By Irving Adler

More than a hundred workouts with solutions and 2 hundred diagrams light up the textual content. lecturers, scholars (particularly these majoring in arithmetic education), and mathematically minded readers will get pleasure from this striking exploration of the position of geometry within the improvement of Western medical thought.

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Then it is clear that m and n are not both even numbers. ) If s = m/n, then m2/n2 = s2 = 2, and m2 = 2n2. That is, m2 must be an even number. This implies that m is an even number, because if m were odd, m2 would also be odd. Consequently there exists a whole number k such that m = 2k. Then m2 = 4k2. But m2 = 2n2. Equating these two expressions for m2, we see that 2n2 = 4k2, or n2 = 2k2. 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.

Therefore a regular polygon can be drawn with n sides for any integral value of n greater than or equal to 3. Let us call a regular polygon with n sides a regular n-gon. It is obvious that a regular n-gon has n angles. It is easy to calculate the number of degrees in each angle of a regular n-gon by first calculating the number of degrees in each exterior angle formed by extending one side. Let us denote by x the number of degrees in the exterior angle. To calculate x we take one exterior angle at each vertex of the n-gon, as shown in the diagram below, and then add them up.

The Theory of Ratio The concept of ratio is easy to define for two magnitudes of the same kind that have a common measure. Suppose, for example, two line segments a and b have a common measure c. That is, there exist positive integers m and n such that a = mc, and b = nc. Then we say that the segments a and b have the ratio m to n. In modern notation, we represent this ratio by the fraction m/n. Using this definition of ratio, there is an obvious and easy way to determine when two pairs of segments have the same ratio: If the ratio of a to b is the fraction m/n, and the ratio of c to d is the fraction p/q, then the ordered pairs (a, b) and (c, d) are in the same ratio if and only if the fractions m/n and p/q are equivalent.