can it be less than the zone; for then it must be less also than (some inscribed surface-less, that is, than the product of AL and the circumference which has the radius C E; which is impossible, because the circumference ADBE is greater than that which has the radius CE. Therefore, it must be equal to the zone; that is, the zone is equal to the product of the circumference AD EB, and the part A L of the diameter.

Next, let HK be any arc, by the revolution of which about the diameter and let HL, KN be drawn perpendiA B a double-based zone is generated; cular to AB. Then, because the whole zone generated by the arc A K is equal to the product of AN and the cir

about the diameter A B a spherical zone is generated, and from K draw KL perpendicular to AB: the zone shall be equal to the product of the whole circumference ADBE by the part A L of the diameter.

The demonstration is in every respect similar to that of prop. 13. For, in the first place, it is evident that the arc AK may be divided into a number of equal arcs, such that, the chords AF, FG, &c. being drawn, their common distance CE from the centre C shall approach to the radius CD within any given difference; and hence it may be shown, as in prop. 15., that there may be inscribed in the zone and circumscribed about it, two surfaces of revolution which differ from each other, and therefore (19.) each of them from the zone, by less than any given difference. Therefore, the product in question cannot be greater than the zone; for then it must be greater also than some circumscribed surface-greater, that is, than the product of MQ and the circumference ADBE; which is impossible, because AL is less than MQ. Neither

If the point Q lies between A and C, MQ will be the difference, not the sum, of M L, and LQ; but, in this case also, A L is less than M Q, because LQ is less than K P, that is, than M A.

cumference ADBE, and the part generated by the arc AH equal to the product of AL and the same circumference, the remainder, that is, the double-based zone in question, is equal to the product of L N and the same circumference.

Therefore, &c.

Cor. 1. If a cylinder, having the axis A B, be circumscribed about the sphere; any zone having the same axis, shall be equal to that portion of the convex surface of the cylinder which is intercepted between the base of the cylinder and the plane of the base of the zone, or between the planes of its two bases, if it be double-based (3.).

Cor. 2. In the same or in equal spheres, any two zones are to one another as the parts of the axis or axes which are intercepted between their respective bases (II. 35.).

duct in question cannot be greater than the sector; for then would it be greater than some circumscribed solid, and therefore (12.) the base of the sector greater than the convex surface of such a solid, which (19.) is impossible: neither can it be less, for then would it be less also than some inscribed solid; which is impossible, because, not only is the base of the sector greater (19.) than the convex surface of such a solid, but the radius C H is likewise greater than the perpendicular CE. Therefore it must be equal to the sector. Therefore, &c.

PROP. 22.

Every spherical segment, upon a single base, is equal to the half of a cylinder having the same base and the same altitude, together with a sphere, of which that altitude is the diameter.

F

D

A

E

G

C

B

Let ADFG be any circular halfsegment, by the revolution of which about the diameter AB, a spherical segment having the altitude AG is generated the spherical segment shall be equal to the half of a cylinder having the same base and altitude, together with a sphere, of which A G is the diameter. Join AF; and from the centre C draw CE perpendicular to A F. Then, because the solid, generated by the segment A D F, is equal to the difference of the solids generated by the sector CADF and the triangle CAF, the former of which (21.) is equal to onethird of the product of CA by its base AGX 2X CA (20.), and the latter (12.) to one-third of the product of CE by the convex surface generated by AF viz. (12.) AG × 2 × CE; the solid, generated by the segment AD F, is equal to≈ × AG (C Ao – CE2), i. e. × AG × AE, or to ×AG × AF, because (I. 36. Cor. 1.) CA CE is equal to A E2, and AE2 (III. 3.) to a fourth of AF. To this add the cone generated by the triangle A FG, which (9.) is equal to x AG FG: therefore, the spherical segment in question is equal to XAG (A F2+2 F G2), i. e. tox AG (3 FG +AG) because AF is equal to AG2+ FG (III.36.). And XAG(3 FG2 + AG) is equal to AGX FG2 +

AG3; the first part of which is (4. Cor. 1.) the half of a cylinder having the altitude AG and the same base

with the segment, and the other part (17. Cor. 4.) a sphere of which A G is the diameter. Therefore, &c.

PROP. 23.

Every double-based spherical segment is equal to the half of a cylinder having the same altitude with the seg ment and a base equal to the sum of its two bases, together with a sphere of which that altitude is the diameter.

Let FH KG be any portion of the semicircle AD B, by the revolution of which about the diameter A B, a doublebased segment, having the altitude G K, is generated: the segment shall be equal to the half of a cylinder having the same altitude and a base equal to the sum of its two bases, together with a sphere of which G K is the diameter.

For, in the first place, because the solid, generated by the revolution of the circular segment FH, is equal to the excess of the difference of the sectors generated by CAH and CAF above the solid generated by the tri

H

K

B

angle CFH, it may be shown by the same steps as in the last proposition, that the solid generated by the revolution of FH is equal to

2

GK x FH2, i. e. if FL be drawn parallel to GK, tox GK (FL2 + LH). But (I. 22.) FL is equal to G K, and LH is equal to the difference of F G and H K (I. 22.): therefore (I. 33.), LH is equal to FG2+HK-2 FGX HK, and GK (FL+LH), or the solid generated by the segment FH, is equal to } ≈ ×G K (FG2+H K2 − 2 FG xHK) + ×G K3. To this add the truncated cone generated by the trapezoid FG KH; which (11.) is equal to × GK (FG2 + H K2 + F G × HK): therefore, the double-based segment in question is equal to GK (3 F G2 + 3 H K2) + ¦ ≈ + G K3; or to GK (F G2 + H K2) + } s × G K3; the first part of which is the half of a cylinder (4. Cor. 1.) having the altitude G K, and a base × FG + H K2, equal to the sum of the bases of the segment, and the other part a sphere of which G K is the diameter (17. Cor. 4.).

Therefore, &c.

Cor. It appears from the demonstrations of this and the preceding proposition, that the solid generated by the re

its altitude equal to the radius of that surface, this pyramid will be equal to the whole sphere (IV. 32. Cor. 1. and 17.). And if, from this, there be cut off, by a plane parallel to the base, a pyramid, having its altitude equal to the radius of the interior surface, the two pyramids will be to one another as the cubes of any two homologous edges (IV. 34. Cor.); or, since it may be shown that their altitudes are to one another in the same ratio with the homologous edges, as the cubes of their altitudes, (IV. 27. Cor. 3.), that is, as the cubes of the radii of the spheres, or (18.) as the spheres. Therefore, because the larger pyramid is equal to the larger sphere, the smaller pyramid is equal to the smaller sphere (II. 18.); and the difference of the two pyramids is equal to the difference of the two spheres, that is, the frustum is equal to the spherical orb. And, because the larger base of the frustum is equal to the surface of the larger sphere, it may be shown that its smaller base is equal to the surface of the smaller sphere, exactly in the same manner as it has been already shown, that the content of the smaller pyramid is equal to the content of the smaller sphere; also the altitude of the frustum is equal to the thickness of the orb. But the frustum is equal to the sum of three pyramids, having the same altitude with it, and for their bases its

AB; and from C let CD, CE be drawn in the planes A D B, A E B, perpendicular to AB (I. 44.): the ungula ADB E shall be to the whole sphere, as the angle D CE to four right angles.

For, since the plane DCE is perpendicular to A B (IV.3.), the angle, which measures the inclination of any two planes passing through AB, may be drawn in that plane at the point C (IV. 17. Schol.); and, if any two of these angles at C be equal to one another, the dihedral angles which they measure will be equal (IV.17.), and therefore the ungulas, which have those dihedral angles, may be made to coincide, and are equal to one another. Now, let the angle D C E be divided into any number of equal angles DCF, FCG, &c.; and therefore the dihedral angle D A B E into the same number of dihedral angles by the planes ACF, ACG, &c. (IV. 17.); and the ungula A DBE into as many equal ungulas A D B F, AD BG, &c. by the same planes. Then, if the angle D C F be contained in the four right angles about C any number of times exactly, or with a remainder, the ungula ADBF will be contained in the whole sphere the same number of times exactly, or with a remainder. Therefore the ungula ADBE is to the whole sphere as the angle DC E to four right angles (II. def. 7.).

We might here add the proportions of similar segments, sectors, orbs, ungulas, and of their convex surfaces. The reader will, however, easily perceive, from the demonstration of prop. 15, that if similar spherical segments and sectors be defined to be such as are generated by similar circular segments and sectors, their surfaces will be as the squares of the radii, and their contents as the cubes of the radii. And the same may be said of similar spherical orbs, defined to be such that the radii of their exterior and interior surfaces are to one another in

the same ratio; and of similar ungulas defined to be such as have their dihedral angles equal to one another.

BOOK VI.

$1. Of great and small circles of the Sphere. § 2. Of Spherical Triangles.-3. Of equal Portions of Spherical Surface, and the Measure of solid Angles.-§ 4. Problems:

§1.-Of great and small Circles of the Sphere.

Def. 1. If a sphere is cut by a plane which passes through the centre, the section is called a great circle of the sphere; the radius of such a section being the greatest possible, the same, namely, with the radius of the sphere.

From this definition it is evident that a great circle may be made to pass through any two points in the surface of a sphere; and that, if the two points be not opposite extremities of a diameter, only one great circle can be made to pass through them, for its plane must pass through the centre of the sphere, and only one plane can be made to pass through three points which are not in the same straight line (IV. 1.) But through

the two extremities of a diameter, any number of great circles may be made to pass, for they are in the same straight line with the centre of the sphere (IV. 1. Cor. 4.).

2. If a sphere is cut by a plane which does not pass through the centre, the section is called a small circle of the sphere; the radius of such a section being less than that of the sphere.

A circle, it is plain, may be made to pass through any three points in the sphere's surface; and it will be a great or a small circle, according as its plane passes through the centre of the sphere, or otherwise.

3. The axis of any circle of the sphere is that diameter of the sphere which is perpendicular to the plane of the circle; and the extremities of the axis are called the poles of the circle.

such as have their planes parallel. 4. Parallel circles of a sphere are

the same axis and poles; for a straight It is evident that parallel circles have line which is perpendicular to one of two other likewise (IV. 11.). It may also be parallel planes is perpendicular to the observed that two parallel circles canof the sphere, that is, they cannot both not both of them pass through the centre be great circles of the sphere.

These four definitions may be illustrated by referring to the figure of prop. 1. in which PAP' is a great circle, ABC a small circle, PO P' the axis,

and P, P' the poles of the circle A B C, and A'B'C', ABC are parallel circles of the sphere.

5. Any portion of the circumference of a great circle is called a spherical

arc.

Two points are said to be joined on the surface of the sphere when the spherical arc between them is described; and this arc is called the spherical distance of the two points, in order to distinguish it from their direct distance, which is the straight line which joins them. The spherical distance of opposite extremities of a diameter of the sphere is evidently half the circumference of a great circle: but the spherical distance of any other two points is less than a semicircumference, being always the lesser of the two arcs into which they divide the great circle which passes through them.

6. The polar distances of any circle of the sphere are the spherical arcs which join any point in the circumference with the two poles of the circle.

according to the opening between its containing arcs: thus the angle BAC is greater than the angle D A C by the angle BAD.

Every spherical angle is measured by the plane angle which measures the inclination of the planes of the containing arcs. For it is easy to perceive, that if this inclination is the same in any two spherical angles, they may be made to coincide, and therefore are equal to one another. If, therefore, the dihedral angle made by the planes of one spherical angle contain any sub-multiple of the dihedral angle made by the planes of another a certain number of times exactly, or with a remainder, the first spherical angle will contain a like submultiple of the other the same number of times exactly or with a remainder; and, therefore, the spherical angles are to one another (II. def. 7.) as the dihedral angles made by their planes, and have the same measures with them.*

8. When one spheri cal arc standing upon another makes the adjacent spherical angles equal to one another, each of them is called a spherical right an

gle, and the arc which stands upon the other is said to be perpendicular, or at right angles to it.

The terms acute and obtuse are likewise applied to spherical angles, in the same sense as in Book I. def. 11.

It is evident that a spherical right angle is measured by a rectilineal right angle, a spherical acute angle by a rectili

Hence a spherical angle has been defined by some writers to be identical with the dihedral angle of its planes; while others have extended to it the general definition of the angle in which two curves cut one another, considering it the same with the plane rectilineal angle of the tangents at the point A; for the latter angle, being contained by perpen

diculars to the common section OA, measures the dihedral angle of the planes.

In the spherical triangles here considered, it is supposed that each of the sides is less than a semicircumference. For, the greatest spherical distance at which two points can be placed is a semicircumference; and if any arc, as PAP', P be taken equal to a semicircumference, its extremities P, P' will be extremities of

a diameter POP' of the sphere, and therefore the same great circle will pass through both of them,

A

and any third point Q on the sphere's surface, so that the arcs Q P and Q P' will be arcs, not of different circles, but of the same circle.

Any three points on the sphere's surface may be assumed for the angles of a spherical triangle (see def. 5.), provided they are not in the same great circle, nor any two of them opposite to one another, that is, opposite extremities of a diameter of the sphere.

10. Two spherical triangles are said to be symmetrical, when the sides of the