass; these vanes are adapted to receive a small telescope. On the centre of the index, where the index-glass of Hadley's quadrant is fixed, is a brass or bell-metal semicircle, two inches in diameter, and one-eighth of an inch thick; this semicircle is screwed fast to the index, in such a manner that the axis of the index is a tangent to it. On the upper centre are two circular brass plates, which revolve concentrically, either together or separately. The under plate has a lever, or part perpendicular to the plane of the instrument, projecting downwards, a little beyond the lower centre; this lever is acted upon by the semicircular plate at the lower centre, to which it is always kept close by a spring on the other side. In the upper of the above mentioned circular plates are two circular perforations or flits, through one of which a screw takes into the head of the instrument, and thro' the other a screw takes into the lower moveable plate. The large speculum is fastened to the upper plate; and by the above mentioned screws the position of this glass may be altered. A circular plate is fixed to the lower centre by three pillars; its centre is a nut to admit a screw, by which the plate carrying the large speculum may be fastened here occasionally.

The scale on the limb is divided into 45 equal parts or degrees, and not into half degrees as is the case in Hadley's quadrant, by reason of the double reflection. These divisions are numbered in a retrograde order; zero being at the extremity of the further radius. Although the limb contains 45 degrees, yet the greatest angle which can be measured, the large speculum remaining fixed to the circular plate, is 10° 18' 21".8; the distance between the two centres being four inches, and the radius of the semicircle one inch. Agreeable to these dimensions, the inventor has given a table exhibiting the value of each primary division on the limb; he hath also given a more ample table, adapted to a distance between the centres of three times the radius of the semicircle, which he says hath been found the most convenient in practice. If an angle greater than 10° 18' is wanted, it may be measured by the method of antiparallelism, as the inventor calls it; which is as follows:

Let the screw which fastens the two circular plates on the upper centre be made fast, and loosen the screw which fastens the upper circular plate to the instrument; now adjust the glasses by the usual method; bring forward the index to any given division on the limb, and make it fast; also fasten the screw which was before loose, and loosen the other screw; then bring the index to zero, and proceed as before.

The inventor gives the following directions for adjusting and using the instrument.

The first thing to be attended to is, to set the horizon-glass perpendicular to the plane of the instrument, which is performed as follows: Hold the instrument with its plane perpendicular to the horizon, and look over backwards into the glass and beyond it. If the limb of the instrument appears in a right line with its reflection, the glass is upright; but if it does not appear so, loosen or tighten the little screw on the foot of the glass until it be adjusted. Then with the instrument, as in taking an altitude, look through the sight vane or telescope at some distant object, with the index fixed in any intended situation; the two screws at the upper Antiparallel centre being loose, turn the glass about till the same object appears nearly in the same part of the horizon-glass: Next hold it in a horizontal position, and adjust the object-glass or large speculum with the screws which are behind and before, on the foot of it, till the object and its reflection are seen in the same horizontal line. Lastly, with the instrument upright, turn the tangent-screw belonging to the horizon-glass at the back of the instrument, until there be a perfect coincidence of the object and its reflection that way, and the adjustments are complete.

Antiparallels, in geometry, are those lines which make equal angles with two other lines, but contrary ways; that is, calling the former pair the first and second lines, and the latter pair the third and fourth lines, if the angle made by the first and third lines be equal to the angle made by the second and fourth, and contrariwise the angle made by the first and fourth equal to the angle made by the second and third; then each pair of lines are antiparallels with respect to each other, viz. the first and second, and the third and fourth. So, if AB and AC be any two lines, and FC and FE be two others, cutting them so,

that the angle B is equal to the angle E, and the angle C is equal to the angle D; then BC and DE are antiparallels with respect to AB and AC; also these latter are antiparallels with regard to the two former. It is a property of these lines, that each pair cuts the other into proportional segments, taking them alternately,

viz. AB : AC :: AE : AD :: DB : EC, and FE : FC :: FB : FD :: DE : BC.

Aperture, in optics, has been defined in the Encyclopaedia, but no rule was given there for finding a just aperture. As much depends upon this circumstance, our optical readers will be pleased with the following practical rule given by Dr Hutton in his Mathematical Dictionary. "Apply several circles of dark paper, of various sizes, upon the face of the glass, from the breadth of a straw to such as leave only a small hole in the glass; and with each of these, separately, view some distant object, as the moon, stars, &c. then that aperture is to be chosen through which they appear the most distinctly.

Huygens first found the use of apertures to conduct much to the perfection of telescopes; and he found by experience (Dioptr. prop. 56.), that the best aperture for an object-glass, for example of 30 feet, is to be determined by this proportion, as 30 to 3, so is the square root of 30 times the distance of the focus of any lens to its proper aperture: and that the focal distances of the eye-glasses are proportioned to the apertures. And M. Auzout says he found, by experience, that the apertures of telescopes ought to be nearly in the sub-duplicate ratio of their lengths. It has also been found by experience, that object-glasses will admit of greater apertures, if the tubes be blacked within side, and their passage furnished with wooden rings.

It is to be noted, that the greater or less aperture of an object-glass, does not increase or diminish the visible area of the object; all that is effected by this is the admittance of more or fewer rays, and consequently the more or less bright appearance of the object. But the largeness of the aperture or focal distance causes the the irregularity of its refractions. Hence, in viewing Venus through a telescope, a much less aperture is to be used than for the moon, or Jupiter, or Saturn, because her light is so bright and glaring. And this circumstance somewhat invalidates and disturbs Azout's proportion, as is shown by Dr Hook, Phil. Trans.