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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 circ. AC; but MS is greater than AB; hence the surface described by this polygon is greater than ABX 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 poly-· gon, since the former envelops the latter on all sides. Hence, in the first place, the diameter of a sphere multiplied 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 DEx 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× 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 envelops 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.

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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 × circle 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 EGx circ. CI, [B. VIII, Prop. vi, Cor. 2.] This quantity is greater than EG× 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 envelops 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>ADx 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 ADx circ. AC+DH× 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 measured by its altitude multiplied by the circumference of a great circle.

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DH and DF; the latter are respectively measured by DQ x circ. CD, and DOx circ. CD: hence the zone described by FH has for its measure

(DQ-DO) x circ. CD, or OQ x circ. CD.

That is, any spherical zone, with one or two bases, is measured by its altitude multiplied by the circumference of a great circle.

Cor. Two zones, taken in the same sphere or in equal spheres, are to each other as their altitude; and any zone is to the surface of the sphere, as the altitude of that zone is to the diameter.

(110.) Let ABCDF be a section of the earth, made by a plane passing through its axis, which is represented by AF; also, let B denote the place where the arctic circle cuts the meridian ABCDF; C and D, the corresponding points for the tropics: then will the line BG, perpendicular to the axis AF, be the radius of the arctic circle, and CH and DK, which are equal, will be the radii of the tropics.

H

K

B

The line AG will denote the altitude of the frigid zone, GH the

altitude of the temperate zone, and HK the altitude of the torrid zone; but surfaces of zones are to each other as their altitudes; therefore the frigid, temperate, and torrid zones are to each other as the lines AG, GH and HK.

If we assume the latitude of the arctic circle to be 66° 32′, and the tropics to be in latitude 23° 28', we shall find the lines AG, GH and HK to be to each other as 1-cos 23° 28', cos 23° 28′-sin 23° 28', 2 sin 23° 28′; or as 0·08271, 0·51907, 0.79644; or nearly as the numbers 4, 25, 38.

(111.) PROBLEM. Suppose a person to be situated h miles above the surface of the earth, and let it be required to find what fractional part of the earth's entire surface is visible to him.

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Hence the fractional part of the earth's surface seen, is

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possible, from any point, however distant, to see half the entire

surface of a sphere.

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