hypothesis, the arch PF is equal to the arch PH, therefore (E. 29. 3.) the chord PF is equal to the chord PH; and two sides of the triangle PFC are equal to two sides of PHC, and a third side CP is common to both the triangles: wherefore, the angle FCP is equal to HCP (E. 8. 1.): again, since the angle FCG is equal to HCG, and that FC, CG are equal to HC, CG, in the triangles GFC, GHC, therefore (E. 4. 1.) GF is equal to GH. And, if E be any other point in the circumference of the circle EFH, and G, E be joined, it may be shewn in the same manner, that GE is equal to GF, or to GH. Therefore (E. 9. 3.) G is the center, and (Art. 22.) CGP is the axis, and P the pole, of the circle EFH.

(32.) COR. If any number of great circles pass through the common poles of two parallel circles, in a sphere, the arches of the great circles, intercepted between the parallels, shall be equal to one another.

(33.) COR. 2.

sphere, bisects the

The pole of any given circle, in a arch of a great circle subtended by the diameter of the given circle (Art. 20. 21. and E. 30. 3.).

(34.) DEF. Any two points having been taken on the surface of a sphere, their Direct Distance is the straight line which joins them: and their Spherical Distance is the arch of a great circle which joins them :

Also, the distance of the pole of any circle, in a sphere,

from any point in the circumference of the circle, measured by an arch of a great circle of the sphere, is called the Polar Distance of that circle.

(35.) DEF. The fourth part of the circumference of a great circle, in a sphere, is called a Quadrant.

PROP. VIII.

(36.) Theorem. The polar distance of a great circle, in a sphere, is a quadrant; and its chord is equal to the side of a square inscribed in a great circle: And if the polar distance of a given circle in a sphere be a quadrant, the circle is a great circle.

For first, since the axis is (Art. 20.) perpendicular to the plane of the great circle, it is manifest that the polar distance subtends a right angle: therefore (E. 33. 6.) it is a quadrant: and (E. 4. 4.) its chord is equal to the side of a square; inscribed in a great circle of the sphere.

Secondly, if the polar distance, which (Art. 33.) is the same at all points of the circumference of the circle, be a quadrant, it is evident that any diameter of the given circle will bisect the great circle, which passes both through its axis, and through that diameter: It must, therefore, itself be a great circle of the sphere.

PROP. IX.

(37.) Theorem. If a point, on a sphere's surface, be at the distance of a quadrant from more than one point

in the circumference of a given great circle of the sphere, it is a pole of that great circle.

Let the point P, on the surface of the sphere PAPD, of which the center is C, be at the spherical distance of

a quadrant, from any two points, A and B, in the circumference of the great circle ABD, that is, let the arches of great circles PA and PB be quadrants: then is P the pole of ABD.

For, join C, A and C, B; and since CA and CB are in the plane of ABD, and that (E. 33. 6.) the angles PCA, PCB are right angles, therefore (E. 4. 11.) PC is at right angles to ABD; and consequently (Art. 20. 21.) P is the pole of the great circle ABD.

PROP. X.

(38.) Theorem. The polar distances of equal circles, in a sphere, are equal to one another: And circles in a sphere, which have equal polar distances, are equal to one another.

If the circles be great circles, the polar distances are equal, because (Art. 36.) each of them is a quadrant.

But, let EH and Al be two lesser circles in the sphere EAH and, first, let the circle EH be equal to

AI: the polar distance of EH is equal to the polar distance of AI:

For, let EH and AI be any diameters of the circles EH and AI: and let EPH and AQI be arches of the great circles of the sphere which pass through the diameters EH and AI respectively. Then (Art. 29.) a pole of EH is in EPH, and a pole of AI is in AQI.

And, since (Art. 7.) all great circles, in a sphere, are equal to one another, and that the diameter EH is equal to the diameter AI, because, by the hypothesis, the circle EH is equal to the circle AI, therefore, (E. 28. 3.) the archEPH is equal to the arch AQI: but (Art. 33.)

the poles of EH and Al are in the bisections of the arches EPH and AQI: since, therefore, the arches EPH and AQI are equal to one another, their halves, that is, the polar distances of the circles EH and Al, aré equal to one another.

The converse proposition is proved, in like manner, by the help of E. 29. 3.

PROP. XI.

(39.) Theorem. If a great circle cut any other circle, in a sphere, at right angles, any two points, in the great circle's circumference, that are equally distant from the two extremities of the common section of the circles, are at equal distances from two points in the circumference of the other circle, that are also equally distant from the two extremities of the common section of the circles and conversely.

Let the great circle ADB cut any other circle in the

same sphere, as AFGB at right angles, and let AB be