Based on the article by M. Beech, Journal of Recreational Mathematics, 20 (3), 1988.
The megalithic standing stones of northern Europe have fascinated researchers for many centuries. Indeed, these spectacular rings, ellipses, and flattened circles, built between 6,000 and 3,500 years ago by a poorly understood race of people, have been slow to yield their secrets [1]. In recent times one of the most prolific surveyors of these ancient monuments was the late Professor Alexander Thom [2]. Thom surveyed and analyzed several hundreds of megalithic sites, and concluded that the construction parameters integral were integer multiples of the megalithic yard. Thom's main source of information was from his, so called, type A and B egg-shaped rings [3]. These, he argued, were designed and built according to the properties of what we would call Pythagorean triangles. Some of the data that Thom offered is reproduced in Table 1.
Side A | Side B | Side C | Remarks |
---|---|---|---|
3 | 4 | 5 | 3^2 + 4^2 = 5^2 |
5 | 12 | 13 | 5^2 + 12^2 = 13^2 |
12 | 35 | 37 | 12^2 + 35^2 = 37^2 |
8 | 9 | 12 | 8^2 + 9^2 = 12^2 + 1 |
11 | 13 | 17 | 11^2 + 13^2 = 17^2 + 1 |
38 | 49 | 62 | 38^2 + 49^2 = 62^2 + 1 |
28 | 30 | 41 | 28^2 + 30^2 = 41^2 + 3 |
Table 1. Sides A, B and C refer to the right-angled triangle used to construct the monument - C is the hypotenuse. See [3].
Table 1 shows the result of seven surveys by Thom. From these surveys Thom argued that while three were exact Pythagorean triangles, the others failed by such a small amount that this would hardly be recognizable on the ground. The inference then is that the ancient builders of stone circles had a knowledge of the properties of Pythagorean triangles. This is not the same, as Thom pointed out, as saying that the constructors knew Pythagoras's theorem. If we accept Thom's idea that the ancient builders had a working knowledge of Pythagoras's theorem (or at least right-angled integer sided triangles) then it implies that we have to attribute a reasonably sophisticated level of mathematical reasoning to what would other wise be considered a primitive culture [4, 5]. The important question, of course, is does the data really support the argument for their being a working knowledge of Pythagoras's theorem.
In an attempt to asses just what knowledge of integer triangles the ancient monument builders might have had, let us consider a set of right-angle triangles with sides A and B, and hypotenuse C. If A and B are integers chosen from the set [1, 30], which covers the typical range of observed values in megalithic yards, then there are 465 possible triangles that can be constructed. The question then becomes, how many of the 465 possible triangles satisfy the condition
A^2 + B^2 = C^2 +/- D ................................. (1)
where both C and D are integers. We allow D to be either possitive or negative. And, recall, it is the integer characteristics that Thom suggests the ancient builders were interested in. From Table 1 we are essentially looking for triangles with small integer (i.e., less than 5, say) values of D. The results summarized in Table 1 would also suggests that the greatest difference between A and B might be taken as 10 megalithic yards.
Table 2 is a summary of our study of equation (1) for all possible triangles with side A and B varying between [1, 30]. The first column gives the range of accepted D values i.e., the integers between [-5,5]. Columns two and three are the number of triangles that satisfy equation (1) for each D value when a) all possible triangles are considered and b) when only those triangles with |A - B| < 10 are considered. Remember, D = 0 corresponds to a true Pythagorean triangle.
D |
Number | Number |
5 | 6 | 6 |
4 | 16 | 11 |
3 | 5 | 3 |
2 | 3 | 2 |
1 | 18 | 12 |
0 | 13 | 9 |
-1 | 5 | 3 |
-2 | 8 | 4 |
-3 | 9 | 5 |
-4 | 11 | 7 |
-5 | 6 | 4 |
total = 100 | total = 64 |
Table 2. An examination of D values. The most common D values are 1, 4, 0, -1.
What do we find? If we are prepared to accept as reasonable the parameter set chosen, then the following experiment can be imagined. Choose integers A and B at random from the set [1, 30] and form a right-angled triangle. The numbers in table 2 indicate that about 22% of the triangles thus constructed would form an integer solution to equation (1). If the difference between A and B is further restricted to be less than or equal to ten then nearly 30% of the triangles constructed would yield integer solutions to equation (1). The implications of this 'numerical' experiment is such that given the megalithic builders always chose integer values for A and B then it should not be too surprising (to us, as later observers) that the resultant right-angled triangle should satisfy a condition of the form of equation (1). The conclusion that follows is, therefore, that the ancient megalithic builders did not understand the properties of Pythagorean triangles (but they may have appreciated the esthetic of integer triangles) and the idea that they might have is really an artifact of modern interpretation, reasoned a posteriori according to our modern-day knowledge of Pythagoras's theorem.
References: [1]. E.C. Krupp, In Search of Ancient Astronomers, McGraw-Hill, New York, 1978. [2]. A. Thom, Megalithic Sites in Britain, Oxford University Press, London, 1967. [3]. A. Thom, Megalithic Geometry in Standing Stones, New Scientist, March 12, 1964. [4]. C. Renfrew, The Social Archaeology of Megalithic Monuments, Scientific American, 249:5, p. 152, November 1983. [5]. C. Ruggles, Prehistoric Astronomy: How Far Did it Go?, New Scientist, June 18, 1981.
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