The Mathematics of the Viereckschanzen of the La Tène Culture

Abstract

The development of script-using ancient civilizations and their achievements in sciences such as astronomy and mathematics have been well researched. Illiterate cultures were able to attain an adequate level of knowledge and transferred this knowledge to succeeding generations via oral tradition and mnemonic artefacts. The constructions of the square ditched enclosures (Viereckschanzen) of the last phase of the La Tène period (LT D, 150-1 BC) are documentations of the mathematical knowledge of this culture and served as a kind of mnemonic artefact. The constructions comprise the application of Pythagorean theorem (in terms of Pythagorean triangles) and square-root approximation triangles. Via these types of triangles, convex quadrangles are constructed (i.e., trapezium, parallelogram, rectangle, kite, lozenge, and square). The constructions were laid out in the terrain on the basis of a consistent ‘Babylonian/Egyptian’ metrology.

Appendices

Appendix

The appendix contains the construction concepts of the investigated enclosures in alphabetical order:

Altheim–Heiligkreuztal–Bannwald enclosure: Fig. 10.

Altheim–Heiligkreuztal–Ruchenholz enclosure: Fig. 10.

Beuren enclosure: Fig. 11.

Bopfingen (palisade) enclosure: Fig. 11.

Burgstallhof–Wilburgstetten enclosure: Fig. 12.

Deisenhofen enclosure: Fig. 12.

Egweil enclosure: Fig. 13.

Ehningen enclosure: Fig. 13.

Esslingen–Oberesslingen enclosure: Fig. 14.

Feldmoching enclosure: Fig. 14.

Gilching–Steinlach enclosure: Fig. 15.

Hardheim–Gerichtstetten enclosure: Fig. 15.

Hechendorf–Güntering enclosure: Fig. 16.

Holzhausen II enclosure: Fig. 16.

Hörstein enclosure: Fig. 17.

Kirchheimbolanden–Donnersberg enclosure: Fig. 17.

Kirchheim–Osterholz enclosure: Figs. 18 and 19.

Königheim–Brehmen enclosure: Fig. 20.

Laibstadt enclosure: Fig. 19.

Langenbach enclosure: Fig. 22.

Manndorf enclosure: Fig. 21.

Markvartice enclosure: Fig. 21.

Maxing enclosure: Fig. 22.

Neuhau–Forst enclosure: Fig. 23.

Niederstetten–Wermutshausen enclosure: Fig. 23.

Niederstotzingen enclosure: Fig. 24.

Nordheim–Bruchhöhe enclosure: Fig. 25.

Nordheim–Kupferschmied enclosure: Fig. 24.

Oberframmering enclosure: Fig. 26.

Oberhaimbuch enclosure: Fig. 26.

Oberschneiding enclosure: Fig. 27.

Papferding enclosure: Fig. 27.

Peterhof enclosure: Fig. 28.

Pfeffenhausen enclosure: Fig. 28.

Plattling–Pankofen enclosure: Fig. 29.

Pliezhausen–Rübgarten enclosure: Fig. 29.

Pocking–Hartkirchen enclosure: Fig. 30.

Poign enclosure: Fig. 30.

Sallach 1 enclosure: Fig. 31.

Sallach 2 enclosure: Fig. 32.

Scheyern enclosure: Fig. 32.

Schöngeising enclosure: Fig. 33.

Teufstetten enclosure: Fig. 33.

Weiltingen enclosure: Fig. 34.

Willmatshofen–Brennburg enclosure: Fig. 34.

Fig. 10

The Altheim-Heiligkreuztal-Bannwald and Altheim-Heiligkreuztal-Ruchenholz enclosures. a The notch lines of the ditches form a trapezium. b The base angles α of the isosceles triangle ACD are taken from the Pythagorean triangle (20, 21, 29). The sides AB and CD are parallel (alternate angles ∠BAC and ∠ACD). c The notch lines of the ditches form a quadrangle (main Viereckschanze) and a rectangle (annexe). d BC/BE = BC/CE ≈ 17/12 is an approximation of √2 (third Smyrna iteration)

Fig. 11

The Beuren and Bopfingen enclosures. a The notch lines of the ditches form a trapezium. b The sides AB and CD are parallel (alternate angles ∠BAF and ∠ECD). c For the original plan of the Bopfingen enclosures, see Fig. 28b. The ditches of the Viereckschanze FGHI are not detected sufficiently; therefore, the vertex F cannot be determined with the required accuracy, and the plan fails to pass the first step of the verification procedure. d BD/AB = BD/AD ≈ 24/17 is an approximation of √2 (third Smyrna iteration)

Fig. 12

The Burgstallhof–Wilburgstetten and Deisenhofen enclosures. a The notch lines of the ditches of the main enclosure and the annexe form two quadrangles. b AC/AD = AC/CD ≈ 140/99 is an approximation of √2 (fifth Smyrna iteration). c EI/AI = 338/239 is an approximation of √2 (sixth Smyrna iteration, Δ = 0.00001). AE/AI ≈ 441/239 is an approximation of √3 (Δ = 0.00017). Vertex H is achieved by Thales’ theorem. d The notch lines of the ditches form a quadrangle. e CF/BF = 71/41 is an approximation of √3 (fifth Smyrna iteration) → BC/BF ≈ 2 = √4. AB/AE = AB/BE ≈ 249/176 is an approximation of √2 (Δ = 0.00056)

Fig. 13

The Egweil and Ehningen enclosures. a The notch lines of the ditches form a quadrangle. b AD/AE = AD/DE ≈ 34/24 = 17/12 is an approximation of √2 (third Smyrna iteration). c The notch lines of the ditches form a quadrangle. d BC/BE = BC/CE ≈ 17/12 is an approximation of √2 (third Smyrna iteration). AD/AF = AD/DF ≈ 41/29 is an approximation of √2 (fourth Smyrna iteration)

Fig. 14

The Esslingen-Oberesslingen and Feldmoching enclosures. a The notch lines of the ditches form a quadrangle. b Application of the primitive Pythagorean triangles (3, 4, 5) and (20, 21, 29) and the Pythagorean triangle (15, 20, 25) [primitive (3, 4, 5)]. c The notch lines of the ditches form a quadrangle. d Plan of the Feldmoching enclosure. e AB/AE = AB/BE ≈ 17/12 is an approximation of √2 (third Smyrna iteration). In general, this geometrical configuration would be over-defined. However, because leg DE (65 double feet₂₄) and leg EB (72 double feet₂₄) add up to 137 double feet₂₄, the construction of both respective triangles DEA and EBA match commensurably with the construction of the primitive Pythagorean triangle DBC of shape (88, 105, 137)

Fig. 15

The Gilching-Steinlach and Hardheim–Gerichtstetten enclosures. a The notch lines of the ditches form a quadrangle. b AD/AE = AD/DE ≈ 17/12 is an approximation of √2 (third Smyrna iteration). The diagonals AC and BD intersect at right angles at point E. c The notch lines of the ditches form a quadrangle. d Application of the primitive Pythagorean triangles (20, 21, 29), (48, 55, 73) and (88, 105, 137)

Fig. 16

The Hechendorf-Güntering and Holzhausen II enclosures. a The notch lines of the ditches form an isosceles trapezium. b Plan of the Hechendorf-Güntering enclosure. c The geometrical configuration demonstrates the formula of the area K of a trapezium: K = ½ (a + b) · h, where a and b are the lengths of the parallel sides AB and CD, h is the height (DD_H or BB_H). d The notch lines of the ditches form a quadrangle—with a deviation at vertex D. The excavator verified five phases of construction, where the displayed plan shows phase four. The deviation at vertex D might be an inaccuracy in the course of the four expansion stages. e AD/AE = AD/DE ≈ 17/12 is an approximation of √2 (third Smyrna iteration)

Fig. 17

The Hörstein (cremation grave—Grabgarten type) and Kirchheimbolanden–Donnersberg enclosures. a The construction concept of the grave is matching with that of the Viereckschanzen. The circular ditch is an exact circle. b The resulting cubit unit of the circle construction is identical to that of the square ditched grave in panel c. c CD/DH = CD/CH ≈ 17/12 is an approximation of √2 (third Smyrna iteration). d The notch lines of the enclosure form a quadrangle. e DE/GE ≈ 17/12 is an approximation of √2 (third Smyrna iteration). DG/GE = 104/60 = 26/15 is an approximation of √3 (fourth Smyrna iteration). DB/AB ≈ 71/41 is an approximation of √3 (fifth Smyrna iteration). In the latter case of the approximation of √3, the triangle ABD is not right-angled

Fig. 18

The Kirchheim–Osterholz (a) and (b) enclosures. a The remains of the palisade structure (a) form a rectangle. b Twofold application of the primitive Pythagorean triangle (88, 105, 137). c The remains of the palisade structure (b) form a parallelogram. d The construction starts with the rectangle EFGH and the twofold application of the primitive Pythagorean triangle (88, 105, 137). The construction also demonstrates the concept ‘the area of a parallelogram = baseline × height’. e The vertices K and N of the parallelogram are constructed via the extreme primitive Pythagorean triangle (21, 220, 221). The vertices L and M are constructed in the same way as vertices K and N

Fig. 19

The Kirchheim–Osterholz (c) and Laibstadt enclosures. a The remains of the palisade structure (c) form an isosceles trapezium with an extension in the south-eastern corner. b The construction starts with the rectangle OPQR. c The vertices S and T of the isosceles trapezium are constructed by extensions with two extreme primitive Pythagorean triangles (49, 1200, 1201). d The course of the southern palisade follows the baseline OP up to the point V and deviates then with a straight line up to the point U. The triangle PUV is right-angled and similar to triangle TRO and SQP. Compared with these triangles, the leg PU underwent a reduction from 49 decisteps to 29 decisteps; this means a factor of 29/49 = 0.5918. An equal reduction provides the leg UV of 1201∙0.5918 = 710.7 ≈ 710 decisteps = 71 steps. e This arithmetic provides the right-angled square root approximation triangle SUV. SU/VU = 123/71 is an approximation of √3 (fifth Smyrna iteration). f The notch lines of the ditches form a quadrangle. g Application of the primitive Pythagorean triangle (65, 72, 97), and the Pythagorean triangle (100, 105, 145) [primitive (20, 21, 29)]

Fig. 20

The Königheim–Brehmen enclosure and its annexes. a The notch lines of the ditches form a quadrangle. b Both triangles BCD and DAB represent Thales´s theorem. In this constellation, we have a representation of the theorem of inscribed quadrilaterals: all vertices lie on a single circle. The construction of vertex A is performed via a primitive Pythagorean triangle (60, 91, 109) with an arbitrary unit of length (e.g., triangle D₁A₁B). c The annexes comprise two quadrangles. d DJ/JH ≈ 97/56 is a high-precision approximation of √3 (sixth Smyrna iteration, upper bound). ID/DK = 168/97 is the corresponding lower bound of this approximation of √3 → IK/DK ≈ 137/97 is an approximation of √2 (Δ = 0.00184)

Fig. 21

The Manndorf and Markvartice enclosures.  a The notch lines of the ditches form a quadrangle. b Plan of the Manndorf enclosure. c CD/CE = CD/DE ≈ 99/70 is a high-precision approximation of √2 (fifth Smyrna iteration). d The notch lines of the ditches form a quadrangle. e AB/BE = 2 = √4 → AE/BE ≈ 78/45 = 26/15 is an approximation of √3 (fourth Smyrna iteration). AG/FG ≈ 52/30 = 26/15 is another approximation of √3 (fourth Smyrna iteration). DH/HF = 33/13 is an approximation of √6 (second Smyrna iteration) → DF/HF ≈ 35½/13 = 71/26 is an approximation of √7 (Δ = 0.08502)

Fig. 22

The Maxing and Langenbach enclosures. a The notch lines of the ditches form an isosceles trapezium. b Plan of the Maxing enclosure. c The construction starts with the square ABCD. The symmetric reduction by extreme Pythagorean triangles provides the isosceles trapezium EFCD. AC/DC = AC/DA = AC/AB = AC/BC ≈ 99/70 is an approximation of √2 (fifth Smyrna iteration). d Application of the extreme primitive Pythagorean triangles (71, 2520, 2521). Triangle ADE is mirror-inverted to triangle BCF. e The notch lines of the ditches form a right trapezium. f Twofold application of the primitive Pythagorean triangle (88, 105, 137). The right angle BCD is a demonstration that the interior angles of a triangle add up to 180° (see also Fig. 19d)

Fig. 23

The Neuhau-Forst and Niederstetten-Wermutshausen enclosures. a The notch lines of the ditches form a trapezium. b DC/DE = DC/CE ≈ 17/12 is an approximation of √2 (third Smyrna iteration, upper bound). AB/AF = AB/BF ≈ 24/17 is another approximation of √2 (third Smyrna iteration, lower bound). c Leg CD is parallel to the straight lines. d Application of the primitive Pythagorean triangles (3, 4, 5) and (48, 55, 73)

Fig. 24

The Niederstotzingen and Nordheim-Kupferschmied enclosures. a The notch lines of the ditches form a quadrangle. b CD/DE = CD/CE ≈ 239/169 is a high-precision approximation of √2 (sixth Smyrna iteration, Δ = 0.00001). c The notch lines of the ditches form a trapezium. The legs of the trapezium A₁B₁C₁D₁ (indicating the inner rims of the banks) are parallel to the legs of the trapezium ABCD. d AD/AE = AD/DE ≈ 17/12 is an approximation of √2 (third Smyrna iteration)

Fig. 25

The Nordheim-Bruchhöhe enclosure and its annexe. a The legs CD and AD are parallel to the straight lines s and t, respectively. b AB/AE ≈ 17/12 is an approximation of √2 (third Smyrna iteration). DF/CF = 58/41 is an approximation of √2 (fourth Smyrna iteration). DC/CF ≈ 71/41 is an approximation of √3 (fifth Smyrna iteration). This combination of approximations of √2 and √3 according to the Smyrna algorithm in a single triangle underscores the knowledge of the Smyrna algorithm; this combination of approximations is also applied in the construction of the triangle IHL of the annexe (panel d) and in the construction of the Plattling–Pankofen enclosure (see Fig. 20b). c The legs IJ and IH are parallel to the straight lines u and v, respectively. d JG/JK = JG/GK ≈ 99/70 is a high-precision approximation of √2 (fifth Smyrna iteration). IL/HL= 58/41 is an approximation of √2 (fourth Smyrna iteration). IH/HL≈ 71/41 is an approximation of √3 (fifth Smyrna iteration)

Fig. 26

The Oberframmering and Oberhaimbuch enclosures. a The notch lines of the ditches form a lozenge. b Fourfold application of the primitive Pythagorean triangle (20, 21, 29). c The notch lines of the ditches form an isosceles trapezium. d Plan of the Oberhaimbuch enclosure. e The construction starts with the rectangle ABC₁D₁ and the twofold application of the primitive Pythagorean triangle (65, 72, 97). f The vertices C and D of the isosceles trapezium are constructed by the extreme Pythagorean triangle (153, 3900, 3903) [primitive (51, 1300, 1301)]

Fig. 27

The Oberschneiding and Papferding enclosures. a The notch lines of the ditches form a rectangle. b Application of the primitive Pythagorean triangle (20, 21, 29). c The notch lines of the ditches form a quadrangle. d The legs AMC and BMD are the diagonals of the quadrangle. The right-angled triangle BCD is a demonstration of Thales’ theorem

Fig. 28

The Peterhof and Pfeffenhausen enclosures. a The notch lines of the ditches form a quadrangle. b BE/AE = 168/97 is a high-precision approximation of √3 (sixth Smyrna iteration). c The notch lines of the ditches form a trapezium. d The right angle BCD is a demonstration that the interior angles of a triangle add up to 180°

Fig. 29

The Plattling-Pankofen and Pliezhausen-Rübgarten enclosures. a The notch lines of the ditches form a quadrangle. b DE/CE = 58/41 is an approximation of √2 (fourth Smyrna iteration). DC/CE ≈ 71/41 is an approximation of √3 (fifth Smyrna iteration). This combination of approximations of √2 and √3 according to the Smyrna algorithm in a single triangle underscores the knowledge of the Smyrna algorithm (see also Fig. 16d). c The legs of the parallelogram A₁B₁C₁D₁ (indicating the upper rim of the inner banks) are parallel to the legs of the parallelogram ABCD. d AD/AF = BC/BE ≈ 187/132 = 17/12 is an approximation of √2 (third Smyrna iteration)

Fig. 30

The Pocking–Hartkirchen and Poign enclosures. a The notch lines of the ditches form a square. b AC/AB = AC/CB = AC/AD = AC/CD ≈ 24/17 is an approximation of √2 (third Smyrna iteration). c Construction of the building in the western corner. d Twofold application of the primitive Pythagorean triangle (88, 105, 137). The resulting cubit units of the Viereckschanze and of the building are identical. e The notch lines of the ditches form an isosceles trapezium. f Application of the primitive Pythagorean triangle (3, 4, 5) to build the rectangle EBFD. g Application of the extreme primitive Pythagorean triangle (49, 1200, 1201). The identical ratio of ropes to decisteps is applied in the construction of the Kirchheim–Osterholz (c) enclosure (see Fig. 10c)

Fig. 31

The Sallach 1 enclosure and its eastern annexe. The documentation of the ditches of the framing structures fail to pass step 1 of the verification procedure. a The notch lines of the ditches of the main earthwork form a quadrangle. b Triangle ABC is similar to the triangle AED. c The legs FG, GH and HI are parallel to the straight lines r, s and t, respectively. d FH/FG = 85/38 is an approximation of √5 (fifth Smyrna iteration)

Fig. 32

The Sallach 2 and Scheyern enclosures.a The notch lines of the ditches form a parallelogram. b DE/FE ≈ 140/99 is a high-precision approximation of √2 (fifth Smyrna iteration). This construction is a demonstration of the property that the diagonals in a parallelogram bisect each other. c The notch lines of the ditches form a trapezium. d The construction is a demonstration of alternate angles

Fig. 33

The Schöngeising and Teufstetten enclosures. a The notch lines of the ditches form a quadrangle. b AD/AE = AD/DE ≈ 17/12 is an approximation of √2 (third Smyrna iteration). DC/BC ≈ 41/29 is an approximation of √2 (fourth Smyrna iteration). BD/BC = 151/87 is an approximation of √3 (Δ = 0.00358). c The notch lines of the ditches form a quadrangle. d BD/BC = BD/DC ≈ 24/17 is an approximation of √2 (third Smyrna iteration)

Fig. 34

The Weiltingen and Willmatshofen-Brennburg enclosures. a The notch lines of the ditches form a parallelogram. b Plan of the Weiltingen enclosure. c Application of the primitive Pythagorean triangles (20, 21, 29) and (88, 105, 137). d The notch lines of the ditches form a quadrangle. e The construction is a demonstration of the theorem of inscribed quadrilaterals (compare with the Königheim–Brehmen enclosure, Fig. 11). AB/AM = AB/BM ≈ 41/29 is an approximation of √2 (fourth Smyrna iteration). The construction of vertex C is performed via a primitive Pythagorean triangle (48, 55, 73) with an arbitrary unit of length (e.g., triangle BM₁C₁); vertex C is the intersection of the extension of leg BC₁ with the circumcircle DAB

References of the Source Data (Plans of the Enclosures)

Altheim–Heiligkreuztal–Bannwald enclosure: Bittel et al. (1990: Teil 2, 6).

Altheim–Heiligkreuztal–Ruchenholz enclosure: Bittel et al. (1990: Teil 2, 7).

Beuren enclosure: Ambs (1999: 63).

Bopfingen (palisade) enclosure: http://www.denkmalpflege-seiten.de/dtsch/viereckschanzen.html.

Burgstallhof–Wilburgstetten enclosure: Schwarz (1959: 90).

Deisenhofen enclosure: Schwarz (1959: 26).

Egweil enclosure: Faßbinder (2005: 15).

Ehningen enclosure: Wieland (1999: 147).

Esslingen–Oberesslingen enclosure: Bittel et al. (1990: Teil 2, 22).

Feldmoching enclosure: Berghausen (2013: 169).

Gilching–Steinlach enclosure: http://www.zeitreise-gilching.de/geschichtegilchings/daten-und-fakten/eisenzeit/keltische-viereckschanzen.html.

Hardheim–Gerichtstetten enclosure: Wieland (1999: 137).

Hechendorf–Güntering enclosure: Berghausen (2013: 157).

Holzhausen II enclosure: Wieland (1999: 196).

Hörstein enclosure: Becker (1987: 100).

Kirchheimbolanden–Donnersberg enclosure: Wieland (1999: 199).

Kirchheim–Osterholz enclosure: Krause (2005: 76).

Königheim–Brehmen enclosure: Bittel et al. (1990: Teil 2, 33).

Laibstadt enclosure: Berghausen (2010: 83).

Langenbach enclosure: Krause (2018: 179–210).

Manndorf enclosure: Berghausen (2013: 199).

Markvartice enclosure: Wieland (1999: 206).

Maxing enclosure: Berghausen (2013: 185).

Neuhau–Forst enclosure: Schwarz (1959: 18).

Niederstetten–Wermutshausen enclosure: Bittel et al. (1990: Teil 2, 49).

Niederstotzingen enclosure: Bittel et al. (1990: Teil 2, 50).

Nordheim–Bruchhöhe enclosure: Neth (Neth and 1997: 80).

Nordheim–Kupferschmied enclosure: Neth (1999: 75ff).

Oberframmering enclosure: Faßbinder and Irlinger (1996: 96).

Oberhaimbuch enclosure: Berghausen (2013: 189).

Oberschneiding enclosure: Husty et al. (2012: 73).

Papferding enclosure: Berghausen and Krause (2007: 80).

Peterhof enclosure: Schneider (1963: 32).

Pfeffenhausen enclosure: Faßbinder and Irlinger (2005: 77).

Plattling–Pankofen enclosure: Wieland (1999: 183).

Pliezhausen–Rübgarten enclosure: Wieland (1999: 127).

Pocking–Hartkirchen enclosure: Wieland (1999: 187).

Poign enclosure: Schwarz (1959: 85).

Sallach 1 enclosure: Schwarz (1959: 66).

Sallach 2 enclosure: Schwarz (1959: 67).

Scheyern enclosure: Schwarz (1959: 30).

Schöngeising enclosure: Schwarz (1959: 14).

Teufstetten enclosure: Berghausen et al. (2006: 64).

Weiltingen enclosure: Berghausen (2013: 151).

Willmatshofen–Brennburg enclosure: Schneider (1963: 33).