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A new reading of Archytas’ doubling of the cube and its implications

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Abstract

The solution attributed to Archytas for the problem of doubling the cube is a landmark of the pre-Euclidean mathematics. This paper offers textual arguments for a new reading of the text of Archytas’ solution for doubling the cube, and an approach to the solution which fits closely with the new reading. The paper also reviews modern attempts to explain the text, which are as complicated as the original, and its connections with some xvi-century mathematical results, without any documented relation to Archytas’ doubling the cube.

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Notes

  1. See Euclide (2007, p. 83): «Sono state proposte varie ricostruzioni di come Archita possa essere giunto a concepire questo complesso intersecarsi di solidi: sono più o meno complicate come l’originale»  («A number of reconstructions of the way in which Archytas made this complex intersection of solids has been suggested: they are more or less as complicated as the original»).

  2. Archimedes (1910–1915, vol. iii, 84.12–88.2). The context is a long section of the commentary in which Eutocius reviews different methods used by Greek authors of antiquity for solving the problem of doubling the cube. The heading of Archytas’ solution for doubling the cube reads: «the discovery of Archytas, as reported by Eudemus». Eudemus was a disciple of Aristotle, who wrote a history of geometry. For full details about the whole textual evidence, see Euclide (2007, pp. 79–81) and Knorr (1989, pp. 100–111).

  3. Not only at a lexical level; I have tried to preserve grammatical categories as well (translating verbs into verbs, nouns into nouns, and so on, and even on lower grammatical levels, e.g. participles into participles). I have tried to preserve the figurae etymologicae, so that words with the same root are translated using the same common root: e.g. , to trace (verb)/trace (noun; its usual translation is «line»); , to draw/to draw around/to draw around in the opposite direction. The translations of Knorr (1989, pp. 102–107) and Thomas (1951, pp. 284–289) [and Euclide (2007, pp. 82–83), in Italian] are also very close to the original text. van der Waerden (1954, pp. 150–151) is a paraphrase, and Heath (1921, vol. i, pp. 246–249) and Knorr (1986, pp. 50–52) are commentaries on the text.

  4. Edited and commented on by Clagett (1964–1984, vol. i, chapter 4, pp. 223–367). There is a detailed comparative analysis of both texts in Knorr (1989, pp. 100–110).

  5. In fact, the Latin text of Gerard of Cremona reads «Mileus»  but the most likely reading from the Arabic text is «Menelaus»  (Clagett 1964–1984, vol. i, p. 365, prop. 16, 12).

  6. In this case, the goal is to find two mean proportionals between V (whose cube root is sought) and 1. If \(m_1,m_2\) are the two mean proportionals, then: \(\frac{V}{m_1}=\frac{m_1}{m_2}=\frac{m_2}{1}\). Therefore, \(\frac{V}{m_1}\cdot \frac{m_1}{m_2}\cdot \frac{m_2}{1}=\left( \frac{m_2}{1}\right) ^3\). So, \(V=m_2^3\), i.e. \(m_2\) is the cube root of V, its side. The setting up of the problem as shown in the Banû Mûsâ’s text is highly unlikely to be of Greek origin, because V is a volume and \(m_2\) is a length; Greek Mathematics never build proportions with non-homogeneous magnitudes (numbers, lengths and volumes). In fact, the presence of the unity in the Arabic text allows us to suggest that the Banû Mûsâ are thinking of the numbers that are the result of the measures, and this approach is impossible from the Greek point of view. At any rate, the Banû Mûsâ only explicitly attribute the finding of two mean proportionals to Menelaus, not its use in the extraction of the «side of a cube».

  7. Proclus (s. v bc), In Primis Euclidis, pp. 203.1–207.25, shows the canonical division into parts of a Greek mathematical proposition (see also Netz 1999; Euclide 2007, pp. 259–312). We do not strictly follow this division, although some of the terms I use are identical to the English translation of Proclus terms. The division into parts I use is based upon the form and the content of Eutocius’ text. Therefore, words such as proof, demonstration, construction and statement have the usual sense and not the technical sense in Proclus’ division. Only the terms proposition and problem are used in its technical sense, and for this reason, they are written in italics in this section.

  8. Henceforth and for clarity, I define explicitly some geometrical configurations or manipulations. I always use terms in italics and initial capital letters to denote them.

  9. This semicircle, like the other geometrical elements, is explicitly mentioned in the text (see line 13R), but does not appear in the diagram of the transmitted text, and probably for that reason scholars have ignored SC and do not include it in their figures.

  10. That is to say, the rotating \(\text {A}\Pi \). An alternative reading could be the «\(<\)trace\(>\) drawn around», i.e. the intersection of the cylindrical and the conical surfaces, as discussed in note 4 (Fig. 4 also shows this trace).

  11. For example, when examining all verbal tokens in Archimedes De Sphaera et Cylindro (see Masià 2012, p. 204), only 5.2 % of them are future verbal tokens. Euclid and Apollonius probably use more or less the same percentage of future verbal tokens. It is usually used in the apodosis of conditional clauses like the one in lines 22R–26R.

  12. «Let the moved semicircle have a position [...] as the \(<\)position\(>\) of the \(<\)semicircle\(>\) \(\Delta \text {KA}\)», line 33R–35R; «let the triangle [...] \(<\)have a position as\(>\) that of \(\Delta \Lambda \text {A}\)», lines 36R–38R.

  13. It is worth noting, again, that the diameter of the rotating semicircle SC is denoted in the same way in both configurations, Initial and Final. But the rotating triangle \(\text {A}\Delta \Pi \) is denoted differently in these configurations: \(\text {A}\Delta \Pi \) and \(\text {A}\Delta \Lambda \). We can assume that the semicircle subsists during the motion and the rotating movement of SC does not generate a surface, while the line \(\text {A}\Lambda \) could be a line on the explicitly defined conical surface; it has different names in the two different positions: \(\text {A}\Lambda ,\,\text {A}\Pi \). Then, the conical surface is also indirectly used to find the solution.

  14. The sentence with the future form is a parenthetical commentary introducing a property of the straight line from K (it will be perpendicular to some plane), justified with a postponed explanation.

  15. I use the same letters \(\text {A, I, K,}\,\Delta \) used in other figures, but the new figure only shows a general right-angled triangle inscribed into a semicircle.

  16. Aristotle, some decades after Archytas, uses this configuration as an example in Metaphysica and Analytica posteriora (see Euclide 2007, pp. 118–119). Euclid’s Elementa discusses the elements of this configuration in books iii and vi.

  17. A great circle of a sphere is the intersection of the sphere with a plane which passes through the centre of the sphere.

  18. Solid geometry is known from very early in Greek geometry, because of its relationship with astronomy. It is also known that the division of the circle into two equal semicircles by a diameter is attributed to Thales of Miletus (s. vi bc). This result can be easily translated to the three-dimensional space: after the creation of a sphere by rotating a circle around one of its diameters, the initial circle is a great circle of the sphere, and it is apparent because of the construction that the great circle divides the sphere into two equal parts. Property 3b) could easily be derived from arguments of symmetry and the application of Thales’ property of the circle.

  19. As we have said, MI is perpendicular to AK, IK to \(\text {A}\Delta \) and \(\Delta \text {K}\) to KA. Therefore, \(\text {A}\Delta \text {K},\,\text {AIK}\) and AIM are similar right-angled triangles (two right-angled triangles are similar when they have a common angle in addition to the right angle). And similar triangles have proportional pairs of corresponding sides.

  20. It is worth noting that Archytas’ solution for doubling the cube is one way of finding in three-dimensional space Archytas’ Solution, i.e. it is one Archytas’ Solution.

  21. See p. 2.

  22. This figure turns out to be equivalent to the diagram of the solution attributed to Plato by Eutocius in the same text where Archytas’ solution for doubling the cube appears (see Knorr 1986, pp. 50–66; Archimedes 1910–1915, vol. iii, 56.13–58.14).

  23. Assuming that \(\text {IM}=x\).

  24. And even the construction behind the solution attributed to Plato by Eutocius for doubling the cube.

  25. As we will see in Sect. 5.1, there are a number of historical and historiographical two-dimensional approaches to the solution of the two mean proportionals. All of them generate a curve: for each curve, every point on it generates an Archytas’ Triangle and only one of them is Archytas’ Solution. Therefore, these approaches are methodologically far from the Archytas’ Solutions that I am ready to explain.

  26. In an Archytas’ Solution KI\(^{\prime }\) and MI should intersect in \(\text {I}=\text {I}'\).

  27. It is evident that the triangle that solves the problem should have a lesser angle \(\Delta \text {AM}\). When making this rotation, the angle is reduced. It should be remembered that the lengths of AM, \(\text {A}\Delta \) are given and this motion keeps AM and the length of \(\text {A}\Delta \) the same.

  28. See this interactive diagram: http://www.geogebratube.org/student/m119505. The use of these interactive diagrams is merely illustrative and facilitates the reading of our arguments; they do not introduce any supplementary (visual) argument.

  29. Or, alternatively, \(\text {K}\) can be moved around the circumference and the new I is found drawing the perpendicular from K to \(\text {A}\Delta \), and cutting off te line \(\text {A}\Delta \).

  30. See this interactive diagram: http://tube.geogebra.org/student/m208427.

  31. And it is also in the background of Becker’s curve (see Sect. 5.1).

  32. The line \(\text {A}\Pi \) and the cone are not shown in the figure to simplify the diagram. In fact, the cone is implicitly shown by the generatrix AM and the circle BMZ.

  33. The sphere is not in the transmitted text and is actually unnecessary for the technical details of the demonstration shown in it (the entire explanation I will develop does not need the sphere). But one of our goals is to find some ideas that may be behind the solution. It seems apparent that the starting point (temporally and logically) of the demonstration is this new Initial Configuration: the three intersecting and perpendicular (semi)circles. From a historical point of view, we cannot go any further. But we can try to explain where this Initial Configuration comes from. An explanation is that an Archytas’ Triangle with given sides could be circumscribed into a sphere, in the search for an Archytas’ Solution. Then, the sphere is only useful for my justification of the Initial Configuration, but not in the development of the rationale. And this justification is based on this argument: it is unlikely that whoever invented Archytas’ solution for doubling the cube did not see the sphere in this new Initial Configuration; in addition, as a seminal idea, it seems to be easier to imagine an Archytas’ Triangle inscribed into a sphere than the three (semi)circles mentioned. I postulate the sphere with the inscribed Archytas’ Triangle to be previous to the figure with the intersection of circle and semicircles. That is, of course, a matter of opinion. But, from this point on, the sphere is irrelevant and could easily be removed from the description of what is happening with Archytas’ construction and the demonstration (although it is more difficult to explain why the inventor of the demonstration performs certain motions). Finally, the large circle of centre A and radius \(\text {A}\Delta \) is added to help the reader to link Figs. 14, 15 and 16.

  34. The rotational motion is suggested by the restriction on I, which must be on the great circle. It is worth noting that the triangle must stay perpendicular to the great circle during the entire process.

  35. See this interactive diagram: http://tube.geogebra.org/student/m210965.

  36. It is worth noting, as mentioned before (see step [4] and Sect. 3.1), that the text explicitly underlines the fact that the last step of the process reads: A, M, K should be aligned.

  37. In other words, the entire process is a sequence of geometrization of all restrictions (in the form of a sphere, a cone, a cylinder and a semicircle), that is to say, restrictions are reduced geometrically:

    • «AM and \(\text {A}\Delta \) are given in length»  and «AMI is a right-angled triangle»  is equivalent to the fact that the triangle is inscribed into a sphere of radius \(\text {A}\Delta \), perpendicular to one great circle, and AM is a side (i.e. generatrix) of a right cone.

    • «\(\text {A}\Delta \text {K}\) is a right-angled triangle»  is equivalent to the fact that it is inscribed on the semicircle \(\text {A}\Delta \text {K}\), in the Initial Configuration.

    • «IK is perpendicular to \(\text {A}\Delta \)»  is equivalent to the fact that IK is on the semicylinder.

    • Finally, «A, M, K are aligned»  is equivalent to the fact that K is on the cone. This is the last step of the process, achieved by rotating the semicircle \(\text {AK}\Delta \).

    This kind of process of geometrization of all restrictions is usual in so-called locus problems (see Euclide 2007, pp. 83–85). It is worth noting that the sphere disappears from the demonstration because it is actually unnecessary once the other conditions have been satisfied.

  38. It is worth noting that there are no such texts (private notes or drawings by mathematicians) in the Ancient Greek mathematical corpus. There are not even any original mathematical texts, but only later editions made by scholars centuries after the mathematician’s death.

  39. With differing degrees: van der Waerden is perhaps the most explicit, while in Euclide (2007) this suggestion is only implicit. The general wonder that produces Archytas’ solution for doubling the cube is due, mainly, to this implicit decision.

  40. Namely Archytas’ Simple Triangle of the Initial Configuration, and the evolution of the two triangles inscribed into the semicircle.

  41. As Burkert remarks (1972, p. 303): «Still and all, in the history of science logical necessity and historical sequence are not always identical».

  42. See this interactive diagram: http://www.geogebratube.org/student/m210935.

  43. A number of them mention the fact that Archytas’ solution is developed in a synthetic way, and the reconstruction is analytical. The discussion about analysis/synthesis in Greek mathematics is a well-known subject (for a complete discussion on this subject, see Euclide 2007, section III B.1, pp. 439–518), but it is avoided in this article because my arguments stand apart from this issue.

  44. Van der Waerden (1954, p. 150, n. 2) use of \(\Delta ^{\prime }\) is «for greater clarity». Diels, the editor of Eutocius’ text, also corrected the text, denoting the rotated point as \(\Delta ^{\prime }\). This is a «inutile»  correction (Euclide 2007, p. 162, n. 258).

  45. I mention this reconstruction only as an outstanding example of a modern reading in the early twentieth century, but useless for understanding Archytas’ solution for doubling the cube.

  46. Van der Waerden even seems convinced of catching on to Archytas’ intuition: «Archytas hit upon the following idea»  (p. 152). As mentioned, van der Waerden and Knorr denote as \(\Delta ^{\prime }\) one of the two \(\Delta \) of the transmitted text and diagram.

  47. Attempting to understand what the scholar is trying to explain and leaving out the error, we could agree that the semicircle AMI will describe half a hemisphere, but not a complete hemisphere.

  48. If we decide to qualify the verb /«draw around»  as a kinematic verb, the verb /«draw»  must be qualified as a kinematic verb too. But this last verb is one of the most common verbs in Greek geometry, used to draw lines. Therefore, we also have to characterize all Greek geometry as kinematic. In fact, I would agree that the lexicon of Greek geometry is kinematic in a very elemental way: it is built with words that describe a drawing activity, which, of course, is a kinematic activity.

  49. In fact, only one kinematic root is not usually found in Greek geometry, with two occurrences: the noun /«movement», and the participle /«moved». It is worth noting that there are also two explicitly static words in the text, unusual in Greek geometry (except that the first one is quite common in Euclid, Data: Def. 4, 6, 8, 13, 14, 15, Prop. 25–41): /«position, place», repeated several times, but sometimes implicit (see beginning of section [4] of the translation).

  50. Underlining the resultative aspect of the verbal action.

  51. And some passages of [4] in which certain elements of step [3] are mentioned.

  52. In addition, following the reading proposed in note 4, both rotating elements (semicircle and triangle) are not used to find the solution, but instead their traces on the cylindrical surfaces, and these traces are static lines. With the usual reading, only the triangle is a moving element in the solution.

  53. The two-dimensional approach and its origin are broadly discussed in Sect. 5.1.

  54. «Sono state proposte varie ricostruzioni di come Archita possa essere giunto a concepire questo complesso intersecarsi di solidi: sono più o meno complicate come l’originale. Cercando anche noi di capire come funziona la dimostrazione [...]»  (Euclide 2007, p. 83).

  55. As we have shown before, only van der Waerden seems aware of this fact, even though the error in the text and in the diagram linked to the solution suggests that his intuition is not accurate.

  56. Or, if there were some relation, it disappeared before the xvi century.

  57. Only Becker’s curve has an indirect relation through an alleged Eudoxus’ solution for doubling the cube.

  58. But Becker, strangely, attributes the discovery to Eudoxus (an alleged disciple or follower of Archytas). Would it not be easier to attribute or link this curve to Archytas, if the curve is found in the «core»  of his solution? In any case, Archytas’ solution does not contain a reference to a curve of this kind, even indirectly, although the construction of the curve is certainly very close to some elements of Archytas’ Construction. Note that nobody has associated the curved traces on the cylinder (of the semicircle and, perhaps, of the triangle) explicitly mentioned in Archytas’ text with the «curved trace»  linked to Eudoxus.

  59. Knorr, Becker and Loria misspell the name of the Spanish Jesuit Juan Bautista Villalpando, mathematician and architect, who wrote, with another Jesuit, Jerónimo Prado, a commentary on the prophet Ezekiel and the Solomon temple, In Ezechielem explanationes et apparatus urbis, ac templi Hierosolymitani commentariis et imaginibus illustratus which includes some chapters devoted to proportional lines (Prado and Villalpando 1596–1604). Probably, Loria only consulted Viviani’s book where Villalpando is mentioned, misspelling the name of Villalpando, Becker copied Loria and Knorr both Becker and Loria. Villalpando would seem to have not been consulted by these scholars.

  60. The curve mentioned is Villalpando’s proportionatrix secunda, although Becker offers a slightly different definition; he calls it “Duplicatrix” because it is the name that Loria (the source of Becker) uses for these kinds of curves.

  61. Loria (1902, p. 317). It is worth noting that the point of view of Loria’s book is completely algebraic, and not geometric. There is no mention of Archytas’ solution for doubling the cube.

  62. I have not found Kepler’s mention of this curve; the Loria citation «Astronomia Nova (Prag, 1609) S. 337»  in Loria (1902, p. 317, n. 1) is not correct, as p. 337 of Kepler’s book does not contain the oval; it is actually the last page of the treatise.

  63. Astronomia Nova was published in 1609 and Villalpando’s third and last volume of his work was published in 1604.

  64. «Vi ò nello stesso tempo osservato che questa curva può segnarsi in altro modo senza bisogno di quel secondo mezzo cerchio»  (Viviani 1674, p. 279).

  65. Only in Becker’s text the context is the discussion of Eudoxus’ curve mentioned by Eutocius. As I have said, he is answering Tannery’s proposal for Eudoxus’ curve, both are without any textual support.

  66. Becker (1966, p. 79) (italics are mine). Becker, as we have said, also associates this curve with Villalpando’s Proportionatrix secunda: «The curve is a beautiful oval, is consequently completely contained within the finite. It is symmetrical around the axis AD. From the 17th c. onwards, this curve is well known. Already Father Villapaudo (sic) used it as “Duplicatrix” (sic)»  (Becker 1966, loc.cit. ). Although this is a clever piece of intuition by Becker, which seems not to have been understood by Knorr (he actually uses Viviani’s curve, and not Becker’s), he does not explain exactly what the «direct tie»  consists of. In fact, the curve is only indirectly tied with «the planimetrical core piece of solution of Archytas», as we will see.

  67. Villalpando’s actual use of this curve to find the two mean proportionals between two assigned straight lines is a bit different, but is still equivalent to the one just explained.

  68. A definition and an interactive visualization of all curves (proportionatrix prima, proportionatrix secunda, Viviani’s (and Knorr’s) approach and Becker’s curve) and my reconstructions discussed before are shown in http://tube.geogebra.org/book/title/id/791183. All curves, except the proportionatrix prima, are in fact the same (as Becker points out, it is \(r=a \cos ^3\theta \)) and Archytas’ Triangle could be shown easily in the final diagram of each one. Tannery’s curve (Tannery 1912–1915, “Sur les solutions du problème de Délos par Archytas et par Eudoxe. Divination d’une solution perdue”, pp. 53–61) is shown in circle \(\text {AB}\Delta \) and not in semicircle \(\text {AK}\Delta \) because the point he is searching for is I.

  69. But following an alternative reading presented in note 4, these elements could be two cylindrical traces that meet at one point.

  70. In fact, no Ancient source places distinction on it.

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    Ramon Masià

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Masià, R. A new reading of Archytas’ doubling of the cube and its implications. Arch. Hist. Exact Sci. 70, 175–204 (2016). https://doi.org/10.1007/s00407-015-0165-9

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