parallel to BC, E L will be equal to BC, (I. 22.), and the triangles LE F, ADG will be similar (I. 18.), because the sides of the one are perpendicular to the sides of the other, each to each; therefore EL or BC is to EF as AD to DG (II. 31.), that is, as the circumference of a circle which has the radius AD to the circumference of a circle which has the radius DG (III. 33. and II. 12.); and therefore (II. 28. Schol. Rule I.) the product of BC and the latter circumference is equal to the product of EF and the former. But the convex surface in question is equal to the product of BC and the circumference of which has the radius D G. Therefore (I. ax. 1.) that surface is likewise equal to the product of E F, and the circumference which has the radius AD. Next, of the solid generated by the triangle ABC. Let CB and FE be produced to meet one another in V. Then the solid in question is the difference of those generated by the triangles A CV and ABV. Now, the solid generated by the triangle ACV is equal to the sum or difference of two cones, having the altitudes AF, VF respectively, and for their common base the circle of which C F is radius-equal, that is (9.), to one-third of the product of A V, which is the sum or difference of the altitudes, by half the radius C F, and the circumference which has the radius CF (III. 32.); or to one-third of the product of AD, half VC, and that circumference (for ADxVC is equal to AVX CF), or lastly, to one-third of the product of AD, and the surface generated by VC (8.). And in the same manner it may be shewn, that the solid generated by the triangle ABV is equal to one-third of the product of AD, and the surface generated by VB. Therefore the difference of these solids, that is, the solid in question, is equal to one-third of the product of A D, and the surface generated by B C. It has been supposed in the above demonstrations, that EF and BC are not parallel. If B C be parallel to E F, the surface generated by BC will be that of a right cylinder having the axis EF, whence the first part of the proposition is manifest; and the solid generated by the triangle ABC will be equal to two-thirds of the cylinder (9.), whence the second part of the proposition. Therefore, &c. Cor. The proof of the second part of the proposition is equally applicable, whether ABC be isosceles, or otherwise. Therefore, if any triangle ABC be made to revolve about an axis which lies in the same plane with it and passes through its vertex A; and if AD be drawn perpendicular to the base, the solid generated by the triangle shall be equal to one-third of the product of AD and the surface generated by the base B C. PROP. 13. G E L C MUN O If the half AFGHKB of any regular polygon of an even number of sides revolve about the diagonal AB; the whole surface of the solid generated by its revolution shall be equal to the product of AB by the circumference of a circle whose radius is the apothem CE of the polygon; and its solid content shall be equal to one-third of the product of this surface by the apothem CE. From the points F, G, H, K, draw FL, GM, HN, KO perpendicular to AB, (I.45.) and join CF, CG, CH, CK. Then, because C is the centre of the polygon, the triangles CAF, CFG, &c. 1 are isosceles triangles, having the common vertex C, and the perpendiculars drawn from C to their respective bases equal each of them to the apothem CE. And, because these triangles revolve about the axis AB passing through C, and that A L, LM, &c. are the parts of the axis intercepted by perpendiculars drawn from the extremities of the base of each; the portions of the whole surface in question, generated by A F, FG, &c., are equal, respectively, to the products of AL, LM, &c., by the circumference of the circle which has the radius CE (12.). Therefore the whole surface is equal to the sum of these products, that is, to the product of A B by the circumference of the same circle. K B 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. 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 H (IV.37.); make AE equal to AB, and complete the cube AF: andin like man- A CB ner upon A C describe the square CG in the same plane with BD, from AE cut off A H equal to A C, and complete the cube AK: and let the faces H K and G K of the latter cube be produced to meet the faces B F, EF, 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, AC; and with respect to the third, its altitude HE is likewise equal to the difference of A B, AC, and tween the squares of AB, A C, because its base H L is a mean proportional beits adjacent sides are equal to AB and AC respectively, and (II. 35.) AB is to AB the difference of the cubes is less than XAC as ABX AC to AC. Therefore a parallelopiped, whose altitude is CB, and its base equal to three times the square of AB. But, because CB 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. Cor. It appears from the demonstration, that the difference of the cubes of two straight lines is equal to the product of the difference of the straight lines by the sum of their squares, and a mean proportional between those squares. any solid circumscribing the sphere. Then, OH since (as in prop. 7.) K B regular polygon may be inscribed in the circle A B D, such that, CE being its apothem, CD2 CE shall be to CE in a ratio less than that of P to Q; let AFGHKB be the half of such a polygon, and let a similar halfpolygon L M N ORV be circumscribed, so that one of its sides LM may touch the circle in D (III. 27. Cor. 2.) Then, if these inscribed and circumscribed half-polygons be made to revolve with the semicircle about the axis AB, they will generate, together with the sphere generated by the semicircle, two figures N 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 If greater, it must be greater also rectangles under AB and the circum- than the surface of some circumscribed ference which has the radius C E, and solid of revolution (15. Cor.), greater, under LV and the circumference which that is (13.), than the product of the diahas the radius CD, and that these rect- gonal LV by the circumference which angles are to one another as the squares has for its radius the apothem CD (see of CE, CD (II. 37. Cor. 1.); the dif- the figure of Prop. 15.); which is imposference of the surfaces is to the surface sible, because the diameter is less than of the inscribed figure as CD2-CE2 LV, and the circumference of the geneto C E2 (II. 20.); that is, in a less ratio rating circle is the same with the cirthan that of P to Q. But the surface of cumference which has the radius CD. the inscribed figure is less than Q : much more, therefore, is the difference of the surfaces less than P (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. 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 CD-CE3 shall be to C E3 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 E (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.). 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. Cor. 1. The surface of a sphere is equal to four times the area of its generating 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 product 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, 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 2 R 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, 3 their solid contents. But (p. 47, but the surface of the sphere is likewise Rule ii.) 4 R2 is to 4r2 as R2 to r2, greater than the surface of the inscribed and R3 is to r3 as R3 to r3. Theresolid (14.). fore, the surfaces are as the squares of the radii, and the solid contents as the cubes of the radii. 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 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 convex 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. 180 The convex surface of the segment is extremities of the arc: then, if the semicircle be made to revolve about the diacalled the base of the sector. meter 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. 13. A spherical orb is a portion of a sphere contained between its surface and that of a lesser sphere, which is concentric (or has the same centre) with it. O 14. A spherical wedge or ungula is a portion of a sphere intercepted between two planes, each of which passes through the centre of the sphere. The convex surface of an ungula is called a lune. E D G 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 same centre C; then, if the whole figure revolve about AB, the parts A EG, 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. 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 DEF, 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 D K, FL (III. 56.) to meet GH in the points K, L respectively; bisect DK in M (I. 43.); through M draw MN parallel to CG (I. 48.) to meet G K in N, and from the points M, N draw Mm, N n 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 DK and the circumference which has the radius Mm (11. Cor.). And, in like manner, since N is the middle point of G K (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 G D 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 DK 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 |