= circumscribed circle; therefore, in the figure there is AC = CB CF AF BF; and CF, and AB, cutting one another in K, perpendicularly: they bisect each other, as perpendiculars of isosceles triangles; by the $ 36, is therefore obtained, according to r radius of the circle: = the side of the equilateral triangle is therefore equal to the radius of its circumscribed circle, multiplied by the square root of three. Corol. 1. If the radius of a circle is considered as unity, the side of the equilateral triangle inscribed in it, will be equal to the square root of three. Corol. 2. The side of the equilateral triangle is incommensurable with the radius of the circumscribed circle, that is they have no common measure. § 79. PROBLEM. To determine the side of a square inscribed in a circle. APPLIC. The radius=r, of the circle ABDE, (fig. 82,) being given to find the side of the square ABDE, inscribed in the circle. Determination. The side of the square inscribed in a circle must be the chord of a right angle at the centre, because each of the sides must subtend the fourth part of four right angles; it is therefore, the hypothenuse of a right angled triangle, of which the two sides about the right angle are equal to the radius, which was here called; thence results by Or the side of the square inscribed in a circle, is equal to the radius multiplied by the square root of two. Corol. 1. This proves also, that the diagonal of a square is incommensurable with its sides; for, what here refers to the radius and the side of the square, holds equally when these radii are sides of a square, and the diagonal of it is represented by the side of this square. Corol. 2. If the radius of the circle is considered as unity, the side of the inscribed square will be equal to the square root of two. § 80. PROBLEM. The side of a regular poligon inscribed in a circle, being given, to find the side of the poligon of double the number of sides, inscribed in the same circle. APPLIC. The regular poligon ABIGFH, being inscribed in a circle, (fig. 83,) of which AB, be one of the sides, it is required to determine AE = BE, the side of the poligon of double the number of sides. Determination. The AB, being a chord of the circle, the CE, bisecting this chord, bisects also the arc AEB, and all the angles at D, the point of intersection of the chord and the radius are right angles; the triangles AED BED, because AD BD, and DE, common, and the angles at D, right angles, thence AE = BE, and (by 36) = = AE2 = AD2 + (CE - CD)2 AE2 — AD2 + CE2 + AC2 — AD2 — 2CE (CA2—AD2)* = and because CE-AC, this becomes AE2 — 2AC2 — 2AC(CA2 — AD2 )2 therefore AE =[2AC (AC — (AC2 . (AC2 —AD2 )})]1 giving the expression of the value of one side of the poligon of double the number of sides by the radius of the circle and the side of the given poligon. The expression of the double radicals shews these sides to be always incommensurable to one another. The radius AC, being made unity, or = 1, and the value of AP, expressed in that unity, being called = a. This formula takes a general form, thus: Scholium 1. By substituting in the expression just found, the value of the side of the regular poligon of six sides, as found above, (§ 56,) to be equal to the radius of the circle, the sides of the poligon of 12 equal sides will be obtained, thus: = r(2. AE' — r ( 2 — (3)1 ) 1 — side of the regular dodecagon. Scholium 2. Substituting for AB, the value of the side of the square inscribed in the circle of the radius =r, the general expression becomes which, therefore, is the side of the regular octagon inscribed in the circle, the radius of which is = r. Corol. The continuance by the substitution of this value of the side of a poligon in the formula from which it is obtained, will again continue the same law for the determination of the value of the poligon next double in number of sides. § 81. PROBLEM. The side of a regular poligon inscribed in a circle being given, together with the radius of the circle to find the side of the regular poligon of equal number of sides, circumscribed to the same circle. = = APPLIC. The side of the regular poligon inscribed in the circle being AB, (fig. 84,) and C, the centre of the circle, the radius of which are, CA CB CE; it is required to determine by these data the side GF, of the poligon of an equal number of sides, the sides of which will touch the circle at E. Determination. The radius CE, being perpendicular to both the chord AB, and the tangent GF, these two lines are parallel, and therefore, the triangles ACB, and FCG, are similar, as are also the two triangles into which each of these is divided by the perpendicular CE; thence is obtained the proportion whence, CD: CA AB: FG, FG = CAXAB The right angled triangle BCD, furnishes an expression for CD, by the radius, and the side of the inscribed poligon, thus: CD2 CA2 - AD2 this can be transformed by making CA2, a common factor, and extracting the root of it, thus: and this inserted in the value just found for FG, gives: This is, therefore, the value of the poligon circumscribed to the circle in which the poligon with the side AB, is inscribed, and the radical expression shews again that these two sides are incommensurable to one another. § 82. PROBLEM. To determine by approximation, the circumference of the circle, in parts of its radius. Determination. By $ 80. The side of a regular poligon inscribed in the circle, being given, the side of the regular poligon of double the number of sices has been determined; the law of the expression being general, as has been shewn; this successive duplication of the number of sides of a regular poligon may be carried to any desired approximation to the circumference of the circle itself, for which it is then accepted. So that having in the general formula of § 80, accepted the radius as unity, and the side of the poligon, with double the number of sides expressed in that unity, being called = a, |