THEOREM XII. If a Circle and a Polygon, Circumscribable about another Circle, are Isoperimeters, the Surface of the Circle is a Geometrical Mean Proportional between that Polygon and a Similar Polygon (regular or irregular) Circumscribed about that Circle. Let c be a circle, P a polygon isoperimetrical to that circle, and circumscribable about some other circle, and p' a polygon similar to P and circumscribable about the circle c: it is affirmed that P: C::C: P. For, P: P: perim2. P :: perim2. P' :: circum2. c: perim2. P' by th. 89, Geom, and the hypothesis. But (th. 11) P: c: per. P': cir. c : : per2. P′ : per. P'x cir. c. Therefore cir2. c per. P' x cir. c PC:: :: cir. c: per. P' :: C: P'. THEOREM XIII. Q. E. D. If a Circle and a Polygon, Circumscribable about another Circle, are Equal in Surface, the Perimeter of that Figure is a Geometrical Mean Proportional between the Circumference of the first Circle and the Perimeter of a Similar Polygon Circumscribed about it. Let c P, and let p' be circumscribed about c and similar to c: then it is affirmed that cir. c: per. P: per. P : per. P'. For, cir. c per. P' :: C: P' :: P: P' :: per2. P : per2. P′. :: per2. P': per. P× per.P'. :: per2. P: per. P× per. Also, per. p': per. P :: per. P: per. P'. Q. E. D. THEOREM XIV. The Circle is Greater than any Rectilinear Figure of the Same Perimeter; and it has a Perimeter Smaller than any Rectilinear Figure of the same Surface. For, in the proportion, P: c:: C: P', (th. 12), since c<P', therefore P< C. And, in the propor. cir. c: per. P:: per. P : per. P' (th. 13), or, cir. c: per. P':: cir2. c: per2. P, cir. c<per. P'; therefore, cir2. c < per2. P, or cir. c < per. P. Q. É. D. Cor. 1. It follows at once, from this and the two preceding theorems, that rectilinear figures which are isoperimeters, and and each circumscribable about a circle, are respectively in the inverse ratio of the perimeters, or of the surfaces, of figures similar to them, and both circumscribed about one and the same circle. And that the perimeters of equal rectilineal figures, each circumscribable about a circle, are respectively in the subduplicate ratio of the perimeters, or of the surfaces, of figures similar to them, and both circumscribed about one and the same circle. Cor. 2. Therefore, the comparison of the perimeters of equal regular figures, having different numbers of sides, and that of the surfaces of regular isoperimetrical figures, is reduced to the comparison of the perimeters, or of the surfaces of regular figures respectively similar to them, and circumscribable about one and the same circle. Lemma 1. If an acute angle of a right-angled triangle be divided into any number of equal parts, the side of the triangle opposite to that acute angle is divided into unequal parts, which are greater as they are more remote from the right angle. C Let the acute angle c, of the rightangled triangle ACF, be divided into equal parts, by the lines CB, CD, CE, drawn from that angle to the opposite side; then shall the parts AB, BD, &c, intercepted by the AB 1 lines drawn from c, be successively longer as they are more remote from the right angle a. For, the angles ACD, BCE, &c, being bisected by CB, CD, &c, therefore by theor. 83 Geom. AC: CD :: AB: BD, and BC: CE :: BD: DE, and DC: CF :: DE: EF. And by th. 21 Geom. CD > CA, CE > CB, CF > CD, and so on: whence it follows, that DB > AB, DE > DB, and so on. Q. E. D. Cor. Hence it is obvious that, if the part the most remote from the right angle A, be repeated a number of times equal to that into which the acute angle is divided, there will result a quantity greater than the side opposite to the divided angle, THEOREM XV. If two Regular Figures, Circumscribed about the Same Circle, differ in their Number of Sides by Unity, that which has the Greatest Number of Sides shall have the Smallest Perimeter. Let CA be the radius of a circle, and AB, AD, the half sides of two regular polygons circumscribed about that circle, of which n+1 1 and the th A which the number of sides differ by unity, being respectively n+1 and n. The angles ACB, ACD, therefore are respectively the part of two right angles: consequently these angles are as n and n+1: and hence, the angle may be conceived divided into n + 1 equal parts, of which BCD is one. Consequently, (cor. to the lemma) (n + 1) BD > AD. Taking, then, unequal quantities from equal quantities, we shall have (n + 1) AD (n + 1) BD < (n + 1) AD - AD, or, (n+1) AB <N. AD. - That is, the semiperimeter of the polygon whose half side is AB is smaller than the semiperimeter of the polygon whose half side is AD: whence the proposition is manifest. Cor. Hence, augmenting successively by unity the number of sides, it follows generally, that the perimeters of polygons circumscribed about any proposed circle, become smaller as the number of their sides become greater. THEOREM XVI. The Surfaces of Regular Isoperimetrical Figures are Greater as the Number of their Sides is Greater: and the Perimeters of Equal Regular Figures are Smaller as the Number of their Sides is Greater. For, 1st. Regular isoperimetrical figures.are (cor. 1 th. 14) in the inverse ratio of figures similar to them circumscribed about the same circle. And (th. 15) these latter are smaller when their number of sides is greater: therefore, on the contrary, the former become greater as they have more sides. : 2dly. The perimeters of equal regular figures are (cor. 1 th. 14) in the subduplicate ratio of the perimeters of similar figures circumscribed about the same circle and (th. 15) these latter are smaller as they have more sides: therefore the perimeters of the former also are smaller when the number of their sides is greater. Q. E. D. SECTION II. SOLIDS. THEOREM XVII. Of all Prisms of the Same Altitude, whose Base is Given in Magnitude and Species, or Figure, or Shape, the Right Prism has the Smallest Surface. For, 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. |