By Irving Adler

More than a hundred routines with solutions and 2 hundred diagrams light up the textual content. lecturers, scholars (particularly these majoring in arithmetic education), and mathematically minded readers will enjoy this impressive exploration of the position of geometry within the improvement of Western clinical thought.

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4999 . . represent the same point. Addition and multiplication of real numbers are defined by means of sums and products of longer and longer terminating decimals. It can be shown that the real number system satisfies conditions 1 to 9 on pages 32–3, so the real number system is a field. * Every point r in the real number system divides the rational numbers into two parts A and B, where A includes all the rational points that are to the left of r, and B includes all the remaining rational points.

4. For each x there exists a member denoted by −x that has the property x + (−x) = (−x) + x = 0. Let us consider next those properties of a field that relate to the operation · (times) alone. Because the number 0 is an exception for condition 8, let us exclude 0 from consideration. Denote by F* the set of all non-zero members of F. If x and y are in F* there is a member of F* denoted by x·y, and the operation · in F* satisfies conditions 5 to 8 of pages 32–3. We repeat these conditions with a slight modification of 8 made possible by the fact that we are now talking about F* instead of F.

That is, n2 must be even, and therefore n must be even. Notice that while m and n are not both even, we have been compelled to say that they are both even. We were led into this absurdity as a result of assuming that the length s may be expressed as a ratio of whole numbers. Therefore this assumption must be false, and we conclude that the side and diagonal of a square do not have a common measure. Greek arithmetic was not sufficiently developed to provide a way of expressing by numbers the ratio of two lengths that do not have a common measure.