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Dan Pedoe, on Geometrical Matters, P 67-74Mathschron009-008.Pdf ON GEOMETRICAL MATTERS Dan Pedoe Dedicated to H.G. Forder on his 90th birthday (received 29 June, 1979) Geometers are sadly aware of the present depressed state of their art. The "new" math, swept elementary geometry aside, in a frantic effort to keep up with the Russians, who had convinced the Western world of their technological superiority by launching Sputnik. The "educators" who led American mathematics disregarded, or were sublimely unconscious of the fact that Russian high schools, then and now, study geometry intensely. Yaglom's new book [16]reveals the extent to which the best high schools in Russia plumb non-Euclidean geometry. In America, publishers demand an assured sale of at least 50,000 copies for any book on elementary mathematics before they will handle it, and only calculus books and such make the grade. Any book on geometry, inadvertently published, only sells to a limited extent, and if the publisher has "inventory problems", (that is, if he suffers from a lack of warehouse space, and who does not?), books are soon pulped. Fortunately there are still University Presses, and the excellent Chelsea and Dover publishers who keep books in print, and Forder's remarkable Calculus of Extension [4] is still obtainable, although some of his early books on geometry have vanished. One of my pulped books, Circles, has been reprinted, with added problems and solutions, by Dover Books [10]. Those who defend the present state of mathematics are quick to point out that a lot of geometry is still being studied and taught, under chapter headings such as "combinatorics", "tesselations", and so on. But too much has been forgotten, and it is the purpose of this note, besides retailing geometrical gossip, to indicate how much has been forgotten. Math. Chronicle 9(1980) 67-73. 67 In a recent paper by Leon Henkin and William A. Leonard the following problem is investigated: Given a parabolic arc, can the vertex be found using only a compass and straight edge? According to the authors, the problem "seems to have passed around in meetings of the California Mathematics Council". But doubts remain with regard to the Henkin - Leonard solution, and the title of the paper expresses the doubts, being A Euclidean Construction? [5]. The doubts arise from the "foundational" difficulties which present themselves when one is faced with an infinite set of points which do not lie on a circle. These are put aside temporarily, but even then the authors are not satisfied that their solution is "truly geometric", and another solution, suggested by an anonymous referee, is printed as being more "truly geometric". Then another doubt is voiced, as to whether solutions of this geometric nature can be found for the ellipse and hyperbola, "where we only have solutions based on analytical formulas". It really does seem that the California Mathematics Council learned their geometry of the conic sections from the few turgid pages devoted to conics in calculus books, and are unaware of the lovely subject called "geometrical conics" which has been in existence for nearly 2,000 years, and can still be studied in books such as F.S. Macaulay's Geometrical Conics [6]. The idea that we only know the properties of the ellipse and hyperbola from "analytical formulas" is indeed a strange one. Shades of Apollonius and Pappus! There is a theorem of more recent vintage, the Pascal Theorem, which has also escaped the notice of the California Mathematics Council, and with the help of this theorem, which should surely be known to all professional mathematicians who venture into geometric exposition, the parabola problem can be solved if we are given a mere 5 points on the curve, and there are no foundational difficulties. 68 The Pascal Theorem says that the opposite sides of a hexagon inscribed in a conic intersect in three collinear points. That is, if the points A, B, C, D, E, F lie on a conic, then the three points L = AB D DE , M = BC fl EF and N = CD fl FA are collinear. Hence, using a straight edge only, we can find the further point of intersection, with the unique conic which passes through five given points, of any line through one of the points. For example, given the points A, B, C, D and E , the point F on AF can be found by the Pascal Theorem. So, let us assume that we are given 5 points which lie on a parabola. We use the known geometrical theorem that the midpoints of parallel chords of a parabola lie on a line parallel to the axis of the parabola. The axis itself is an axis of symmetry for the curve, so that once we have its direction, all that we need do to find the axis itself is to determine where a line through a point on the curve which is perpendicular to the axis direction meets the curve again, and the axis itself is found by bisecting a segment at right angles. Some rather more modern geometry [11, p.346] displays the parabola as a conic which touches the line at infinity at a point ft , the axis being VS , where V is the vertex. Applying the Pascal Theorem to the inscribed hexagon VABCDSi , where A, B, C and D are any 4 given points on the curve, V is rapidly found. We assume, of course, that if P is any finite point, then P2 is parallel to the axis. Details of the construction can be found in a forthcoming issue of Crux Mathematioorum [1], which is a Canadian magazine under the editorship of L6o Sauv6. The editor, and a devoted international circle of readers, are enthusiasts for geometry. Given 5 points on an ellipse or hyperbola, the axes, vertices and foci can be found by Euclidean constructions. An obvious approach to geometers of the old school, such as myself, is via the theory of involution pencils [11, p.332]. There are certainly constructions given, without explanation, in books on engineering drawing. Years 69 ago, when I taught at the University of Birmingham, one of my geometry students also took a course in engineering drawing. She asked me about a process she had been taught, without any explanation, which turned out to be plane perspective, a method for finding the map of a given plane figure which is produced by three-dimensional uniocular perspective when the object plane and the image plane are folded together (rabatted) about the axis of intersection of the two planes. I described this in a book [12], pulped long ago, and I also give a description in a recent book, Geometry and the Liberal Arts [13]. To see that sections of a circular cone can indeed be ellipses, parabolas or hyperbolas, and that they can be produced by simple drawing with a ruler, compass and setsquare is, I think, an exciting educational experience, especially for American students, who often pass through their one year of geometry at high school without ever handling geometrical instruments. Geometrical drawing is one of the simple pleasures of life, and as a pastime it can lead to new theorems. A proof that this is so can be seen in The Seven Circles Theorem and other new theorems 3],.if this is still obtainable. The two first-named authors discovered new and fascinating theorems in circle geometry by accurate drawing, and neither is a professional mathematician. I hear that the first printing of their book sold out rapidly. As a final piece of geometrical gossip, attention should be directed to two elaborate bibliographies, which have recently appeared, which list the publications to date on the remarkable Morley Theorem. This theorem states that if we draw the trisectors of the angles of a triangle, then the intersections of adjacent trisectors are the vertices of an equilateral triangle. The first bibliography, of nearly 200 items, appeared in the Canadian magazine Eureka [2], the original name of Crux Mathematicorum [1]. This was followed by a bibliography in the American Mathematical Monthly [9]. In Inversive Geometry [8] by Frank Morley and his son F.V. Morley, the theorem is proved by considering the locus of the 70 centre of a cardioid which touches the sides of a given triangle. A similar proof was given 9 years earlier in a paper which Frank Morley wrote for a Japanese Mathematical Journal designed for secondary school students [7]. Since the proof, although delightful and full of geometrical insights, is as difficult to understand as most of Inversive Geometry, I transliterated it (no other term is applicable!) for the special Morley issue of Eureka [2], When the issue of the American Mathematical Monthly with the Morley bibliography and article [9] appeared, it contained an interview with Morley's son F.V. Morley, in which he refers to the proof of the Morley Theorem given by his father and himself in Inversive Geometry [8]. This proof appears on p.244, with a diagram to illustrate the theorem. F.V. Morley says [p.741 of 9]: "it was the cardioids which led him to, and provided for him the most elegant proof of, the trisector theorem. Proof and theorem were pleasing in their togetherness." But on p.738 of the same Morley article the following astonishing statement appears: the reference is to Frank Morley. "And so he never enunciated, in print, just the simple theorem, nor did he ever publish a direct verification of it." I have written to the senior author of the Monthly paper to point out that the proof in Inversive Geometry, the proof in the Japanese paper [7] and the statement made by Morley's son make this ex cathedra assertion that Frank Morley never enunciated his theorem in print ridiculous (I was rather more polite, in fact!), but I received a dusty answer, and have now been informed by another Morley bibliographer that the dispute "has reached an impasse"! 71 The history of mathematics is, of course, of great concern to all mathematicians, and nobody should rest comfortably when truth is subverted.
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