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In this review I shall criticise a specific aspect of this book, namely Mueller's arguments against the geometrical algebra (GA) hypothesis, i.e., the hypothesis that some parts of Euclid's Elements are essentially about algebra, only expressed in geometrical form.
In terms of Euclid's Book II Mueller argues that the proofs of the first few propositions suggest his ignorance of algebra. The first two propositions may be expressed thus in algebraic terms, as Mueller notes:
II.1. If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments.
II.2. If a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole.
(x+y)x+(x+y)y = (x+y)^2 (II,2a)
Mueller notes that II,2a can easily be derived from II,1a. But Euclid does not prove II.2 in this manner (he instead proves it from scratch, without reference to II.1). According to Mueller:
"The fact that he does not do so is an indication that he does not perceive the relation between these propositions in the way in which a modern algebraist would. For Euclid each of II,1--3 states an independent geometric fact." (p. 46)
On the basis of this and similar examples Mueller generalises thus:
"However one wishes to describe the results proved in book II, the proofs themselves show no sense of the connection between the propositions involved.Read more ›
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