*Linear geometry*(1965) and

*Geometry. An algebraic approach*(1992). We present below extracts from some reviews of these two works.

**1. Linear geometry (1965), by Rafael Artzy.**

**1.1. Publisher's description.**

Most linear algebra texts neglect geometry in general and linear geometry in particular. This text for advanced undergraduates and graduate students stresses the relationship between algebra and linear geometry. It begins by using the complex number plane as an introduction to a variety of transformations and their groups in the Euclidean plane, explaining algebraic concepts as they arise. A brief account of Poincaré's model of the hyperbolic plane and its transformation group follow. Succeeding chapters contain a systematic treatment of affine, Euclidean, and projective spaces over fields that emphasizes transformations and their groups, along with an outline of results involving other geometries. An examination of the foundations of geometry starts from rudimentary projective incidence planes, then gradually adjoins axioms and develops various non-Desarguesian, Desarguesian, and Pappian planes, their corresponding algebraic structures, and their collineation groups. The axioms of order, continuity, and congruence make their appearance and lead to Euclidean and non-Euclidean planes. Lists of books for suggested further reading follow the third and fourth chapters, and the Appendix provides lists of notations, axioms, and transformation groups.

**1.2. Review by: Harold Scott MacDonald Coxeter.**

*Mathematical Reviews* MR0188842 **(32 #6274)**.

Chapter 1 begins with a description of Euclidean isometries and similarities as linear transformations of a complex variable *z*. The author uses the well-chosen name "dilative rotation" for the spiral similarity *z* ↦ *az*, but he refrains from mentioning by name the "dilative reflection" *z* ↦ *az*. Although he expresses the general isometry as a product of reflections, he does not stress the fact that the general homography (or "bilinear transformation") is a product of inversions. He gives a good account of isometries in the hyperbolic plane, using Poincaré's half-plane model. Chapter 2 deals largely with affine geometry, using vectors and linear transformations. The passage from affine space to Euclidean space is made by means of an inner product and a positive definite quadratic form. The classification of isometries in Euclidean space is illustrated by particularly well-drawn figures, but similarities (or "similitudes") are scarcely mentioned. In Chapter 3, projective *n*-space is derived from an (*n* + 1)-dimensional vector space. ... A point is said to be interior to a conic if it lies on no tangent. This definition prepares the way for the Cayley-Klein model of the hyperbolic plane. Elliptic 3-space is treated by means of quaternions, and there is an interesting introduction to geometries in which the coordinate field is replaced by the corpus of quaternions or by the non-associative algebra of octaves. The author mentions the theorem of Bruck and Ryser which restricts the possible numbers of points on a line in a non-Desarguesian plane. In Chapter 4, coordinates are rigorously introduced into an axiomatic projective plane. The treatment resembles that of D Pedoe's *An introduction to projective geometry* (1963), but the reader is helped by a more generous supply of figures and a very fine collection of exercises.

**1.3. Review by: Alfred Aeppli.**

*Amer. Math. Monthly* **74** (6) (1967), 755-756.

This is a fine introduction to linear geometry, suitable at the advanced undergraduate and graduate levels. It is "modern" in several ways. The vocabulary and the notations are often of 20th cenrury origin; a certain amount of algebra and algebraic methods developed in the last fifty years are presented; and the treatment has most of the time an axiomatic, hence in some sense an "abstract" and "most general" character. The first three chapters contain an introduction to projective, affine, euclidean, elliptic, hyperbolic geometries. Coordinates over given fields are used and transformation groups play a central role in the spirit of Klein's Erlangen program. The fourth and last chapter develops geometries from geometric axioms in the spirit of Hilbert's classic "Foundations of Geometry." ... Many references are given, and I hope the interested student will be eager to use them after the study of Artzy's stimulating text. The limitations of the book is obvious. The bulk of the work is done in two dimensions. Analysis exists (the reals are introduced) but does not flourish. The geometry is linear, in some places bilinear, and the conics serve principally as sets of pointswhich are infinitely far away, hence Pascal, Brianchon and Steiner have nothing to say. These restrictions, however, are well justified. The author offers a book of manageable size to be used as a text in a course on linear geometry treated in an algebraic way. Finally, I cannot help but remark on point which struck me and which might be of general interest apart from this particular book on linear geometry. The author refers in the text to "the German mathematician C F Gauss," "the British mathematician Arthur Cayley," "the French mathematician Henri Poincaré, " etc. ... Should a man's nationality be mentioned in a text book of this type? In my opinion, in most cases it is not necessary and I would like to suggest not doing it at all. The repeated use of national labels might generate micro-gaullistic feelings of all sorts.

**1.4 Review by: Mark Hunacek.**

*Mathematical Association of America Reviews* (2017).

https://www.maa.org/press/maa-reviews/linear-geometry

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. ... 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. ... 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. ... 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.

**2. Geometry. An algebraic approach (1992), by Rafael Artzy.**

**2.1. Review by: Eric John Fyfe Primrose.**

*Mathematical Reviews*MR1188640

**(93i:51001)**.

The principal approach of this book is that of "linear geometry", with special emphasis on the ideas of transformation geometry, particularly reflection geometry in the sense of F Bachmann [*Aufbau der Geometrie aus dem Spiegelungsbegriff* (1959)]. Chapter 1 deals with affine spaces and their groups of affine transformations, and then Euclidean spaces and their motion groups. Chapter 2 gives a detailed account of motions in the Euclidean plane, and the generation of its motion group by reflections. Chapter 3 is concerned with projective spaces, particularly the real projective plane. Finally, Chapter 4 lays the axiomatic foundations for various metric planes.