Mathematical Methods in Hellenistic Times

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Read Chapter 5 that starts on page 133 and answer the 3 questions

3.5 Ratio and Proportion 71

FIGURE 3.22

Diagonals of inner pentagon

of a pentagram

then

= a. But then AE = AF, and since by construction AE = FC, it follows that triangle

AFC is isosceles and that d = ? . Finally, a = ß + ? = d + d = 2d, as desired. To show that

ß = d, circumscribe a circle around triangle AFC. Since the rectangle contained by CE,

FE, equals the square on FC, it follows that this rectangle also equals the square on AE.

Proposition III–37 then asserts that under these conditions on the lines AE and CE, AE is

tangent to the circle. Proposition III–32 then allowed Euclid to conclude that ß =d as desired,

completing the proof of the construction.

Given the isosceles triangle with base angles double the vertex angle, the inscribing of

the regular pentagon in a circle is now straightforward. Euclid first inscribed the isosceles

triangle ACE in the circle. Next, he bisected the angles at A and E. The intersection of these

bisectors with the circle are pointsD and B, respectively. Then A, B, C, D, E are the vertices

of a regular pentagon.

Euclid completed Book IV with the construction of a regular hexagon and a regular 15-

gon in a circle, but did not mention the construction of other regular polygons. Presumably, he

was aware that the construction of a polygon of 2nk sides (k = 3, 4, 5) was easy, beginning

with the constructions already made, and even that, in analogy with his 15-gon construction,

it was straightforward to construct a polygon of kl sides (k, l relatively prime) if one can

construct one of k sides as well as one of l sides. Whether he was aware of a construction for

the heptagon, however, is not known. In any case, that construction, the first record of which

is in the work of Archimedes, would for Euclid be part of advanced mathematics, rather than

part of the “elements,” because it requires tools other than a straightedge and compass.

3.5 RATIO AND PROPORTION

The regular pentagon is part of the pentagram, evidently one of the symbols used by the

Pythagoreans. Thus, it is believed that the Pythagoreans worked out a construction of the

pentagon, although more likely their construction used similarity rather than the method

described above. It is therefore plausible that the property of the pentagram in reproducing

itself when one connects the diagonals of the inner pentagon (Fig. 3.22) could well have

been an alternative path to the discovery of incommensurability, rather than the one described

earlier. To explain this, we need to move to Book VII, the first of the three books of number

theory in the Elements.

Book VII, like all the number theory books, deals with what we call the positive integers in

contrast to the geometrical magnitudes of the earlier books. And the first item of business for

Euclid here is the familiar process for finding the greatest common divisor of two numbers.

This algorithm, usually called the Euclidean algorithm although certainly known long

before Euclid, is presented in Propositions VII-1 and VII-2. Given two numbers, a, b, with

a >b, one subtracts b from a as many times as possible; if there is a remainder, c, which of

course must be less than b, one then subtracts c from b as many times as possible. Continuing

in this manner, one eventually comes either to a number m, which “measures” (divides) the

one before (Proposition VII–2), or to the unit (1) (Proposition VII–1). In the first case, Euclid

proved that m is the greatest common measure (divisor) of a and b. In the second case, he

showed that a and b are prime to one another. For example, given the two numbers 18 and

80, first subtract 18 from 80. One can do this four times, with remainder 8. Next subtract 8

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