But

© PQ : © AB :: PI2 : AC2 :: TI. ID : TC.CD. (2)

(546. Similar figures are to each other as the squares on their corresponding

sects.)

(522. If from any point in a circle a perpendicular be dropped upon a diameter, it will be a mean proportional between the segments of the diameter.)

By hypothesis and construction, in proportions (1) and (2), the first, third, and fourth terms are respectively equal,

645. The sects joining its pole to points on any circle of a sphere are equal.

P

A

P

PROOF. (600. If through the center of a circle a line be passed perpendicular to its plane, the sects from any point of this line to points on the circle are equal.)

646. COROLLARY. Since chord PA equals chord PB, therefore the arc subtended by chord PA in the great circle PA equals the arc subtended by chord PB in the great circle PB.

Hence the great-circle-arcs joining a pole to points on its circle are equal.

So, if an arc of a great circle be revolved in a sphere about one of its extremities, its other extremity will describe a circle of the sphere.

647. One-fourth a great circle is called a Quadrant.

648. The great-circle-arc joining any point in a great circle with its pole is a quadrant.

649. If a point P be a quadrant from two points, A, B, which are not opposite, it is the pole of the great circle through A, B ; for each of the angles POA, POB, is right, and therefore PO is perpendicular to the plane OAB.

650. The angle between two intersecting curves is the angle between their tangents, at the point of intersection.

When the curves are arcs of great circles of the same sphere, the angle is called a Spherical Angle.

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).

A

H

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.

A

B

A

B

F

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.