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have the angles at their bases all equal: consequently, the angles of the polygon, which are each made up of two angles at the bases of two contiguous isosceles triangles, will be equal to one another. Q. E. D.
Of all Figures having the Same Number of Sides and Equal Perimeters, the Greatest is Regular.
For, the greatest figure under the given conditions has all its sides equal (th. 2 cor.). But since the sum of the sides and the number of them are given, each of them is given therefore (th. 6), the figure is inscribable in a circle: and consequently (th. 8) all its angles are equal; that is, it is regular. Q. E. D.
Cor. Hence we see that regular polygons possess the property of a maximum of surface, when compared with any other figures of the same name and with equal perimeters.
A Regular Polygon has a Smaller Perimeter than an Irregular one Equal to it in Surface, and having the Same Number of Sides.
This is the converse of the preceding theorem, and may be demonstrated thus: Let R and I be two figures equal in surface and having the same number of sides, of which R is regular, I irregular: let also R be a regular figure similar to R, and having a perimeter equal to that of 1. Then (th. 9) R>1; but I = R; therefore R' > R. But R' and R are similar; consequently, perimeter of R'> perimeter of R; while R' per. I (by hyp.). Hence, per. I > per. R. Q. E. D. per.
The Surfaces of Polygons, Circumscribed about the Same or Equal Circles, are respectively as their Perimeters*. Let the polygon ABCD be circumscribed about the circle EFGH; and let this polygon be divided into triangles, by lines drawn. from its several angles to the centre o of the circle. Then, since each of the tangents AB, BC, &c, is perpendicular to its
*This theorem, together with the analogous ones respecting bodies circumscribin cylinders and spheres, were given by Emerson in his Geometry, and their use in the theory of Isoperimeters was just suggested: but the full application of them to that theory is due to Simon Lhuillier.
corresponding radius OE, OF, &c, drawn to the point of contact (th. 46 Geom.); and since the area of a triangle is equal to the rectangle of the perpendicular and half the base (Mens. of Surfaces, pr. 2); it follows, that the area of each of the triangles ABO, BCO, &c, is equal to the rectangle of the radius of the circle and half the corresponding side AB, BC, &c: and consequently, the area of the polygon ABCD, circumscribing the circle, will be equal to the rectangle of the radius of the circle and half the perimeter of the polygon. But, the surface of the circle is equal to the rectangle of the radius and half the circumference (th. 94 Geom.). Therefore, the surface of the circle, is to that of the polygon, as half the circumference of the former, to half the perimeter of the latter; or, as the circumference of the former, to the perimeter of the latter. Now, let P and P' be any two polygons circumscribing a circle c: then, by the foregoing, we have
surf. c surf. P:: circum. c: perim. P,
surf. c: surf. p' :: circum. c: perim. P.
But, since the antecedents of the ratios in both these proportions, are equal, the consequents are proportional: that is, surf. P: surf. P':: perim. P: perim. P'. Q. E. D.
Cor. 1. Any one of the triangular portions ABO, of a polygon circumscribing a circle, is to the corresponding circular sector, as the side AB of the polygon, to the arc of the circle included between AO and BO.
Cor. 2. Every circular arc is greater than its chord, and less than the sum of the two tangents drawn from its extremities and produced till they meet.
The first part of this corollary is evident, because a right line is the shortest distance between two given points. The second part follows at once from this proposition: for EA + AH being to the arch EIH, as the quadrangle AEOH to the circular sector HIEO; and the quadrangle being greater than the sector, because it contains it; it follows that EA+AH is greater than the arch EIH*.
Cor. 3. Hence also, any single tangent EA, is greater than its corresponding arc EI.
This second corollary is introduced, not because of its immediate connection with the subject under discussion, but because, notwithstanding its simplicity, some authors have employed whole pages in attempting its de monstration, and failed at last.
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
:: cir. c: per. P' :: C: P'.
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.
Q. E. D.
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 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.
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.
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
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,
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
and the th
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.
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.
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.