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Abstract
This paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians. This link went unnoticed for well over 1500 years until the publication of two groundbreaking but again ostensibly unrelated works by two German mathematicians at the close of the nineteenth century. In this, the second and final part of the paper, we continue our examination of the chain of mathematical events and the related development of mathematical disciplines, without which the connection might never have been noticed in the first place.
Notes
1 For more information on the various algebraic, combinatorial and topological relationships of the (7, 3, 1) triple system see Brown Citation2002.
2 To quote Robin Wilson (Citation2003, 271): ‘this lack of awareness probably arises from the fact that the Cambridge and Dublin Mathematical Journal, though well known in Britain, was little known on the Continent.’
3 This famously prompted Kirkman's sarcastic retort: ‘… how did the Cambridge and Dublin Mathematical Journal, Vol. II, p. 191, contrive to steal so much from a later paper in Crelle's Journal, Vol. LVI, p. 326, on exactly the same problem in combinations?’—Kirkman Citation1887.
4 The discovery for which Kirkman is best remembered today, known as ‘Kirkman's schoolgirls problem’, arose from his work on the (15, 3, 1) block design—see Biggs Citation1981, Wilson Citation2003, Brown and Mellinger Citation2009.
5 For more detail on the history of geometry during the nineteenth century, see Gray Citation2010.
6 ‘Wir erhalten hiernach eine zwölffache Zusammenstellung, der, wenn wir die neun Wendungspuncte durch P, Q, R, P1, P2, Q1, Q2, R1 und R2 bezeichnen, das folgende Schema entspricht’.
7 ‘Nicht jede Anzahl von Elementen ist von der Art, dass sie sich so zu drei gruppieren lassen, dass in den verschiedenen Gruppen alle Combinationen zweier Elemente vorkommen und jede derselben nur ein einziges Mal. Wenn m die Anzahl solcher Elemente und n irgend eine ganze Zahl, Null nicht ausgeschlossen, bedeutet, so überzeugt man sich leicht, dass m von der Form 6n + 3 sein muss. Die Anzahl der verschiedenen Gruppen beträgt alsdann ein Drittel der Anzahl der Combinationen von m Elementen zu zwei, mithin , und jedes Element kommt in
verschiedenen Gruppen vor.’
8 For more on Pieri, see Marchisotto Citation2006 and especially Marchisotto and Smith Citation2007.
9 ‘In den folgenden Zeilen will ich zeigen, dass dieses nur in den Fällen n = 2, 4, 8 möglich ist ...’
10 ‘Durch diesen Nachweis wird dann insbesondere auch die alte Streitfrage, ob sich die bekannten Produktformeln für Summen von 2, 4 und 8 Quadraten auf Summen von mehr als 8 Quadraten ausdehnen lassen, endgültig, und zwar in verneinendem Sinne entschieden.’
11 ‘Roberts und Cayley haben sich im 16. und 17. Bande des Quarterly Journal mit dem Nachweis beschäftigt, dass ein Produkt von zwei Summen von je 16 Quadraten nicht als Summe von 16 Quadraten darstellbar sei. Ihre äusserst mühsamen, auf Probieren beruhenden Betrachtungen besitzen indessen keine Beweiskraft, weil ihnen bezüglich der bilinearen Formen z1, z2… spezielle Annahmen zugrunde liegen, die durch nichts gerechtfertigt sind.’
12 For an expanded and more detailed exposition of Hurwitz's proof, see Dickson Citation1919.
13 For a proof of this, see Curtis Citation1963.
14 ‘...das commutative Gesetz der Multiplikation zweier Strecken auch hier nichts anderes als den Pascalschen Satz’. Note that throughout the Grundlagen Hilbert referred to Pappus's theorem by its more general name of Pascal's theorem.
15 Although the connection of Pappus's theorem with commutativity of multiplication had also been considered by Friedrich Schur in the first edition of his 1898 Lehrbuch der Analytischen Geometrie (Schur Citation1898, 11), the link was not stated explicitly until the second edition of 1912 (Schur Citation1912, 11), thirteen years after the publication of Hilbert's Grundlagen.
16 Of course, this was far from the end of the matter. The Grundlagen also contained Hilbert's construction of a geometry in which Desargues’ Theorem fails to hold (Hilbert Citation1899, 51–55), corresponding to an absence of associativity in multiplication. Thus the real numbers, complex numbers, and quaternions give rise to Arguesian geometries, but the octonions do not. In other words, Desargues’ theorem can only hold in a geometric system whose axioms are equivalent to those of ,
, or
, that is, where Euler's four-squares theorem holds. As a consequence of this discovery, the early twentieth century saw the creation of a variety of non-Arguesian and non-Pappian geometries by F R Moulton Citation(1902) and Oswald Veblen and Joseph Wedderburn (Citation1907) among others. This culminated in the work of Ruth Moufang in the 1930s (Moufang Citation1933), inaugurating the systematic study of non-Arguesian planes and their associated non-associative algebras (or ‘Moufang loops’), which via their connections with ternary rings, cohomology sets, and Jordan algebras, continue to exert an influence on mathematics to this day (see Weibel Citation2007).