CIRCLE. 



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, A,) screwed at 

 the top of the pillars, and holding 1 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, n, 

 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 hollow 

 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. O 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, and a screw for moving it slow- 

 ly a small quantity after clamping. A simi- 

 lar screw, for occasionally attaching the 

 index plate, If, 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 

 circle and telescope, and take the bearing 

 from the pivots at the end of the axis. R 

 is a spirit level hung to the two horns w, 

 n, and adjustable by a screw at its end. 

 S is a telescope beneath the instrument, 

 which is set to any distant object when 

 the instrument is in use, and serves to 

 shew thkt the instrument does not change 

 its position. See OBSERVATORY and SUR- 

 VEYING. 



( 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 of the 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, 2535, 89793, 25846, 26433, 

 83279, 50288, 41971, 69399, 37510, 

 5^209, 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. 



CIRCLE, the quadrature of the, or the 

 manner of making a square, whose sur- 

 face is perfectly and geometrically equal 

 to that of a circle, is a problem that has 

 employed the geometricians of all ages. 



Many maintain it to be impossible ; 

 Des Cartes, in particular, insists on it, 

 that a right line and a circle being of dif- 

 ferent natures, there can be no strict 

 proportion between them: and in effect 



