SECTION III. AREA ON A SPHERICAL SURFACE

279. Generation of the Surface of a Sphere. It has been shown (§ 188) that the rotation of a semicircle about its diameter generates a sphere. It follows that the surface generated by the semicircumference is a spherical surface, and from this fact the area of a spherical surface can be found.

280. Zones. That part of the surface of a sphere which is included between two parallel planes that meet the surface is called a zone. Either of the planes may be a tangent or a secant plane, so the zone can be bounded by two circumferences on the surface, by but one circumference when one plane is tangent, or it can include the whole surface. The perpendicular between the planes is the altitude of the zone.

The zones on the surface of the earth are familiar examples.

281. Generation of a Zone. The rotation of any arc of a semicircle about its diameter generates a zone, for all points of the surface so generated are equidistant from the center of the circle, and the surface lies between planes perpendicular to the diameter.

282. If, in the generating arc of a zone, a broken line is inscribed whose vertices divide the arc into equal parts, then, as the number of these parts is increased indefinitely, the area generated by the broken line approaches the area of the zone as its limit.

This is practically an axiom of the sphere, although it is evident from the plane geometry discussion of the regular polygon inscribed in a circle. It may be called a proposition, but in any case it is assumed here without proof.

283. Theorem XVII. The area of a zone is equal to the product of its altitude and the circumference of a great circle. Area = 2 πrh.

For, if a broken line is inscribed, as explained in § 282, the area generated by any one sect will be 2 πah', where a is the apothem to the broken line, and h' is the altitude of the frustum generated by that sect. (See § 171, the perpendicular in that formula being the apothem in this figure). On adding these areas for all the sects, and representing the sum of the altitudes, or the altitude of the zone, by h, the total area generated by the broken line is 2 Tah. But if the number of parts is increased indefinitely, the surface generated by the broken line approaches the zone, and the apothem approaches the radius of the sphere. Therefore the area of the zone is 2 πrh.

284. COR. The area of a spherical surface is the product of its diameter and the circumference of a great circle. Area = 4 πr2.

For it is a zone of what altitude?

257. Zones on the same sphere are proportional to their altitudes.

258. Find the area of a sphere of radius 2 inches. How does it compare with the area of a great circle? How does the area of any zone compare with the area of a great circle?

259. Find the area of a zone on a sphere of radius 1 ft. if the polar distance of one of its circles is 30° and of the other is 60°.

tions, and these angles are equal. (Why?) In the figure, two lunes are shown, one bounded by PXP' and PYP', the other by PYP' and PZP'; their angles are marked A and B.

286. Addition and Subtraction of Lunes. Two lunes are added by placing them so that they lie on opposite sides of a common semicircumference, the entire figure so formed being their sum. They are subtracted by placing them on the same side of a common semicircumference, that part of the larger not contained in the smaller being their difference.

The sum or the difference of two lunes is also a lune, – since it is still bounded by semicircumferences, as is also any multiple of a lune, for it could be formed by successive additions of equal lunes.

287. Equality of Lunes. If, on the same spherical surface, one lune is placed on another with one bounding semicircumference in common, the other circumferences either coincide, or one of them lies entirely within the lune bounded by the other. There is, therefore, no difference between equivalence and congruence of lunes, and the word equal will be used for them.

288. Notation for Lunes. In dealing with lunes, the lune is designated by the letter L with a subscript denoting the spherical angle at the point of intersection of the

semicircumferences; for example, LA denotes the lune whose spherical angle is A; L180 denotes the lune whose spherical angle is 180°, — in other words, a hemispherical surface. That this is a satisfactory way in which to denote a lune will be evident from the following paragraphs.

289. Relations between Lunes and their Spherical Angles. From the foregoing discussion, and the fact that a spherical angle has the same measure as the dihedral angle between the circles whose arcs are its arms, the following propositions about lunes follow.

* 290. Lunes are equal if their spherical angles are equal.

LA=LB if A= B.

Superpose.

* 291. If two lunes are equal, their spherical angles are equal.

If LA LB, then A = B.

=

*292. The spherical angle of the sum, or of the difference, of two lunes is the sum, or the difference, respectively, of their spherical angles.

LALB=L(A+B)*

*293. The spherical angle of any multiple of a lune is the same multiple of its spherical angle.

nLA= LnA

*294. Lunes are proportional to their spherical angles.

=

LA A

LB

B

Use the ordinary method of proof for the commensurable

case.

295. Theorem XVIII.

The area of a lune is the same part of the area of the sphere that its spherical angle is of 360°.

260. Find the area of a lune of angle 20° on a sphere of radius 18 inches.

261. Find the angle of a lune that is one fifth of the spherical surface.

262. Find the angle of a lune that is equivalent to a trirectangular triangle (one whose angles are all right angles).

263. A lune on a sphere of radius 10 in. is equivalent to the whole surface of a sphere of radius 4 in. What is its angle?

296. Theorem XIX. A spherical triangle is equivalent to a lune whose angle is one half the spherical excess of the triangle.

Call the triangle ABC, T. Let its spherical excess be E, and letter the intersections of the great circum

ferences that form its sides as in the figure. Then the semisurface ACA'C (or L180°) is composed of four triangles, whose areas are as follows:

▲ ABC = T; ▲ AC'B = Lc

C

B

Α

Β'

T; ▲ A'CB = LA

T;

A A'BCA AB'C (opposite triangles), .. Lg — T.

=

LB

Therefore on adding these values of the four triangles,