Chap. VIII.

GEOMETRY.

747

Fig. 700.

The Sphere is divided then into two equal portions, asshown by the sections H of the globe K.

The sphere is described by the revolution of a semi-circle about its diameter, or it may be defined as a bodybounded by a surface of which every point is equally dis-tant from the centre. The curve surface of either ofthese zones or half globes is equal to twice the area ofone of its great circles, or rather of the section made bythe plane passing through the centre: the curve sur-face of a zone or portion contained between two parallelplanes is equal to the curved surface of a cylinderof the same height with the height of the zone, or the dis-tance between the planes, and of the same diameterwith the sphere, from whence we learn that the wholesurface of the sphere is equal to the curved surface of thecircumscribing cylinder.

The solid content of a globe is equal to that of acone whose altitude is the radius, and whose base isequal to the surface of the sphere ; hence the content ofthe sphere is one-third of the product of its radius intoits surface, and the sphere is also equal to two-thirds ofits circumscribing cylinder. Hence the cone, the sphere,and the cylinder, whose diameters and heights are equal,are in the proportions of 1, 2, and 3, or the cylinderis equal to the sphere and cone taken together; the coneis equal to a third part of the cylinder, and the sphere isdouble the cone.

In the Globe P the great circle dotted at M, the otherat N, and the outer circumference Q IIP, are all equalto one another, because the planes by which it is cutpass through its centre.

The globe is by this means cut into four equal parts,and the content of each, as well as their superficies, maybe easily ascertained.

A small Circle , inscribed on a globe, as that shown at15 on the globe A, must be struck from another centre.

Globular projection, or the representation of theboundaries of planes which pass through the glol>e atright angles with its diameter, belongs more immediatelyto spherical geometry : a circle whose plane passesthrough its poles is called the meridian, and which cutsthe planes of the equator, and all circles parallel to it atright angles.

The plane of the horizon of any place touches theearth’s surface, and divides the whole expanse of theheavens into two hemispheres. The earth’s surface wasby the ancient astronomers divided into five zones,founded on the different lengths of the longest day, as weproceed from the equator towards either of the poles :these were also denominated climates, and were each ofsuch a breadth that the longest day at the boundarynearest the pole exceeded the longest day at the boundarynearer the equator by a certain space of time, as half anhour or an hour: within the polar circle the climateswere supposed of such a breadth as to make the longestday at the opposite sides differ by a month.

The Circle B of the globe C has its centre in the middleof the plane D, and the section made is common to bothfigures.

Small equal Circles of the same globe have theircentres equidistant from its centre. In the globe E the circles F and G are equal to oneanother, because they are equally distant from the middle of the sphere. II and I arealso so in the figure HIKE , and the two portions cut off are equal.

An Arc is that portion of a circle which is less than a quarter of its periphery ; thedotted portion EF is an arc of the circle EFGH, as is AD, CB of the other circledrawn in lines.

The Toint K of the arc I K 1/ is the summit, and the line M N, as it touches the pointT ot the arc O Q, is also its summit.

Fig. 701.

Fig. 704.

Fig 705.