(T. XI., C. III.). Hence at the limit we have, for the measure of the spherical sector generated by the plane sector CDF, 3 x CA2 × LM' = 2 × CA × LM × CA. But 2 × CA × LM is the measure of the surface of the zone forming the base of the spherical sector; hence the volume of a spherical sector is measured by its surface into one third of its radius.

Scholium. Let R be the radius of a sphere: its surface will be 4R2; its solidity, 47R2 × R, or . R3. If the diameter is named D, we shall have R = }D, and R3 = {D3: hence the solidity may likewise be expressed by . D3, or «D3.

THEOREM XIII.

The surface of a sphere is to the whole surface of the circumscribed cylinder (including its bases), as 2 is to 3; and the volumes of these two bodies are to each other in the same ratio.

Let MPNQ be a great circle of the sphere, and ABCD the circumscribed square. If the semicircle PMQ and the half square PADQ are at the same time. made to revolve about the diameter PQ, the semicircle will generate the sphere, while the half square will generate the cylinder circumscribed about that sphere.

C

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The altitude AD of that cylinder is equal to the diameter PQ; the base of the cylinder is equal to the great circle, its diameter AB being equal to MN: hence (T. I.), the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter. This measure (T. IX.) is the same as that of the surface of the sphere: hence, the surface of the sphere is equal to the convex surface of the circumscribed cylinder.

But the surface of the sphere is equal to four great circles; hence the convex surface of the cylinder is also equal to four great circles; and adding the two bases, each equal to a great circle, the total surface of the circumscribed cylinder will be equal to six great circles: hence the surface of the sphere is to the total surface of the circumscribed cylinder as 4

is to 6, or as 2 18 to 3, which is the first branch of the proposition.

In the next place, since the base of the circumscribed cylinder is equal to a great circle, and its altitude to the diameter, the volume of the cylinder (T. II.) will be equal to a great circle multiplied by its diameter. But (T. XII.) the volume of the sphere is equal to four great circles multiplied by a third of the radius; in other terms, to one great circle multiplied by of the radius, or by of the diameter. Hence the sphere is to the circumscribed cylinder as 2 to 3, and consequently the volumes of these two bodies are as their surfaces.

EIGHTH BOOK.

SPHERICAL GEOMETRY.

DEFINITIONS.

I. A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.

Those arcs, named the sides of the triangle, are always supposed to be each less than a semi-circumference; the angles, which their planes form with each other, are the angles of the triangle.

II. A spherical triangle takes the name of right-angled, isosceles, equilateral, in the same cases as a rectilineal triangle. III. A spherical polygon is a portion of the surface of a sphere, terminated by several arcs of great circles.

IV. A lune is that portion of the surface of a sphere, which is included between two great semicircles meeting in a common diameter.

V. A spherical wedge, or ungula, is that portion of the solid sphere which is included between the same great semicircles, and has the lune for its base.

VI. A spherical pyramid is a portion of the solid sphere, included between the planes of a polyedral angle whose vertex is the centre; the base of the pyramid is the spherical polygon intercepted by the same planes.

VII. The pole of a circle is a point on the surface of the sphere equally distant from all the points of the circumference.

THEOREM I.

In every spherical triangle, any side is less than the sum of the other two.

Let O be the centre of the sphere; and draw the radii OA, OB, OC. Imagine the planes AOB, AOC, COB; those planes will form a polyedral angle at the point 0; and the angles AOB, AOC, COB will be measured by AB, AC, BC, the sides of the spherical triangle. But (B. V., T. XIX.) each of the three plane triangles composing a polyedral angle is less than the sum of the other

B

A

two; hence, any side of the triangle ABC is less than the sum of the other two.

THEOREM II.

The sum of all the three sides of a spherical triangle is less than the circumference of a great circle.

Let ABC be any spherical triangle; produce the sides AB, AC till they meet again in D. The arcs ABD, ACD will be semi-circumferencessince (B. VII., T. VI., C. II.) two great circles always bisect each other. But in the triangle BCD we have (T. I.) the side BC <BD+CD: add AB+ AC

D

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to each; we shall have AB+ AC+ BC < ABD + ACD, that is to say, less than a circumference.

THEOREM III.

The sum of all the sides of any spherical polygon is less than the circumference of a great circle.

Let us take, for example, the pentagon ABCDE. Produce the sides AB, DC till they meet in F; then, since BC is less

AFG, which last is itself less than the circumference of a great circle (T. II.): hence the perimeter of the polygon ABCDE is less than this same circumference.

THEOREM IV.

If a diameter be drawn perpendicular to the plane of a great circle, its extremities will be the poles of that circle, and also of all small circles parallel to it.

B

I

F

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For, DC being perpendicular to the plane AMB, is perpendicular to all the straight lines CA, CM, CB, etc., drawn through its foot in this plane; hence all the arcs DA, DM, DB, etc., are quarters of the circumference. So likewise are all the arcs EA, EM, EB, etc.; hence the points D and E are each equally distant from all the points of the circumference AMB; therefore (D. VII.) they are the poles of that circumference.

E

Again, the radius DC, perpendicular to the plane AMB, is perpendicular to its parallel FNG; hence (B. VII., T. VI., C. IV.) it passes through O, the centre of the eircle FNG; therefore, if the oblique lines DF, DN, DG be drawn, they will be equal (B. V., T. V.); but, the chords being equal, the arcs are equal: hence the point D is the pole of the small circle FNG; and, for like reasons, the point E is the other pole.

Cor. I. Every arc DM drawn from a point in the arc of a great circle AMB to its pole, is a quarter of the circumference, which, for the sake of brevity, is usually named a quadrant; and this quadrant, at the same time, makes a right angle with the arc AM. For (B. V., T. XVI.), the line DC being perpen