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Geometry and arithmetic in the medieval traditions of Euclid’s Elements: a view from Book II

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Abstract

This article explores the changing relationships between geometric and arithmetic ideas in medieval Europe mathematics, as reflected via the propositions of Book II of Euclid’s Elements. Of particular interest is the way in which some medieval treatises organically incorporated into the body of arithmetic results that were formulated in Book II and originally conceived in a purely geometric context. Eventually, in the Campanus version of the Elements these results were reincorporated into the arithmetic books of the Euclidean treatise. Thus, while most of the Latin versions of the Elements had duly preserved the purely geometric spirit of Euclid’s original, the specific text that played the most prominent role in the initial passage of the Elements from manuscript to print—i.e., Campanus’ version—followed a different approach. On the one hand, Book II itself continued to appear there as a purely geometric text. On the other hand, the first ten results of Book II could now be seen also as possibly translatable into arithmetic, and in many cases even as inseparably associated with their arithmetic representation.

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Notes

  1. Beginning in the late nineteenth century, this view was promoted by prominent scholars such as Paul Tannery (1843–1904), Hieronymus Georg Zeuthen (1839–1930), Sir Thomas Little Heath (1861–1940), and Otto Neugebauer (1899–1990). Book II became a pivotal focus in the elaboration of the details of this historiographical perspective.

  2. The issue of the adequate use of direct and indirect sources in order to establish an authoritative version of the Euclidean text is a significant historiographical question still under debate nowadays. See (Rommevaux et al. 2001). I do not deal at all with this issue here, and I simply follow the widely accepted English version (Heath 1956 [1908]).

  3. Proposition II.5 has also been used as focal point in other accounts about Greek mathematics or about changing views on geometry through history; see, e.g., Neal 2002, 123 ff.; Netz 1999, 9–11; Vitrac and Caveing 1990, 370–372.

  4. Recent scholarship has devoted increased attention to the ways in which diagrams appearing in critical editions of the Elements differ from those in extant manuscripts. See, e.g., (Saito and Sidoli 2012). Although considerations of this kind may be relevant to our analysis here, in this article I will refer only to diagrams as they appear in the available critical editions.

  5. The distinction between the two kinds of proofs has efficiently been used in (Oaks 2011) for the case of Islam mathematics.

  6. See Doron Zeilberger, Opinion 82: “A Good Lemma is Worth a Thousand Theorems” (written: August 14, 2007; downloaded May 02, 2012: http://www.math.rutgers.edu/~zeilberg/Opinion82.html): “Theorems are nice, but they are usually dead-ends. A lemma may be ‘trivial’, or easy to prove once stated, but if it is good, its value far surpasses even the deepest theorems.”

  7. And for an English translation, see (Lo Bello 2009, 32).

  8. (Vitrac 2005, 6 ff.) speaks about two contrasting styles of geometrical proof in Greek mathematics: “demonstrative” versus “algorithmic.” Without wanting to make too much of word choice, and without the benefit of the much broader scope of Vitrac’s analysis, I think that for the case of Book II, at least, the contraposition of “constructive” versus “operational” is more adequate to encapsulate the difference between Euclid’s and Heron’s proofs.

  9. A Latin version is extant which dates from the fourteenth century (Sesiano 1993). I will be referring to this Latin text, which most likely reflects what was available to the European mathematicians we shall be discussing below. There is also a Hebrew version with comments by Mordechai Finzi (died 1475) which seems to have been a translation from a Spanish version, but no such Spanish version has been preserved. See (Levey 1966; Weinberg 1935).

  10. See also Oaks 2011, 255–256.

  11. “... sicut dixit Euclides in secundo tractatu libri sui”.

  12. Besides this method of solution for finding the thing, whose validity is proved in two different ways, Abū Kāmil also introduced a second method of solution directly yielding the value of the square. Although highly interesting in itself, it is beyond the scope of the present article. See (Sesiano 1993, 329–330; Moyon 2007, 310).

  13. This text in its various extant manuscripts and translations has had a somewhat convoluted history (Brentjes 2001b), which I will not delve into here. I will refer to the text as rediscovered and published in Latin translation in (Curtze 1899). For details, see also (Lo Bello 2003a, 2009; Tummers 1994).

  14. In Curtze’s late-nineteenth-century edition of this text, Al-Nayrīzī’s commentary was accompanied by a footnote in which he formulated the original Euclidean text together with an algebraic rendering of the proposition (Curtze 1899, 94). Even here this algebraic rendering is anachronistic. Reinterpreting a geometric result in arithmetic terms (as Al-Nayrīzī did) is not yet the same as reinterpreting it algebraically.

  15. Neither Curtze nor—to the best of my knowledge—any other later commentator has called attention to the remarkable fact that al-Nayrīzī’s IX.16 is an addition to, and indeed a generalization of, Euclid’s IX.15, and that, incidentally, an interesting feature of Euclid’s proof of IX.15 is its reliance on what seem to be arithmetic versions of II.3 and II.4 (Heath 1956 [1908]), Vol. 2, 404–405). Campanus, following al-Nayrīzī, included arithmetic versions of propositions of Book II as additions to the same IX.16. I discuss this point in greater detail in [LC2].

  16. One interesting point of his Latin version is that in Al-Nayrīzī’s additions to IX.16, products of numbers are called “areas” (superficiales).

  17. The introduction to (Busard 2005) contains a clear and well-organized summary of the rich and scholarly impressive literature that has dealt over the last thirty years with the issue of the Latin versions of the Elements. The reader may also consult the following: (Brentjes 1997/1998, 2001a); (Busard 1983, 1984, 1996a; Busard and Folkerts 1992; Folkerts 2006—Section III); (Murdoch 1968, 1971).

  18. Murdoch (1966) called attention to a very accurate twelfth-century translation of the Elements by an anonymous translator who also translated Ptolemy’s Almagest from the Greek. However, there is no evidence of any substantial use of this translation. See also (Busard 1987).

  19. Unless otherwise stated, translations from Latin, Hebrew, or German are mine.

  20. The Hebrew printed version of Bar Ḥiyya’s text is accompanied by an introduction in Hebrew written by Guttmann, and it contains further comments, also in Hebrew, written by Zvi Hirsch Yaffe (1853–1927). The Guttmann edition also served as the basis for a translation of the text to Catalán (Millás-Vallicrosa 1931).

  21. The paragraphs are numerated in the edition I am using here. I add the initials BH, for ease of reference.

  22. Read from the point of view of current Hebrew usage, Bar Ḥiyya’s terminology may give rise to some confusion, as was pointed out in (Sarfatti 1968, 82). I am following here Sarfatti’s interpretation (e.g., multiplication for “

    figure y

    ”, addition for “

    figure z

    ”, root for “

    figure aa

    ”). For a brief account of the differences between ancient and modern Hebrew mathematical terminology, see (Corry and Schappacher 2010, 449–457).

  23. Actually, the text in Guttmann (1912–1913), p. 15, says: “

    figure ae

    ” (on 4 and on 5). But this seems to be a typo.

  24. (Guttmann 1912–1913, 15) [In the quotation, I have replaced the Hebrew letters appearing in the original diagram with the letters I am using here in my diagram]:

    figure ag

  25. These include, in the order in which they appear, versions of: III.35, I.33, VI.4, I.37, I.38, I.41, I.35, I.36, and VI.1.

  26. In Sect. 49: “

    figure ai

    ”. In the text Bar Ḥiyya sometimes calls the number by its name and sometimes uses the alphabetic characters ...

    figure aj

    to indicate them. For the side of the square or for the unknown Bar Ḥiyya uses alternatively

    figure ak

    and

    figure al

    even within the same sentence. The text has many interesting linguistic aspects that I do not discuss here, starting with the very use of the word “thisboreth” (

    figure am

    ).

  27. Like with the Hebrew text, also here there are some open issues concerning the manuscripts. See (Curtze 1902, 4).

  28. A detailed analysis of this problem and its roots in Babylonian mathematics appears in (Sesiano 1987).

  29. It would also be of particular interest to compare Fibonacci’s proof of II.5 with all the others discussed here, but in consideration with the length of this article I will focus on II.9 alone.

  30. (Høyrup 2010, 16) has argued that the potential impact of the Arithmetica materialized only to a very limited extent. Here I want to focus on the way in which, via its reading by Campanus, it did have an important influence in the specific issue of the increasingly arithmetic-algebraic conceptions of Book II.

  31. The Stapulensis printed version of 1496 does include numerical examples. In this as well as in other respects, it is interesting to compare the printed with the original Jordanus’ version. We shall leave this comparison for a future opportunity.

  32. It must be stressed, however, that Jordanus included in his treatise another proposition, A-X.3, which is in fact equivalent to A-I.19 and that embodies the so-called Regula Nicomachi: given three numbers in arithmetic progression, \(a-b=b-c\), then we have \(b^{2}-a\cdot c = (a-b)^{2}\). Boethius has stated in his Arithmetica that this rule was discovered by Nicomachus (Busard 1991, 14). Jordanus’ very short proof of A-X.3 simply invokes A-I.19. This apparent repetition reflects, so it seems to me, Jordanus’ conscious view of A-I.19 as naturally belonging to the context of this preliminary type of results, where a given number is separated in two parts and the relations between the partial products, even if an equivalent form of it could appear in the context of a later book as well.

  33. It is pertinent to refer the reader in this regard to (Puig 1994). This seldom-cited, but highly original article makes a remarkable connection between the arithmetic results presented in Jordanus’ De numeris Datis and the diagrammatic aspects underlying the arguments of the proofs. I discuss this in some detail in [LC2].

  34. See also Campanus’ comments to VII.6 (p. 236), which is compared with V.13 (i.e., Euclid V.12).

  35. All of these are very interesting in themselves, but discussing them would be beyond the scope of this article. See (Rommevaux 1999, 93–100).

  36. The only other place where we find such additions to the diagram is in the very last arithmetic proposition, IX.39.

  37. See (Busard 2005, 291), and also, proposition A-IV.19 of Jordanus in (Busard 1991, 103). Campanus’ proof is somewhat different from (and more detailed than) Jordanus’.

  38. These are comments 4–12. Comments 1–3 correspond to Jordanus’ A-I.9–A-I.11, whereas comment 13 corresponds to Euclid’s V.11. See (Busard 2005, 33).

  39. The difference is directly noticeable by comparing steps (l.1)–(l.5) here below with steps (c.1)–(c.7) for Heron, with (k.1)–(k.5) for Jordanus, or with the algebraic interpretation that (Curtze 1899, 96) provides in a footnote for Al-Nayrīzī.

  40. Though I would not like to lay excessive stress on names, I feel it is improper to claim, as in (Busard 2005, 39), that “the algebraic method which Campanus used for proving Campanus IX.16 add. 4–12 agrees with the method which Anairitius used for proving II.2–II.10.” For one thing, I already indicated that both the “method” and the line of argumentation actually differ from that of Al-Nayrīzī. But also, using the term “algebra” in this context may be misleading since nothing here is manipulation of abstract symbols according to formal rules, not even in the manner intended in Islamic mathematics. Rather, Campanus simply operates on numbers (which are indicated with letters that serve as names) according to general, arithmetic rules.

  41. Here, I refer to the text as is appears in (Lange 1909), which is where the Hebrew text first appeared in print in modern times, together with a German translation. See also (Simonson 2000a, b) for a description of the contents, for some important remarks concerning the existing manuscripts and versions, and for additional parts not appearing in the Lange edition.

  42. This is the version appearing in (Lange 1909, 5). It is remarkable that the proposition was not included in the second edition. See (Simonson 2000a, b, 297).

  43. The full original Greek text appears in (Heiberg and Menge 1883–1893, Vol. 5), as Appendix Scholiorum 4. I thank Michael Fried for help on translating parts of it.

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Acknowledgments

I want to pay my debt of gratitude to several friends and colleagues who read previous versions of this text, or parts of it, answered queries and sent me reading material, raised questions, and suggested ideas and advanced critical views: Fabio Acerbi, Sonja Brentjes, Karine Chemla, Menso Folkerts, Jan Hogendijk, Jens Høyrup, Tony Levi, Anthony Lo Bello, Mark Moyon, Jeffrey Oaks, Sabine Rommevaux, Ken Saito, Shai Simonson, and Roy Wagner. Special thanks I owe to my friends Michael Fried and Miki Elazar for help and patience with translations of difficult passages from Latin and Greek, and to Veit Probst in Heidelberg, without whose kind mediation I would have had a much harder time in gathering all the material needed to complete this work. Last, but not least, hearty thanks go to Len Berggren, to whom I submitted this article for publication. His very detailed reading and thoughtful editorial suggestions helped me significantly improve (and somehow shorten) the entire text.

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    Leo Corry

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Communicated by: Len Berggren.

Dedicated to my dear friend Sabetai Unguru on his 82th birthday.

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Corry, L. Geometry and arithmetic in the medieval traditions of Euclid’s Elements: a view from Book II. Arch. Hist. Exact Sci. 67, 637–705 (2013). https://doi.org/10.1007/s00407-013-0121-5

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