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By Donald S. Passman

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1637, La Géométrie [one of the appendixes to Discours de la méthode], Leiden: Maire. [Repr. ] Euler, L. 1748, Introductio in analysin infinitorum, 2 vols, Lausanne: Bousquet. [Also Opera omnia, Series 1, Vols 8 and 9. L. 1921, A History of Greek Mathematics, 2 vols, Oxford: Clarendon Press. L. 1961, Science Awakening (transl. Dresden), New York: Oxford University Press. 3 Regular polyhedra BRANKO GRÜNBAUM The idea of singling out certain polyhedra with unusual symmetry properties first occurred during the heyday of Greek geometry.

Figure 3 The conchoid of Nicomedes Figure 4 The cissoid of Diocles Page 863 2 CURVES IN CARTESIAN GEOMETRY Such elaborate constructions make it plain that very few curves can be introduced in this way. Much less did they elicit general concepts; the idea of a tangent to a curve does not seem to have been defined in general, although tangents to the circle, the conic sections, and the spiral were discussed. What is lacking is the concept of one quantity varying with another, so that the two between them define a moving point tracing out a curve.

The angle DAR1 is one-third of the angle DAR0. For this reason the curve is sometimes called the ‘trisectrix’. Figure 2 The trisection of angle DAR0. The trisectrix has another use: it helps to solve the problem of squaring the circle. It meets the side AB at the point S in Figure 2, for which Page 862 AS=2a/π. A circle of radius r has an area of πr2; for a circle of unit radius this reduces to π. With a=1/2, the trisectrix gives a length of 1/π, which can easily be made to yield a length of π, thus squaring the circle.

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