CIRCLE, the name of various astrono nomical instruments for observing right ascensions, declinations, azimuths, alti tudes, and likewise for .the purposes of the most improved theodolite.
Plate Circular Instrument" is a re presentation of an instrument made by Mr. Troughton, and of which lie liberally permitted our draughtsman to take a drawing. It is an instrument which mea sures both horizontal and vertical angles. with great accuracy, and is equally adapt ed for astronomical purposes and survey ing.
The instrument is supported on three screws, two of which, x, y, are shewn in the figure ; the three arms through which these pass meet in the centre, and hold a strong, vertical steel axis, truly turned, and very exactly fitted into two sockets, one at the top and the other at the bot tom of a cone, A : upon this axis the up per part of the instrument turns. B is the azimuth circle, laying upon the three arms of the tripod, and capable of turn ing round on the steel axis before men tioned it is held by a screw, g, which moves the circles slowly round when turn ed: this motion is to adjust the circle, so that the plane of the vertical circle, P, shall be in the meridian when the index is set to zero. The circle is divided into degrees and every five minutes, and the microscope subdivides them into seconds. Another similar microscope is fixed di ametrically opposite, upon the circular plate H, and turns round upon the verti cal axis with the rest of the instrument. (For the constructions of these micro. scopes, see that article.) I, I, are two hollow conical pillars, screwed on the in dex plate to support the axis of the ver tical circle, P, by means of two bars (one only. of which can be seen, h,) screwed at the top of the pillars, andholding at their outer ends tubes, which contain angular bearings for the pivots of the axis: these bearings, or Y's, as they are called, from resembling that letter, can be elevated or depressed by screws e, beneath them, to bring the axis parallel to the plane of the azimuth circle. m, m, are two crook ed hollow tubes, screwed to the upright pillars, holding two microscopes, n, to, reading divisions diametrically opposite to each other on the vertical circle P. The vertical circle is composed of two circles, each cut from a solid plate, and attached to two flanches on a Iloilo* conical axis E; they are firmly braced together by short pillars, as in the figure; between the circles the telescope F is fixed, it is 30 inches long and 2 in dia meter. 0 is a thin plate of metal, screwed to the further main pillar, I, by its lower end, and its upper end supporting a clamp for fixing the circle, when set at any elevation, ands screw for movingit slow ly a small quantity after clamping. A simi hr screw, for occasionally attaching the index plate, II, to the azimuth circle, B, is seen at p. a is a small roller pushed up wards by a spring, I: it acts against a ring upon the conical axis E, and its use is to support part of the weight of the circe and telescope, and take the bearing from the pivots at the end of the axis. It
is a spirit level hung to the two horns m, on, and adjustable by a screw at its end. S is a telescope beneath the instrument, is set to any distant object when the instrument is in use, and serves to strew that the instrument does not change its position. See OBSERVATORY and Snit VEYING.
c CIRCLE, in geometry, a plane figure comprehended by a single curve line, call ed its circumference, to which right lines or radii, drawn from a point in the mid dle, called the centre, are equal to each other.
The area of a circle is found by multi plying the circumference by the fourth part of the diameter, or half the circum ference by half the diameter : for every circle may be conceived to be a polygon of an infinite number of sides, and the semidiameter must be equal to the per pendicular of such a polygon, and the cir cumference of the circle equal to the pe riphery of the polygon : therefore half the circumference multiplied by half the dia meter gives the area of the circle.
Circles, and similar figures inscribed in them, are always as the squares of the diameters ; so that they are in a duplicate ratio of their diameters, and consequently of their radii.
A circle is equal to a triangle, the base of which is equal to the periphery, and its altitude to its radius : circles therefore are in a ratio compounded of the periphe ries and the radii.
To find the proportion of the diameter of a circle to its circumference. Find, by continual bisection, the sides of the in scribed polygon, till you arrive at a side subtending any arch, however small; this found, find likewise the side of a similar circumscribed polygon ; multiply each by the number of the sides of the polygon, by which you will have the perimeter of each polygon. The ratio ofthe diameter to the periphery of the circle will be greater than that of the same diame ter to the perimeter of the circumscribed polygon, but less than that of the in scribed polygon. The difference of the two being known, the ratio of the diame ter to the periphery is easily had in num bers, very nearly, though not justly true. Thus Archimedes fixed the proportion at 7 to 22.
Wolfius finds it as 10000000000000000 to 31415926535897932: and the learned Mr. Machin has carried it to one hundred places, as follows: if the diameter of a circle be 1, the circumference will be 3,14159, 26535, 89793, 23846, 26433, 83279, 50288, 41971, 69399, 37510, 58209, 74944, 59230, 78164, 05286, 20899, 86280, 34825, 34211, 70679 of the same parts. But the ratios generally used in practice are that of Archimedes, and the following ; as 106 to 333, as 113 to 355, as 1702 to 5347, as 1815 to 5702, or as 1 to 3.14159.