747. If two angles of a spherical triangle be equal, the sides opposite these angles are equal, and the triangle is isosceles.

B

In the spherical ▲ A B C, let B = C.

(in two polar & each side of one is the supplement of the lying opposite to

(in an isosceles spherical ▲, the opposite the equal sides are equal).

PROPOSITION XXVII. THEOREM.

748. In a spherical triangle the greater side is opposite the greater angle; and, conversely, the greater angle is opposite the greater side.

I.

In the ▲ A B C, let ▲ A B C > ≤ C.

We are to prove

ACA B.

Draw the arc BD of a great circle, making ≤ C B D = ≤ C.

Then

DC=DB,

§ 747

(if two of a spherical ▲ be equal the sides opposite these are equal).

$722

(the sum of two sides of a spherical ▲ is greater than the third side).

But both of these conclusions are contrary to the hypothesis.

..ZABC > < C.

Q. E. D.

749. On unequal spheres mutually equiangular triangles are similar.

C

From 0, the common centre of two unequal spheres, draw the radii O A, O B and O C cutting the surface of the smaller sphere in a, b and c. Draw arcs of great circles, A B, AC, BC, ab, a c, bc. AABC similar to ▲ abc.

We are to prove

A, B, C are equal respectively to a, b, c,

(since the corresponding dihedrals in each case are the same).

In the similar sectors A O B and a Ob,

AB:ab:: A0: a0;

§ 385

and in the similar sectors AOC and a Oc,

AC ac: AO: a 0.

§ 385

.. AB: ab :: AC: a c.

In like manner, A B : a b :: B C : b c.

That is, the homologous sides of the two A are proportional, and their homologous are equal.

.. AABC is similar to ▲ a b c.

Q. E. D.

750. SCHOLIUM. The statement that mutually equiangular spherical A are mutually equilateral, and equal, or symmetrical and equivalent, is true only when limited to the same sphere, or equal spheres. But when the spheres are unequal, the spherical ▲ are similar, but not equal. Hence, to compare two similar spherical A, it is necessary to know the linear extent of two homologous sides; or, what is equivalent, to know the radii of the spheres. And, as in the case of plane A, two similar spherical A have the same ratio as the squares of the linear measures of any two homologous sides, and therefore as the squares of the radii of the spheres.

ON COMPARISON AND MEASUREMENT OF SPHERICAL SURFACES.

751. DEF. A Lune is a part of the surface of a sphere included between two semi-circumferences of great circles.

752. DEF. The Angle of a lune is the angle included by the semi-circumferences which forms its boundary. Thus C AB is the angle of the lune.

753. DEF. A Spherical Ungula, or Wedge, is a part of a sphere bounded by a lune and two great semicircles.

754. DEF. The Base of an ungula

is the bounding lune.

755. DEF. The Angle of an ungula is the dihedral of its bounding semicir

B

cles, and is equal to the angle of the bounding lune.

Α

E

756. DEF. The Edge of an ungula is the edge of its angle. 757. DEF. The Spherical Excess of a spherical triangle is the excess of the sum of its angles over two right angles.

C

D

758. DEF. Three planes which pass through the centre of the sphere, each perpendicular to the other two, divide the surface of the sphere into eight tri-rectangular triangles. Thus B the three planes CADB, CEDF and AEBF divide the surface of the sphere into the eight tri-rectangular triangles C E B, D E B, C B F, D B F,

etc.

As in Plane Geometry the whole angular magnitude about any point in a plane is divided by two straight lines perpendicular to each other into four right angles, and each right angle is measured by a quadrant, or fourth part of a circumference described about that point as a centre with any given radius; so, if, through a point in space, three planes be made to pass perpendicular to one another, they will divide the whole angular magnitude about that point into eight solid right angles, each of which is measured by an eighth part of the surface of a sphere described about that point with any given radius.

And, as in Plane Geometry, each quadrant which measures a right angle is divided into 90 equal parts called degrees, so each of the eight tri-rectangular spherical triangles is divided into 90 equal parts called degrees of surface. Hence, the whole surface of the sphere is divided into 720 degrees of surface.

PROPOSITION XXIX. LEMMA.

759. The area of the surface generated by the revolution of a straight line about another line in the same plane with it as an axis, is equal to the product of the projection of the line on the axis by the circumference whose radius is perpendicular to the revolving line erected at its middle point and terminated by the axis.

Let the straight line A B revolve about the axis Y Y' in the same plane; let EF be its projection on the axis; and CO the perpendicular to A B at its middle point C, and terminated in the axis.

We are to prove area A B

The surface generated by A B is the lateral surface of the frustum of a cone of revolution.

Draw CHL, and A D I, to Y Y'.

area A B АВХ 2 п СН,

$ 662

(the lateral area of a frustum of a cone of revolution is equal to the slant height multiplied by the circumference of a section equidistant from its

bases).

The ABD and COH are similar;

.. AD : ABCH: CO.

But CH : СО :: 2 п С Н : 2 п СО,

§ 287

§ 375

(circumferences of have the same ratio as their radii).
.. ADAB :: 2 π CH: 2T CO.

760. SCHOLIUM. If either extremity of A B be in the axis YY', AB generates the lateral surface of a cone of revolution; and if AB be parallel to the axis Y Y', it generates the lateral area of a cylinder of revolution. In either case the formula holds good.