Geometry and algebra in ancient civilizations

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Springer, 1983 - History - 223 pages
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Great Exposition of Math: Not a History of Science Book
Geometry and Algebra in Ancient Civilizations is a very accurate mathematical text that incorporates the mathematical traditions of many
Old-World civilizations. If each chapter is read for only the mathematical correctness, one will learn that ancient peoples from all cultures are highly capable of abstract thinking and problem solving. Nevertheless, there is one problem that is highly pervasive throughout the text, and this problem is independent from the historical exposition: Van der Waerden’s theory.
Amazingly, the book is quite accurate with the archival aspects of ancient mathematical traditions, despite the absence of a bibliography. However, Van der Waerden’s theory of cross-cultural exchange of knowledge is highly improbable for the timeframe in which this book is centred (1800 BC – 500 AD). The idea that there exists a common origin to ancient mathematical knowledge is too farfetched a theory to be taken seriously. For example, Van der Waerden claims that the Pythagorean theorem found across many cultures is descendent from the Neolithic oral tradition of the henge builders of Northern Europe.
Van der Waerden seems to ignore that mathematics is independent to cultural background; hence, any sufficiently advanced society, with the desire to investigate and exploit the language of mathematics is capable of independent discovery. (I highly doubt the ancient Mayans were influenced by Indian mathematics for the conception of the number zero. These two cultures were independently open to interpret zero as a number.)
As for the latter chapters comparing the knowledge of Apollonius, Liu Hui and Aryabhata, Van der Waerden fails to explain the obvious linguistic barriers of Greek, Sanskrit and Chinese. There are also several contradictions to this particular transmission-of-knowledge theory.
1)Apollonius did not compute pi to be 3.1416; however, Ptolemy did arrive at the fraction 377/120 (in decimal 3.14167). Liu Hui and Aryabhata did, but each mathematician arrived at the result very differently.
2)The translation of Greek mathematical texts to Chinese was highly unlikely. Not only was there a significant language barrier from the lack of contact; the mathematical style of both cultures is very different. The converse is equally valid.
3)Liu Hui should not have gotten the box-lid solid from the Greeks because he conceived of the shape for a very different purpose. Archimedes compared the volume of the box-lid solid with the cube, while Liu Hui compared the volume of the box-lid with the sphere.
This book presents a ludicrous theory of the history of mathematics in antiquity, nevertheless, the mathematics presented is accurate. The guy knows his math, but his idea of mathematical origins is flawed.


Pythagorean Triangles
Pythagoras and the Ox
Euclids Proof

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