498. Scholium. The poles P and Q might lie within the triangles ABC, DEF: in which case it would be requisite to add the three triangles DQF, FQE. DQE together, in order to make up the triangle DEF; and in like manner, to add the three triangles APC, CPB, APB together in order to make up the triangle ABC: in all other respects, the demonstration and the result would still be the same.

THEOREM.

499. If the circumferences of two great circles intersect each other on the surface of a hemisphere, the sum of the opposite triangles thus formed is equivalent to the surface of a lune whose angle is equal to the angle formed by the circles.

Let the circumferences AOB, COD, intersect on the hemisphere OACBD; then will the opposite triangles AOC, BOD be equal to the lune whose angle is BOD.

For, producing the arcs OB, OD on the other hemisphere, till they meet in N, the arc OBN will be a semicircumference, and AOB one also; and taking

B

OB from both, we shall have BN=AO. For a like reason, we have DN=CO, and BD=AC. Hence the two triangles AOC, BDN have their three sides respectively equal; besides, they are so placed as to be symmetrical; hence (496.) they are equal in surface, and the sum of the triangles AOC, BOD is equivalent to the lune OBNDO whose angle is BOD.

500. Scholium. It is likewise evident that the two spherical pyramids, which have the triangles AOC, BOD for bases, are together equivalent to the spherical ungula whose angle is BOD.

THEOREM.

501.

The surface of a spherical triangle is measured by the excess of the sum of its three angles above two right angles.

Η

I

G

Let ABC be the proposed triangle: produce its sides till they meet the great circle DEFG drawn at pleasure without the triangle. By the last Theorem, the two triangles ADE, AGH are together equivalent to the lune whose angle is A, and which is measured (495.) by 2A. Hence we have ADE+AGH=2A; and for a like reason, BGF+BID= B, and CIH+ CFE=2C. But the sum of those six triangles exceeds the hemisphere by twice the triangle ABC, and the hemisphere is represented by 4; therefore twice the triangle ABC is equal to 2A+2B+2C-4; and consequently, once ABC=A+B+ C-2; hence every spherical triangle is measured by the sum of all its angles minus two right angles.

D

502. Cor. 1. However many right angles there be contained in this measure, just so many tri-rectangular triangles, or eighths of the sphere, which (494) are the unit of surface, will the proposed triangle contain. If the angles, for example, are each equal to of a right angle, the three angles will amount to 4 right angles, and the proposed triangle will be represented by 4-2 or 2; therefore it will be equal to two tri-rectangular triangles, or to the fourth part of the whole surface of the sphere.

503. Cor. 2. The spherical triangle ABC is equivalent A+B+C to the lune whose angle is -1; likewise the spheri2 cal pyramid, which has ABC for its base, is equivalent to the

spherical ungula whose angle is

A+B+C

2

-1.

504. Scholium. While the spherical triangle ABC is compared with the tri-rectangular triangle, the spherical pyramid, which has ABC for its base, is compared with the tri-rectangular pyramid, and a similar proportion is found to subsist between them. The solid angle at the vertex of the pyramid is, in like manner compared with the solid angle at the vertex of the tri-rectangular pyramid. These comparisons are founded on the coincidence of the corresponding parts. If the bases of the pyramids coincide, the pyramids themselves will evidently coincide, and likewise the solid angles at their vertices. From this, some consequences are deduced.

First. Two triangular spherical pyramids are to each other as their bases and since a polygonal pyramid may always be divided into a certain number of triangular ones, it follows that any two spherical pyramids are to each other, as the polygons which form their bases.

Second. The solid angles at the vertices of those pyramids are also as their bases; hence, for comparing any two solid angles, we have merely to place their vertices at the centres of two equal spheres, and the solid angles will be to each other as the spherical polygons intercepted between their planes or faces.

The vertical angle of the tri-rectangular pyramid is formed by three planes at right angles to each other: this angle, which may be called a right solid angle, will serve as a very natural unit of measure for all other solid angles. And if so, the same number, that exhibits the area of a spherical polygon, will exhibit the measure of the corresponding solid angle. If the area of the polygon is, for example, in other words, if the polygon is of the tri-rectangular polygon, then the corresponding solid angle will also be of the right solid angle.

THEOREM.

505. The surface of a spherical polygon is measured by the sum of all its angles, minus two right angles multiplied by the number of sides in the polygon less two.

From one of the vertices A, let diagonals AC, AD be drawn to all the other vertices; the polygon ABCDE will be divided into as many triangles minus two as it has sides. surface of each triangle is measured by the sum of all its angles minus two right angles; and the sum of the an

But the

D

A

B

gles in all the triangles is evidently the same as that of all the angles in the polygon; hence the surface of the polygon is equal to the sum of all its angles diminished by twice as many right angles as it has sides minus two.

506. Scholium. Let s be the sum of all the angles in a spherical polygon, n the number of its sides; the right angle being taken for unity, the surface of the polygon will be measured by s-2 (n-2), or s-2n+4.

BOOK VIII.

THE THREE ROUND BODIES.

Definitions.

507. A cylinder is the solid generated by the revolution of a rectangle ABCD, conceived to turn about the immoveable side AB.

In this movement, the sides AD, BC, continuing always perpendicular to AB, describe equal circles DHP, CGQ, which are called the bases of the cylinder, the side CD at the same time describing the convex surface.

The immoveable line AB is called the axis of the cylinder.

E

M

P

N

G

Every section KLM, made in the cylinder, at right angles to the axis, is a circle equal to either of the bases; for, whilst the rectangle ABCD turns about AB, the line KI, perpendicular to AB, describes a circle, equal to the ase, and this circle is nothing else than the section made perpendicular to the axis at the point I.

Every section PQGH, made through the axis, is a rectangle double of the generating rectangle ABCD.

508. A cone is the solid generated by the revolution of a right-angled triangle SAB, conceived to turn about the immoveable side SA.

In this movement, the side AB describes a circle BDCE, named the base of the cone; the hypotenuse SB describes its convex surface.

The point S is named the vertex of the cone, SA the axis or the altitude, and SB the side or the apothem.

Every section HKFI, at right angles to the axis, is a circle; every section SDE, through the axis, is an isosceles triangle double of the generating triangle SAB.

509.

If from the cone SCDB, the cone SFKH be cut off by a section parallel to the base, the remaining solid CBHF is called a truncated cone, or the frustum of a cone.

We may conceive it to be generated by the revolution of a trapezoid ABHG, whose angles A and G are right, about the side AG. The immoveable line AG is called the axis or altitude of the frustum, the circles BDC, HFK, are its bases, and BH is its side.

510. Two cylinders, or two cones, are similar, when their axes are to each other as the diameters of their bases.

511.

If in the circle ACD, which forms the base of a cylinder, a polygon ABCDE is inscribed, a right prism, con- F structed on this base ABCDE, and equal in altitude to the cylinder, is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism.

The edges AF, BG, CH, &c. of the prism, being perpendicular to the plane of the base, are evidently included in the convex surface of the cylinder; hence A the prism and the cylinder touch one another along these edges.

512. In like manner, if ABCD is a polygon, circumscribed about the base of a cylinder, a right prism, constructed on this base ABCD, and equal in altitude to the cylinder, is said to be circumscribed about the cylinder, or the cylinder to be inscribed in the prism.

Let M, N, &c. be the points of contact in the sides AB, BC, &c.; and through A the points M, N, &c. let MX, NY, &c. be drawn perpendicular to the plane of the base those perpendiculars will evi

:

W

F

dently lie both in the surface of the cylinder, and in that of the circumscribed prism; hence they will be their lines of

contact.

Note. The Cylinder, the Cone, and the Sphere, are the three round bodies treated of in the Elements of Geometry.