For, let Ac' and c'B be any other parts into which the given line AB may be divided; and let AC, AC', be bisected in D, D', respectively. Then shall Ac2. CB = 4AD. DC, CB (cor. to theor. 31 Geom.) > 4AD'. D'C. CB, or greater than its equal C'A2. C'в, by the preceding theorem. THEOREM XXV. Of all Right Parallelopipeds Given in Magnitude, that which has the Smallest Surface has all its Faces Squares, or is a Cube. And reciprocally, of all Parallelopipeds of Equal Surface, the Greatest is a Cube. For, by theorems 19 and 21, the right parallelopiped having the smallest surface with the same capacity, or the greatest capacity with the same surface, has a square for its base. But, any face whatever may be taken for base: therefore, in the parallelopiped whose surface is the smallest with the same capacity, or whose capacity is the greatest with the same surface, any two opposite faces whatever are squares: consequently, this parallelopiped is a cube. THEOREM XXVI. The Capacities of Prisms Circumscribing the Same Right Cylinder, are Respectively as their Surfaces, whether Total or Lateral. For, the capacities are respectively as the bases of the prisms; that is to say (th. 11), as the perimeters of their bases; and these are manifestly as the lateral surfaces: whence the proposition is evident. Cor. The surface of a right prism circumscribing a cylinder, is to the surface of that cylinder, as the capacity of the former, to the capacity of the latter. 'Def. The Archimedean cylinder is that which circumscribes a sphere, or whose altitude is equal to the diameter of its base. THEOREM XXVII. The Archimedean Cylinder has a Smaller Surface than any other Right Cylinder of Equal Capacity; and it is Greater than any other Right Cylinder of Equal Surface. Let c and c' denote two right cylinders, of which the first is Archimedean, the other not then, 1st, If... cc', surf. c < surf. c': = For, A For, having circumscribed about the cylinders c, c', the right prisms P, P', with square bases, the former will be a cube, the second not: and the following series of equal ratios will obtain, viz, C : P :: surf. c: surf. P :: base c: base p :: base c' base p':: c': P' :: surf c': surf. P'. Then, 1st when cc'. Since c: P:: c': P', it follows that PP'; and therefore (th. 25) surf. P < surf. P'. But, surf. c surf. P:: surf. c': surf. P'; consequently surf. c < surf. c'. Q. E. ID. 2dly when surf. c surf. c'. Then, since surf. c: surf. P: surf. c': surf. p', it follows that surf. P = surf. P'; and therefore (th. 25) P> P'. But C: P:: c': P'; consequently c> c. Q. E. 2D. THEOREM XXVIII. Of all Right Prisms whose Bases are Circumscribable about Circles, and Given in Species, that whose Altitude is Double the Radius of the Circle Inscribed in the Base, has the Smallest Surface with the Same Capacity, and the Greatest Capacity with the Same Surface. This may be demonstrated exactly as the preceding theo rem, by supposing cylinders inscribed in the prisms. Scholium. If the base cannot be circumscribed about a circle, the right prisin which has the minimum surface, or the maximum capacity, is that whose lateral surface is quadruple of the surface of one end, or that whose lateral surface is two-thirds of the total surface. This is manifestly the case with the Archimedean cylinder; and the extension of the property depends solely on the mutual connexion subsisting between the properties of the cylinder, and those of circumscribing prisms. THEOREM XXIX. The Surfaces of Right Cones Circumscribed about a Sphere, are as their Solidities. For, it may be demonstrated, in a manner analogous to the demonstrations of theorems 11 and 26, that these cones are equal to right cones whose altitude is equal to the radius of the inscribed sphere, and whose bases are equal to the total surfaces of the cones: therefore the surfaces and solidities are proportional. THEOREM THEOREM Xxx. The Surface or the Solidity of a Right Cone Circumscribed about a Sphere, is Directly as the Square of the Cone's Altitude, and Inversely as the Excess of that Altitude over the Diameter of the Sphere. Let VAT be a right-angled triangle which, by its-rotation upon VA as an axis, generates a right cone; and BDA the semicircle which by a like rotation upon VA forms the inscribed sphere: then, the surface or the solidity of the cone varies as For, draw the VA2 VB B D T radius CD to the point of contact of the semicircle and VT. Then, because the triangles VAT, VDC, are similar, it is AT: VT:: CD ; VC. And, by compos. AT; AT+VT: CD : CD + CV = VA; Therefore AT2: (AT+VT) AT :: CD: VA, by multiply ing the terms of the first ratio by AT. But, because VB, VD, VA are continued proportionals, it is VB VA :: VD2 : VA3 :: ́CD2: AT2 by sim. triangles. But CD VA:: AT2: (AT + VT)AT by the last; and these mult. give CD. VB : VA2 ; : CD2 : (AT + VT)AT, 2 VA? or VB: CD :: VA2 : (AT + VT)AT = CD• But the surface of the cone, which is denoted by T. AT2 + *.AT. VT*, is manifestly proportional to the first member of this equation, is also proportional to the second member, or, since CD is constant, it is proportional to -, or to a third proportional to BV and AV. And, since the capacities of these circumscribing cones are as their surfaces (th. 29), the truth. of the whole proposition is evident. Lemma 2. AV2 BV The difference of two right lines being given, the thirdproportional to the less and the greater of them is a minimum when the greater of those lines is double the other. For, since AP AV :: AV; BV; AP-AV :: AV : AV-BV; AP VP Hence, VP. AV = AP . AB. But VP. AV is either and th. 23 of this chapter). :: AV AB. or < AP2 (cor. to th. 31 Geom. Therefore AP ABAP2: whence 4ABAP, or AP4AB. Consequently, the minimum value of AP is the quadruple of AB; and in that case PV VA = 2AB. THEOREM XXXI. Q. E. D*. Of all Right Cones Circumscribed about the Same Sphere, the Smallest is that whose Altitude is Double the Diameter of the Sphere. VA2 For, by th. 30, the solidity varies as (see the fig. to that theorem): and, by lemma 2, since VA - VB is given, the third proportional is a minimum when VA=2AB. Q.E.D. VA2 VB Cor. 1. Hence, the distance from the centre of the sphere to the vertex of the least circumscribing cone, is triple the radius of the sphere. Cor. 2. Hence also, the side of such cone is triple the radius of its base. THEOREM XXXII. The Whole Surface of a Right Cone being Given, the Inscribed Sphere is the Greatest when the Slant Side of the Cone is Triple the Radius of its Base. For, let c and c' be two right cones of equal whole surface, the radii of their respective inscribed spheres being *Though the evidence of a single demonstration, conducted on sound mathematical principles, is really irresistible, and therefore needs no corroboration; yet it is frequently conducive as well to mental improvement, as to mental delight, to obtain like results from different processes. In this view it will be advantageous to the student, to confirm the truth of several of the propositions in this chapter by means of the fluxional analysis. Let the truth enunciated in the above Jemma be taken for an example: and let AB be denoted by a, av by x, вV by x-a. Then we shall have x-a:x:: x : the third proportional; which is to be a minimum. Hence, the fluxion of this fraction will be equal to zero (Flux. art. 51). That is (Flux. = o. Consequently, aa—2ax = 0, and x=2α, x-a x2x-2axx (x-a)2. er AV = 2AB, as above. arts. 19 and 30), denoted denoted by R and R'; let the side of the cone c be triple the radius of its base, the same ratio not obtaining in c; and let c' be a cone similar to c, and circumscribed about the same sphere with c'. Then, (by th. 31) surf. c'< sarf. c': therefore surf. c'‹ surf c. But c" and c are similar, therefore all the dimensions of c" are less than the corresponding dimensions of c: and consequently the radius R' of the sphere inscribed in c" or in c', is less than the radius R of the sphere inscribed in c, or R > R ́. Q. E. D. Cor. The capacity of a right cone being given, the inscribed sphere is the greatest when the side of the cone is triple the radius of its base. For the capacities of such cones vary as their surfaces (th. 29). THEOREM XXXIII. Of all Right Cones of Equal Whole Surface, the Greatest is that whose Side is Triple the Radius of its Base: and reciprocally, of all Right Cones of Equal Capacity, that whose Side is Triple the Radius of its Base has the Least Surface. For, by th. 29, the capacity of a right cone is in the compound ratio of its whole surface and the radius of its inscribed sphere. Therefore, the whole surface being given, the capacity is proportional to the radius of the inscribed sphere: and consequently is a maximum when the radius of the inscribed sphere is such; that is, (th. 32) when the side of the cone is triple the radius of the base*. Again, reciprocally, the capacity being given, the surface is in the inverse ratio of the sphere inscribed therefore, it, is the smallest when that radius is the greatest; that is (th. 32) when the side of the cone is triple the radius of its base. Q.E.D. * Here again a similar result may easily be deduced from the method of fluxions. Let the radius of the base be denoted by 1, the slant side of the cone by z, its whole surface by a2, and 3.141593 by π. Then the circumference of the cone's base will be 27x, its area 1a, and the convex surface The whole surface is, therefore, a2 + пiz: and this being a2 = · α, we have z=—-X. But the altitude of the cone is equal to the square root of the difference of the squares of the side and of the radius of the base; that a4 2a2 is, it is W- -). And this multiplied into of the area of the base, T |