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| Keywords: Michael Stifel's Arithmetica Integra (1544) p225.tif This is page 225 of the Arithmetica Integra 1544 by Michael Stifel 1487-1567 Stifel is one of the best-known German cossists of the sixteenth century Stifel's work covered the basics of algebra using the German symbols for powers of the unknown and also considering negative exponents for one of the first times in a European book He also presented the Pascal triangle as a tool for finding roots of numbers and was one of the first to present one combined form of the algorithm for solving quadratic equations http //mathdl maa org/mathDL/ pa content sa viewDocument nodeId 2591 bodyId 3752 The diagram here on p 255 represents the solution to the pair of simultaneous equations x2 + y2 - x + y 78 xy + x + y 39 Here the two unknowns are represented by AC and BC while the sum AB is called B by Stifel Also the script z is Stifel's notation for the square of the first unknown namely x2 Note that therefore the smaller square on the upper right is labeled with the script z the two rectangles are labeled 39 - 1B since their areas are each xy which is equal to 30 - x + y and the larger square which is equal to y2 is labeled 78 + B - z that is 78 + x + y - x2 Stifel completes the problem as follows The sum of the areas of all four regions of the diagram is equal to 156 - B and this equals B2 It follows that B 12 Therefore the larger square has area 90 - x2 and the two rectangles each have area 27 But either of those rectangles is the mean proportional between the larger square and the smaller square Therefore 90 - x2 27 27 x2 It follows that 90x2 - x4 729 So x2 9 and x 3 Then y 9 and the problem is solved http //mathdl maa org/mathDL/ pa content sa viewDocument nodeId 2591 bodyId 3752 PD-old Algebra Michael Stifel | ||||