The triangle as a geometric variable

Extract

Among Schubert's many experiments in the application of a symbolic calculus to problems of enumerative geometry, some special attention is due to his long memoir entitled “Anzahlgeometrische Behandlung des Dreiecks” [1]. For one thing, he is dealing here with a simple, though not elementary, kind of geometric variable, the triangle in a fixed plane, so that the paper gives a clear insight into his general method; and, for another, there is contained in this paper, as was recently suggested by Freudenthal ([2], p. 19), an apparently miraculous device, the introduction of “infinitesimal triangles”, which we can now recognize (§4) as having had the effect of desingularizing the triangle domain in which the calculus was to operate. The principal target of Schubert's investigations was the discovery of Bézout-type formulae for the number of triangles common to two algebraic systems Σ_r and Σ_6-r (r = 1, 2, 3) of complementary dimensions, the systems being supposed to intersect in only a finite number of triangles, and the multiplicities of these triangles being assumed to be suitably defined. His systems, also, had to be “normal” i.e. they could only contain such sub-systems of degenerate triangles as were of the dimensions he regarded as normal. He found, by his methods, that “normal” system Σ₁ and Σ₅ are each characterized (in so far as intersection numbers are concerned) by 7 projective characters, systems Σ₂ and Σ₄ by 17 such characters, and systems Σ₃ by 22 such characters.