# Amazing Traces of a Babylonian Origin in Greek Mathematics by Joran Friberg

By Joran Friberg

A sequel to unforeseen hyperlinks among Egyptian and Babylonian arithmetic (World medical, 2005), this e-book is predicated at the writer s in depth and flooring breaking reports of the lengthy background of Mesopotamian arithmetic, from the overdue 4th to the past due 1st millennium BC. it truly is argued within the booklet that a number of of the main well-known Greek mathematicians seem to have been acquainted with numerous elements of Babylonian metric algebra, a handy identify for an tricky mixture of geometry, metrology, and quadratic equations that's recognized from either Babylonian and pre-Babylonian mathematical clay pills. The publication s use of metric algebra diagrams within the Babylonian type, the place the aspect lengths and components of geometric figures are explicitly indicated, rather than entirely summary lettered diagrams within the Greek type, is vital for a stronger figuring out of many fascinating propositions and buildings in Greek mathematical works. the writer s comparisons with Babylonian arithmetic additionally bring about new solutions to a few very important open questions within the historical past of Greek arithmetic

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C = sq. a + sq. b – 2 b · q B3b : B3a : sq. p – sq. q = D, p – q = b, sq. p – sq. q = D, p + q = b, with D = sq. c – sq. a with D = sq. c – sq. a 2 b · q = D – sq. b, etc. 2 b · q = sq. b – D, etc. Fig. 1. Interpretations of the diagrams in El. 12 and El. 13. 8. El. 12-13, Constructive Counterparts to El. 8 23 With the notations introduced in Fig. 1, left, the proof of El. 12 proceeds as follows: sq. p = sq. b + sq. q + 2 b · q sq. p + sq. h = sq. b + sq. q + sq. h + 2 b · q sq. c = sq. b + sq.

The very convenient approximation / = appr. 3 is used in all Babylonian mathematical texts. More precisely, the area A and the diameter d of a circle are expressed as follows in terms of the arc (circumference) a: A = 5 · a, where ‘5’ means ;05 = 5/60 = 1/12 = appr. 1/4/, d = 20 · a, where ‘20’ means ;20 = 20/60 = 1/3 = appr. 1//. In addition to sexagesimal fractions, such as the circle constants ‘5’ and ‘20’, there are also two other kinds of fractions of numbers that appear in Babylonian mathematical texts.

5. El. 8 and the Two Basic Subtractive Quadratic-Linear Systems of Equations El. 8 If a straight line is cut at random, four times the rectangle contained by the whole and one of the segments, together with the square on the remaining segment, is equal to the square described on the whole and the mentioned segment as on one straight line. A C B D G M T K O S E U Q R N P H L F Fig. 1. The diagram in El. 8. 5. El. 8 and the Two Basic Subtractive Quadratic-Linear Systems of Equations 15 In Fig. 2 below, the given straight line AB is called u and the two segments into which it is cut are called s and q.