651. If from the vertices, A and F, of any two angles in a sphere, as poles, great circles, BC and GH, be described, the angles will be to one another in the ratio of the arcs of these circles intercepted between their sides (produced if necessary).

PH

For the angles A and F are equal respectively to the angles BOC and GOH.

(591. Parallels intersecting the same plane are equally inclined to it.)

But BOC and GOH are angles at the centers of equal circles, and therefore, by 506, are to one another in the ratio of the arcs BC and GH.

652. COROLLARY. Any great-circle-arc drawn through the pole of a given great circle is perpendicular to that circle.

FG is 1 GH. For, by hypothesis, FG is a quadrant; therefore the great circle described with G as pole passes through F, and so the arc intercepted on it between GF and GH is, by 648, also a quadrant. But, by 651, the angle at G is to this quadrant as a straight angle is to a half-line.

Inversely, any great-circle-arc perpendicular to a great circle will pass through its pole.

For if we use G as pole when the angle at G is a right angle, then FH, its corresponding arc, is a quadrant when GF is a quadrant; therefore, by 649, F is the pole of GH.

THEOREM VIII.

653. The smallest line in a sphere, between two points, is the great-circle-arc not greater than a semicircle, which joins them.

HYPOTHESIS. AB is a great-circle-arc, not greater than a semicircle, joining any two points A and B on a sphere.

FIRST, let the points A and B be joined by the broken line ACB, which consists of the two great-circle-arcs AC and CB.

CONCLUSION. AC + CB > AB.

PROOF. Join O, the center of the sphere, with A, B, and C.

AOC+ COB > AOB.

(603. If three lines not in the same plane meet at one point, any two of the angles formed are together greater than the third.)

But the corresponding arcs are in the same ratio as these angles,

.. AC+ CB > AB.

SECOND, let P be any point whatever on the great-circle-arc AB. The smallest line on the sphere from A to B must pass through P. For by revolving the great-circle-arcs AP and BP about A and B as poles, describe circles.

These circles touch at P, and lie wholly without each other; for let F be any other point in the circle whose pole is B, and join FA, FB by great-circle-arcs, then, by our First,

FA FB AB,

:. FA> PA, and F lies without the circle whose pole is A.

Now let ADEB be any line on the sphere from A to B not passing through P, and therefore cutting the two circles in different points, one in D, the other in E. A portion of the line ADEB, namely, DE, lies between the two circles. Hence if the portion AD be revolved about A until it takes the position AGP, and the portion BE be revolved about B into the position BHP, the line AGPHB will be less than ADEB. Hence the smallest line from A to B passes through P, that is, through any or every point in AB; consequently it must be the arc AB itself.

654. COROLLARY. A sect is the smallest line in a plane between two points.

BOOK IX.

TWO-DIMENSIONAL SPHERICS.

INTRODUCTION.

655. Book IX. will develop the Geometry of the Sphere, from theorems and problems almost identical with those whose assumption gave us Plane Geometry. In Book VIII., these have been demonstrated by considering the sphere as contained in ordinary tri-dimensional space. But, if we really confine ourselves to the sphere itself, they do not admit of demonstration, except by making some more difficult assumption: and so they are the most fundamental properties of this surface and its characteristic line, the great circle; just as the assumptions in our first book were the most fundamental properties of the plane and its characteristic line.

So now we will call a great circle simply the spherical line; and, whenever in this book the word line is used, it means spherical line. Sect now means a part of a line less than a half-line.