For, the area of each face of the prism is proportional to its height; therefore the area of each face is the smallest when its height is the smallest, that is to say, when it is equal to the altitude of the prism itself:, and in that case the prism is evidently a right prism. Q. E. D. THEOREM XVIII. Of all Prisms whose Base is Given in Magnitude and Species, and whose Lateral Surface is the Same, the Right Prism has the Greatest Altitude, or the Greatest Capacity. This is the converse of the preceding theorem, and may readily be proved after the manner of theorem 2. THEOREM XIX. Of all Right Prisms of the Same Altitude, whose Bases are Given in Magnitude and of a Given Number of Sides, that whose Base is a Regular Figure has the Smallest Surface. For, the surface of a right prism of given altitude, and base given in magnitude, is evidently proportional to the perimeter of its base. But (th. 10) the base being given in magnitude, and having a given number of sides, its perimeter is smallest when it is regular: whence, the truth of the proposition is manifest. THEOREM XX. Of Two Right Prisms of the Same Altitude, and with Irregular Bases Equal in Surface, that whose Base has the Greatest Number of Sides has the Smallest Surface: and, in particular, the Right Cylinder has a Smaller Surface than any Prism of the Same Altitude and the Same Capacity. The demonstration is analogous to that of the preceding theorem, being at once deducible from theorems 16 and 14. THEOREM XXI. Of all Right Prisms whose Altitudes and whose Whole Surfaces are Equal, and whose Bases have a Given Number of Sides; that whose Base is a Regular Figure is the Greatest. Let P, P, be two right prisms of the same name, equal in altitude, and equal whole surface, the first of these having a regular, the second an irregular base; then is the base of the prism P, less than the base of the prism r'. For, let p" be a prism of equal altitude, and whose base is equal to that of the prism P' and similar to that of the prism P. Then, Then, the lateral surface of the prism p" is smaller than the lateral surface of the prism p' (th. 19): hence, the total surface of p" is smaller than the total surface of P', and therefore (by hyp.) smaller than the whole surface of P. But the prisms P and P have equal altitudes and similar bases; therefore the dimensions of the base of p" are smaller than the dimensions of the base of P. Consequently the base of p", or that of p', is less than the base of P; or the base of P greater than that of P'. Q. E. D. THEOREM XXII. Of Two Right Prisms, having Equal Altitudes, Equal Total Surfaces, and Regular Bases, that whose Base has the Greatest Number of Sides, has the Greatest Capacity. And, in particular, a Right Cylinder is Greater than any Right Prism of Equal Altitude and Equal Total Surface. The demonstration of this is similar to that of the preceding theorem, and flows from th. 20. THEOREM XXIII. The Greatest Parallelopiped which can be contained under the Three Parts of a Given Line, any way taken, will be that constituted of Equal length, breadth, and depth. For, let AB be the given line, and, if possible, let two parts AE, ED, be unequal. Bisect AD in C, then will A CE D the rectangle under AE (AC+CE) B and ED (AC-CE), be less than Ac2, or than AC. CD, by the square of CE (th. 33 Geom.). Consequently, the solid AE. ED. DB, will be less than the solid AC. CD. DB; which is repugnant to the hypothesis. Cor. Hence, of all the rectangular parallelopipeds, having the sum of their three dimensions the same, the cube is the greatest. THEOREM XXIV. The Greatest Parallelopiped that can possibly be contained under the Square of one Part of a Given Line, and the other Part, any way taken, will be when the former Part is the Double of the latter. 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. |