Pappus of Alexandria

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<BR> <br>Pappus was born in approximately 920 in Alexandria, Egypt. He was the last of the great Greek geometers and one of his major theorems is considered to be the basis of modern projective geometry ("Pappus"). Pappus flourished in the fourth century, writing his key work, the Mathematical Collection, as a guide to Greek geometry ("Biography"). In this work, Pappus discusses theorems and constructions of over thirty mathematicians including Euclid, Archimedes and Ptolemy ("Biography"), providing alternatives of proofs and generalizing theorems. <br>The Collection is a handbook to all of Greek geometry and is now almost the sole source of history of that science (Thomas 564). The separate books of the Collection were divided by Pappus into numbered sections. In the fourth section, Pappus discusses an extension on the Pythagorean Theorem (Thomas 575) now known as Pappus Area (Williams). Pappus drew parallelograms on two sides of a triangle, extended the external parallels to intersection, connected the included vertex of the triangle and the intersection point, used the direction and length of that segment to construct a parallelogram adjacent to the third side of the triangle, and proved that the sum of the areas of the first two parallelograms is equal to the area of the third parallelogram (Williams, Thomas 578-9). <br>Section five of book five of the Collection discusses regular solids with equal surfaces and their varying sizes (Heath 395). Pappus's conjecture was that the solid with the most faces is the greatest (Heath 396). He proved this using the pyramid, the cube, the octahedron, the dodecahedron, and the icosahedron of equal surfaces. Pappus noted that some of the other major Greek geometers had already worked out the proof of this conjecture using the analytical method, but that he would give a method of his own by synthetical

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