SPHERE. 
ed on its surface, by the motions of tlie ex- 
tremities of the chords E D, F G, I H, &c. 
at right angles to AB; the diameters of 
which circles are equal to those chords. 
3. Tlie poles of a circle on the sphere, are 
those points on its surface, equally distant 
from tlie circumference of that circle : thus 
A and B are the poles of the circles de- 
scribed on the sphere by the ends of the 
chords E D, F G, I H, &c. 4. A great cir- 
cle of the sphere is one equally distant from 
both its poles ; as that described by the ex- 
tremities of the diameter E D, which is 
equally distant from both its poles A and 
B. 5. Lesser circles of the sphere are those 
which are unequally distant from both their 
poles; as those described by the extre- 
mities of the chords F G, H I, &c. because 
unequally distant from their poles A and B. 
See Circle. 
Axioms. 1. The diameter of every great 
circle passes through the centre of the 
sphere : but the diameters of ail lesser cir- 
cles do not pass through the same centre : 
hence also the centre of the sphere is the 
common centre of all the great circles. 
2. Every section of a sphere by a plane, is 
a circle. 3. A sphere is divided into two 
equal parts, or hemispheres, by the plane 
of every great circle ; and into two un- 
equal parts, called segments, by the plane 
of every lesser circle. 4. The pole of 
every great circle is 90° distant from it on 
the surface of the sphere ; and no two great 
circles can have a common pole. 5. The 
poles of a great circle are the two extre- 
mities of that diameter of the sphere, 
which is perpendicular to the plane of that 
circle. 6. A plane passing through three 
points on the surface of the sphere, equally 
distant from any of tire poles of a great cir- 
cle will be parallel to the plane of that 
great circle. 7. The shortest distance be- 
tw'een two points, on the surface of a 
sphere, is the arch of a great circle passing 
through these points. 8. If one great cir- 
cle meets another, the angles on either side 
are supplements to each other ; and every 
spherical angle is less than lSO”. 9. If two 
circles intei-sect each other, the opposite 
angles are equal. 10. All circles 'on the 
sphere, having the same pole, are cut into 
similar arches, by great circles passing 
through that pole. 
Sphere, properties of the. 1. All spheres 
are to one another as the cubes of their 
diameters. 2. The surface of a sphere is 
equal to four times the area of one of its 
great circles, as is demonstrated by Archi- 
medes in his book of the Sphere and Cy- 
linder, lib. i. prop. 37. hence, to find the 
superfices of any sphere, we have this easy 
rule; let the area of a great circle bo mul- 
tiplied by 4, and the product will be the 
superficies; or, according to Euclid, lib. vi. 
prop. 20. and lib. xii. prop. 2. the area of a 
given sphere, C E B D (fig 3) is equal to 
that of a circle, whose radius is equal to 
the diameter of the sphere B C. There- 
fore, having measured the circle described 
with the radius B C, this will give the sur- 
face of the sphere. 3. The solidity of a 
sphere is equal to the surface multiplied 
into one third of the radius : or, a sphere is 
equal to two thirds of its circumscribing 
cylinder, having its base equal to a great 
circle of the sphere. Let A B E C (fig. 4 
and 5), be the quadrant of a circle, and A B 
D C the circumscribed square, equal twice 
the tiiangle ADC: by the revolution of 
the figure about the right line A C, as an 
axis, a hemisphere will be generate^ by the 
quadrant, a cylinder of the same base and 
height of the square, and a cone by the tri- 
angle : let these three be cut any how by 
the plane H F, parallel to the base A B ; 
and the section of the cylinder will be a 
circle, whose radius is F H ; in the hemi- 
sphere, a circle wdiose radius is F E ; and 
in the cone, a circle of the radius FG. 
But EAM = HF^) = EF^ + FA^": but 
A F^ = F G% because A C = C D ; and 
therefore H F^ = E F^ -{- F G* ; or the cir- 
cle of the radius H F, is equal to a circle of 
the radius E F, together with a circle of the 
radius GF: and since this is true every 
where, all the circles together described by 
the respective radii HF, that is the cy- 
linder, are equal to all the circles described 
by the respective radii E F and F G, that 
is, to the hemisphere and cone taken to; 
gether. But by Euclid, lib. xii. prop. 10. 
the cone generated by the triangle D A C, 
is one third part of the cylinder, generated 
by the square B C, whence it follows, that 
the hemisphere generated by the rotation 
of the quadrant A B E C, is equal to the re- 
maining two thirds of the cylinder, and that 
the whole sphere is two thirds of the cy- 
linder circumscribed about it. Hence it 
follows, that a sphere is equal to a cone 
whose height is equal to the semi-diameter 
of the sphere, and its base equal to the su- 
perficies of the sphere, or to the area of 
tour great circles of the sphere, or to that 
of a circle, whose radius is equal to the 
diameter of the sphere. 
Sphere, in astronomy, that concave orb. 
