By Joran Friberg

A sequel to unforeseen hyperlinks among Egyptian and Babylonian arithmetic (World clinical, 2005), this e-book is predicated at the writer s extensive and floor breaking stories of the lengthy heritage 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 recognized Greek mathematicians seem to have been acquainted with a variety of facets of Babylonian metric algebra, a handy identify for an intricate mix of geometry, metrology, and quadratic equations that's identified from either Babylonian and pre-Babylonian mathematical clay drugs. The e-book s use of metric algebra diagrams within the Babylonian type, the place the part lengths and parts of geometric figures are explicitly indicated, rather than utterly summary lettered diagrams within the Greek sort, is key for a far better knowing 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 background of Greek arithmetic

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**Sample text**

2 below: Note that in § 1b the computed value of u is again ‘30’, but when s = 30, then in Fig. 2, left, the going-out q = ‘1’ cannot possibly have the value 1 00, which is greater than 30. Indeed, in a geometric interpretation like the one in Fig. 2, the difference s – q must be a (positive) length number. For this reason, the author of BM 13901 apparently chose to interpret ‘the going-out is 1’ in § 1 b as meaning that q = 1, not q = 1 00! 12. Old Babylonian Solutions to Metric Algebra Problems s–q q/2 A q A s – q/2 s – q/2 q/2 s 39 sq.

Fig. ) This equation, in its turn, can be reformulated in the following way: 4 A = D where A = u · s, p = u + s, q = u – s, and D = sq. p – sq. q. In other words, if sq. p – sq. q = D, then D = 4 A = 4 u · s = 2 u · 2 s. (Note that 2 u and 2 s can be interpreted as the length of the mid-line and the width, respectively, of the square corner formed by removing a square of side p from a square of side p, as in Fig. ) Therefore, any rectangular-linear system of equations of type B3a: sq. p – sq. q = D, p + q = 2 u can be solved by use of El.

P/2 = A + sq. q/2 where A = u · s, p = u + s, and q = u – s. Therefore, the purpose of El. 6 may have been, essentially, to demonstrate that any rectangular-linear system of equations of type B1b: u · s = A, u–s=q can be solved as follows: (u + s)/2 = p/2 = sqs. (A + sq. q/2), u = p/2 + q/2 = sqs. (A + sq. q/2) + q/2, s = p/2 – q/2 = sqs. (A + sq. q/2) – q/2. As mentioned above, the solution to a quadratic-linear system of equations of type B2a can be obtained by use of El. 4 in combination with El.