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Invitation to Geometry

Dover Publications
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I very much enjoyed this book and recommend it highly for the mathematical enrichment and recreation of the technical student and professional. Melzak based the book on lectures he used for non-technical students, and suggests that it can be a suitable foundation for such a course. I, however, question the utility of the book in this role for the average college student and instructor. I suggest that it would be far more suitable as a supplementary text for a course for college and university students on technical tracks, including those headed into high school teaching of math or physics.

Before proceeding further, let me confess my background and limitations. I am an applied mathematician specializing in operations research, mathematical modeling and simulation. Only in the distant pass have I taught college mathematics. Geometry is not my strong suit and I can recall only taking one college class in the subject. I asked to review this book, not as an expert but as a technical student. It has served that purpose well and whetted my appetite for more.

Invitation to Geometry consists of ten chapters, plus Notes, Bibliography and Index. Each chapter concludes with a dozen or so exercises, without answers.

Chapter 1. Heron’s Formula. With this chapter it becomes clear that this is no elementary geometry textbook. Melzak jumps straight into Heron’s Formula for the area of a triangle of sides a, b, and c, a topic surprisingly absent in many basic geometry books. Melzak eschews or postpones the more common introductory topics: logic, congruence, similarity, polygons, isometry, and Euclidean space and postulates. So it becomes clear why the title is Invitation to Geometry rather than Introduction to Geometry. In this compact book, Melzak is whetting the reader’s appetite rather than serving a full course meal.

Chapter 2. Triangle Transversals. This challenging chapter addresses geometry useful in analyzing perspective in art and stresses and strains in mechanics. Melzak proves a two-part theorem of Routh and goes on to prove theorems of Ceva, Menelaus, and Desargues. As he does often in this book, Melzak not only proves results, but comments on the method of proof being applied. For instance, in proving the general Routh result Melzak employs the method of special cases and invokes and justifies the statement, “without loss of generality it may be assumed...”

Chapter 3. Rotations and Rolling. This chapter includes an entertaining discourse on the mathematics and applications of the cycloid, the curve traced by a point on a rolling wheel. Did you know that a pendulum constrained by a double cycloid-shaped arch will have a constant period, even for large amplitudes (tautochrone property)? Or that the track that minimizes the time for a frictionless object to slide between two points in a gravitational field is a cycloid (brachistochrone property)? Melzak presents these and other results arising from rotations of linked circles, a subject, we find from the Notes, which was particularly important for ancient astronomers trying to explain planetary and lunar orbits in terms of circles only.

Chapter 4. Oblique Sections of Certain Solids. Melzak opens this chapter by giving the reader the insight that certain non-physical transformations in one dimension become physical transformations in a higher dimensioned space. For instance, a classic quadrilateral kite can be rotated in two-dimensional space, but a reflection of the “east” and ”west” corners can be physically achieved only by going to three-dimensional space. One of Melzak’s stated aims for this text is to help the reader improve spatial visualization skills. He advances this goal in “the belief that the modern superabundance of paper, book pages, screens, blackboards, and other surfaces has rendered us in effect two-dimensional in our thinking and visualizing.” For instance, he challenges the reader to envision slicing a 3D cube by a plane in a certain way, and to describe the intersection of that slice in the plane as the plane is moved. The reader may find that surprisingly difficult. But Melzak guides the reader through the process. He proceeds to demonstrate surprising properties of the cube, such as the fact that, “a bona-hole of square cross section can be punched through a solid cube so that another cube, bigger than the original one, can be pushed through the hole.” He also discusses planar cross sections of a torus.

Chapter 5. Conic Sections and Pascal’s Theorem. The chapter begins with some fascinating history of conic sections and why they were of such interest to the ancient Greeks, who did not have our tools of algebra and coordinate systems. Melzak gives examples of practical problems addressed by such tools and then launches into a discussion of Pascal’s Theorem: If a hexagon is inscribed into a conic, the pairs of its opposite sides intersect in three points that are collinear. When I understood that the hexagon was arbitrary, not necessarily regular or even convex, I was doubtful it could be true. But I was persuaded by the lovely out-of-left-field proof in which the theorem drops out of a seemingly unrelated lemma. Such proofs are joys to the mathematician and can inspire a student, as Melzak hopes.

Chapter 6. Examples of Geometric Extrema. This chapter presents eight problems in which the perimeter, area or volume are maximized or minimized. Melzak intends, he writes, “(1) to illustrate and supplement standard techniques of calculus, (2) to show the benefit of physical, especially mechanical, analogs, (3) to exhibit the occurrence and importance of boundary extrema, and (4) to introduce certain new geometrical ideas.” He succeeds. Problems include practical problems, such as the house builder’s problem of finding the largest rectangle that will fit in a give triangle, to theoretically important problems, such as minimization of the packing fraction for spheres. Discussions and proofs are appealing, whether lengthy or short. One problem was stated and solved in a third of a page. I, who have generally turned to calculus-based approaches for optimization problems in my work, have been inspired to dust off the pure geometrical approaches languishing at the bottom of my tool bag.

Chapter 7. Simple Geometry and Trigonometry on the Sphere. This chapter is anchored on some interesting Grecian mathematical history, such as the Archimedean proposition that the area of a sphere is equal to that of its wrapping cylinder. Melzak presents a lovely proof of that proposition by first proving a more general theorem, and then using some calculus-like thinking to prove the proposition. The chapter then goes into some more standard spherical trigonometry, but mentions applications, such as the three-dimensional random walk process, that are quite nice.

Chapter 8. Introduction to Graphs. In this chapter Melzak presents and explores certain classic problems concerning undirected graphs, including the Bridges of Königsberg problem, the find-the-Minotaur maze problem, and the Steiner graph problem. As with many of the other chapters we get some nice history as well as compelling discussions and proofs. As an operations researcher I particularly enjoyed the algorithmic approaches for the maze exploration problem.

Chapter 9. Elements of Convexity. This chapter, too, I enjoyed as an operations researcher since many OR algorithms require that a problem to be decomposed into convex volumes for mathematical optimization. Beyond simply defining convexity, Melzak presents the concepts of convex hulls, convex hubs, and star-shaped volume-point sets.

Chapter 10. Curves in Space and Curves on Surfaces. This chapter presents the Frenet-Serret equations for dealing with parametrically defined curves in K-space. It applies these equations to certain physics problems that employ the concept of minimum potential energy to gain a solution, e.g., the problem of stretching a rubber band along a sphere or other surface. Unlike most of the other chapters, this chapter definitely assumes that the reader understands the concepts of first and second derivatives.

Notes. The Notes section at the end of the book is well worth reading before, after and during the chapter exploration. Notes include selection rationale, general approach, and historical background.

Several decades ago I taught basic mathematics to non-technical college students. I cannot imagine that those students would have been able to handle a book as challenging as this. Yet Melzak has used the book’s foundation notes over the years for a full year course in geometry for just such students. I can only imagine that those were exceptional students and that Melzak is an exceptional instructor.

Melzak also suggests that chapter sets of Invitation to Geometry may be used for one-semester courses, as follows: basic geometrical core, 1, 4, 5, 6, 10; slant on pure geometry, 1, 2, 5, 8, 9, 10; slant on applications, 1, 3, 4, 6, 7, 8; for collateral work, e.g., to calculus, 1, 3, 4, 5, 6, 8; for more historical background ,1, 3, 5, 7, 8. I would recommend that the audience for such classes be primarily technical students and only the exceptionally prepared non-technical one. I would think that for ordinary students and instructors a more basic geometry textbook would be preferred, with Invitation to Geometry used only as a collateral resource.

The real audience for this book, I suggest, is the technical student and professional, reading for enrichment and entertainment. I highly recommend this book to technical professionals like myself who do not have strong backgrounds in geometry or have let those skills atrophy. They, like me, will find the book entertaining, engrossing and challenging. With its small physical size, I find it makes an ideal companion for long subway rides. I found this led to an amazing time dilation, with my trip seemingly cut in half, except when I was so deep into one of Melzak’s discourses proofs that I missed my subway stops.

H. Ric Blacksten is a Principal Analyst with Innovative Decisions, Inc. He has worked as a physicist, mathematician and, for the latter part of his career, an operations research analyst specializing in mathematical modeling and simulation. His email address is

Date Received: 
Tuesday, May 26, 2009
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Z. A. Melzak
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H. RIc Blacksten
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1. Heron's Formula and Related Ones
2. Triangle Transversals
3. Rotation and Rolling
4. Oblique Sections of Certain Solids
5. Conic Sections of Certain Solids
6. Examples of Geometrical Extrema
7. Simple Geometry and Trigonometry on the Sphere
8. Introduction to Graphs
9. Elements of Convexity
10. Curves in Space and Curves on Surfaces
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Tuesday, September 8, 2009