Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas. Part I

Abstract

This two-part paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians, a link that went unnoticed for well over 1500 years. In this, the first installment of the paper, we examine the initial chain of mathematical events that would ultimately lead to the discovery of this remarkable link between two seemingly distinct areas of mathematics, encompassing contributions by a variety of mathematicians, from the most distinguished to the relatively unknown.

Notes

¹ For a good introduction to Pappus's Theorem and some of its ramifications, see Marchisotto 2002.

² For recent analysis of Diophantus's algebra and methods of solution, see Christianidis 2007, Bernard and Christianidis 2012, and Christianidis and Oaks 2013.

³ ‘Invenire quatuor numeros compositi ex omnibus quadratus, singulorum tam adiectione quam detractione faciat quadratum.’ Heath 1910, 166 gives this as Book III, Problem 19.

⁴ ‘Adhuc autem suapte natura numerus 65 dividitur bis in duos quadratos, nempe in 16 49 et rursus in 64 1, quod ei contingit quia fit ex multiplicatione mutua 5 13 quorum vterque in duos dividitur quadratos.’

⁵ ‘Quand les droites HDa, HEb, cED, lga, lfb, HlK, DgK, EfK, soit en divers plans soit en un mesme, s'entrerencontrent par quelconque ordre ou biais que ce puisse estre, en de semblables points; les points c, f, g sont en une droite cfg’.

⁶ The one apparent exception to this is Book V, Problem 14 (‘To divide unity into three parts such that, if we add the same number to each of the parts, the results are all squares’), which is equivalent to finding three square numbers, each bigger than a given number n, that sum to 3n + 1.

⁷ ‘Hanc marginis exiguitas non caperet’.

⁸ ‘Theorema, quod quicunque numerus sit summa quatuor quadratorum, demonstrare non possum ...’ (Fuss 1843, vol 1, 30).

⁹ Euler's four-squares formula also appeared in a later paper on orthogonal substitutions (Euler 1771, 311).

¹⁰ Degen's formula contains a misprint in which the sign of Rt is given as ±. This error was first pointed out by Dickson 1919, 164.

¹¹ A good survey article on the history of n-square identities is Hollings 2006.

¹² In doing so, he was presumably ignorant of Euler's proof that iⁱ = e^{−(π/2) ± 2nπ} (Euler 1751, 130–133).

¹³ Those unfamiliar with it may wish to consult the accounts given in Crowe 1985, Hankins 1980, and van der Waerden 1976.

¹⁴ Graves mistakenly wrote Lagrange, although he meant Legendre 1798.

¹⁵ Euler's formula (2) appears in Legendre 1798, 200, where Legendre remarks in a footnote: ‘On peut s'assurer qu’il n'existe aucune formule semblable pour trois quarrés, c'est-à-dire que le produit d'une somme de trois quarrés par une some de trois quarrés, ne peut pas être exprimé généralement par une somme de trois quarrés. Car si cela étoit possible, le produit (1 + 1 + 1)(16 + 4 + 1), qui est 63, pourroit se décomposer en trois quarrés’.

¹⁶ Graves’ notation was actually 1, i, j, k, l, m, n, o. We use the notation employed in Cayley 1845.

¹⁷ See Addendum to Young 1848 in Transactions of the Royal Irish Academy, 21 (Part II) (1848), 338–341.

¹⁸ John Graves’ younger brother Robert.

¹⁹ See, for example, Baez 2002, and Conway and Smith 2003. It must also have been disappointing for Graves when he eventually discovered that even the eight-squares formula had first been published by someone else. As he wrote to Hamilton on 4 December 1852: ‘The theorem of eight squares, which I communicated to you some years ago, had, I find, been previously discovered by C. F. Degen, “Adumbratio Demonstrationis Theorematis Arithmetici maxime generalis.” Mémoires de l’Académie Impériale des Sciences de St. Petersburg, tom. viii. p. 207, St. Petersburg, 1822. Conventui exhibuit die 7 Oct. 1818.’—Graves 1882–89, vol 2, 577n

²⁰ It should be noted that Hamilton's biquaternions are very different from the algebra of the same name introduced by Clifford in 1873, which are of the form p + ωq, where p and q are real quaternions, ω commutes with every real quaternion, and ω² = 0 or ω² = 1.