When I first studied linear algebra as an undergraduate, I learned, as do most if not all similarly situated students, that many of the ideas of the subject (linear independence, span, inner products, etc.) have strong geometric content and can be motivated by reference to that geometry. What I did not then realize, and would not learn for another year or so, is that the process can be reversed and that geometric ideas can be studied by reference to linear algebra.

So, for example, if we define “point” as an element of a two-dimensional vector space, and “line” as any coset of a one-dimensional subspace, we get one version (a little simplified, as we’ll soon see) of plane affine geometry, and even with this simple machinery we can prove a lot of interesting geometric theorems. It is trivial to prove, for example, that two points lie on a unique line, but we can also prove less obvious things like the concurrency of the medians of a triangle, or the theorems of Ceva and Menelaus. We can then generalize things by letting our underlying vector space be *n*-dimensional instead of two-dimensional, or by adding an inner product to our underlying vector space, in which case we get theorems involving distance and angles rather than just incidence and parallelism. It is also possible to define projective planes and higher dimensional spaces in terms of vector spaces. Anybody wanting a brief but very elegant exposition of these ideas can likely do no better than to turn to chapter 3 of Kaplansky’s *Linear Algebra and Geometry: A Second Course*; the review of that book lists several other related references.

It is this approach to geometry that Rafael Artzy, the author of the book now under review, refers to as “linear geometry”. That’s not a term that seems to be commonly used these days, but Artzy is not alone in its use; another book, by Gruenberg and Weir, also has this title and is apparently still available from Springer-Verlag as an entry in their *Graduate Texts in Mathematics* series.

Artzy’s book is pitched at a somewhat less demanding level than is the book by Gruenberg and Weir and is, with the likely exception of the last chapter, accessible to good undergraduates, particularly if they have had a prior course in linear algebra.

The book has four chapters. The first isn’t really on linear geometry, but is a prefatory chapter to help motivate the ideas that follow. In it, Artzy introduces the complex number system (from scratch, but at a rapid pace and without proofs), and then, identifying the points in the Euclidean plane with complex numbers, uses the structure of that number system to discuss various transformations of the Euclidean plane. Möbius transformations (here called bilinear transformations) are also discussed, and, in a final section, the Poincaré half-plane model of hyperbolic geometry is defined and transformations on it are examined.

Linear geometry *per se* begins in chapter 2. The author begins with a fairly extensive review of vector spaces and linear transformations, including dual spaces, and then proceeds to a definition of affine *n*-space that is a bit more sophisticated and abstract than the one formulated above; he works with an arbitrary set (I’ll call it \(A\); Artzy calls it \(A^n\) to emphasize the dimension) on which an \(n\)-dimensional vector space \(V\) operates, subject to certain conditions. Very informally, any two elements \(P\) and \(Q\) of \(A\) define a vector \(\vec{PQ}\) which acts on \(P\) to produce \(Q\). (Think of this as an arrow from \(P\) to \(Q\).) This is a more standard definition (see, for example, *An Algebraic Approach to Geometry* by Borceux and Tarrida’s *Affine Maps, Euclidean Motions and Quadrics*) than the simpler one sketched above (which can be viewed as a special case of this), and avoids giving the zero element of a vector space undue prominence. It also allows us to say that a subspace of an affine space is an affine space, which isn’t the case under the more simplistic definition given above.

With this definition in hand, the author then discusses affine transformations and, towards the end of the chapter, introduces an inner product to get Euclidean geometry. Again, the transformation approach is emphasized, with isometries in the plane and three-space being classified. (In the case of the Euclidean plane, these results are all anticipated by the discussion in chapter 1.) The final section of this chapter looks briefly at affine planes defined over finite fields.

This is an enjoyable chapter, but one thing that seems like a missed opportunity is the failure to discuss some of the interesting theorems of affine geometry (such as the ones mentioned in the second paragraph above); the primary focus here seems to be on transformations.

Projective geometry is the subject of chapter 3. To define a projective plane over a field \(F\), simply take the plane to be a three-dimensional vector space \(V\) over \(F\); “points” and “lines” are one-dimensional and two-dimensional subspaces of \(V\), respectively. A trivial exercise in linear algebra then shows that any two distinct lines intersect in exactly one point; this of course is the hallmark of projective plane geometry. If we fix a basis for \(V\), then any point can be identified by a coordinate vector with three scalar components, not all \(0\), and where multiplying by a scalar does not change the point. These are, of course, the homogenous coordinates of a point. All of this generalizes naturally to projective \(n\)-space, where we start with an \((n+1)\)-dimensional vector space. Artzy begins the chapter directly with projective \(n\)-space and then segues immediately into a discussion of projective transformations. We also see in this chapter, among other things, the cross-ratio and conics defined and discussed, as well as the statements and proofs of the theorems of Desargues and Pappus.

The fourth chapter of the book is the most difficult and probably one that is not easily accessible to most undergraduates. In this chapter, the author begins by departing from the linear-algebraic approach to geometry and instead looks at the axiomatic foundations of affine and projective (plane) geometry. There are relatively simple axioms for both affine and projective planes, and along with proving some standard theorems, one can ask the following interesting question: which (axiomatically defined) projective planes are the “field planes” discussed above? The answer turns out to be nontrivial but very beautiful: an axiomatic projective plane is coordinatized by homogenous coordinates over a field if and only if Pappus’s Theorem is true in that plane; it is coordinatized by homogenous coordinates over a division ring if and only if Desargues’s theorem is true in it. (So, as a consequence: if Pappus’s theorem is true in a projective plane, then so is Desargues’s theorem, and if Desargues’s theorem is true in a *finite* projective plane, then — by Weddeburn’s theorem on finite division rings — so is Pappus’ theorem. The former fact can be proved without algebra, but I have never seen a purely geometric proof of the latter.)

It also turns out that even if Desargues’s theorem is not true in an axiomatic projective plane, the plane can still be coordinatized, but the scalars now come from something called a “ternary ring”; these are not really “rings” in the usual sense, but are sets on which a ternary operation is defined satisfying certain conditions.

The fourth chapter of the book is devoted to a discussion of these ideas. It is a difficult and time-consuming enterprise, but the author does it skillfully. He starts with an axiomatic projective plane (he calls it a “rudimentary projective plane”) and develops ternary rings to deal with them. He then gradually adds axioms, including the results of Desargues and Pappus, and shows how the algebraic coordinatizing structure changes with these new axioms. After establishing the relationship between fields and Pappus’s theorem, additional axioms are added to produce the real field. The collineation groups of these structure are also discussed.

By and large, the book is quite successful. The material is interesting and is well-presented. The writing style is clear, attention is paid to motivation, and there are a good selection of exercises. I do, however, have a few nits to pick.

First and foremost, the author writes functions on the right instead of the left, a convention that I have never liked and which is (thankfully) not seen very much in modern textbooks.

Second, there are a couple of statements in the book that are inaccurate, though one is simply due to the passage of time. That one is the statement that the question of the existence of projective planes of order 10 is unsettled; this was definitely correct when it was written, but in 1989 Clement Lam (using a computer) answered this question in the negative. Another incorrect statement, one that cannot so easily be excused, is the comment on page 125 that “there exist elements in every Galois field that do not have a square root.” This is simply false: in finite fields of characteristic 2 (which are explicitly considered in the text), every element has a square root. (Quick proof: in such fields, the mapping \(x\mapsto x^2\) is one-to-one, and hence onto.) For finite fields of characteristic \(>2\), however, the statement is true, by pretty much the same proof: the mapping \(x\mapsto x^2\) is now not one-to-one, hence not onto.

I also was not a big fan of the first chapter of the book. The material here is covered nicely but, as previously noted, doesn’t really seem to me to fit in with the rest of the text. The chapter seemed almost forced, a way of filling out the book. The analysis of Euclidean isometries in the second chapter seemed duplicative.

The introduction of the hyperbolic plane in chapter 1 struck me as somewhat abrupt; there is a lot of wonderful history associated with the realization that geometries exist that do not satisfy the Euclidean parallel postulate, and the students I have had in my upper-level geometry courses have generally found this history fascinating. None of that is gone into here. My quibble here may be a matter of individual taste; I have felt, for many years now, that giving students a sense of history is extremely valuable in an upper-level mathematics course. Not everyone shares this view, of course, but those who do may find the introduction of hyperbolic geometry here rather sudden and without the kind of fanfare that I think it deserves. Contrast, for example, the sense of historical adventure and excitement engendered by the book *Euclidean and Non-Euclidean Geometries: Development and History* by Greenberg, which, in the author’s words, “presents the discovery of non-Euclidean geometry and the subsequent reformulation of Euclidean geometry as a suspense story”.

These issues aside, this is a valuable book. I don’t think that the material matches very many courses that are currently taught in American universities, but faculty members who are interested in algebra and geometry should definitely have this book on their shelves as a useful reference.

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