By Joran Friberg

A sequel to unforeseen hyperlinks among Egyptian and Babylonian arithmetic (World clinical, 2005), this booklet relies at the writer s in depth and flooring breaking stories of the lengthy heritage of Mesopotamian arithmetic, from the overdue 4th to the overdue 1st millennium BC. it really is argued within the publication that a number of of the main well-known Greek mathematicians seem to have been conversant in a variety of points of Babylonian metric algebra, a handy identify for an difficult mixture of geometry, metrology, and quadratic equations that's recognized from either Babylonian and pre-Babylonian mathematical clay capsules. The booklet s use of metric algebra diagrams within the Babylonian sort, the place the aspect lengths and components of geometric figures are explicitly indicated, rather than thoroughly summary lettered diagrams within the Greek type, is key for a stronger knowing of many fascinating propositions and structures in Greek mathematical works. the writer s comparisons with Babylonian arithmetic additionally result in new solutions to a couple very important open questions within the background of Greek arithmetic

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**Example text**

Their lengths I heaped, b. The equalsides are what? (several problems missing) 7d. 7e. 7f. ····· 8a. 8b. 8c. 9a. 9b. 9c. (badly preserved) D is the field between, d the distance. The outer and inner equalsides are what? D is the field between. c (times) the outer equalside is the inner equalside. The inner equalside is what? p is the outer equalside, q the middle, r the inner equalside. The field between between is what? The field between between is E. Their lengths I joined, b. The equalsides are what?

Let u = p/2 + q/2 = p/2 + sqs. (sq. p/2 – A), s = p/2 – q/2 = p/2 – sqs. (sq. p/2 – A). 7. El. 14*, Constructive Counterparts to El. 5-6 21 Then u + s = p, u – s = q, and it can be proved as above that the lengths u and s constructed in this way with departure from the given quantities p and A are solutions to a rectangular-linear system of equations of type B1a: u · s = A = sq. h, u + s = p. What does this result have to do with El. 14, where Euclid shows how to “construct a square equal to a given rectilineal figure”?

C – sq. a. Then it follows from the result in El. II. 13 that 2 b · q = sq. b – D, so that q = (sq. b – D)/(2 b), p = (p + q) – q = b – q = (sq. b + D)/(2 b). It may seem a bit strange that in El. 12-13 the case of the obtuseangled triangle precedes the case of the acute-angled triangle. The reason can be that, as pointed out above, the proof of El. 12 makes use of El. 4, while the proof of El. 13 makes use of the later proposition El. 7. 9. Summary. The Three Parts of Elements II The discussion above aimed to demonstrate that Elements II can be divided into three distinct parts with obvious relations to the nine basic equations or systems of equations in OB metric algebra: A.