SPHERE, 



tlie middle, called the etntre, whence all lines, drawn to the 

 furface, are equal. 



The fphere is fuppofed to be gfenerated by the revolution 

 of a femicircle ABC [Plate XIV. Geometry, Jig. 2.) about 

 its diameter A C, which is alfo called the axis oj the fphere ; 

 and the extreme points of the axis, A and C, are caUed the 

 poles of the fphere. 



Sphere, Properties of the. I. A fphere is equal to a 

 pyramid, whofe bafe is equal to the furface, and its height 

 to the radius of the fphere. 



Hence, a fphere being efteemed fuch a pyramid, its cube, 

 or folid content, is found like that of a pyramid. 



2. A fphere is to a cyhnder, Handing on an equal bafis, 

 and of the fame height, as 2 to 3. Hence, alfo, may the 

 cube, or content of the fpiiere, be found. 



%. A fphere is equal to four times the cone, the bafe of 

 which is equal to the generating circle, and the height of 

 which is equal to the r:idius. Or, a hemifphere is equal to 

 twice the cone of the fame bafe and height. And a cylinder, 

 of the fame height and bafe, being a triple of the cone, it 

 follows, that the hemifphere is two-thirds of the cylinder, 

 and confequently the wliole fphere two-thirds of the circum- 

 fcribing cyhnder. Archir.i. de Sph. et Cycl. Apud 

 Opcraper Rivaltum, p. 67. 



4. The cube of the diameter of a fphere is to the folid 

 content of the fphere, nearly as 300 to 157 : and thus, alfo, 

 may the content of the fphere be meafured. 



5. The fnrface of a fphere is quadruple the area of a 

 circle defcribed with the radius of the fphere. For fince a 

 fphere is equal to a pyramid, whofe bafe is the furface, and 

 its altitude the radms of the fphere ; the furface of the 

 fphere is had, by dividing its folidity by a tiiird part of its 

 femi-diameter, or one-fixth ot the whole diameter. 



But the folidity of the fphere is the produft of ids of 

 the greatefl circle by the diameter ; and if this product be 

 divided by ^th of the diameter, the quotient, or \' of the 

 greatell circle, i. e. four times the greateft circle, will be the 

 lurface of the fphere. 



Otherwife : the furface of a fphere is equal to the pro- 

 duct of its diameter, and the periphery of its generating 

 circle ; but the area of this circle ts equal to the produA 

 oi tlie femi-diameter and half the periphery, or one-fourth 

 of the produiSl of the diameter and periphery ; and, there- 

 fore, the furface of the fphere is equal to four times the 

 area of its generating circle. 



Hence, 6. The furfaces of fpheres are to one another as 

 the fquares of their diameters ; becaufe thefe furfaces, being 

 four times their generating circles, are as thefe circles, i. e. 

 as the fquares of their diameters. 



7. Spheres being ;d parts of cylinders of equal bafe and 

 altitude, and cylinders being as the cubes of their alti- 

 tudes, are as the cubes, or in the triplicate ratio of their 

 diameters. 



8. The furface of any portion of a fphere, greater or iefs 

 than the hemifphere, is equal to the area of a circle, whofe 

 radius is a line drawn from the vertex of that portion to the 

 circumference of the circle which is its bafe. Arcliimedes, 

 ubifupra, pp. 84, 85. 



Spiieuk, the diameter of a, bei?!g given, to find its furface and 

 fulidity. Find the periphery of the circle defcribed by tlie 

 radius of the fphere, or of the generating circle. 



Multiply this, found, into the diameter, or the fquare of 

 the diameter, by 3.1416, the produft is the furface of the 

 fphere. Multiply the furface by a fixth part of the dia- 

 meter, or the area of the generating circle by jds of its dia- 

 meter j or, again (becaule the area of fuch circle is to the 

 Square of the diameter as .7854 to i), the cube of the 



diameter by .5236 {= 4 of -7854) ; and the produft la the 

 folidity of the fphere. 



Thus, fuppofing the diameter of the fphere 56, the peri- 

 phery will be found 175.9, which multiplied by the dia- 

 meter, the produtt 9852 is the furface of the fphere ; which 

 multiplied by ^th part of the diameter, gives the folidity 

 91623.6. Or thus : 



Find the cube of the diameter 175616 ; then to 300,157, 

 and the cube found, find a fourth proportional. This is the 

 folidity of the fphere required. 



For finding the furface and folidity of the fphere by the 

 method of fluxions, fee Superficies and Solidity. For 

 fegments and feftors of fpheres, fee Frustum, Segment, 

 and Sector. 



Sphere, DoBrine of the. See Spherics. 



Sphere, Projeaion of the. See Projection. 



Sphere of Jlaivity. See Activity. 



Sphere, in AJlronomy, that concave orb or expanfe which 

 invelts our globe, and in which the heavenly bodies, "uiz. 

 fun, itars, planets, and comets, appear to be fixed at an 

 equal diilance from the eye. 



This is alio called the fphere of the world ; and is the fub- 

 jeft of the fpherical aftronomy. 



This fphere, as it includes the fixed ftars, whence we alfo 

 occafionally call it the fphere of the fixed Jlars, is immenfely 

 great. Tlie diameter of the earth's orbit is fo fmall, in re- 

 fpeft to the diameter of this, that the centre of the fphere 

 is not fenfibly changed by any alteration of the fpeftator's 

 place in the feveral parts of the orbit ; but Itill, in all the 

 points of the earth's furface, and at all times, the uihabit- 

 ants have the fame appearance of the fphere ; that is, the 

 fixed liars feem to poifefs the fame points in the furface of 

 the fphere. For our way of judging of the places, &c. 

 of the liars, is to conceive right lines. drawn from the eye, 

 or the centre of the earth, through the centre of the ftars, 

 and continued thence till they cut the forefaid fphere ; the 

 points where thefe lines terminate in it are the apparent 

 places of thofe ftars. 



The better to determine the places of the heavenly bodies 

 in the fphere, feveral circles are imagined to be defcribed 

 in the furface thereof, hence called circles of the fphere. 



Sphere, in Geography, &c. denotes a certain difpofition 

 of the circles on the iurface of the earth, with regard to one 

 another, which varies in various parts of it. 



The circles originally conceived on the furface of the 

 fphere of the world, are almoil all transferred, by analogy, 

 to the furface of the earth, where they are conceived to be 

 drawn direftly underneath thofe of the fphere, or in the fame 

 planes with them ; fo that, if the planes of thofe of the 

 earth were continued to the fphere, they would coincide 

 with the refpeftive circles upon it. Thus we have an ho- 

 rizon, meridian, equator, &c. on the earth. 



As the equator in the heavens divides the fphere into two 

 equal parts, the one north, and the other fouth, fo does the 

 equator on the furface of the earth divide the globe in the 

 fame manner. 



And as the meridians in the heavens pafs through the 

 poles of the horizon, fo do thofe of the eartli, &c. 



With regard, then, to the pofition of fome of the circles 

 in refpecl of others, we have a right, a parallel, and an oblique 

 fphere. 



Sphere, Armillary, or Artificial, is an aftronomical inllru- 

 ment, reprefenting the feveral circles of the fphere in their 

 natural order ; ferving to give an idea of the office and po- 

 fition of each of them, and to folve various problems relating 

 to them. 



It is thus called, as confifting of a number of fafciz or 



rings 



I 



