PROP. 14. If within and about a semicircle there are inscribed and circumscribed any two half-polygons, the one having the diameter of the semicircle for its diagonal, and the other the diameter produced; and if these figures are made to revolve together with the semicircle about the diameter; the sphere generated by the semicircle shall be greater than the inscribed solid of revolution, and less than the circumscribed solid also the surface of the sphere shall be greater than the surface of the inscribed solid of revolution, and less than that of the circumscribed solid. For, with respect to the solid contents, the sphere contains the inscribed solid, and is itself contained in the circumscribed solid and, with respect to the surfaces, the surface of the sphere envelops the surface of the inscribed solid, and is enveloped by that of the circumscribed solid (Lemma 2.). Therefore, &c. Lemma 3. : If there be two straight lines, of which one is given, and the other may be made to approach to it within any given difference; the cube of the latter may also be made to approach to the cube of the former within any given difference. Let AB, AC be the two straight lines, of which AB is given. Upon AB (I. 52.) describe the square BD; from A draw A E perpendicular E to the plane BAE н (IV.37.); make AE D CB equal to AB, and complete the cube AF: and in like man- A ner upon A C describe the square CG in the same plane with BD, from A E cut off A H equal to A C, and complete the cube AK: and let the faces HK and G K of the latter cube be produced to meet the faces B F, E F, of the former. Then the difference of the two cubes is equal to the sum of the three parallelopipeds GF, CL, and LE. Of these, the first has its base DF equal to the square of AB (IV. 22.), and its altitude D G equal to the difference of AB, AC; the second has its base C K equal to the square of AC (IV. 22.), and its altitude C B likewise equal to the difference of AB, A C; and with respect to the third, its altitude HE is likewise equal to the difference of A B, A C, and its base H L is a mean proportional between the squares of AB, A C, because its adjacent sides are equal to AB and AC respectively, and (II. 35.) AB2 is to AB the difference of the cubes is less than × AC as ABX AC to AC2. Therefore a parallelopiped, whose altitude is CB, and its base equal to three times the square of AB. But, because C B may be made less than any given line, this parallelopiped may be made less than any given solid. Much more, therefore, may the difference of the cubes be made less than any given solid, that is, than any given difference. Therefore, &c. of revolution, the one inscribed in the sphere, the other circumscribed about it. And because the surfaces of these figures are equal respectively (13.) to the rectangles under AB and the circumference which has the radius C E, and under LV and the circumference which has the radius CD, and that these rectangles are to one another as the squares of CE, CD (II. 37. Cor. 1.); the difference of the surfaces is to the surface of the inscribed figure as CD2-CE2 to CE (II. 20.); that is, in a less ratio than that of P to Q. But the surface of the inscribed figure is less than Q : much more, therefore, is the difference of the surfaces less than P (II. 18. Cor.). Next, let S be the given difference of contents; and let T be the content of any solid circumscribing the sphere. Then, since a regular polygon may be inscribed in the circle A B D, such that, CE being its apothem, CD-CE shall be less than any given difference, and therefore also such that CDS-CE3 shall be to CE3 in a ratio less than that of S to T (Lemma 3.); let A FG HKB be the half of such a polygon, and let the figures of revolution be inscribed in the sphere, and circumscribed about it, as before. Then, because the contents of these figures are (13.) equal respectively to the thirds of two parallelopipeds (IV. 25. Schol.), having their bases equal to the surfaces, and their altitudes equal to CE, CD, and that these parallelopipeds are to one another as the cubes of CE, CD, for their bases are, as was shown in the former part of the proposition, as the squares of CE, CD; the difference of the contents is to the content of the inscribed figure as CD C Es to C Es (II. 20.), that is, in a less ratio than that of S to T. But the content of the inscribed figure is less than T: much more, therefore, is the difference of the contents less than S (II. 18. Cor.). For if this product be not equal to the surface of the sphere, it must either be greater or less than it. If greater, it must be greater also than the surface of some circumscribed solid of revolution (15. Cor.), greater, that is (13.), than the product of the diagonal LV by the circumference which has for its radius the apothem CD (see the figure of Prop. 15.); which is impossible, because the diameter is less than LV, and the circumference of the generating circle is the same with the circumference which has the radius CD. If less, it must also be less than the surface of some inscribed solid of revolution (15. Cor.)-less, that is (13.), than the product of the diameter A B, which is the same with the diameter of the generating circle, by the circumference which has for its radius the apothem CE; which is impossible, because the circumference of the generating circle is greater than the circumference which has the radius C E. Therefore the product in question is neither greater nor less than the surface of the sphere; that is, it is equal to it. Therefore, &c. gene Cor. 1. The surface of a sphere is equal to four times the area of its rating circle. For the area of this circle is equal to half the product of the radius and circumference (III. 32.). Cor. 2. If a right cylinder be circumscribed about a sphere; the surface of the sphere shall be equal to the convex surface of the cylinder. For the latter is equal to the pro duct of its altitude, and the circumference of its base (3.); and its base is equal to the generating circle of the sphere, and its altitude to the diameter. Cor. 3. The surface of a sphere is equal to two-thirds of the whole surface of the circumscribing cylinder. Cor. 4. If D is the diameter of a sphere, its whole surface is equal to & D2 (III. 34. Schol.). PROP. 17. The solid content of a sphere is equal to one-third of the product of the radius by the surface. For the third part of this product cannot be greater than the content of the sphere; since then it would be greater also than the content of some circumscribed solid of revolution (15 Cor.)greater, that is, than one-third of the product of the same radius by the surface of that solid (13.); which is impossible, because the surface of the sphere is less than that of the solid (14.). the radii), in the triplicate ratio of the Nor, on the other hand, can it be less Let D, d be the diameters of two than the content of the sphere, for then spheres, and R, their radii. Then would it be less than some inscribed (16. Cor. 4.) * D2, de will represent solid of revolution (15. Cor.), less, that is, their surfaces, and (17. Cor. 4.) D3, than one-third of the product of the apo-d3 their solid contents; or, since D them by the surface of that solid (13.); is equal to 2R and d to 2r, 4 ≈ R2, 4 ≈ r2 which is impossible, because not only will represent their surfaces, and R3, is the radius greater than the apothem, but the surface of the sphere is likewise greater than the surface of the inscribed solid (14.). Therefore the product in question is equal to the solid content of the sphere. Therefore, &c. Cor. 1. The solid content of a sphere is equal to one-third of the product of the radius by four times the area of the generating circle (16. Cor. 1.). Cor. 2. The solid content of a sphere is two-thirds of the solid content of the circumscribing cylinder. For the latter is equal to twice the product of the radius, and the area of the generating circle (4). Cor. 3. If any solid contained by planes be circumscribed about a sphere, the content of the sphere will be to the content of the solid as the surface of the sphere to the surface of the solid. For the solid may be divided into pyramids, having the centre of the sphere for their common vertex, and their altitudes equal each to the radius of the sphere; and since each of these pyramids is equal to a third of the product of its base and altitude, their sum is equal to a third of the product of the convex surface of the solid and the radius of the sphere: also, the sphere is equal to a pyramid, having the same altitude, and its base equal to the surface of the sphere (IV. 32.). Cor. 4. If D is the diameter of a sphere, its whole solid content is equal to x D (16. Cor. 4.). PROP. 18. (Euc. xii. 18.). The surfaces of spheres are as the squares of the radii, and their solid contents are as the cubes of the radii. For the surfaces are equal respectively to four times the areas of the generating circles (16. Cor. 1.), and these areas are as the squares of the radii (III. 33.). And the solid contents are to one another in a ratio which is compounded of the ratios of the surfaces and of the radii; that is (because the surfaces are to one another in the duplicate ratio of T T their solid contents. But (p. 47, Rule ii.) 4 R2 is to 4 r2 as R to r2, and R3 is to r3 as R3 to r3. Therefore, the surfaces are as the squares of the radii, and the solid contents as the cubes of the radii. Therefore, &c. SECTION 3.-Surfaces and contents of certain portions of the sphere. In order to have a clear apprehension of the figures intended in the following definitions, it is necessary to keep in mind, that every section of a sphere which is made by a plane is a circle, the centre of which is the foot of the perpendicular drawn to the plane from the centre of the sphere (IV. 8. Cor.). Def. 10. A segment of a sphere is any portion of it which is cut off by a plane, and the circle in which the plane cuts the sphere is called the base of the segment. When the plane passes through the centre, the two segments into which the sphere is divided are equal to one another, and are therefore each of them called a hemisphere. The convex surface of a segment is called a zone. 11. A double-based spherical segment is a portion of a sphere intercepted between two parallel planes; and the circles in which these planes cut the sphere are called the bases of the segment. The con vex surface of a double-based segment is likewise called a zone. 12. A sector of a sphere is the solid figure contained by the convex surface of a segment, and that of a right cone, which has the same base with the segment, and for its vertex the centre of the sphere. D N Let ADB be a semicircle, and from the points D, E of the semicircumference, let the straight lines DF, EG be drawn at right angles to the diameter AB; join CE, and let KNL be a second semicircle, having the centre C; then, if the whole figure_revolve about AB, the parts AEG, DE GF, AEC, and ADBLNK will generate a spherical segment, a doublebased spherical segment, a spherical sector, and a spherical orb respectively. And if the semicircle A D B, instead of making a complete revolution, revolve only through a certain angle, it will generate a spherical wedge or ungula. same extremities of the arc: then, if the semicircle be made to revolve about the diameter A B, the zone which is generated by the arc DF, shall be greater than the surface generated by the chord DF, and less than the surface generated by the tangent GH. From the points D, F draw the straight lines Dd, Ff, each of them perpendicular to AB (I. 45.). Then, in the supposed revolution of the figure, these straight lines will generate two circles which have the points d,f for their centres, and d D,fF for their radii respectively (IV. 3. Cor. 2.). And, because the zone generated by the arc DE F, together with these two circles, forms a convex surface which envelops, and therefore (Lemma 2.) is greater than the convex surface consisting of the surface generated by the chord DF and the same two circles, the zone generated by the arc DEF is greater than the surface generated by the chord D F. In the next place, from the points D, F draw the tangents DK, F L respectively; bisect DK in M (I. 43.); (III. 56.) to meet GH in the points K, L through M draw MN parallel to CG (I. 48.) to meet G K in N, and from the points M, N draw Mm, Nn perpendicular each of them to A B; and, lastly, through m draw mp parallel to M N to meet N n inp. Then, because the middle point of D K, in the supposed revolution of the figure about the axis AB, generates the circumference which has the radius Mm, the surface generated by DK is equal to the product of D K and the circumference which has the radius Mm (11. Cor.). And, in like manner, since N is the middle point of GK (II. 29.), the surface generated by GK is equal to the product of G K and the circumference which has the radius N n. But, because (III. 2. Cor. 1.) the angle KDG is a right angle, and therefore (I.8.) the angle K GD less than a right angle, that is, than KD G, DK is less than G K (I. 9.); and, because (I. 22.) Mm is equal to Np, which is less than Nn, the circumference which has the radius M m is less than the circumference which has the radius Nn (III.33.). Therefore, upon both accounts, the surface generated by D K is less than the surface generated by G K. And in the same manner it may be shown that the surface generated by L F is less than the surface generated by LH. Therefore, the whole convex surface generated by the three straight lines D K, KL, LF is 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 CE; which is impossible, because the circumference ADB E 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 A B: the zone shall be equal to the product of the whole circumference A D BE 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 Lis less than M Q, because LQ is less than K P, that is, than M A. B cumference ADBE, and the part generated by the arc A H 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 LN 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.). PROP. 21. one-third of the product of its base and Every spherical sector is equal to the radius of the sphere. is demonstrated are similar to those inThe steps by which this proposition dicated in the preceding. In the first place, it may be shown that two solids of revolution may be, one inscribed in the sector, the other circumscribed about it, which approach each of them to the sector more nearly than by any given difference. Hence, the pro |