This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1860 Excerpt: ... BC in E (III. 30); (Dero.) therefore BE and EC are each of them the fifteenth part of the whole circumference ABCD; therefore if the straight lines BE and EC be drawn, and (IV. 1) straight lines equal to them be placed around in the whole circle, an equilateral and equiangular quindecagon shall be inscribed in it. Cor ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1860 Excerpt: ... BC in E (III. 30); (Dero.) therefore BE and EC are each of them the fifteenth part of the whole circumference ABCD; therefore if the straight lines BE and EC be drawn, and (IV. 1) straight lines equal to them be placed around in the whole circle, an equilateral and equiangular quindecagon shall be inscribed in it. Cor.--In the same manner, as was done in the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon shall be described about it. And likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it. Schol.--Whatever regular polygon can be inscribed in a circle, another of double the number of sides may be inscribed in it, by bisecting the arc subtended by a side of the former. It has been proved by M. Gauss, that it is practicable by means of plane geometry--namely, by the intersections of the straight line and circle, or by the resolution of simple and quadratic equations--to inscribe in a circle any regular polygon, provided the number of its sides be a prime number; that is, a number which has nocommon measure except 1--and be also some power of 2 increased, by 1. He has proved more generally--in his Disquisitiones Arithmeticce--if n, the number of sides of the regular polygon, exceeds by 1 the number which is the product of the prime factors 2," 3&, 5,.".. that the division of the circle into n equal parts, and consequently the possibility of inscribing in it a regular polygon of n sides, can be reduced to the solution of a equations of the second degree, b of the third, c of the fifth, and so on. EXERCISES. 1. Through a given point within or without...
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