The mathematics of the structure of Stonehenge

Meca \8SS ea CMA A = Rave Arch. Hist. Exact Sci. (2011) 65:67-97 DOI 10. 1007/s00407-0 10-007 1-0 The mathematics in the structures of Stonehenge Albert Kainzinger Received: || February 2010 / Published online: 4 November 2010 © Springer-Verlag 2010 Abstract The development of ancient civilizations and their achievements in sciences such as mathematics and astronomy are well researched for script-using civilizations. On the basis of oral tradition and mnemonic artifacts illiterate ancient civilizations were able to attain an adequate level of knowledge. The Neolithic and Bronze Age earthworks and circles are such mnemonic artifacts. Explanatory models are given for the shape of the stone formations and the ditch of Stonehenge reflecting the circular and specific non-circular shapes of these structures. The basic mathematical concepts are Pythagorean triangles, thus adopting the construction procedures of the Neolithic circular ditches of Central Europe in the fifth Millennium BC and later earthworks and stone circles in Britain and Brittany. This knowledge was extended with new elliptical concepts. Approximations for the values of x and the square root of 2 are encoded in the henge. All constructions were performed using a standardized “Babylonian” metrology that shows a remarkable consistency and comprehensible development over some 14 centuries. 1 Introduction In ancient script-using civilizations, the achievements in mathematics and astronomy have been investigated to an adequate degree (e.g.. Neugebauer 1969, 1975; van der Waerden 1954, 1973, 1983). In illiterate ancient European civilizations, there is Communicated by Menso Folkerts. Electronic supplementary material The online version of this article (doi: 10, 1007/s00407-01 0-007 | -0} contains supplementary material, which is available to authorized users. A. Kainzinger (59) Elektrasir. 46a. 81925 Munich, Germany e-mail: akainzinger@ googlemail.com a Springer

Arch. Hist. Exact Sci. (2011) 65:67–97 DOI 10.1007/s00407-010-0071-0 The mathematics in the structures of Stonehenge Albert Kainzinger Received: 11 February 2010 / Published online: 4 November 2010 © Springer-Verlag 2010 Abstract The development of ancient civilizations and their achievements in sciences such as mathematics and astronomy are well researched for script-using civilizations. On the basis of oral tradition and mnemonic artifacts illiterate ancient civilizations were able to attain an adequate level of knowledge. The Neolithic and Bronze Age earthworks and circles are such mnemonic artifacts. Explanatory models are given for the shape of the stone formations and the ditch of Stonehenge reflecting the circular and specific non-circular shapes of these structures. The basic mathematical concepts are Pythagorean triangles, thus adopting the construction procedures of the Neolithic circular ditches of Central Europe in the fifth Millennium bc and later earthworks and stone circles in Britain and Brittany. This knowledge was extended with new elliptical concepts. Approximations for the values of π and the square root of 2 are encoded in the henge. All constructions were performed using a standardized “Babylonian” metrology that shows a remarkable consistency and comprehensible development over some 14 centuries. 1 Introduction In ancient script-using civilizations, the achievements in mathematics and astronomy have been investigated to an adequate degree (e.g., Neugebauer 1969, 1975; van der Waerden 1954, 1973, 1983). In illiterate ancient European civilizations, there is Communicated by Menso Folkerts. Electronic supplementary material The online version of this article (doi:10.1007/s00407-010-0071-0) contains supplementary material, which is available to authorized users. A. Kainzinger (B) Elektrastr. 46a, 81925 Munich, Germany e-mail: akainzinger@googlemail.com

some evidence of basic astronomical knowledge due to prehistoric monuments and earthworks (Thom and Thom 1980), but there are almost no secured findings about the level of mathematical knowledge. Because there are many identical solutions for mathematical problems in the ancient civilizations of Mesopotamia, Egypt, India, China and Greece, van der Waerden assumed a common origin for the basic mathematical concepts, which he placed in Neolithic Europe (van der Waerden 1983). Two examples should highlight the basis of this supposition: (a) in all the abovementioned civilizations, the Pythagorean Theorem was used, and a common set of Pythagorean Triangles was known and/or calculated in the same way; (b) in India and in Greece, comparable ritual geometric constructions were performed (generally for the shapes of altars). This assumption was supported by earlier articles of Seidenberg (Seidenberg 1961, 1978, 1981). A figure summarizing these findings of van der Waerden is included as Electronic Supplementary Material (ESM Fig. 1). In illiterate civilizations, knowledge is generally transferred by oral tradition to subsequent generations. Mathews proposed an ancient core of a Neolithic oral tradition of mathematics (Mathews 1985). In addition, the knowledge can be documented by the depiction of illustrating pictures or symbols on mnemonic objects (e.g., Vansina 1985; Haarmann 1991) or by the construction of appropriate devices and monuments. The majority of the Neolithic and Bronze Age earthworks and stone/wood rings are object of this mnemonic procedure. The encoding of the mathematical concepts in these monuments was performed by the appropriate construction of ditches and rings which are composed by circles and circular arcs. With their centers and endpoints, these elements determine the geometrical models. This procedure is distinct from the documentation of geometric problem solutions nowadays. With the present method, linear geometric models are depicted directly by their line segments, while in the ancient monuments, the linear geometric models are represented exclusively by the respective vertices. The principal objective of the investigation of ancient earthwork and stone/wood rings now is to determine the original construction concept. So much effort in the construction of these outstanding monuments could not have been expended without a comprehensible and consistent plan. A determination procedure comprising six steps was established to work out highly probable concepts of the ancient plans (see Sect. 8, Fig. 13). Two steps have to be explained at this point to familiarize the reader with the often-used terms “backward construction” and “forward construction.” A backward construction is the deduction of a potential construction concept from the available data of the plan of the ancient monument’s present remains. There might be several attempts and different results in this step for the search of the original concept. The six-step determination procedure aims to reduce these options to preferably one plausible plan. The forward construction is the exact construction based on the selected mathematical concept and the established metrology. This article is concerned exclusively with the construction concepts of the Stonehenge structures. Some efforts have been there to uncover the principles of the Stonehenge constructions, e.g., the early attempt by the use of computers in the early 1960s (Hawkins 1965). However, this statistical approach and other methods produced no satisfactory results. A new, sound approach was published by Johnson

The mathematics in the structures of Stonehenge (2008), although the author cannot agree with the proposed mathematical explanation models (Johnson 2008). 2 Z and Y holes We begin with the last phases of Stonehenge. Although the construction of the Z and Y holes document in some way a decline of the mathematical knowledge of the Stonehenge society, the construction concepts of these ring-like structures reflect the beginning in an especially descriptive manner. The Y and Z holes obviously do not form an exact circle. We encounter this characteristic in nearly all early rondels (circular ditches) and wood and stone “circles” in Central Europe and Britain. In the case of the exact circles in Stonehenge (Aubrey holes, Sarsen circle), we will show a clear reasoning for these constructions. The basis of the construction of these ringlike structures is always a mathematical concept. In the majority the construction starts with a Pythagorean triangle. The knowledge of Pythagorean triangles traces back to the fifth millennium bc (rondels/“Kreisgräben” in Central Europe). 2.1 Z holes The construction of the Z holes is an excellent example of this procedure. To exemplify the a.m. determination procedure, we will depict the results of the backward construction for the Z holes (Fig. 1a) and also the forward construction as well (Fig. 2). (For the remaining structures, a combination of the sampling points and the resulting forward construction is displayed, with the exception of the Bluestone horseshoe.) At first, an isosceles triangle (Z 31 Z 32 Z 33 ) composed of two mirrored Pythagorean triangles of the shape (3,4,5) is constructed by sharing the leg “4.” Then, another isosceles triangle, (Z 31 Z 33 Z 34 ), also composed of two mirrored Pythagorean triangles of the shape (3,4,5) is attached, but in this case sharing the leg “3.” The outline of the overall geometry is another single Pythagorean triangle (Z 32 Z 33 Z 34 ) of the shape (3,4,5) (Fig. 1c). From a mathematical point of view, the basis for a Pythagorean triangle is a Pythagorean triple of three integers a, b, and c fulfilling the Pythagorean theorem a 2 + b2 = c2 . If these integers have no common factor, then they are called “primitive Pythagorean triples.”1 In the depiction of the construction concepts by Pythagorean triangles, the integers i of the corresponding primitive Pythagorean triples are denoted by squared brackets: [i]. The position of the vertices Z31–Z34 (which are the centers of the respective circular arcs) are calculated by a nonlinear regression analysis (see Sect. 8); the sampling points for the regression analysis are depicted in Fig. 1a by a small cross symbol. The construction was performed on the basis of a consistent metrology (see also Sect. 7). The concrete length of the legs of the respective Pythagorean triangles are 6, 8, and 10 cubits; the resulting overall shape is a Pythagorean triangle with legs of 12, 16, and 20 cubits. The position of vertex Z31 is near the generic center of the Stonehenge structures. 1 For example, (6,8,10) is a Pythagorean triple; (3,4,5) is the corresponding primitive Pythagorean triple.

2.2 Y holes As we will show later, the construction concept of the Z holes is somewhat simpler than that of the preceding phases 3ii and 3iv of Stonehenge. The Y holes show a further decline of the construction complexity. The base of the construction is only one Pythagorean triangle (Y31 Y32 Y33 ) of the type (5,12,13); this is the simplest construction concept of all the ring-like structures of Stonehenge (Fig. 3b). The concrete lengths of the legs are 10, 24, and 26 cubits, thus starting the construction with a Pythagorean triangle of the shape (10,24,26). Once again the vertices of the triangle serve as the centers of the three circular arcs on which the stones were set. Vertex Y31 again is very near to the general center of Stonehenge. The length of the radius y1 of the dominant circular arc with center Y31 seems to be set arbitrarily to 50 cubits which equals five poles. The leg Y31 Y32 of the Pythagorean triangle and the radius Y32 Y 3 of the circular arc with center Y32 form a straight line; as the leg Y31 Y32 is 10 cubits, the length of this latter radius is exactly 40 cubits which equals four poles. The length of the radius with center Y33 results from the smoothing procedure at hole Y15. In the backward construction (Fig. 3a), these two arcs meet each other at hole Y15 with a small gap, while the arc with center Y32 misses the main arc in the segment between the holes Y2 and Y3. In general, the holes/stones of the Y ring were not set with the accuracy of the Z ring. The shapes of three holes are outside the exact circular arc (Y20, Y25, Y29), and others do not hit the arc centrically. For the position of the hole/stone Y8, we have no mathematical concept. It could be an arbitrary determination to fill the gap between the holes Y7 and Y9. This additionally reflects the decline in the geometrical methods and accuracy for ring constructions in this last phase of the Stonehenge constructions. For the forward construction, the mean of the measures of the radii y1 and y2 were used; the radius y1 provides the cubit measure with the least tolerance. In this forward construction, the position of vertex Y33 by an exact construction of the Pythagorean triangle (10,24,26) becomes a bit higher compared with the backward construction. This forward construction now results in a smooth curve at hole Y15 and hole Y3 as well (Fig. 3c). 2.3 Deduced measures ‘ The above mentioned cubit measures are ideal values according to the selected mathematical concept—in these cases, the underlying Pythagorean triangles. The actual length values extracted from the plan are summarized in Table 1A. For the Z holes, the resulting five cubit measures vary between 49.9003 and 51.4553 cm and have a mean of 50.7228 cm. The mean cubit measures of the Y holes are 52.3558 cm (backward construction) and 53.7403 cm (forward construction). The arbitrary radius z 1 of the circular arc with center Z31 with the supposed length of 36 cubits (= 6 reeds = 3 rods) results in a cubit measure of 50.4836 cm which is very close to the above established mean value for the Z holes. The radius y1 of the circular arc with center Y31 seems to be set arbitrarily to 50 cubits (= 10 poles) and results in a cubit measure of 53.9227 cm which is again very close to the above established mean value for the Y holes (forward construction).

Table 1 Measures of the Y and Z holes, the ditch, and the Aubrey holes. “length figures” are the measures of the respective segments in the computer program. “length henge” results from the scale in the figure. “result. cubit measure” is calculated as “length henge” divided by “no. cubits” Segment Length figure ( mm) Length henge (m) No. cubits Result. cubit measure (cm) A: Y holes and Z holes measures Measure 58.4667 20.00000 Z 31 –Z 32 15.0421 5.14553 10 51.4553 Z 31 –Z 33 14.8960 5.09555 10 50.9555 Z 32 –Z 33 17.6184 6.02682 12 50.2235 Z 31 –Z 34 14.5884 4.99033 10 49.9033 Z 33 –Z 34 23.8901 8.17221 16 51.0763 Z holes Mean: 50.7228 z1 53.129 18.17411 36a 50.4836 z2 66.471 22.73807 45 50.5290 z3 72.916 24.94275 50 49.8855 z4 73.462 25.12952 50 50.2590 Y31 –Y32 15.2718 5.22410 10 52.2410 Y32 –Y33 36.9047 12.62418 24 52.6007 Y31 –Y33 39.6949 13.57864 26 52.2255 Y holes Mean: 52.3558 53.9227 y1 78.8170 26.96133 50b y2 62.9270 21.42313 40c 53.5578 y3 114.7840 39.26474 75 52.3530 d Y31 –Y32 15.7101 5.37403 10 53.7403 52.0045 B: ditch and Aubrey Holes measures Measure 30.9000 20.00000 Ditch D1 D2 8.0347 5.20045 10 D1 D3 7.6574 4.95625 10 49.5625 D2 D3 12.3383 7.98595 16 49.9122 Mean: 50,4931 S1 90.528 58.59417 120e S2 79.408 51.39676 100 51.3968 S3 83.412 53.98835 110 49.0803 S4 83.624 54.12557 110 49.2051 68.9710 44.64142 90f 49.6016 48.8285 Aubrey Holes

The mathematics in the structures of Stonehenge Table 1 Continued Segment Length figure ( mm) Length henge (m) No. cubits Result. cubit measure (cm) C: Station stones, Heel stone measures 91-93 139.5024 90.29282 182 49.6114 92-94 139.0235 89.98285 182 49.4411 91-92 53.8535 34.85663 70 49.7952 94-93 52.2805 33.83851 70 48.3407 91-94 129.2255 83.64110 168 49.7864 92-93 128.2930 83.03754 168 49.4271 91-96 119.3471 77.24731 159 48.5832 94-96 120.3418 77.89113 159 48.9881 Mean: 49.2467 a 36 cubits = 6 reeds = 3 rods b 50 cubits = 5 poles c 40 cubits = 4 poles d forward construction e 120 cubits = 20 reeds = 10 rods = 2 ropes f 90 cubits = 9 poles = 1 1/2 ropes 3 Ditch and Aubrey Holes The ditch and the Aubrey Holes belong to Stonehenge phase 1 (ca. 2950 bc). 3.1 Ditch The ditch was excavated by Hawley in the early 1920s in the eastern and southern parts. As far as the original course and shape of the ditch is concerned, the excavation results have to be questioned. The excavated cross sections and the irregularities of both the width and the depth along the ditch ring are not in line with earlier and coeval rondels and ditches. In general, the bottom of the ditch should define a comprehensible line or small band that reflects the original construction concept. The eastern ditch terminal at the entrance causeway is a good example for this critical analysis. Johnson assumed that “it is more than likely therefore that the workmen had mistakenly cut a little beyond the actual end of the ditch, slicing through the postholes on the edge of the causeway” (Johnson 2008, p. 103). It seems that the workmen had overdone their task in the rest of the ditch also. This open question can be resolved by an adequate investigation and cautious excavation of the remaining part of the ditch including non-destructive methods. For the generation of a potential construction concept, we have to cope with an unsettled database as far as the excavated ditch is concerned. The situation for the remaining non-excavated part of the ditch is at the present time somewhat better but not ideal; here, we can assume the construction curve in the middle of the still evident depression of the ditch. However, this can be an indication only due to the unknown refilling conditions. Despite this soft database, the backward construction provides a comprehensible mathematical construction concept (Fig. 4).

For each of the two sections of the non-excavated part of the ditch, eight reading points were chosen. The reading points of the two sections of the excavated part are sampled on the basis of the sampling theorem at equidistant spaces of 5◦ as shown in Fig. 4c. This procedure provides 17 and 15 sampling points, respectively. The construction starts with the isosceles triangle D1 D2 D3 composed of two mirrored Pythagorean triangles of the shape (3,4,5) in this case sharing the leg “3” (Fig. 4b). This geometric figure (including the one sharing the leg “4”) can be assessed as one with the highest appreciation and is therefore quite probable for the starting phase of the Stonehenge structures. These shapes of isosceles triangles were also the basis of many explanation models for the construction of stone rings in Britain and Brittany by Thom and Thom (1980). The construction of center D4 is not apparently due to the soft database for the respective circular arcs; this center could be identical either to the point D1 or the center A1 of the Aubrey Holes (see below). The lengths of the legs of the triangle D1 D2 D3 are 6, 8, and 10 cubits. The lengths of the radius s1 are the most reliable data for the deduction of a cubit measure. This length can be assumed 120 cubits which equals 20 reeds or 10 rods and results in a cubit measure of 48.8285 cm that fits to the overall measuring scheme (see Sect. 7). However,owing to the a.m. database situation, the determination procedure according Fig. 13 has to be assessed as non-successful. As both the construction concept and the deduced measuring unit fit in the overall explanation model of the Stonehenge structures, the ditch results are worthwhile to be documented. 3.2 Aubrey Holes The 56 Aubrey Holes define an exact circle (Fig. 4a, c). The circle was calculated by circular regression analysis involving all the 56 holes. The center of this circle A1 is inside the triangle D1 D2 D3 (Fig. 4b). The construction of the 56 holes/stones is a good example for the encoding of mathematical concepts in earthworks by illiterate societies; thereby an approximation of the circle constant π is represented in the henge: radius of the Aubrey holes circle rA = 90 cubits = 9 poles (Table 1B) → diameter dA = 180 cubits = 18 poles; circumference of the Aubrey holes circle cA = 56 poles; 28 56 poles cA = = 3.1111 . . . . = circle constant π := dA 18 poles 9 The mathematical concept is based on a regular polygon of 56 sides inscribed in a circle. The respective circular area AC could be established by the method described by Johannes KEPLER (Wußing 2008, p. 439). This method also starts with an inscribed polygon composed of isosceles triangles providing the formula AC = circumference × radius/2

Ditch p Y Stönehole 97 y Y Heelstone . Approximate limit of excavation North Barrow Y Holes., o è o Z Holes|

This formula is equivalent to AC = r 2 π but does not need π explicitly. We get of the algebraic AC = 92 × 28/9 = 252 pole2 = 36 × 7 pole2 . We have no knowledge √ 7 could, however, be achievements of the Stonehenge people. An approximation of √ established by the formula a 2 ± b ≈ a ± b/2a, which was used in Babylonian, Chinese, and Indian mathematics. “It seems that the [a.m.] approximations . . . were already known in pre-Babylonian √ mathematics” der Waerden 1983, p. 47); q.v. √ (van √ 7, we get 7 = 32 − 2 ≈ 2 2/3 and thereby (Folkerts 2006, p. II 29–33). For √ 252 ≈ 6 × 2 2/3 = 16. Therefore, we have a solution for the problem “squaring the circle” in whole numbers: the area of a circle with the diameter 18 equals the area of a square with the sides 16. This proportion 18/16, alternatively, 9/8 is also applied in problem 50 of the Rhind papyrus (ca. 1650 bc, copied from an earlier papyrus of ca. 1860–1814 bc) for the tasks “squaring the circle” and “circling the square” (van der Waerden 1983, p. 170). The deduced value 28/9 is the most accurate approximation for π at the beginning of the third millennium bc known so far. The establishment of this figure is an admirable mathematical achievement for this time horizon. “The Babylonians . . . always used π = 3. This is also the value given by Vitruvius; and is found again in the Chinese literature.” (van der Waerden 1954, p. 32). We will meet the approximation π = 3 in the structures of Stonehenge also (see Sect. 5.1.). 3.3 Deduced measures The mean cubit measure for the ditch is 50.4931 cm (Table 1B). The arbitrary radius s1 of the circular arc with center D1 with the supposed length of 120 cubits or 2 ropes results in a cubit measure of 48.8285 cm. The radius r A of the Aubrey holes was set arbitrarily to 90 cubits or nine poles and results in a cubit measure of 49.6016 cm. Of all the measures deduced from the structures of Stonehenge, this measure is the one with the lowest range of error or the highest confidence as we have a complete circle with 56 sampling points all lying on an exact circle. 4 Station stones and Heelstone The Station stones and/or holes display no circular or ring-like structure (Fig. 5a). The Station stones 91-94 determine a rectangle that is composed of two Pythagorean triangles (91-92-93 and 91-93-94 or vice versa 91-92-94 and 92-93-94); these triangles have the primitive shape (5,12,13) and are combined with the leg “13.” The concrete length of the legs are 70, 168, and 182 cubits thus having a Pythagorean triangles of the real shape (70,168,182) and a multiplication factor of 14 compared to the primitive triangle. Another isosceles triangle composed of two mirrored Pythagorean triangles of the primitive shape (28,45,53) is added at the leg 91-94. This isosceles triangle has therefore the shape (56,53,53); the Heelstone 96 was placed at the top of this triangle. The concrete lengths of the legs of the Pythagorean triangles (28,45,53) are 84, 135, and 159 cubits; thus, the base leg of the isosceles triangle 91-94-96 is 168 cubits. This geometry represents the algebraic concept of the least common multiple (LCM) as 168 is the LCM of the integers 12 and 56.

Approximate limit of excavation Approximate limit of excavation aad Aubrey Holes

Presently, the Heelstone 96 is leaning considerably toward SSE so we can assume an ancient construction point at the northwestern rim of the Heelstone’s present image. The exact forward construction was rotated adequately to cope with this fact (Fig. 5b). The bearing from the center of the rectangle to the Heelstone SC -96 has an angle to East of 39.65◦ . The deduced cubit measure is 49.2467 cm (Table 1C). 5 Sarsen stones The mathematics involved in the construction of the Sarsen stones shows a great leap forward compared with the preceding phases of Stonehenge. In this stage (ca. 2550 bc) as well as in the later phase of the Bluestones, elliptical concepts were adopted. 5.1 Sarsen circle The Sarsen circle is a completely exact circle. For the circular regression analysis, all the 17 still standing stones were incorporated. The reading points were taken at the innermost edge of the stones. This approach provided a best fit of the regression circle with the exception of stones 10 and 11 which are not as exactly on the circle line as do all others (Fig. 6c). The radius of this circle with center S1 is 30 cubits which equals five reeds. This supports the assumption that the inner faces of the stones were the basis for the construction concept as well as the fact that the inner faces of the uprights were made considerably smoother than the outer faces. By the construction of the 30 Sarsen circle uprights, we have another approximation of the circle constant π encoded in the Stonehenge structures. We have identified two options of constructions: (a) if we apply the concept of the Aubrey holes circle, then the circumference of the Sarsen circle was taken as 30 reeds: radius of the Sarsen circle rS = 30 cubits = 5 reeds (Table 2A) → diameter dS = 60 cubits = 10 reeds = 1 rope; circumference of the Sarsen circle cS = 30 reeds = 3 ropes; 3 ropes cS 30 reeds = = 3. circle constant is π := = dS 10 reeds 1 rope The mathematical concept bases again on a regular polygon inscribed in a circle, in this case with 30 sides, resulting in an integer value for π . For the area of the circle, 2 we get AC = 52 × 3 = 75 for the calculation √ √reed . Applying the√a.m. approximation of a square root, we get 3 ≈ 1 3/4 and for 75 = 52 × 3 ≈ 5 × 1 3/4 = 8 3/4. The solution for the problem “squaring the circle” in this case is: the area of a circle with the diameter 10 equals the area of a square with the side 8 3/4. This Sarsen circle approximation for π is worse than that established by the Aubrey holes circle some 400 years ago, but in accordance with the practiced standards in Babylonia. Either the Aubrey holes’ value of 28/9 was not communicated by oral tradition and it had fallen into oblivion, or it was a conscious decision to operate from now on with this simple integer value. The latter assumption could be motivated by having a more

Fallen sarsen

practical figure for circle calculations or because this change was related to ritual traditions; Seidenberg et al. showed that in Greece and India “geometrical constructions were regarded important for ritual purposes” (Seidenberg 1961, 1978, 1981; van der Waerden 1983). (b) In the above mentioned concept, the angle of a singular isosceles triangle with base 1 reed and legs 5 reeds has a central angle of 11.478◦ which is significantly smaller than the theoretical value of 360◦ /30 = 12◦ ; relating to the full circle, we get a difference by the 30 triangles of 15.65◦ which could doubtless be recognized by the builders of the Sarsen circle. A construction which fits the 30-side polygon better into the full circle could be based on an isosceles triangle with base 1 reed + 2 palm = 38 palms: radius of the Sarsen circle rS = 30 cubits = 5 reeds (Table 2A) → diameter dS = 60 cubits = 360 palms circumference of the Sarsen circle cS = 30 × 38 palms = 1140 palms; 57 1140 palms cS = = 3.1666 . . . . = circle constant is π := dS 360 palms 18 This value approximates π better than the value 28/9 = 56/18 established by the Aubrey holes, and can be seen as a direct improvement of the latter value by raising the numerator from 56 to 57. For the area of the circle, we get AC = 52 × 57/18 = 475/6 = 2 92 − 11/6 ≈ 92 − 2 reed √ . Applying the a.m. approximation for the calculation of a square root, we get 92 − 2 ≈ 9 − 2/18 = 8 8/9. The solution for the problem “squaring the circle” in this case is: the area of a circle with the diameter 10 equals the area of a square with the side 8 8/9. To settle for one of the cases (a) or (b), we need more secured data about exact and regular stone or wood circles in coeval earthworks. 5.2 Sarsen horseshoe The seven Trilithons 51–54, 57, 58, and 60 form an ellipse. Five points in a plane always form one and only one conic section—in this convex situation exactly one ellipse. Hence, the accuracy of the seven reading points on the ellipse is remarkable (Fig. 6c). Once again the Trilithons were smoother at the inner faces, and therefore the reading points were also taken at the innermost edge of the stones. The fallen and broken Trilithon 59 was not incorporated in the computing as there are not sufficient data on the original position. However, the remaining eventually displaced base matches with the computed ellipse. The resulting major and minor axes of the ellipse generate the rhombus T2 T4 T3 T5 and the rectangle T6 T7 T8 T9 (Fig. 6a). Both geometric objects are composed of Pythagorean triangles of the shape (28,45,53). In the case of the triangle T6 T8 T9 (and the congruent triangles T6 T7 T8 , T6 T7 T9 and T7 T8 T9 alike), this fact is very impressive: these triangles have the concrete legs of 28, 45, and 53 cubits (Fig. 6b). The results of the elliptical regression analysis including the standard deviation values are provided as ESM. The Trilithon 56 was not obviously constructed at a position on the elliptical curve. This upright was straightened in 1901, and its partner stone 55 has fallen and broken.

The mathematics in the structures of Stonehenge For a later comment, we want to state here that the depiction of Trilithon 56 in Fig. 8a shows the alignment of the smoother outer face not perpendicular to the major axis of the ellipse but rotated some 10◦ anticlockwise. If we assume that stone 55 has fallen in a right angle in relation to its original alignment, then this statement holds for stone 55 also. There are still reasonable and debatable arguments about the original position of both stones (Johnson 2008, pp. 136–140). Therefore, the determination procedure according to Fig. 13 breaks down after step 1. The exact mathematical forward construction of the ellipse matches with that of the plan with a high degree of precision (Fig. 6c). Surprisingly, the centers of the Sarsen circle and the Sarsen ellipse are not identical. However, there is a comprehensible mathematical concept to explain this difference (Fig. 6d). The a.m. centers and the virtual point S2 configure the Pythagorean triangle S1 S2 T1 of the shape (3,4,5). The lengths of the legs are 3, 4, and 5 feet (Table 2A). The deduced foot measure is 29.8918 cm. The ancient foot measures vary considerably, even on the basis of a stable cubit measure depending on different subdivisions of the cubit and the palm (Rottländer 1979).2 The deduced foot measure is therefore absolutely within the deduced overall measuring scheme. The displacement of the center T1 of the ellipse might be because the Sarsen horseshoe was thereby placed better in the middle of the Sarsen circle. 5.3 Deduced measures Once again, the above mentioned cubit measures are ideal values according to the selected mathematical concept—in these cases, the underlying circle and the Pythagorean triangles. The actual length values extracted from the plan are summarized in Table 2A. The cubit measure for the Sarsen circle is 48.8733 cm and that for the Sarsen horseshoe, 49.3194 cm; the mean of all Sarsen stones’ cubit measures is 49.2079 cm. 6 Bluestones The Bluestone circle and horseshoe were constructed some 450 years after the setting of the Sarsen stones. 6.1 Bluestone circle The present record of the remains of the Bluestone circle comprises 27 stones of which 12 have fallen. Hence, we can involve 15 stones in the determination procedure for the original construction plan. The original amount of stones generating the Bluestone circle is estimated between 44 and 62 uprights (Johnson 2008, p. 158); thus, we can imagine a very dense ringlike structure. Despite there being only 15 secured original positions, the nonlinear regression analysis provides a clear picture (Fig. 7c). We obtain four circular arcs with centers B21 –B24 with 5, 6, 3, and 3 reading points, 2 Typical variations: foot on the statue of Gudea = 26.45 cm; 18 digit foot from the Nippur cubit = 33.199 cm.

Table 2 Measures of the Sarsen stones and the Bluestones Segment Length figure ( mm) Length henge (m) No. cubits Result. cubit measure (cm) A: Sarsen stones measures Measure 52.0250 10.00000 76.279 14.66199 Sarsen circle Radius 30a 48.8733 Sarsen horseshoe/Trilithons T6 T9 71.222 13.68996 28 48.8927 T8 T9 116.204 22.33618 45 49.6360 T6 T8 136.2936 26.19771 53 49.4296 Mean: Mean of all Sarsen stones: 49.3194 49.2079 Circle–horseshoe relation No. feet Foot meas. S1 S2 4.7230 0.90783 3 30.2611 S2 T1 6.1575 1.18357 4 29.5891 S1 T1 7.7583 1.49126 8 29.8253 Mean: 29.8918 B: Bluestones measures Bluestone circle Measure 52.0250 B21 B22 5.2062 10.00000 1.00071 12 50.0356 B21 B23 14.7238 2.83014 35 48.5167 B22 B23 15.9908 3.07368 37 49.8434 B21 B24 37.3092 7.17140 84 51.2243 B23 B24 39.8863 7.66676 91 50.5500 Mean: 50.0340 50.9071 v1 66.2110 12.72677 25 v2 54.1300 10.40461 20 52.0231 v3 62.6860 12.04921 24b 50.2050 v4 98.1440 18.86478 36 52.4022 5 49.3889 Bluestones horseshoe Measure 65.3750 10.00000 B1 B6 16.1440 2.46945 B1 B11 16.3149 2.49559 5 49.9117 B1 B2 39.1920 5.99495 12 49.9579 B2 B6 42.0928 6.43867 13 49.5282 B2 B11 43.0451 6.58434 13 50.6487 50.2176 B3 B6 42.6787 6.52829 13 B3 B11 41.8508 6.40165 13 49.2435 B1 -79 39.4354 6.03218

The mathematics in the structures of Stonehenge Table 2 Continued Segment Length figure ( mm) Length henge (m) No. cubits Result. cubit measure (cm) Mean: Circle and horeseshoe mean: 49.9506 49.9923 B2 -79 55.4680 8.48459 B6 -79 55.5642 8.49930 Forward construction a 30 cubits = 5 reeds → diameter = 10 reeds = 1 rope b 24 cubits = 4 reeds = 2 rods correspondingly; the stones 40g and 47 are connecting points of the adjacent arcs. The circular arcs with centers B23 and B24 are calculated on the basis of three stone positions. As three points in a plane define one and only one circle, we get an exact solution for the respective centers. However, this accuracy is deceptive because the regression analysis is calculated without redundancy. In this case, a defective position of one reading point (stone) results directly in a defective position of the center, while the regression calculation of a circle on the basis of more than three points compensates much better for slight position errors. Therefore, the circular arc with center B24 has the lowest probability in our assessment procedure as this arc has additionally a very small central angle from stone 47 to stone 31. Nevertheless, the results in combination with the deduced measure units pass our determination procedure according to Fig. 13—with reservations as far as the center B24 is concerned. The construction starts with the Pythagorean triangle B21 B22 B23 of the shape (12,35,37) and the size 12, 35, and 37 palms (Fig. 7b). Again, this primitive form of the Pythagorean triangle is a strong support in favor of this explanation model. Another Pythagorean triangle of the shape (5,12,13) is added to the first at the leg “35.” As 35 is a multiple of 5, we get this triangle in terms of palms in the concrete shape of (35,84,91). The calculated radius of the circle with center B23 is 24 cubits which equals 4 reeds or 2 rods (see ESM Table 2). The construction of the ring on which the stones of the Bluestone circle were set started probably with this circular arc. The additional arcs were attached in a way to get a smooth curve; the ring does not close smoothly between the stones 31 and 32c (Fig. 7a). The exact forward construction (Fig. 7c) matches with the plan perfectly: the connection of the adjacent arcs at stone 40g is now totally smooth compared with the backward construction where the calculation provided a minor gap (Fig. 7a); the gap between stone 31 and 32c remains. 6.2 Bluestone horseshoe As far as mathematics is concerned, the complexity of Stonehenge’s construction concepts reaches its point of culmination with the Bluestone horseshoe. In combination with the holes 73–78 the stones/holes of the Bluestone horseshoe create a ring-like structure. The nonlinear regression analysis provides three ellipses (Fig. 8a): a large ellipse with center B1 , a smaller ellipse in SW with center B6 and another smaller

The mathematics in the structures of Stonehenge B19 B C B5 B5 B14 [53] B2 B13 d ro 1 ] = 8 c ] [2 12 [12 [13] B9 B18 B11 13 c B12 B10 B4 B6 5 B3 c ] [5 c 5 [13 ] 13 c 5] [4 B15 B7 ] [5 [53] B6 B1 B11 13 c d [13] 1 ro ] 8 = c ] [2 12 [12 B1 B16 5] [4 13 c [1 3 ] B8 [53] B2 [53] B3 B4 B17 Fig. 8 Bluestone horseshoe. a The ellipses with center B1 (large ellipse), center B6 (small ellipse SW), and center B11 (small ellipse NE) are calculated by elliptical regression analysis. b In the depiction of the construction concept, the small ellipse NE was shifted perpendicular to the major axis of the large ellipse so that the center B11 rests on this major axis; additionally, this small ellipse was rotated anticlockwise so that the axes coincide. c The basis of the construction concept of the Bluestone horseshoe is the lozenge B2 B6 B3 B11 composed of four Pythagorean triangles of the shape (5,12,13); each of these Pythagorean triangles has concrete legs of 5, 12, and 13 cubits. The large ellipse with center B1 defines the rectangle B16 B17 B18 B19 (see Fig. 13b) and a regular lozenge B2 B4 B3 B5

ellipse in NE with center B11 . In the depiction of the construction concept in Fig. 8b, the center B11 of the small ellipse NE was shifted perpendicular to the major axis of the large ellipse so that this center rests on this major axis (length of shift 9.2 cm); additionally this small ellipse was rotated anticlockwise so that the axes coincide (rotation 5.67◦ ). The slight shift and rotation are within the respective confidence intervals of the regression analysis due to the input data situation: the original position of the now disappeared stones 73–78 cannot be established with the accuracy of the other elements of the Bluestone horseshoe. The basis of the construction concept of the Bluestone horseshoe is the lozenge B2 B6 B3 B11 composed of four Pythagorean triangles of the shape (5,12,13); each of these Pythagorean triangles has concrete legs of 5, 12, and 13 cubits. The large ellipse with center B1 defines the rectangle B16 B17 B18 B19 and a regular lozenge B2 B4 B3 B5 composed of four triangles similar to Pythagorean triangles of the shape (28,45,53). The proportion of the legs “45” and “28” is 45/28 = 1.607 . . ., and therefore also a good approximation for the Golden Ratio of 1.618 . . . (Fig. 8b, c). The small ellipse SW with center B6 defines the regular lozenge B7 B10 B8 B9 composed of four right-angled triangles being similar to a Pythagorean triangle of the shape (48,55,73) (Fig. 9a). The ellipse B7 B10 B8 B9 is significantly rotated anticlockwise by 16.362◦ in relation to the major axis of the large ellipse. We had stated above that the alignment of the great Trilithons of the Sarsen horseshoe might have been rotated the same way. This rotation of in the inner segment of both horseshoes may have a reason for astronomical or ritual purposes. The small ellipse NE with center B11 defines the regular lozenge B12 B15 B13 B14 composed of four right-angled triangles also being similar to a Pythagorean triangle of the shape (48,55,73) (Fig. 9b). Both small ellipses are tangential to the larger ellipse—the small ellipse SW with one tangent point and the small ellipse NE with two tangent points; hence, we can assume that the intention of the Bluestone horseshoe designers was to demonstrate this geometrical achievement. The positioning of hole/stone 79 gives another insight to the mathematics applied by the builders of Stonehenge in phase 3iv. (a) The line segment B1 -79 has the same length as the semi-minor axis B1 B2 (12 cubits = 1 rod) thus giving with segment B2 -79, the √ construction of the diagonal of a unit square or a geometrical representation of 2 (Figs. 8b, 9c). On the other hand, this diagonal equals the line segment B6 -79 with a difference of 1.471 cm as established in the forward construction (Table 2B). B1 B6 equals five cubits, and so we get an approximation for the square root of 2 by 17/12 = 1.41666 . . . (exact value 1.41421 . . .). “This approximation (1;25 in the Babylonian sexagesimal notation) frequently occurs in Babylonian texts” (van der Waerden 1983, p. 47). (b) We have shown above that the proportion of the major and minor semi-axes of the large Bluestone ellipse is an approximation for the Golden Ratio; the point 79 therefore divides the major semi-axis B1 B5 in the Golden Section (Fig. 8b). We had assumed that the large ellipse was constructed on the basis of a regular lozenge based upon Pythagorean triangles of the shape (28,45,53). The construction of a Golden Section geometry as a first step cannot be excluded totally either. A construction method for the Golden Section as also described in the Elements by EUCLID is depicted in Fig. 9d. This open question could be resolved if the remains of the respective construction pegs would ever be found in the soil of Stonehenge. (c) For the special case of an ellipse with the axes in the proportion of the Golden Ratio,

B 10 Fig. 10 Forward construction for the Bluestone horseshoe. The small ellipses with corresponding centers B6 and B11 are constructed similar to a Pythagorean triangle of the shape (48,55,73) and in the way that the indicated tangent points are achieved construction of an ellipse being tangent to another given ellipse can be achieved by an approximation procedure. We can interpret the structures of ancient rings and earthworks as large mnemonic artifacts. Fortunately, there are also handy ancient artifacts discovered in the proximity of Stonehenge which underscore the layout concept of the Bluestone horseshoe: the Bush Barrow gold lozenges. The large lozenge displays the underlying lozenge of the two small Bluestone horseshoe ellipses based upon a Pythagorean triangle of the shape (48,55,73). For the investigation, we use the scaled graphic rendering in Fig. 11a. The angles 41.121◦ in the determining triangle L 1 L 3 L 2 matches the respective angles 41.112◦ in the Pythagorean triangle (48,55,73) nearly exactly (Fig. 11b). However, there is another way to confirm this explanatory model: when applying the determination procedure according to Fig. 13 to the gold lozenge also, we have to establish a consistent metrology for the manufacture of the lozenge. The lozenge is now 18.55 cm

The mathematics in the structures of Stonehenge C L2 L3 L5 B L4 L7 L2 D L6 L12 3] [7 ] [53 L3 [55] L1 3] [7 corn 3 7 [55] L5 L13 [45] [53 [53 L11 ] [45] [53 ] L15 ] L14 mean ( L11L13L12, L4 L8 [55] 55 corn L11L13L14, L11L15L12, L11L15L14) = 31.304° Pythagorean triangle (28,45,53) = 31.891° L9 L9L8L6 = L1L3L2 = 41.121° Pythagorean triangle (48,55,73) = 41.112° Fig. 11 Bush Barrow gold lozenges. a Bush Barrow large lozenge. The lozenge defines the regular rhombus L 2 L 3 L 4 L 5 . b The rhombus is composed of four Pythagorean triangles of the shape (48,55,73). Hence, the triangle L 8 L 9 L 6 is a Pythagorean triangle of the shape (48,55,73) also, and this triangle has concrete legs of 48, 55, and 73 corns as well as the axes of the rhombus L 2 L 3 L 4 L 5 having the lengths of 48 and 55 corns. c Bush Barrow small lozenge. The lozenge defines the regular rhombus L 12 L 13 L 14 L 15 . d The rhombus is composed of four Pythagorean triangles of the shape (28,45,53) long; it is considerably compressed at the corners, and the original size should be slightly larger due to the crumbling of the plate. Let us, therefore, assume as working hypotheses that the original distance between the vertices L3 and L5 was 19 cm. 19 cm divided by 55 provides 0.34545 cm; taking this measure as a unit for the “corn” measure, we have to multiply by 144 and get a cubit measure of 49.7455 cm. This cubit is equivalent to the established cubit for the Bluestone horseshoe of 49.9506 cm (2B). Hence, the triangles L 8 L 9 L 6 (and L 6 L 7 L 8 alike) are Pythagorean triangles with the concrete legs of 48, 55 and 73 corns and the axes of the lozenge have a length of 48 and 55 corns, respectively. The small Bush Barrow gold lozenge was not manufactured with the precision and artistry of the large one. The production of this tiny plate may date back to the early

phases of the manufacture of gold plates. This plate could have been passed over generations to the man to whom it was attached in his tomb. The marvelous large lozenge seems to reflect the culmination of that kind of plate. Nevertheless, the small gold lozenge could have played a comparable role. For the backward construction, we use a photo as no adequate scaled rendering is available (Fig. 11c); therefore, we have to take into account potential slight photographic distortions. The backward construction is very close to a regular rhombus composed of Pythagorean triangles of the shape (28,45,53) (Fig. 11d). This Pythagorean triangle is the basis of the construction of the large ellipse of the Sarsen horseshoe, and the large ellipse of the Bluestone horseshoe as well. The length of the major axis of the small lozenge is 3.1 cm; the division by 45 yields a measure unit of 0.06888 cm—far below the unit of a corn. We have no secured sources for a subdivision of the unit corn in antiquity. Following the old English unit “(shoe) iron” of 0.0529167 cm (1/48 inch), let us denote this potential unit an “iron”. A subdivision of a corn by six irons yields a cubit measure of 59.5199 cm and a subdivision by five irons of 49.5999 cm. The latter measure matches with the established cubit measure scheme for Stonehenge, although this can only be an indication. 6.3 Deduced measures The actual length values extracted from the plan are summarized in Table 2B. The mean cubit measure for the Bluestone circle is 50.0340 cm, and the mean for the Bluestone horseshoe 49.9506 cm. As both values match to a high degree, we have established a mean cubit measure for all Bluestones of 49.9923 cm. 7 Metrology All measurements of the Stonehenge structures are based on a consistent metrology. This metrology is in line with practiced standards in Babylonia concerning both the principal system of measures and the concrete measures. Verifications of mathematical constructions without written sources get an additional affirmation if they are based on a consistent and coherent metrology and correspond with coeval verified systems of measures. In the construction of the Stonehenge structures, length measures were used as we encounter them in Babylonia. A preferred unit to compare ancient measures of length is the cubit. In the common Babylonian system of measures, a cubit was subdivided into six palms, a palm into four (st. five) fingers, and a finger into six corns; six cubits equal one reed and 12 cubits equal one rod; 10 cubits equal one pole, and 10 reeds equal one rope (Trapp and Wallerus 2006).3 The unit 10 palms was also a used measure; this unit is commonly denoted as Megalithic yard (my) (see also ESM Fig. 2). The system of measures used in the geometrical construction of Stonehenge conforms to this Babylonian system. We have identified respective cubit measures for the phases of Stonehenge under investiga3 The Babylonian corn measure differs from the old English barleycorn measure. In the latter case, the basis for the measure is the length of a barleycorn while a Babylonian corn (še) is about the third of this measure and seems to represent the width of a corn.

The mathematics in the structures of Stonehenge development of the measuring unit according to the phases of Stonehenge (mean values) 56 55 cubit measure [cm] 54 53 3vi 52 51 3vi 50 49 3iv 1 3i 3ii 48 47 46 -3000 -2800 -2600 -2400 -2200 time [year] -2000 -1800 -1600 measure units Aubrey Holes 1 cubit measure -2950 49.6016 Station stones, Heel stone 3i -26501) 49.2467 structure 1) phase time Sarsen circle and horseshoe 3ii -2550 49.2079 Bluestone circle and horseshoe 3iv -2100 49.9923 Z holes 3vi -1900 50.7228 Y holes 3vi -1600 53.7403 currently undated, 'before the central stones' (Johnson 2008, p. 136) Fig. 12 System of measures. The system of measures corresponds with practiced standards in Babylonia tion. The length of the cubit has a comprehensible development scheme from phase 1 (ca. 2950 bc) to the end of phase 3vi (ca. 1600 bc) (Fig. 12). A considerable increase in length measures in times of decline is an often-noticed fact. The cubit measure 49.99 cm established for Stonehenge phase 3iv (ca. 2100 bc) corresponds with the coeval Babylonian cubit of Lagash of 49.61 cm.4 This Lagash value is identical to the cubit measure of the Aubrey Holes of 49.60 cm, which we had identified as the cubit measure with the lowest range of error. Hence, the consistency of the Stonehenge measuring system is a strong support favoring the established mathematical concepts encoded in the structures of Stonehenge. 4 The cubit of Lagash was documented on a statue of Gudea of Lagash. The current status of research does not provide exact data for the ruling time of Gudea. New approaches vary from 2140 to 2060 bc.

8 Methods The construction of exact circles in ancient earthworks and stone/wood rings can be verified easily. The deduction of construction concepts of noncircular, ringlike structures without written sources is a complex task with a considerable number of pitfalls. To avoid misinterpretations to a great extent and to work out a highly probable solution a six-step determination procedure was established (Fig. 13). In terms of science theory, this procedure should be taken as final conclusions as “scientific proofs” for concepts of illiterate societies are hardly to be established. An explanatory model for a potential construction concept can be accepted only if all steps of the procedure are passed successfully. The procedure can break down at every step—except step 4. Step 1 If the plan of the present situation of the monument does not contain sufficient data, then the procedure should be canceled at this point. Secured correction can be made, e.g., if the original position of a leaning stone is obvious. Step 2 To establish circular or elliptical arcs in the backward calculations in a reproducible manner, nonlinear regression analysis methods are inevitable. Step 3 If the selected construction concept does not match with a consistent system of measurements, the procedure should be canceled at this point (exception: similar transformations). Step 4 The forward construction yields an exact mathematical construction model. Step 5 Owing to both the decay of ditches/stones/holes and the potential imprecise construction of individual elements of the original monument, the concept of step 2 is no exact model in general. Therefore, slight modifications of the measure, position original layout of the earthwork original construction concept ancient secured corrections decay plan of present situation present 6 concluding assessment of the result 1 backward calculation potential 2 concepts of construction forward calculation other sources/ research results matching 5 of plan and construction slight modification of measure, position & heading exact 4 construction model 3 potential system of measurements Fig. 13 Determination procedure. The determination of the original construction concept and potential earth work layout is performed on the basis of a six-step procedure. The procedure can break down at all steps except step 4

The mathematics in the structures of Stonehenge and alignment of the step 4 model can be made in step 5. If this forward construction model results in a worse matching compared to the backward construction, then the procedure should be canceled at this point in general. Step 6 The potential solution of step 5 should pass an overall assessment taking into account also superordinate arguments. The circles and circular/elliptical arcs in the backward calculation were computed by a nonlinear regression analysis. For the analysis of line elements (ditch, bank), preferable eight or more points of the respective section of the curve should be identified. In case of particular irregularities of curved lines, the reading points should be taken on the basis of the sampling theorem, as performed in the line determination of the excavated part of the ditch. In case of stones and holes, the number of reading points is predefined. Centers of stones and holes were established by a visual approximation. The center of gravity of shapes should be taken if the shape has an irregular noncircular image and no additional information about the original center is available. The results of the regression analysis including the standard deviations are provided as Supplementary Material (ESM Tables 1–4). 9 Summary All the stone structures of the Stonehenge enclosure and probably also the ditch were designed, constructed, and set on the basis of mathematical concepts. We could identify only one exception: in times of an obvious decline of the mathematical traditions, Table 3 List of the first 12 primitive Pythagorean triangles. “rope length” denotes the sum of the lengths of the three legs First 12 primitive Pythagorean triangles (ordered by rope length) Primitive Pythagorean triangle Rope length Used in constructions Element (3,4,5) (5,12,13) 12 II (I) Sarsen circle/horseshoe, Z holes, (ditch) 30 IIII Station stones, Bluestone circle, Bluestone horseshoe, Y holes (8,15,17) 40 I Bluestone circle III Heelstone, Sarsen horseshoe, Bluestone horseshoe II Bluestone horseshoe (II) (7,24,25) 56 (20,21,29) 70 (12,35,37) 84 (9,40,41) 90 (28,45,53) 126 (11,60,61) 132 (16,63,65) 144 (33,56,65) 154

the stone indicated by hole Y8 of the Y holes was set presumably arbitrarily. The major mathematical concept in the design of the Stonehenge structures is the application of Pythagorean triangles. This is in line with earlier and coeval earthworks and stone rings. Among the outstanding Pythagorean triangle (3,4,5), specific Pythagorean triangles were repeatedly applied in the construction concepts of Stonehenge: (5,12,13), (28,45,53), and (48,55,73)—the last two with considerable large rope lengths (see Table 3). The Pythagorean triangle (28,45,53) is applied for the design of both the ellipse of the Sarsen horseshoe and the large ellipse of the Bluestone horseshoe. The Pythagorean triangle (48,55,73) seems to be discovered in the forefront of the construction of the Bluestone horseshoe, and the design of the two smaller ellipses was dedicated to this discovery, while the shape of the large ellipse was designed in the tradition of the Sarsen horseshoe ellipse, indicating a practice bequeathed over centuries. The dedication of the Pythagorean triangle (48,55,73) to Stonehenge is underscored by the identical design of the Bush Borrow large lozenge. In general, Pythagorean triangles were composed to form rectangles, isosceles triangles, rhombi, and general polygons. The standard concept for the design of ringlike structures was the composition of circular arcs or—in the case of the horseshoes—elliptical arcs to predominantly smooth lines; we could notice the total closing of the line (mathematical continuity) in the Bluestone horseshoe only. In this phase, the application of mathematics with the involvement of elliptical concepts was at its peak. As far as geometry is concerned the construction of ellipses being tangent to a given ellipse is the highlight of the Stonehenge mathematical designs. The concrete construction method of the ellipses could not be disclosed. The encoding of approximations for the circle constant π (28/9, 3 and 57/18 respectively) for the task “squaring the circle” and of the approximation 17/12 for the square root of 2 in the construction of Stonehenge is another important achievement as is the application of the law of similitude. The a.m. concepts are encoded in the structures of Stonehenge. It can be assumed that the general level of mathematical achievements in the Stonehenge society was remarkably of a high standard—with a distinct culmination in phase 3iv and an obvious time of decline in phase 3vi (Y holes). The used length metrology corresponds with common Babylonian systems of measurement based on the unit of a cubit, broken down till the measure of a corn and summed up to the measure units reed, pole, and rope. Systems of measurement change rarely; we can, therefore, assume a common origin of these metrologies noticeable before the time under consideration. In addition, this consistent metrology establishes high confidence in the described mathematical concepts. References Folkerts, Menso. 2006. The development of mathematics in Medieval Europe. Aldershot: Ashgate. Haarmann, Harald. 1991. Universalgeschichte der Schrift, 38–50. Frankfurt-New York: Campus. Hawkins, Gerald S. 1965. Stonehenge decoded. Garden City: Doubleday. Johnson, Anthony. 2008. Solving Stonehenge. London: Thames & Hudson. Mathews, Jerold. 1985. A neolithic oral tradition for the van der Waerden/Seidenberg origin of mathematics. Archive for History of Exact Sciences 34: 193–220. Neugebauer, Otto. 1969. The exact sciences in antiquity, 2nd ed. Mineola: Dover Publication. Neugebauer, Otto. 1975. A history of ancient mathematical astronomy part one-three. New York: Springer.

The mathematics in the structures of Stonehenge Rottländer, Rolf C.A. 1979. Antike Längenmaße, 40–43. Braunschweig/Wiesbaden: Vieweg. Seidenberg, Abraham. 1961. The ritual origin of geometry. Archive for History of Exact Sciences 1: 488–527. Seidenberg, Abraham. 1978. The origin of mathematics. Archive for History of Exact Sciences 18: 301–342. Seidenberg, Abraham. 1981. The ritual origin of the circle and square. Archive for History of Exact Sciences 25: 270–327. Thom, Alexander, and A.S. Thom. 1980. Megalithic rings: plans and data for 229 Monuments in Britain. British Archaeological Reports, Br. Series 81. Trapp, Wolfgang, and Heinz Wallerus. 2006. Handbuch der Maße, Zahlen, Gewichte und der Zeitrechnung, 193–194. Ditzingen: Reclam. van der Waerden, Bartel L. 1954. Science awakening. Groningen: Nordhoff. van der Waerden, Bartel L. 1973. Science awakening II. In Birth of Astronomy. New York: Springer. van der Waerden, Bartel L. 1983. Geometry and algebra in ancient civilizations. New York: Springer. Vansina, Jan. 1985. Oral tradition as history, 44–45. Wisconsin: University of Wisconsin Press. Wußing, Hans. 2008. 6000 Jahre Mathematik. Berlin: Springer.