The step technique was probably known earlier than Cardano, because it seems to have been described briefly by Leonardo da Vinci (1452-1519) in one of his notebooks. On Foglio 798a (volume 9) of the Codex Atlanticus, Leonardo sketched a long thin rectangle, in which he drew three light lines parallel to its length and two light lines parallel to its width, so that the rectangle was divided into twelve equal rectangles. He then outlined in bold the zigzag cut that would create the two pieces for the step. He labeled this drawing with the following (in Italian):Il modo a fare crescere e scemare un quadro o un tondo altrettanto.which translates as:The way to make a rectangle or also a circle larger and smaller.I do not understand how his remark applies to circles. See the updates to chapter 1 for the full reference to the facsimile reproduction and transcription of Codex Atlanticus.
David Singmaster brought to my attention the fact that Luca Pacioli described a step dissection in his circa 1500 manuscript, "De Viribus Quantitatis". (This was belatedly published by Castello Sforzesco (Milan) in 1997.) On page 190v, Pacioli described how to cut up a 4 x 24 rectangle into two pieces that can form a 3 x 32 rectangle. Leonardo's diagram (mentioned above) completely conforms to Pacioli's description. This is not surprising, since the two men lived and worked together for a time.
Erik Demaine has sent me a copy of Wakoku Chiyekurabe (Mathematical Contests), by Kan Chu Sen, 1721 (in Japanese). It contains an illustration of the two-piece dissection of a (3a x 4b)-rectangle to a (4a x 3b)-rectangle.
My attribution to Brahmagupta of the explicit formulation of the methodx = m2 - n2, y = 2mn, and z = m2 + n2was muddied by my use of the phrase "every basic solution", implying that Brahmagupta had proved that every basic solution can be so generated. But a proof of this came much later. Thanks to Sherman Stein for pointing out my lack of clarity.
There are a number of different techniques for 4-piece dissections of squares in the class of Pythagoras and class of Plato. I identified four different ones for the class of Pythagoras in a talk that I gave at the Gathering for Gardner in 2004. However Edo Timmermans has identified a number of more rather complicated ones in his article, "Pythagorese Dissections", Cubism For Fun, no. 73 (July 2007), pages 12-13.
Copyright 1998-2007, Greg N. Frederickson.
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Last updated December 30, 2007.