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Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.
In H. Eves Return to Mathematical Circles., Boston: Prindle, Weber and Schmidt, 1988.
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Amazing Traces of a Babylonian Origin in Greek Mathematics
Amazing Traces of a Babylonian Origin in Greek Mathematics. Jöran Friberg, 2007, xx + 476 pp illustrations, diagrams, and indices. ISBN-13 978-981-270-452-8, ISBN-10, 981-270-452-3. Hackensack, NJ: World Scientific Publishing Co. US $98 / £56.
Jöran Friberg, Professor of History of Science and Technology at Chalmers University of Technology (Gothenburg, Sweden), presents not a story but eighteen short stories about plausible connections between Old Babylonian mathematics and classical Greek mathematics, plus two appendices. While his stated audience lists mathematicians, historians of science, Greek scholars, and Assyriologists, anyone interested in the intercultural relationship of ideas will find here fields ready for the harvest. The analytic table of contents outlines each chapter, just enough information to pique one’s interest. Take for instance the nine pages that focus on the Pythagorean Theorem and its environment: from Euclid’s proof through Pappus’s generalization that reverses direction to a study of the original Old Babylonian diagonal rule for rectangles with further implications for chains of triangles, trapezoids, or rectangles. To switch metaphors: an enjoyable feast!
Babylonian Algebra and Elements are the focus of four chapters: (1) that looks at Book II, (5) that considers Book X and Babylonian metric algebra, and (10) that collects metric algebra in Books I, VI and Data, not to overlook (13) which traces metric algebra in Diophantus’s Arithmetica. Chapters (7) and (8) study regular Babylonian polygons with those identified in Elements VI and XIII. Euclid’s “Lost Book On Divisions and Babylonian Striped Figures” engage Chapter 11, and “Elements IV and Old Babylonian Figures Within Figures” fill Chapter 6. The Pythagorean triplets are met in Chapter 3 that leads easily to (4) that investigates the lemma in Elements X 32/33 and Old Babylonian geometric progressions. The remaining chapters, 12 to 18, require an hour or more each for serious thought. They compare and contrast Babylonian geometry with corresponding topics from Hippocrates, Theon of Smyrna, Theodorus of Cyrene, the works of Heron, Ptolemy, and Brahmagupta, and common ideas from Heron, Ptolemy, and Archimedes, with concluding remarks on the Pseudo-Heronic Geometrica. The two appendices, “A Chain of Trapezoids with Fixed Diagonals” and “Catalog of Babylonian Geometric Figures,” speak for themselves.
Anyone chairing a weekly seminar in the history of Greek or Old Babylonian mathematics will find ample material for like minded students.
Barnabas Hughes, O.F.M., Professor Emeritus, California State University, Northridge