Cor. 2. Two great circles always bisect each other; for their common intersection, passing through the centre, is a diameter.

Cor. 3. Every great circle divides the sphere and its surface into two equal parts; for, if the two hemispheres were separated, and afterwards placed on the common base, with their convexities turned the same way, the two surfaces would exactly coincide, no point of the one being nearer the centre than any point of the other.

Cor. 4. The centre of a small circle, and that of the sphere, are in the same straight line perpendicular to the plane of the little circle.

Cor. 5. Small circles are the less, the farther they lie from the centre of the sphere; for the greater CO is, the less is the chord AB, the diameter of the small circle AMB.

Cor. 6. An arc of a great circle may always be made to pass through any two given points in the surface of the sphere; for the two given points and the centre of the sphere make three points, which determine the position of a plane. But if the two given points were at the extremities of a diameter, these two points and the centre would then lie in one straight line, and an infinite number of great circles might be made to pass through the two given points.

Now, if we conceive the semicircle NES to revolve about the diameter NS, the point E will describe the equator, while the points A and B will describe parallels of latitude; and as all circumferences are divided into 360°, it follows that the length of a degree of longitude at the latitude of A, is to a degree of longitude at the latitude B, as AC is to BD; and, in general, degrees of longitude at different places of the earth, are as their distances from the axis of revolution. The length of a degree of longitude at latitude 60o, is just one half the length of a degree of longitude at the equator.

PROPOSITION IX.

THEOREM. The surface of a sphere is equal to its diameter multiplied by the circumference of a great circle.

It is first to be shown, that the diameter of a sphere, multiplied by the circumference of its great circle, can not measure the surface of a larger sphere. If possible, let

D

MA

P

BS E

AB X circ. AC be the surface of the sphere whose radius is CD.

About the circle whose radius is CA, circumscribe a

regular polygon having an even number of sides, so as not to meet the circumference whose radius is CD : let M and S be the two opposite vertices of this polygon; and about the diameter MS, let the half polygon MPS be made to revolve. The surface described by this polygon will be measured (B. VIII, Prop. vii, Cor. 3) by MS X circ. AC; but MS is greater than AB; hence the surface described by this polygon is greater than AB X circ. AC, and consequently greater than the surface of the sphere whose radius is CD; but the surface of the sphere is greater than the surface described by the polygon, since the former envelopes the latter on all sides. Hence, in the first place, the diameter of a sphere multipled by the circumference of its great circle cannot measure the surface of a larger sphere.

Neither can this same product measure the surface of a smaller sphere. For, if possible, let DE × circ. CD be the surface of that sphere whose radius is CA. The same construction being made as in the former case, the surface of the solid generated by the revolution of the half polygon will still be equal to MS × circ. AC; but MS is less than DE, and circ. AC is less than circ. CD; hence, for these two reasons, the surface of the solid described by the polygon must be less than DE X circ. CD, and therefore less than the surface of the sphere whose radius is AC; but the surface described by the polygon is greater than the surface of the sphere whose radius is AC, because the former envelopes the latter. Hence, in the second place, the diameter of a sphere multiplied by the circumference of its great circle cannot measure the surface of a smaller sphere.

Therefore the surface of a sphere is equal to its diameter multiplied by the circumference of its great circle.

Cor. The surface of the great circle is measured by multiplying its circumference by half the radius, or by a fourth of the diameter; hence the surface of a sphere is four times that of its great circle.

PROPOSITION X.

THEOREM. The surface of any spherical zone is equal to its altitude multiplied by the circumference of a great circle.

Let EF be any arc less or greater than a quadrant, and let FG be drawn perpendicular to the radius EC; the zone with one base, described by the revolution of the arc EF about EC, will be measured by EG X circ. EC.

For, suppose, first, that this zone is measured by something less; if possible, by EG X circ. CA. In the arc EF, inscribe a portion of a regular polygon EMNF, whose sides shall not reach the circumference described with the radius CA; and draw CI perpendicular to EM. The surface described by the polygon EMF turning about EC, will be measured by EG X eirc. CI [B. VIII, Prop. vi, Cor. 2]. This quantity is greater than EG X circ. AC, which by hypothesis is the measure of the zone described by the arc EF. Hence the surface described by the polygon EMNF must be greater than the surface described by EF the circumscribed arc; whereas this latter surface is greater than the former, which it envelopes on all sides. Hence, in the first place, the measure of any spherical zone with one base cannot be less than the altitude multiplied by the circumference of a great circle.

Secondly, the measure of this zone cannot be greater than its altitude multiplied by the circumference of a great circle. For, suppose the zone described by the revolution of the arc AB about AC to be the proposed one; and, if possible, let zone AB > AD X circ. AC. The whole surface of the sphere composed of the two zones AB, BH is measured by AH X circ. AC [B. VIII, Prop. Ix], or by AD X circ. AC + DH X circ. AC; hence, if we have zone AB > AD × circ. AC, we must also have zone BH DH X circ. AC; which cannot be the case, as is shown above. Therefore, in the second place, the measure of a spherical zone with one base cannot be greater than the altitude of this zone multiplied by the circumference of a great circle.

Hence, finally, every spherical zone with one base is