By Gabor Toth

Prior variation bought 2000 copies in three years; Explores the delicate connections among quantity concept, Classical Geometry and sleek Algebra; Over one hundred eighty illustrations, in addition to textual content and Maple documents, can be found through the net facilitate knowing: http://mathsgi01.rutgers.edu/cgi-bin/wrap/gtoth/; includes an insert with 4-color illustrations; contains a number of examples and worked-out difficulties

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A little more about this later. We now return to cubics in Weierstrass form. A good example on which to elaborate is given by the equation y2 x3 − 3x + 2. 8. Here, (1, 0) is a double point. ) Color Plate 1b shows the intersection of the graph of the polynomial f(x, y) x3 − 3x + 2 − y2 in R3 with equidistant (horizontal) planes. 9). This indeed shows the three basic types discussed earlier! ♦ Let us try our machinery for the Weierstrass form. Since f(x, y) P(x) − y2 , we have ∂f ∂x P (x) and ∂f ∂y −2y.

Although π is irrational, it is not only tempting but very important to ﬁnd good approximations of π by rational numbers7 such as the Babylonian 25/8 (a clay tablet found near Susa) or the Egyptian Irrationality of π was ﬁrst proved by Lambert in 1766. For an interesting account on π, see P. Beckmann, A History of π, The Golem Press, 1971. For a recent comprehensive treatment of π, see L. Berggren, J. Borwein, and P. Borwein, π: A Source Book, Springer, 1997. 6 7 2. “. . ). The Babylonians obtained the ﬁrst value by stating that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle was “equal” to 57/60 + 36/602 .

Since r is a root of P with multiplicity two, P(x) a(x − r)2 (x − s), r s, where a, r, and hence s, are rational. Thus, x s + m2 /a and y m(s − r + m2 /a), and these are rational iff m is. As before, varying m on Q , we obtain all rational points on our cubic curve. In the elusive case where P has three distinct roots, the method of rational slopes does not work. A cubic rational curve in Weierstrass form y2 P(x), where the cubic polynomial P has no double or triple roots, is called elliptic. (♦ The name “elliptic” comes from the fact that when trying to determine the circumference of an ellipse one encounters elliptic integrals of the form R(x, y)dx, 3.