Elements of geometry and trigonometry . ian a circumscribed regular polygon,kaviiw double the number of sides. PROrOSITION VIII. THEOREAÎ. Two regular polygons, of the same number of sides^ can alwayshe formed, the one circumscribed about a circle, the other in-scribed in it, whiclt shall differ from each other by less thanany assignable surface. Let Q be the side of a squareless than the given AC, a fourth part ofthe circumference, and then bisect the half of this fourth, andproceed in this manner, alwaysbisecting one of the arcs formedby the last bisection, until anarc is foun
Elements of geometry and trigonometry . ian a circumscribed regular polygon,kaviiw double the number of sides. PROrOSITION VIII. THEOREAÎ. Two regular polygons, of the same number of sides^ can alwayshe formed, the one circumscribed about a circle, the other in-scribed in it, whiclt shall differ from each other by less thanany assignable surface. Let Q be the side of a squareless than the given AC, a fourth part ofthe circumference, and then bisect the half of this fourth, andproceed in this manner, alwaysbisecting one of the arcs formedby the last bisection, until anarc is found whose chord AB isless than Q. As this arc willbe an exact part of the circum-ference, if we apply chords AB,BC, CD, &c. each equal to ABand there will bethe circle. Next, describe about the circle a similar polygon abcde &c.(Prop. VI.) : the difference of these two polygons will be lessthan the square of Q. For, from the points a and b, draw the lines aO, bO, to thecentre O : they will pass through the points A and B, as was. the last will terminate at A,formed a regular polygon ABCDE &c in BOOK V. 117 shown in Prop. VI. Draw also OK to the point of contactK : it will bisect AB in 1, and be perpendicular to it (Book VI. Sell.). Produce AO to K, and draw BE. Let P represent the circumscribed j)olygon, and j) t!ic in-scribed polygon : then, since the triangles «06, AOB, are likeparts of P and p, we shall have aOh : AOB . ? : p (Book II. Prop. XI.) :But the triangles being similar, aOh : AOB : : Ocl : OA-, or , P : ;? : : Oa- : 0K-. Again, since the triangles Oa\\, EAB are similar, havingtheir sides respectively parallel, Oa- : 6k- : : AE- : EB-, hence,V : p : : AE- : EB-, or by division, P : P—p : : AE- : AE-—EFA or P is less than the square described on the diameter AE(Prop. VII. Cor.) ; therefore P—p is less than the square de-scribed on AB : that is, less than the given square on Q : hencethe difference between the circumscribed and inscrib
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