## Heinz Prüfer's reviews

We give short extracts from reviews. |

**1. Review by: Leonard M Blumenthal.**

This book arose from lectures that the author (deceased) gave at Münster University. Synthetic projective geometry is presented based upon an axiomatic structure comprised of postulates of connection (including the parallel postulate, in Playfair's form) for three dimensions, postulates of order and continuity. The operations of projection and section are defined in terms of the primitive notions of the system, and the first five chapters are concerned with developing in a rigorous manner the projective geometry of forms of the first and second orders in one, two, and three dimensions. ... This is a carefully written book. The figures are plentiful and well drawn, the style and typography are pleasing, and an adequate index is provided. The book contains no exercises.

**2. Review by: Ott-Heinrich Keller.**

After the death of Prüfer in 1934, G Fleddermann and G Kothe took on the task of adding to his manuscript the necessary supplements. Now there is a second, substantially unchanged, edition. ... The book is very carefully thought out in every detail. It skillfully introduces the beginner to the facts and ideas of the subect. It can be used with advantage both in conjunction with a lecture course or for self-study thanks to its clear and easy to understand presentation. But the further one progresses, the more one notices the actual depth of the work; it gives an insight into the context and in the clear ordering of the facts and the proofs always lead towards general aspects yet having regard to the essentials. Even the most mature mathematician can learn from this book and still have his fun with it.

**3. Review by: G Feigl.**

The author of this book died on the 7th April 1934 leaving it almost completed. The editors have had to add a few paragraphs to complete the last two chapters. While projective geometry is currently no longer purely synthetic, but usually presented as mixed analytic-synthetic with greater reliance on the analytical, is given here as a closed purely synthetic construction. Coordinates are only introduced after the completion of the theory in the last chapter, in a certain sense to round off the material. Furthermore, the author is restricted to real elements. The presentation combines perfectly brevity of expression, clarity of geometrical discussions, rigour of proofs and clarity of the logical structure.

**4. Review by: Virgil Snyder.**

The first half of this book is devoted to the usual topics of synthetic geometry of the first and second order in one, two and three dimensions, including polarity. It is based on the operations projection and section, but independently of intuition. Each step is rigorously defined and explained in terms of the axioms used, including continuity. But that space is three dimensional is tacitly assumed without an axiom of closure. The Playfair statement is the form adopted for the parallel axiom. An unusually large amount of material is satisfactorily discussed in these 150 pages. No exercises are provided for the student. No use is ever made of imaginary elements.Then follows a chapter on metrical geometry, mostly confined to two dimensions. This is particularly well done. The concept of perpendicularity is introduced by axioms; the involution of pairs of perpendicular lines of a pencil and a polarity having no curve of incident elements are the only new ideas needed. These are applied to prove a number of metrical theorems of plane geometry, connected with triangles. For the sake of logical completeness, now follows a chapter on non-euclidean geometry. This is much harder reading; it is logically consistent, but pedagogically is less successful.

Throughout the book figures are used freely, but only as suggestions, never as an essential part of the proof. A chapter on descriptive geometry is hardly more than a sketch; it discusses so many principles in the short space available that a reader would be helpless in trying to apply them to any other than the simplest problems.

The final chapter is on coordinates in one and two dimensions. It is based on the fundamental theorem that three pairs of corresponding elements fix a one-dimensional projectivity, so that the correspondent of any fourth element is uniquely determined. The idea of cross-ratio is not explicitly introduced. After defining addition and multiplication geometrically, it is shown that the rules of ordinary algebra apply. In passing from homogeneous to nonhomogeneous coordinates, the statements are frequently too inclusive; as given they include division by a vanishing coefficient. After showing that loci represented by linear equations in point coordinates are straight lines, the analytic formulation of projectivity is discussed, and also correlation. The determination of the fixed elements is not taken up except in a few particular cases.

The style is on the whole pleasing; the book is easy to read.

JOC/EFR January 2015

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