
Re: RMP # 48
Posted:
Oct 6, 2007 5:37 AM


I think it's a popular misunderstanding among math historians that the Egyptians had no idea of irrational numbers. What they had was a different idea of such numbers: we can't point point them out exactly, but we can calculate with a 'cloud' or 'halo' of rational numbers around those elusive numbers ... The supreme god Ra, one of whose emanations was the sun, and whose hieroglyph was a small circle with a dot in the center, had a hundred names (if memory serves), but nobody knew his real name, not even his own daughter. This idea survives in Islam: Allah got 99 names but nobody knows the true name ...
Also the Greeks were calculating with rational numbers only, and we are doing the same: all our values for the square root of 2, and for pi, are rational, even the value of pi Yasumada Kanada calculated to over fifty million places  just a rational number.
The Egyptians accepted this basic fact: we can only handle rational numbers. And at the same time they went beyond: we can approximate irrational numbers with algebraic and geometric methods.  Calculate the square root of 2 using the trial and error method. You'll inevitably find better and better approximations that reveal a simple pattern, the pattern of a number column I reconstructed in 1979 while examining the geometry of Leonardo da Vinci's _Last Supper_ at Milan. This number column allows to find ever better values for the square root of 2 without reaching the true value (the numbers are endlessly increasing, and soon not even a computer can process them any more). What about the circle? Picture a square measuring 10 by 10 royal cubits or 70 by 70 palms. The diagonals measure practically 99 palms. Draw a circle around the center of the grid. The radius measures 5 royal cubits, while the circumference will pass the ends of the four axes, plus eight points of the grid according to the 'sacred triangle' 345, all in all twelve points of the grid, which divide the circumference into as many arcs. Measure them and you'll find eight arcs measuring pratically 90 fingers each, and four arcs measuring practically 40 fingers each, yielding a circumference of practically 880 fingers. Divide this length by the diameter 10 royal cubits or 70 plams or 280 fingers and you get 22/7 or 3 1/7 or 3 '7 for pi ... Then you can go a step further. Subdivide the grid 10 by 10 cubits into a grid of 50 by 50 smaller units. Now the circumference will pass four more points of the grid which are defined by the triple 72425 ... Go on to the grids 250 by 250, 1250 by 1250 ... ever smaller units, and the circumference of the circle will each time pass four more points of the grid. By connecting the points with straight lines you get a sequence of polygons that grow ever closer to the cirlce, and their calculation requires nothing more than the approximations of the square roots of 2 and 5, which can be done with the help of analoguous number colums. The full method including drawings of the polygons can be found here: http://www.seshat.ch/home/rhind1.htm I seriously believe that the Egyptian mathematicians of the Old Kingdom were the first to calculate the circumference, area, and number of the circle in a systematic way, over two millennia prior to Archimedes.
Regards, Franz Gnaedinger

