SPHERE, or GLOBE, a solid body, the surface of which is every where equally distant from a given point or centre within it. This distance of each point from the centre is called the radius. In the article MENSURATION will be found the feminize which connect the surface and solidity of a sphere with the radius : we shall here add that the weight of a sphere of pure water is found in ounces avoirdu pois, by multiplying the cube of the number of inches in the radius by ; and in pounds avoirdupois by multiplying the cube of the number of feet in the radius by 26P05. These results multiplied by the specific gravity give the weight of a sphere of any other substance.
A section made by a sphere and plane is always a circle. When the cutting plane passes through the centre of the sphere, this proposition is obvious from the definition of a circle. When the plane does not pass through the centre the assertion follows so soon as it is shown that a plane curve having all its points equidistant from a given point not in the plane is a circle. A section passing through the centre is called a great circle, and one which does not pass through the centre a small circle. These terms are incorrect, since a small circle may be in zz use as nearly as we please equal to a great circle : the words sreekr and smaller would be more correct.
The centre of a circular section is found by drawing a perpendicular from the centre of the sphere to the phuie of the section. All sections whose planes are parallel have their centres on one straight line, namely, the perpendicular to the planes which passes through the eentre of the sphere. The great circle in such a system (ou•n) is called the prig:stare, the common perpendicular (coq) the axis, all the small circles (DE YO, K L MN, fie) parallels, the extremities of the axle (r and Q) polo, and all great circles passing through the axis and poles (Poen, roe, PA Q, &c.) secondaries.
By the angle made by two great circles is always understood the angle made by their planes, which is also that made by their tangents at the point of intersection, and that made by the intersections of the two circles with the third circle to which both are secondary. It is
also the angle made by the axes of the two circles. Thus the spherical angle E Pr is the angle made by the planes l'EQ and PFQ, or the angle made by tangents to the circles drawn through F, or the angle uo A.
The angle made by two straight lines drawn from the centre (as o A and on) is often confounded with the arc (An) which that angle marks out on the sphere. When this causes any confusion, which at first it will sometimes do, instead of each arc mentioned, read its angle : thus for the arc An read the " angle subtended by the arc AB' or A011. Thus when we say that the angle made by two great circles is the arc intercepted between their poles, we mean not to equate the angle to the length of an arc, but to the angle which that arc subtends at the centre.
The following propositions are essential to the doctrine of the sphere in geography and astronomy; they may be easily proved, and will serve as exercises in the meaning of the preceding terms : L If the poles of a first circle lie upon a second, the poles of the second will also lie upon the first.
2. If a sphere be made by the revolution of a semicircle round its diameter, the diameter will be an axis, the middle point of the semi circle will describe the primary, all other points will describe parallels, and every position of the generating circle will be a secoudary.
3. If a point on a sphere be distant from each of two other points (not opposite) by a quadrant of a great circle, the firea point must be a pole of the great circle which joins the and third.
4. The arc of a parallel (as ee) is found from the corresponding arc of the primary (Au) by multiplying the latter by the cosine of the angle (r oA) which is subtended by the intercepted arc (Ay) of the secondary.