TRIGONOMETRY. P7 
the centre, connecting two points of the surface. The section of a sphere 
by a plane is a circle, which is smaller as the distance of the plane of 
intersection from the centre is greater (pl. 3, fig. 103). If the plane pass 
through the centre, the circle thus formed whose diameter is that of the 
sphere, is called a great circle. All others are small circles. A line 
connecting the centre of a spnere with that of a circle of intersection, is 
perpendicular to the plane of the latter. If two or more circles therefore 
are parallel to each other, their centres will all be in a diameter of the 
sphere, perpendicular to their planes; this is called their axis, and its 
extremities their poles. Every great circle bisects the sphere; two great 
circles mutually bisect each other, and divide the surface into four parts. 
If one great circle pass through the poles of another, their planes will be 
perpendicular. The angle between two great circles is measured by the 
are of a circle they intercept, whose plane is perpendicular to that of the 
two circles (pl. 3, figs. 108, 110). Two parallel circles include a part of 
the sphere called a spherical segment, and a part of the surface called a 
zone. If one of the circles be tangent to the sphere, the zone has only one 
base. The altitude of a zone or spherical segment is the perpendicular 
distance between the planes of the bases. The area of a zone is obtained 
by multiplying its altitude by the circumference of a great circle (fig. 102). 
The surface of a sphere is equal to the area of four great circles. The 
solidity of a sphere is obtained by multiplying the third power of the 
diameter by w (3°1415926) and dividing by 6. If we take a cone, 
hemisphere, and cylinder, of the same base and altitude (the altitude equal 
to a radius of the hemisphere), the solidities of these three bodies will be to 
each other as 1, 2, 3, that is. the cone will be one half the hemisphere, 
and this, two thirds of the cylinder; a cone, sphere, and cylinder will be in 
the same proportion, if the first and last have for bases, a great circle of the 
sphere, and for altitudes, a diameter (pl. 3, fig. 104). 
Ill. TRIGONOMETRY, OR THE MEASUREMENT OF 
TRIANGLES. 
1. PLANE TRIGONOMETRY. 
Plane Trigonometry teaches how to obtain all the parts of a plane 
triangle, three numerically expressed parts being given, one of which must 
always be a side. Since every rectilineal figure may be divided into 
triangles, trigonometry serves for the determination of all rectilineal 
figures. Geometry gives directly but a single example, viz. the deter- 
mination of the third side of a right angled triangle, knowing the other two. 
To obtain this result we square the numbers expressing the lengths of the 
known sides, add them together, if the hypothenuse is desired, or subtract 
the less from the greater, for one of the legs. The square root of the 
result will be the length of the third side. 
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