Archimedes

The Works of Archimedes

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. 1897 Excerpt: ... Proposition 3. Given two unequal magnitudes and a circle, it is possible to inscribe a polygon in the circle and to describe another about it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than tliat of the greater magnitude to the less. Let A, B represent the given magnitudes, A being the greater. Find Prop. 2 two straight lines F, KL, of which F is the greater, such that F: KLA: B (1). Draw LM perpendicular to LK and of such length that KM=F. In the given circle let CE, DG be two diameters at right angles. Then, bisecting the angle DOC, bisecting the half again, and so on, we shall arrive ultimately at an angle (as NOG) less than twice the angle LKM. Join NG, which (by the construction) will be the side of a regular polygon inscribed in the circle. Let OP be the radius of the circle bisecting the angle NOG (and therefore bisecting NC at right angles, in H, say), and let the tangent at P meet OG, ON produced in S, T respectively. Now, since, L CON 2 Z. LKM, /. HOC/. LKM, and the angles at H, L are right; therefore MK: LK OC: OH OP: OH. Hence ST: CN MK: LK F: LK; therefore, a fortiori, by (1), ST: CNA: B. Thus two polygons are found satisfying the given condition. Proposition 4. Again, given two unequal magnitudes and a sector, it is possible to describe a polygon about the sector and to inscribe another in it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than the greater magnitude has to the less. The "inscribed polygon" found in this proposition is one which has for two sides the two radii bounding the sector, while the remaining sides (the number of which is, by construction, some power of 2) subtend equal parts of the arc of the sector; ...