SPHERE

, in Geometry, a solid body contained under one single uniform surface, every point of which is equally distant from a certain point in the middle called its centre.

The Sphere may be supposed to be generated by the revolution of a semicircle ABD about its di meter AB, which is also called the axis of the Sphere, and the extreme points of the axis, A and B, the poles of the Sphere; also the middle of the axis C is the centre, and half the axis, AC, the radius.

1. A Sphere may be considered as made up of an infinite number of pyramids, whose common altitude is equal to the radius of the Sphere, and all their bases form the surface of the Sphere. And therefore the solid content of the Sphere is equal to that of a pyramid whose altitude is the radius, and its base is equal to the surface of the Sphere, that is, the solid content is equal to 1/3 of the product of its radius and surface.

2. A Sphere is equal to 2/3 of its circumscribing cylinder, or of the cylinder of the same height and diameter, and therefore equal to the cube of the diameter multiplied by .5236, or 2/3 of .7854; or equal to double a cone of the same base and height. Hence also different Spheres are to one another as the cubes of their diameters. And their surfaces as the squares of the same diameters.

3. The surface or superficies of any Sphere, is equal to 4 times the area of its great circle, or of a circle of the same diameter as the Sphere. Or

4. The surface of the whole Sphere is equal to the area of a circle whose radius is equal to the diameter of the Sphere. And, in like manner, the curve surface of any segment EDF, whether greater or less than a hemisphere, is equal to a circle whose radius is the chord line DE, drawn from the vertex D of the segment to the circumference of its base, or the chord of half its arc.

5. The curve surface of any segment or zone of a Sphere, is also equal to the curve surface of a cylinder of the same height with that portion, and of the same diameter with the Sphere. Also the surface of the whole Sphere, or of an hemisphere, is equal to the curve surface of its circumscribing cylinder. And the curve surfaces of their corresponding parts are equal, that are contained between any two places parallel to the base. And consequently the surface of any segment or zone of a Sphere, is as its height or altitude.

Most of these properties are contained in Archimedes's treatise on the Sphere and cylinder. And many other rules for the surfaces and solidities of Spheres, their segments, zones, frustums, &c, may be seen in my Mensuration, part 3, sect. 1, prob. 10, &c.

Hence, if d denote the diameter or axis of a Sphere, s its curve surface, c its solid content, and a = .7854 the area of a circle whose diam. is 1; then we shall, from the foregoing properties, have these following general values or equations, viz, .

Doctrine of the Sphere. See Spherics.

Projection of the Sphere. See Projection.

Sphere of Activity, of any body, is that determinate space or extent all around it, to which, and no farther, the effluvia or the virtue of that body reaches, and in which it operates according to the nature of the body. See Activity.

, in Astronomy, that concave orb or expanse which invests our globe, and in which the hea- | venly bodies, the sun, moon, stars, planets, and comets, appear to be fixed at an equal distance from the eye. This is also called the Sphere of the world; and it is the subject of spherical astronomy.

This Sphere, as it includes the fixed stars, from whence it is sometimes called the Sphere of the fixed stars, is immensely great. So much so, that the diameter of the earth's orbit is vastly small in respect of it; and consequently the centre of the Sphere is not sensibly changed by any alteration of the spectator's place in the several parts of the orbit: but still in all points of the earth's surface, and at all times, the inhabitants have the same appearance of the Sphere; that is, the fixed stars seem to possess the same points in the surface of the Sphere. For, our way of judging of the places &c of the stars, is to conceive right lines drawn from the eye, or from the centre of the earth, through the centres of the stars, and thence continued till they cut the Sphere; and the points where these lines so meet the Sphere, are the apparent places of those stars.

The better to determine the places of the heavenly bodies in the Sphere, several circles are conceived to be drawn in the surface of it, which are called circles of the Sphere.

, in Geography, &c, denotes a certain disposition of the circles on the surface of the earth, with regard to one another, which varies in the different parts of it.

The circles originally conceived on the surface of the Sphere of the world, are almost all transferred, by analogy, to the surface of the earth, where they are conceived to be drawn directly underneath those of the Sphere, or in the same positions with them; so that, if the planes of those of the earth were continued to the Sphere of the stars, they would coincide with the respective circles on it. Thus, we have an horizon, meridian, equator, &c, on the earth. And as the equinoctial, or equator, in the heavens, divides the Sphere into two equal parts, the one north and the other south, so does the equator on the surface of the earth divide its globe in the same manner. And as the meridians in the heavens pass through the poles of the equinoctial, so do those on the earth, &c. With regard then to the position of some of these circles in respect of others, we have a right, an oblique, and a parallel Sphere.

A Right or Direct Sphere, (fig. 4, plate 26), is that which has the poles of the world PS in its horizon, and the equator EQ in the zenith and nadir. The inhabitants of this Sphere live exactly at the equator of the earth, or under the line. They have therefore no latitude, nor no elevation of the pole. They can see both poles of the world; all the stars do rise, culminate, and set to them; and the sun always rises at right-angles to their horizon, making their days and nights always of equal length, because the horizon bisects the circle of the diurnal revolution.

An Oblique Sphere, (fig. 5, plate 26), is that in which the equator EQ, as also the axis PS, cuts the horizon HO obliquely. In this Sphere, one pole P is above the horizon, and the other below it; and therefore the inhabitants of it see always the former pole, but never the latter; the sun and stars &c all rise and set obliquely; and the days and nights are always varying, and growing alternately longer and shorter.

A Parallel Sphere, (fig. 6, plate 26), is that which has the equator in or parallel to the horizon, as well as all the sun's parallels of declination. Hence, the poles are in the zenith and nadir; the sun and stars move always quite around parallel to the horizon, the inhabitants, if any, being just at the two poles, having 6 months continual day, and 6 months night, in each year; and the greatest height to which the sun rises to them, is 23° 28′, or equal to his greatest declination.

Armillary or Artificial Sphere, is an astronomical instrument, representing the several circles of the Sphere in their natural order; serving to give an idea of the office and position of each of them, and to resolve various problems relating to them.

It is thus called, as consisting of a number of rings of brass, or other matter, called by the Latins armillæ, from their resembling of bracelets or rings for the arm.

By this, it is distinguished from the globe, which, though it has all the circles of the Sphere on its surface; yet is not cut into armillæ or rings, to represent the circles simply and alone; but exhibits also the intermediate spaces between the circles.

Armillary Spheres are of different kinds, with regard to the position of the earth in them; whence they become distinguished into Ptolomaic and Copernican Spheres: in the first of which, the earth is in the centre, and in the latter near the circumference, according to the position which that planet obtains in those systems.

The Ptolomaic Sphere, is that commonly in use, and is represented in fig. 6, plate 2, vol. 1, with the names of the several circles, lines, &c of the Sphere inscribed upon it. In the middle, upon the axis of the Sphere, is a ball T, representing the earth, on the surface of which are the circles &c of the earth. The Sphere is made to revolve about the said axis, which remains at rest; by which means the sun's diurnal and annual courses about the earth are represented according to the Ptolomaic hypothesis: and even by means of this, all problems relating to the phenomena of the sun and earth are resolved, as upon the celestial globe, and after the same manner; which see described under Globe.

Copernican Sphere, fig. 7, plate 26, is very different from the Ptolomaic, both in its constitution and use; and is more intricate in both. Indeed the instrument is in the hands of so few people, and its use so inconsiderable, except what we have in the other more common instruments, particularly the globe and the Ptolomaic Sphere, that any farther account of it is unnecessary.

Dr. Long had an Armillary Sphere of glass, of a very large size, which is described and represented in his Astronomy. And Mr. Ferguson constructed a similar one of brass, which is exhibited in his Lectures, p. 194 &c.