The quadrature of the circle discovered ... / Submitted to the consideration of the Royal Society.

  • Parsey, Arthur, 1791-1857.
Date:
1832
Catalogue details

Licence: Public Domain Mark

Credit: The quadrature of the circle discovered ... / Submitted to the consideration of the Royal Society. Source: Wellcome Collection.

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    square inscribed will be */ 2, that of the square circum¬ scribed will be equal to the diameter 2, therefore the surface of the square inscribed ~ \/ 2 x / 2 - and that of the square circumscribed 2 X 2 =: 4. Now if we make A — 2 and B = 4, we find the figure of double the number of sides (octagon) inscribed — A! — A x B — 2 x 4< — 8 — 2A x B 2,8284271, and the octagon circumscribed = B' = -+- Af 4 x_4_= 3 3137085, 2 + y/ 8 Knowing these last polygons inscribed and circumscribed, we find by their means the polygons of double the number of sides (16) from the new supposition A = 2,8284271, B — 9,3137085, the other A' — ^/ Ax B“ 3,0614674 and Bf = 2 A. x I I — 3,1825979, and by assuming A— 3,0614674, and B = 3,1825979, we shall find the polygons of 32 sides, from which we perceive that the greater the number of sides, the nearer the polygons approach each other ; and by proceed¬ ing in this manner, we shall find ultimately that the differ¬ ence between the polygon inscribed and the polygon circum¬ scribed will be so small, that we may conclude they coincide. The calculations of the polygons are here prolonged till there is no difference between them to about the seventeenth place of decimals. Number of sides. 4 - 8 - 16 - 32 - 64 - 128 -- 256 - 512 _ Polygon inscribed. 2 2,8284271 - 3,0614674 - 3,1214451 - 3,1365485 - 3,1403311 - 3,1412772 - 3,1415138 - Polygon circumscribed. 4 3,3137085 3,1825979 3,1517249 3,1441184 3,1422236 3,1417504 3,1416321