. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. Ill] GEOMETRICAL RECREATIONS 47 Hence, by Euc. n. 13, AB* + BC* AB' + BD' BG ~ BJD ' y)Di A Ra .'.j^ + BC-2BE = ^L + BD-2BE. AB' JtT)_AB» B0 . AB' AB* BG BD .-. BG = BD, a result which is impossible. Third Fallacy*. To prove that the sum of the lengths of two sides of any triangle is equal to the length of the third side. AK , , , 7d. Let ABC be a triangle. Complete the parallelogram of which AB and BG are sides. Divide AB into
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. Ill] GEOMETRICAL RECREATIONS 47 Hence, by Euc. n. 13, AB* + BC* AB' + BD' BG ~ BJD ' y)Di A Ra .'.j^ + BC-2BE = ^L + BD-2BE. AB' JtT)_AB» B0 . AB' AB* BG BD .-. BG = BD, a result which is impossible. Third Fallacy*. To prove that the sum of the lengths of two sides of any triangle is equal to the length of the third side. AK , , , 7d. Let ABC be a triangle. Complete the parallelogram of which AB and BG are sides. Divide AB into n + 1 equal parts, and through the points so determined draw n lines parallel to BG. Similarly, divide BG into n + 1 equal parts, and through the points so determined draw n lines parallel to AB. The parallelogram ABGD is thus divided into (n + 1)' equal and similar parallelograms. I draw the figure for the case in which n is equal to 3, then, taking the parallelograms of which AG is a diagonal, as indicated in the diagram, we have AB + BG = AG + EJ + KL + MN + GH + JK + LM + NO. A similar relation is true however large n may be. Now let n increase indefinitely. Then the lines AG, GH, &c. will * The Canterbury Puzzles, by H. E. Dudeney, London, 1907, pp. 26— Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Ball, W. W. Rouse (Walter William Rouse), 1850-1925. London, Macmillan
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