The odd-looking title of this Dover reprint reflects the fact that it is, in fact, a compilation of two books (or perhaps “booklets” would be a better term, as both are less than 60 pages long), one titled *Proof in Geometry* and the other titled *Mistakes in Geometric Proofs*. English translations of these were published separately by D.C. Heath in 1963 (my university library has both books as separate items, and identifies Fetisov as the author of the former and Dubnov as the author of the latter); the original Russian editions from which these translations were made were published about ten years earlier. I was unable to determine whether, in the original Russian, they were also published as separate texts.

Here, however, they are bundled together in one slim volume, but are kept entirely independent of one another; the table of contents for the latter book does not even appear until the former has ended, and the page-numbering begins anew with the second book. It seems appropriate, therefore, to discuss each of these two book(let)s separately.

*Proof in Geometry* starts as an introduction to proof, done mostly in the context of Euclidean geometry at roughly the level of high-school (or, at least, high school as it was taught back then; nowadays, my students tell me, a typical high school geometry course deemphasizes proof). After a discussion of what a proof is (chapter 1) and why proofs are necessary (chapter 2), there is a chapter on basic proof ideas (e.g., the difference between a statement and its converse) and common student errors in the construction of a proof (e.g., assuming what you want to prove). The final chapter of the booklet (“What Propositions May be Accepted without Proof?”) is a quick look at what axioms can be used to develop Euclidean geometry.

The particular set of axioms chosen here are essentially the ones developed by Hilbert at the turn of the 20^{th} century, though unfortunately I didn’t see Hilbert’s name mentioned anywhere. These axioms are divided into groups: incidence (although the author refers to these as “axioms of connection”), betweenness, congruence (although the statement of the axioms themselves refers to segments and angles as being “equal” rather than congruent), line separation, continuity, and of course the parallel axiom. Naturally the author does not engage in a rigorous construction of Euclidean geometry via these axioms — that would, of course, require an entire book hundreds of pages longer than this one — but he does manage, in the course of less than 20 pages, to give an informative overview of how geometry is developed from these axioms; one or two results are proved precisely to illustrate the general idea.

This chapter was, I thought, the most interesting one of the four. Nowadays, many college mathematics departments offer an “introduction to proof” course, and as a result a contemporary reader has many books on this subject from which to choose. (As best as I can recall from my youth, this was *not* the case back in the early 1960s.) Consequently, I don’t think that the first three chapters offer the reader a lot that can’t be found elsewhere, and in greater detail. But the fourth chapter, even 50-odd years after it was written, does seem to offer something that is not quite as easily found in the literature; the explanation of how the axioms are organized and what they say is quite well done. (One quibble, though: although the author states that some of the axioms can be changed slightly, he does not really indicate that there are any substantively different ways to axiomatize Euclidean geometry. By the time the Russian volume was first written in the 1950s, for example, Birkhoff’s axiom system (based on the real numbers) was fairly common currency, having been developed about twenty years earlier; this approach involved only four axioms.)

This brings us to *Mistakes in Geometric Reasoning*. This booklet is divided into four chapters, the first and third of which (with the fairly clunky titles “Mistakes in Reasoning Within the Grasp of the Beginner” and “Mistakes in Reasoning Connected with the Concept of Limit”, respectively) contain a number of false “proofs” — 15 in all — of geometric results; the second and fourth chapters then explain the errors in the proofs. (Some of the results that are claimed to be proved are obviously false; others are, in fact, true theorems in Euclidean geometry, but are given false proofs. More on this shortly.)

Anybody who has been around a while will surely recognize at least some of the false proofs presented here; the old chestnut “all triangles are isosceles”, for example, is one of them. (It occurs to me, however, that I may be doing the author an injustice: considering that the English edition of this book was written in 1963, it is quite possible that the reason some of these results *became* old chestnuts is that they were taken from this book.) However, there were several fallacious proofs here that I had not seen before, and while finding the error in any of them is not something that a professional would have to spend any time on, they certainly would make interesting fodder for class discussion or homework assignments.

There is one aspect of some of these results that, depending on your viewpoint, may be a good or a bad thing. Some of the false proofs tie in nicely with a discussion of non-Euclidean geometry, although this connection is not discussed in detail in the book. For example, many books that develop non-Euclidean geometry do so by looking first at “neutral” geometry — i.e., the body of results obtained by assuming all the axioms of Euclidean geometry except the famous (and controversial) fifth postulate. For centuries, mathematicians attempted to prove the fifth postulate as a theorem in neutral geometry (see Saccheri’s *Euclid Vindicated from Every Blemish*), but all such attempts ended in failure; we now know that they *had* to end in failure, because models of non-Euclidean geometry within the framework of Euclidean geometry can be shown to exist.

It can be shown that the statement “the sum of the angles of a triangle is 180 degrees” is, in neutral geometry, equivalent to the fifth postulate, so proving the quoted statement is tantamount to proving the fifth postulate, and is therefore impossible without making some assumption that is equivalent to the fifth postulate itself. Several of the false proofs in chapter 1 purport to prove results like this — results that actually are true in Euclidean geometry, but not true in neutral geometry. (In fact, the Euclidean parallel postulate itself is the subject of one false proof.) The error in these proofs is the assumption of a result in neutral geometry that one has no right to make.

Examples like this are nice for people who, like me, get to teach this stuff, and I certainly intend to give some of these examples the next time I teach non-Euclidean geometry. A beginning geometry student without the more sophisticated perspective described above, however, might wonder what the fuss is all about. In view of the fact that the student was brought up learning that the sum of the angles of a triangle *is* 180 degrees, he or she may wonder what’s the harm in assuming, for example, that the angle sum is a constant. Why, the student may wonder, is this an “error” at all? To the author’s credit, he does make some attempt to address this issue, stating, for example, in the statement of some proofs, that they should not be based on the fifth postulate. But I’m not sure that this brief discussion is sufficient for students who have not had the chance to grapple with some of the fairly subtle ideas involved here, and as a result some of the real significance of these false proofs may get lost in the shuffle. On the other hand, with some supplementary lecture, false proofs like this might prove very useful.

Conclusion: I don’t see this book getting much use as an actual text for a course, but instructors who teach geometry might derive some benefit from using it as a desk reference, as a source of ideas for lectures or assignments.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.