Review of A Course in the Geometry of n Dimensions

The title of the book, A Course in the Geometry of n Dimensions, is a misnomer on two accounts. First, the book is too small -- 63 pages in all -- for even a 1-semester course. Second, the book is not about geometry per se. A longer title, say, Basic Elements of the Geometry of n-Space and Their Application to the Advanced Theoretical Statistics: An Intuitive Approach, would better relate the contents of the book to a potential reader. But that, of course, would be too long a title for such a small book.

The book consists of two parts that split it almost exactly in a 2-to-1 ratio. The larger of the two is the first part that serves as an introduction to the geometric elements of n-space: varieties, flats, lines, planes, coordinate transformations, simplex, angles and solid angles, tangents, quadratic forms, content (n-dimensional volume). Whenever possible, the author relies on analogy with the low (1-3) dimensional spaces, but, as often, he is forced to directly employ much heavier machinery of matrix algebra and n-dimensional calculus.

The book is unmistakably lecture notes in printed form. In a preface to the first edition (1908) of the recently republished Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis (see a brief review by Fernando Gouvêa), Felix Klein pinpoints the difficulty in publishing lecture notes:

  For it is a far cry from the spoken word of the teacher, influenced as it is by accidental conditions, to the subsequently polished and readable record. In precision of statement and in uniformity of explanations, the lecturer stops short of what we are accustomed to consider necessary for a printed publication.

The lack of uniformity is very noticeable in the book under review. For example, an n-dimensional parallelotope is defined (p. 9) by analogy to the parallelogram as being bounded by pairs of parallel (n-1)-flats. Indeed, the notion of flats has been introduced earlier (p. 6). However, no condition of them being parallel could be found in the book. An orthotope is defined (p. 10) by analogy to the rectangle as a parallelotope, in which bounding (n-1)-flats are perpendicular, and a hypercube "as an orthotope in which the parallel bounding (n-1)-flats are all the same distance apart." However, a condition for the orthogonality of the flats may only be surmised as implicit in the definition of the angle between flats on p. 21, while the notion of the distance between the flats appears to be assumed to be known to the reader. The author uses freely n-dimensional integration, change of variables, and Jacobians, although finds it necessary to define orthogonal transformations (LL' = I, in matrix form.) One the other hand, length preservation under orthogonal transformations is claimed but never deduced.

Other similar examples can be found throughout the book, which makes it obviously unsuitable for a casual or an undergraduate reader. In fact, the book has been published with a different audience in mind.

In author's experience, most students of "statistics at the advanced level, even those with a good mathematical background, encounter serious difficulty with proofs depending on n-dimensional systems." In the second part of the book, the author applies shortcuts based on geometric insights gained previously to demonstrate several statistical results. For example, the joint distribution of independent samples from a population obeying the normal distribution has hypershperes as the sets of constant density. From here it follows that the distribution of the Euclidean length is governed by the same formula as the normal distribution times a dimensionality factor. Further on, correlation coefficients are interpreted as normalized scalar products (although this particular term is not used in the book), and partial correlations are realized as the scalar products of projections on certain flats. With a suitable orthogonal transformation the author then concludes that correlations and partial correlations are distributed similarly albeit in flats of different dimensions.

Other topics in the second part of the book include Student's t-distribution, Wishart's distribution, bivariate normal distribution, regression and multiple correlation, canonical correlations, and component analysis.

The book was published for the benefit of teachers of advanced statistics. It summarizes in a very concise form an attempt by a very well known colleague of theirs to bridge the gap between the formal requirements of such courses and a frequently inadequate level of mathematical sophistication of the attending students. Others may well find his endeavor helpful.

A Course in the Geometry of n Dimensions, by M. G. Kendall. Dover Publications, 2004. Softcover, 63 pp., $7.95. ISBN 0-486-43927-5.

Recently (18 May, 2008) I received a letter from Steve Hellinger:


Good Morning Alex,

I read your review of Dover's re-issue of Maurice Kendall's 1961 monograph "A Course in the Geometry of N Dimensions". I do agree in general with your critique. However I recently worked my way through the entire monograph (my copy is the 1961 Hafner version, which I started to read years ago as a graduate student) and I found a large number of issues that potential readers should be aware of, as per the list below. I wrote to Dover to inquire whether the editors might be interested in having me prepare a set of explanatory notes to accompany the monograph but Dover did not respond (no surprise there!). But I would like to provide readers with my comments on the monograph and the "Book Review" section of the "Interactive Mathematics Miscellany and Puzzles" website seems a good place. However I cannot find a facility at the website that would enable me to post my comments. Is there such a facility?

Thanks for your time,
Steve Hellinger

  1. Many mistakes in the statement of equations and relations
  2. Inconsistent and confusing use of notation from one section to the next, and even from one paragraph to the next
  3. A serious flaw in a major proof
  4. Inadequate definitions of terms in quite a few places
  5. Very numerous typographical errors
  6. A lack of references to the works from which much of the statistical section is drawn

My site, not having a wiki facility, is trailing the times, so that we had to be content with a manual exchange. Steve's review is much more detailed and objective than mine. I have placed it in a separate page.

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