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College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle

Nathan Altshiller-Court
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Elena Anne Marchisotto
, on

In College Geometry, Nathan Atshiller-Court focuses his study of the Euclidean geometry of the triangle and the circle using synthetic methods, making room for notions from projective geometry like harmonic division and poles and polars. The book has ten chapters: 1) Geometric Constructions, using a method of analysis (assuming the problem is solved, drawing a figure approximately satisfying the conditions of the problem, analyzing the parts of the figure until you discover a relation that may be used for the construction of the required figure), construction of the figure and proof it is the required one; and discussion of the problem as to the conditions of its possibility, number of solutions, etc; 2) Similitude and Homothecy; 3) Properties of the Triangle; 4) The Quadrilateral; 5) The Simson Line; 6) Transversals; 7) Harmonic Division; 8) Circles; 9) Inversions; 10) Recent Geometry of the Triangle (e.g., Lemoine geometry; Apollonian, Brocard and Tucker Circles, etc.).

There are as many as nine subsections within each chapter, and nearly all sections have their own exercises, culminating in review exercises and the more challenging supplementary exercises at the chapters’ ends. Historical and bibliographical notes that contain references to original articles and sources for the materials are provided. These notes (absent from the first 1924 edition) are valuable resources for researchers.

From the description of the content of the text, certain questions may arise: What precisely is the “geometry of the triangle and circle” and does its study merit a 2007 re-publication of a 1952 text? What are “synthetic methods”? For an answer to the two-part first question, this reviewer can do no better than to refer the reader to [Davis 1995], for several historical definitions of triangle geometry, and detailed discussion of its emergence in the 1870s as a “distinguished subfield of mathematics” through the writings of Emile Lemoine (1840–1912) to its subsequent rise and fall from grace. Davis identifies several remarkable properties associated with triangles that convey the spirit and the content of College Geometry. See, for example, the discussion in [Davis 1995, p. 205] of the nine point circle theorem (In a triangle the midpoints of the sides, the feet of the altitudes and the Euler points lie on the same circle), which is Theorem 207 in [Altshiller-Court 2007 , p 103 ]. The geometry of the circle in College Geometry is reminiscent of the classic treatment in [Coolidge 1916].

But the results of triangle and circle geometry are not only interesting in the classical sense. They continue to generate interest among mathematicians, who often seek different proofs for them. For example Ptolemy’s theorem (In a cyclic quadrilateral the product of the diagonals is equal to the sum of the products of the opposite sides) which appears on page 128 of College Geometry , recently enjoyed a new proof by vectors in [Just and Schaumber, 2004]. Branko Grunbaum observed that even mathematicians themselves may be “…unaware of the fact that the elementary, intuitive approach to geometry continues (and will continue) to generate mathematically profound and interesting problems and results” [1981, p. 234]. The synthetic approach in College Geometry is an elementary and intuitive one — which brings us to the second question posed above.

Atshiller-Court gives the following description of “the synthetic method of solution of construction problems” in College Geometry:

Some construction problems are direct applications of known propositions and their solutions are almost immediately apparent. Example: Construct an equilateral triangle. If the solution of a problem is more involved, but the solution is known, it may be presented by starting with an operation, which we know how to perform, followed by a series of operations of this kind, until the goal is reached. This procedure is called the synthetic method of solution of problems. It is used to present the solutions of problems in textbooks. However, this method cannot be followed when one is confronted with a problem the solution of which is not apparent, for it offers no clue as to what the first step shall be, and the possible first steps are too numerous to be tried at random [p. 3]

He then proceeds to describe an “analytic method” where one starts with the construction of a figure approximately satisfying the conditions of the problem, and then proceeds to analyze it with the intention of discovering a relationship that may be used for the construction of the required figures. All of Atshiller-Court’s construction problems follow this “analytic” method. Virtually every chapter has exercises that include construction problems. For this reason, this book is an excellent resource for student exploration using Geometer’s Sketchpad .

However this reviewer’s characterization of College Geometry as a “synthetic” approach conforms to the Courant and Robbins definition [1996, p. 165], referring to the way that theorems are proved, deduced from given assumptions, independent of algebra and the concept of the number continuum. Synthetic geometry, in this sense, is not the visual geometry as described by Davis: “a visual theorem is the graphical or visual output from a computer program — usually one of a family of such outputs — which the eye organizes into a coherent, identifiable whole and which is able to inspire mathematical questions of a traditional nature or which contributes in some way to our understanding or enrichment of some mathematical or real world situation” [1993, p. 333]. But the treatment in Altshiller-Court’s book does appropriate this visual idea, with its focus on constructions, ample use of diagrams, and questions designed to promote a deeper spatial understanding of the geometry of the plane.

The synthetic approach of College Geometry is nicely captured in the following description by Davis: “Playing around synthetically, in coordinate-free fashion, the way that Euclid is written up; but also playing around with algebra, trigonometry, and rectangular, oblique, homogeneous, barycentric, trilinear, complex, conjugate, projective coordinates….” [1995, p. 206]. In Altshiller-Court’s time, this approach to geometry was called “modern”.

When did the “synthetic” geometry of Euclid become “modern” geometry of the twentieth century? A broad (but not comprehensive) sweep of history reveals that synthetic geometry was studied by the Greeks (although they did not have that name for it). For centuries it reigned supreme, until the seventeenth century saw the birth of the analytic geometry of René Descartes (1596–1650). It wasn’t until the latter part of the next century that synthetic methods experienced a resurgence, one which continued well into the nineteenth century: in the projective geometry, inspired by the work of Gaspard Monge (1746–1818) and his disciples, and studied by geometers like Georg Karl Christian von Staudt (1798–1867), Mario Pieri (1860–1913), Oswald Veblen (1880–1960) and John Wesley Young (1865–1948); in the hyperbolic geometry of Carl Friedrich Gauss (1777–1855), Nikolai Ivanovich Lobachevsky (1792–1856), Eugenio Beltrami (1835–1900), and others; in the transformational geometry of Felix Klein (1849–1925); and in the “modern” geometry of the book under review.

It was natural for modern geometry to find its way to the classroom. It emerged from the synthetic geometry of Euclid , which historically dominated the teaching of geometry. In 1892, R. J. Aley of Indiana University wrote:

For more than two thousand years Euclid has held almost undisputed sway in the field of synthetic geometry. So strong a hold has it on school men that few American colleges dare offer anything else to freshmen. Is this because of tradition, or is there something in Euclid that makes it intrinsically better than anything mathematics has produced in modern times? [p. 297].

According to the back cover of the 2007 reprint, Altshiller-Court’s first edition of College Geometry served as a standard university-level text for a quarter of a century, until it was revised and enlarged by the author in 1952. Its competitors included analogous texts by R.A. Johnson [1929] and Levi [1939]. The second edition of College Geometry would see its use in the classroom decline, not merely because newer texts replaced it, but because introductory geometry courses lost their place in college curriculums. In 1988, Sherk lamented the plight of geometry in the schools:

It would be an interesting study to follow the fortunes of geometry as a discipline within the broader framework of all mathematics. There was a time when geometry was dominant: every person with pretensions to being educated had to know geometry, and Euclid was king… But as the present century proceeded, interest in geometry fell off; it began no longer to be classified in the "mainstream" of mathematics. Today, in North America at least, an undergraduate student majoring in mathematics will be lucky to have even one geometry course included in his or her program. High school has provided the student with the rudiments of Euclidean geometry — synthetic and analytic — and there may be some offerings of differential geometry in college. But it is rare to find other college undergraduate geometry courses, and still more rare to find substantial enrollments in those that do exist [p.893].

The situation seems no different today. Can synthetic geometry resume a prominent place in twenty-first century college classrooms? This reviewer would hope so. On the subject of the role proofs in teaching, Reuben Hersh noted: “What a proof should do is to provide insight into why the theorem is true” [1993, p. 396] The synthetic approach is an intuitive one that promotes such insights. See, for example, Altshiller-Court’s comparisons [1964, p. 338] of an analytic solution (from a Putnam Competition) and synthetic ones to the following problem: If M, N are points on the sides AC, AB of a triangle ABC and the lines BM, CN intersect on the altitude AD, show that AD is the bisector of the angle MDN [Altshiller-Court 2007, pp. 32, Exercise 58]. See also [Schuster 1963, pp. 81–83]

Recent books (relative to Altshiller-Court’s) that include synthetic treatments of geometry to varying degrees have been written by Coxeter, Coxeter and Greitzer, Eves, Greenberg, Hartshorne, Meserve, Moise, Noronha, Prenowitz and Jordan, Silvester, Smith, and Usiskin et al. Altshiller-Court’s book would serve as a supplement not only for these, but especially for those who treat the subject from purely algebraic/analytically points of view. The Dover edition of College Geometry costs less than $17.00 and Amazon is selling it used for about $10.00. For example, Fenn’s [2001] geometry text includes a section (3.9) on triangles and special cases of concurrence, but fails to include Ceva’s theorem (discussed in the Altshiller-Court book [2007, Chapter 6, pp. 158 ff.]) of which Fenn’s concurrence results are special cases. Instructors choosing Pedoe’s [1970] geometry text will find an extensive treatment of Apollonius’ problem almost exclusively from an algebraic point of view. Altshiller-Court’s book would provide the beautiful synthetic arguments for this problem, as well as treatment of theorems such as that of Desargues, which is missing from the Pedoe text.

College Geometry presents an instructor with a wide selection of classic topics in geometry from which to choose — many more than could be incorporated in a one semester course. The notes set a historical context for the material, providing interesting information on the origins of ideas and propositions. We learn, for example, that the term “transversal” was introduced by L. N. M. Carnot (1753–1823). As indicated earlier, the treatment invites the modern technologies like the Geometer’s Sketchpad . The quality and extensive number of the exercises makes College Geometry a good resource for those interested in plane geometry (although no solutions are given). Some of these exercises continue to enjoy discussion in the literature. For example, a solution to following problem was solved in a 2001 edition of The American Mathematical Monthly [van Lamoen and Meyer, 569]: Show that the Euler lines of the three triangles cut off from a given triangle by the sides of its orthic triangle have a point in common, on the nine point circle of the given triangle [Altshiller-Court 2007, Exercise 23, p. 120]. The Davis article [1995] alludes to many more.

There are no modern “real-life” or computer applications in College Geometry, but Davis discusses a modern focus for its content:

Hundreds of elementary and not so elementary theorems that were in the literature have now been proved by computer. Many new theorems have been discovered, again in a variety of ways. Triangle geometry always was a practice ground for strategies of proof in the spirit of Euclid, and it has now become a testing ground for strategies of decidability, proof, and theorem discovery. These strategies have run from naive schemes to the employment of deep and abstract results of modern algebra and differential algebra. But there is yet more that emerges from the change of focus: I believe that the experience gained in this change can become a prime source of raw material for philosophical discussions on the nature of proof, methodologies of research, the role and nature of intuition, educational values, etc. [1995, p. 211].

Indeed, the book contains a plethora of geometric problems and classic synthetic proofs that unfortunately often do not find their way into college classrooms. Prospective secondary school teachers in particular (as well as practicing high school teachers) would benefit from its content and its methodology. Its treatment of the geometry of circles and triangles, including historical problems and geometric constructions deepens its readers’ understanding of plane geometry, without requiring prior knowledge other than high school algebra and geometry. Its numerous exercises (at different levels) provide opportunities for students to hone both geometric and algebraic skills. A detailed Table of Content and Indexes of names and topics (which cover both the text and exercises) make it easy to navigate the text. The book would serve as an excellent source of synthetic methods of proof in a first course in proofs for mathematics majors. In lieu of freshman courses in college algebra often for non-majors, why not courses in college geometry? And if college geometry, why not the synthetic approach as intuitive study of properties of geometric figures constructed from points, lines, planes, circles, etc. by means of direct consideration of them?

The use of Altshiller-Court’s book in college classrooms, to supplement geometry texts that adopt algebraic/analytic approaches, and even those that include synthetic treatments, is a good idea. That it should be part of any college library goes without saying, both for its content, for its longevity as a college textbook, and for its continual value as a resource for mathematicians and mathematics educators. (e.g., see [Beauregard and Zelator, 2002, p. 139, Davis 1995, p. 207 and its extensive bibliography])

Biographical information Nathan Altshiller-Court is not easy to find. He was born on January 22, 1881 in Warsaw Poland . He studied at the University of Ghent in Belgium . He taught at Columbia University , and the University of Washington , before becoming assistant professor in 1917 at the University of Oklahoma , then Associate Professor in 1923, and Professor in 1935. With the exception of the year 1924–25 when he was at the Sorbonne in Paris , Atshiller-Court taught at the University of Oklahoma continuously from 1916 until his retirement in 1951. He died July 20, 1968. He wrote 3 books and more than 100 papers. The nicest source for information on Altshiller-Court is probably the text of the N. A. Court Lecture at an MAA sectional meeting on April 4, 1975, delivered by John C. Brixey of University of Oklahoma , at Central State University, Oklahoma, entitled “Memories of Professor Court and the Early Days of the Oklahoma Section”. Brixey’s talk shows that Altshiller-Court was a much revered mathematician and mathematics educator. See


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Elena A. Marchisotto is Professor of Mathematics at California State University, Northridge. She is the author, with James T. Smith, of The Legacy of Mario Pieri in Geometry and Arithmetic.

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