vex surfaces, or of their solid con- (1. 26. Cor. and I. 12. Cor. 2.), and theretents, shall be less than any given dif- fore by much more less than Q. Much ference. more, therefore, is the difference of the Let V be the vertex of a right cone, convex surfaces less than P (II.18.Cor.); A B D its base, and C the centre of the and more still is the difference of the base. And, first, let P be the given dif- surfaces of the inscribed and circumference of the surfaces; and let Q be the scribed pyramids, which have the comconvex surface of some circumscribed mon vertex V, less than P. pyramid. Then, because (as in 111.31.) Next, let S be the given difference of a regular polygon may be inscribed in the solid contents; and let T be the the circle ABD, the apothem CE of solid content of some circumscribed pywhich approaches to the radius C D ramid. As before, let the regular polywithin any given difference, a polygon gon A BFG H be inscribed in the circle may be inscribed, such that CD2 – ABD, such that C D2 – C Emay be CE2 may be to CE? in a ratio less to C Ez in a less ratio than that of S to than any assigned; less, therefore, T; let a similar polygon be circumthan that of P to Q. Let A BFGH scribed, and the inscribed and circumbe such a polygon; and let a similar scribed pyramids completed. Then, polygon KLMNO be circumscribed because these pyramids have the same altitude, their solid contents are to one another as their bases, that is, as CE, C D2 (IV.32.): therefore, the difference of the contents is to that of the inscribed pyramid as CD2-C E’ to CE? (II. 20.) that is, in a less ratio than that of S to T. Therefore, because the content of the inscribed prism is less than T, much more is the difference of the contents less than S (II. 18. Cor.). Therefore, &c. Cor. 1. Any right cone being given, M a regular pyramid may be inscribed (or F circumscribed), which shall differ from H the cone in convex surface, or in solid В! content, by less than any given differ ence; for the difference between the L cone and either of the pyramids, wheK ther in surface or in content, is less than about the circle, so that the side KL the difference of the two pyramids (6.). may touch the circle in D (III. 27. Cor. 2. Any two similar right cones Cor. 2.). Join VA, V B, &c., VK, being given, similar regular pyramids VL, &c., and through E draw EU may be inscribed (or circumscribed), parallel to D V, to meet CV in U, and which shall differ from the cones in join UA, UB, &c.* Then, it may convex surface, or in solid content, by easily be shown, that the convex sur- less than any the same given difference. faces of the two pyramids, which have the points U, V, for their vertices, and PROP. 8. the inscribed and circumscribed poly- The convex surface of a right cone is gons for their bases, are made up of equal to half the product of its slunt similar triangles, which are to one an- side by the circumference of its base. other in the same ratio, each to each, For, if half this product be not equal viz. that of C E2 to C D2 (II. 42. Cor.). to the convex surface of the pyramid, it Therefore, the convex surfaces of the must either be greater or less than that pyramids are to one another in the same surface. If greater, it must also be ratio (II. 23. Cor. 1.); and the differ- greater than the convex surface of some ence of their convex surfaces is to that circumscribed pyramid (7. Cor. 1.)of the lesser pyramid as C D2-C E? to greater, that is, than half the product CE? (II. 20.), that is, in a less ratio than of the slant side of the cone by the perithat of P to Q. But the convex surface meter of the circumscribed polygon of the lesser pyramid is less than the (I. 26. Cor.), which is the base of the surface of the pyramid VABFGH pyramid; for the triangles which form * The lines UA, U B, &c, are omitted in the the convex surface of the pyramid have figure. for their bases the sides of the circum scribed polygon, and the lines drawn solid content of the cone is į R2 A from their common vertex to the points (III. 34. Scholium.). of contact (which lines are each a slant Cor. 2. (Euc. xii. 10.). If a right cylinder side of the cone) perpendicular to those and a right cone have the same base and bases respectively (III. 2. Cor. 1. and the same altitude, the cone shall be a IV. 4.). But this is impossible, be. third part of the cylinder (4.). cause the circumference of the base of Cor. 3. If a right cone and a pyramid the cylinder is less than the perimeter have equal bases and altitudes, the cone of the circumscribed polygon. Again, shall be equal to the pyramid (IV. 32. if half this product be less than the con Cor. 1.). vex surface of the cone, it must be less Cor. 4. (Euc. xii. 11 and 14.) Right also than the convex surface of some cones which have equal altitudes are to inscribed pyramid (5. Cor. 1.)-less, that one another as their bases; and right is, than half the product of the perpen- cones which have equal bases are to one dicular drawn from the vertex to a side another as their altitudes : also, any of the inscribed regular polygon which is two right cones are to one another in the the base of the pyramid, by the perimeter ratio which is compounded of the ratios of that polygon; which is impossible, of their bases and altitudes (IV. 32. because not only is the slant side greater Cor. 2.). than the perpendicular (IV. 8.), but the PROP. 10. (Euc. xii. 12.). circumference of the base of the cylinder The surfaces of similar right cones is also greater than the perimeter of the are as the squares of the axes; and inscribed polygon. their solid contents are as the cubes of Therefore, half the product in question the axes. is neither greater nor less than the con- For, in the first place, there may be vex surface of the pyramid ; that is, it is inscribed in the cones similar pyramids, equal to it. the convex surfaces of which approach Therefore, &c. more nearly to the convex surfaces of Cor. If R is the radius of the base of the cones than by any the same given a right cone, and S its slant side, the difference (7. Cor. 2.); and the convex convex surface of the cone is w RS surfaces of these pyramids are, to one (III. 34. Scholium.). another, always in the same ratio, viz. PROP. 9. as the squares of the sides of their bases (II. 42. Cor. and II. 23. Cor. 1.), that The solid content of a right cone is is, as the squares of the radii of the cirequal to one-third of the product of its cumscribing circles (III. 30.), or (II. base and altitude. 37. Cor. 4.) as the squares of the axes This proposition is demonstrated in of the cones, for the axes of the cones the same way as the preceding. If a are to one another as the radij of their third of the product in question exceed bases (def. 6.); therefore the convex the content of the cone, it must likewise surfaces of the cones are to one another exceed the content of some circum- in the same ratio (II. 28.), viz. as the scribed pyramid (7. Cor. 1.); but this is squares of their axes. impossible, because the latter (IV. 32. And, in the same manner, because Cor. 1.) is equal to a third of the pro- there may be inscribed in the cones duct of its altitude, which is the same similar pyramids, the solid contents of with that of the cone, by its base, which which approach more nearly to the solid is greater than the base of the cone. And if, on the other hand, it be less contents of the cones than by any the than the content of the cone, it must because the solid contents of these pyra same given difference (7. Cor. 2.); and likewise be less than the content of some mids are to one another always in the inscribed pyramid (7. Cor. 1.); but this same ratio, viz. as the cubes of the sides is impossible, because the latter (IV.32. of their bases (IV. 35.), or (IV. 27. Cor. 1.) is equal to a third of the pro- Cor. 3.) as the cubes of the axes of the duct of its altitude, which is the same cones; the solid contents of the cones with that of the cone, by its base, which are to one another in the same ratio is less than the base of the cone. (II. 28.), viz, as the cubes of their axes. Therefore, a third of the product in question is equal to the content of the Otherwise : Let A, a represent the axes of two Therefore, &c. similar right cones; R, r the radii of their Cor. 1. If R is the radius of the base bases; and S, s their slant sides. Then of a right cone, and A its altitude, the (8. Cor.) ~RS and #rs will represent cone. 2 n d in 2 their convex surfaces respectively, and equal to the trapezoid A E ea. And (9. Cor. 1.) 3-R2 A, Tr? a their solid because the trapezoid A E ea is equal contents. But, because the cones are to the product of its altitude A a by half similar (def. 6.) R:A:: 1:a; and be the sum of its parallel sides A É, ae cause the slant sides S, s are the hypo- (I. 28.), the convex surface of the frustenuses of right-angled triangles, which tum is equal to the produet of its slant have the sides A, R and a, r about the side A a, by half the sum of the cirright angles proportional (11. 32.) S : A cumferences A BD, a bd. :.8: a: therefore (II. 37. Cor. 3.), RS In the next place, with regard to the : A::rs: a’, alternando (II. 19.) RS solid content. Let KLMN be a triangular :rs :: A?: a, and (p. 47, Rule ii.) к *RS: 78::A? : a?, that is, the convex surfaces of the cones are as the squares of their axes. Again, because R: A:r: a, R? : A2 : : ;? : a? (II. 37. Cor. 4.), and (p. 47, Rule ii.) R’A : A3 : p2 a : as: therefore L N B alternando (II. 19.) R’A : p? a :: A3 M : as, and (p. 47, Rule ii.) 1. R? A: r2 a :: A3 ; as, that is, the solid pyramid, having its base L M N equal to contents of the cones are to one another the base ABD, and in the same plane with as the cubes of their axes. it, and its vertex K in the same parallel Therefore, &c. to the base with the vertex of the cone. PROP. 11. Then, because the cone and pyramid The convex surface of a frustum of a have equal bases, and the same altitude, right cone is equal to the product of its they are equal to one another (9. Cor. 3.). slant side by half the sum of the circum; the pyramid in the triangle i mn: then, ferences of its two bases ; and its solid because I m n is similar to LMN content is equal to the sum of the solid (IV. 12. and IV. 15.), they are to one contents of three cones, which have the another as I m2 and L M2 (III. 42. Cor.), same altitude with the frustum, and for that is, (II. 37. Cor. 4.) as Kl2 and their bases its two bases, and a mean K L’, or (IV. 14.) as Va2 and V A2: proportional between them. Let A BD, abd, be the bases of the but the bases abd and ABD are to frustum of a cone, which has the vertex and II. 31.), and LMN is equal to one another in the same ratio (III. 33. ABD: therefore, also, l m n is equal to a b d (II. 12. and II. 18.), and E (9. Cor. 3.) the cone Vabd is equal to the pyramid Kim n. Therefore (I. ax. 3.) the frustum of the cone is equal to the frustum of the pyramid, A But the latter (IV. 33.) is equal to the V. Draw any slant side V a A; from A sum of three pyramids, having, the draw any straight line AE perpendicular same altitude with the frustum, and to VA (Î. 44.); suppose AE to be taken for their þases the bases of the frustum, equal to the circumference A BD, and and a mean proportional between them; join VE, and through a draw a e paralleland (9. Cor. 3.) each of these pyrato AE (Í. 48.), to meet VE in e. Then, mids is equal to a cone having the same because the circumferences A B D, abd, altitude and an equal base. Therefore, are as their radii (III. 33.), that is, as also, the frustum of the cone is equal to VA, V a (11. 31.), that is, again, as A E, the sum of three cones, having the same ae (II. 31.), and that AE is equal to altitude with it, and for their bases the ABD, ae is equal to abd (II. 18.). bases of the frustum, and a mean proNow the triangle V AE is equal to the portional between them. convex surface of the cone VABD Therefore, &c. (6.), because (I. 26. Cor.) it is equal to Cor. If a straight line A a be made to half the product of V A the slant side, revolve about any axis, VC, in the same and A E, which is equal to the circum- plane with it, the surface generated by ference ABD; and, for the like reason, such straight line shall be equal to the the triangle Vae is equal to the convex product of the straight line and the cirsurface of the cone V abd; therefore, cumference generated by its middle the convex surface of the frustum is point F, F B the sphere. For the generated surface is that of a from a given point, leads to another cylinder if the line be parallel to the axis; property of those surfaces, from which and, in every other case, that of a frus- the measures assigned in Props. 2. and tum of a right cone. In the former 6. may be very readily inferred. This case, the reason is sufficiently manifest property is, that they are developable, (3.). In the latter, it may be shown, that is, they may be conceived to be that if FG be drawn parallel to A E in unfolded and spread out upon a plane. the first figure of the proposition) to Now, it is easy to perceive, that if the meet V E in G, FG will be equal to the surface of a right cylinder be devecircumference generated by the point F: loped, it will form a rectangle, which also, because V F is equal to half the has for its base the circumference of the sum of VA and Va, FG is equal to half circle, which is the base of the cylinder, the sum of AE and a e (II. 30. Cor. 2.); and for its altitude the altitude of the therefore the circumference generated by cylinder; whence it follows, that the the point F is equal to half the sum of convex surface of a right cylinder is the circumferences ABD, abd; and equal to the product of its altitude by hence by the proposition, the convex the circumference of its base. In like surface of the frustum, that is, the con- manner the developed surface of a right vex surface generated by the line A a is cone will form a circular sector, the arc equal to the product of A a and the cir- of which is equal to the circumference cumference generated by its middle of the base of the cone, and its radius point F. to the slant side; whence it follows, that the convex surface of a right Scholium. cone is equal to half the product of its Although the propositions of this sec- slant side by the circumference of its tion have, for greater brevity and sim- base. plicity, been stated and demonstrated only with regard to the right cylinder and Section 2.-Surface and content of right cone, it will be found that Props. 2. and 7. apply equally to the oblique cylin PROP. 12. der and oblique cone, to which the demonstrations may be without difficulty If an isosceles triangle ABC be made adapted, and hence it may be demon- to revolve about an axis which lies in strated, almost in the words of Props. the same plane with it and passes 4. 9. 11.5. and 10. that the solid content through the vertex A, and if a perpenof an oblique cylinder is equal to the dicular AD be drawn from the vertex product of its base and altitude; the to the base, and E F be that portion of solid content of an oblique cone to one the axis which is intercepted by perthird of the product of its base and pendiculars drawn to it from the exaltitude; the solid content of a truncated tremities of the base; the convex surface oblique cone to the sum of the solid generated by the base shall be equal to contents of three cones, having the same the product of E F by the circumference , altitude with it, and for their bases its of a circle having the radius AĎ; and two bases, and a mean proportional be- the generated by the triangle shall tween them; and, lastly, that the sur- be equal to one-third of the product of faces of similar oblique cylinders and this surface by the perpendicular AD. cones are to one another as the squares First, of the surface generated by of their axes, and their solid contents as the base B C. This is evidently the the cubes of the axes. With regard to convex surface of a truncated cone the convex surface of the oblique cylin- having the axis E F, and EB, FC der, it may likewise be shown in a simi. for the radii of its lar manner (see IV. 29. Scholium) to bases; and is therebe equal to the product of its side (or fore (11. Cor.) equal axis) by the perimeter of a plane sec- to the product of BC, tion perpendicular to it. and the circumferIt remains to observe, with regard to ence generated by its Props. 3. and 8., that the remarkable pro- middle point D, that B perty by which the convex surfaces of is, (if D G be drawn DA the cylinder and cone have been defined, parallel to BE to meet viz. that of containing, the first, straight EF in G) the circum A lines parallel to a given straight line, ference which has and the other straight lines diverging the radius DG. Now, if E L be drawn a E F A L parallel to BC, E L will be equal to BC, the proposition is equally applicable, (I. 22.), and the triangles LEF, AD G whether A B C be isosceles, or otherwill be similar (I. 18.), because the sides wise. Therefore, if any triangle A BC of the one are perpendicular to the sides be made to revolve about an axis which of the other, each to each; therefore lies in the same plane with it and passes EL or B C is to EF as A D to DG through its vertex A; and if AD be (II. 31.), that is, as the circumference drawn perpendicular to the base, the of a circle which has the radius A D to solid generated by the triangle shall be the circumference of a circle which has equal to one-third of the product of the radius DG (III. 33. and II. 12.); AD and the surface generated by the and therefore (II. 28. Schol. Rule I.) base B C. the product of BC and the latter cir PROP. 13. cumference is equal to the product of EF and the former. But the convex If the half A FGH KB of any regusurface in question is equal to the pro- lar polygon of an even number of sides duct of B C and the circumference of revolve about the diagonal A B; the which has the radius D G. Therefore whole surface of the solid generated by (I. ax. 1.) that surface is likewise equal its revolution shall be equal to the proto the product of E F, and the circum- duct of A B by the circumference of a ference which has the radius AD. circle whose radius is the apothem CE Next, of the solid generated by the of the polygon; and its solid content triangle ABC. Let C B and FE be shall be equal to one-third of the product produced to meet one another in V. of this surface by the apothem CE. Then the solid in question is the dif- From the points F, G, H, K, draw ference of those generated by the trian- FL, GM, HN, KO gles ACV and A B V. Now, the solid perpendicular to AB, generated by the triangle ACV is (1.45.) and join CF, CG, equal to the sum or difference of two CH, C K. Then, becones, having the altitudes AF, VF cause C is the centre of G M. respectively, and for their common base the polygon, the trianthe circle of which C F is radius—equal, gles CAF, CFG, &c. Il that is (9.), to one-third of the product are isosceles triangles, of A V, which is the sum or difference having the common verof the altitudes, by half the radius CF, tex C, and the perpen K B and the circumference which has the diculars drawn from C radius C F (III. 32.); or to one-third to their respective bases equal each of of the product of AD, half VC, and that them to the apothem CE. And, because circumference (for ADXV.C is equal these triangles revolve about the axis AB to A VxC F), or lastly, to one-third of passing through C, and that A L, LM, the product of AD, and the surface &c. are the parts of the axis intercepted generated by VC (8.). And in the by perpendiculars drawn from the exsame manner it may be shewn, that tremities of the base of each; the porthe solid generated by the triangle tions of the whole surface in question, A B V is equal to one-third of the pro- generated by A F, FG, &c., are equal, duct of AD, and the surface generated respectively, to the products of AL, LM, by VB. Therefore the difference of &c., by the circumference of the circle these solids, that is, the solid in question, which has the radius CE (12.). Thereis equal to one-third of the product of fore the whole surface is equal to the A D, and the surface generated by BC. sum of these products, that is, to the It has been supposed in the above product of A B by the circumference of demonstrations, that E F and B C are the same circle. not parallel. If B C be parallel to E F, Again, because the portions of the the surface generated by B C will be whole solid which are generated by the that of a right cylinder having the axis triangles CAF, CFG, &c. are the third EF, whence the first part of the pro- parts, respectively, of the products of the position is manifest; and the solid ge. portions of surface generated by A F, nerated by the triangle A B C will be FG, &c. by the apothem C E (12.); the equal to two-thirds of the cylinder (9.), whole solid is the third part of the sum of whence the second part of the propo- these products, that is, the third part of sition. the product of the whole surface by the Therefore, &c. apothem CE. Cor. The proof of the second part of Therefore, &c. 02 |