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Topics in Elementary Geometry

O. Bottema
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
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The first edition of this book appeared in wartime Holland (1944) when, according to its author, ‘it was written during the repressive reality of occupation, darkness and sadness’ — a fact which is in no way reflected by the book’s contents. A second expanded edition was published in 1987, and this recent English version of that has been made possible by the excellent translational skills of Reine Erné of Leiden University. In addition, there is a foreword by Robin Hartshorne that nicely summarises the style and contents of the book and provides a brief biography of the author, which seems to be unavailable from other sources.

Oene Bottema (1901–1992) is described by Hartshorne as being relatively unknown outside Holland, where he has been described as ‘the great geometer’. Specifically, he is said to have had an encyclopaedic knowledge of 19th-century geometry, imbued with the rigor of 20th-century mathematics (a Dutch Coxeter perhaps?).

And this reminds me of the book by Coxeter and Greitzer, Geometry Revisited (MAA New Mathematical Library, 1967), in so far as the general aims, the intended readership are the same as Bottema’s, but the respective contents, although overlapping to some degree, diverge to the extent that the two books could be said to be complementary.

Anyway, this book consists of 27 short chapters and 127 pages. Seventeen of the chapters concern results associated with triangles (Ceva’s theorem, Euler line, Simson line, Morley’s triangle, Pedal triangles, and so forth). Amongst the remaining chapters there is treatment of Poncelet polygons, inversion, isogonal conjugation and the theorems of Desargues, Pappus and Pascal etc.

The distinguishing feature of the book is, to my mind, the author’s innovative treatment of his material. Chapter 1, for example, provides novel insights into Pythagoras’ theorem — both mathematically and historically. It examines the theorem in terms of its overall role in mathematics and generalises it to any similar shapes on the sides of a right-angled triangle. Also, although I knew that there are over 600 proofs of this theorem, I didn’t know that a proof by US President James Garfield was amongst them.

As examples of lesser known results, chapter 17 shows that, for triangles in which the centre of mass of the perimeter lies on the incircle, the following relation between side-lengths is true:

a–1 + b–1 + c–1 = 10(a + b + c)–1

Then, in chapter 18 (only two pages in length) barycentric coordinates are employed to establish and interesting analogue of the Euler line.

Although there are no exercises provided in this book, the very process of reading it is a developmental process in itself, and this is facilitated by the provision of an appendix that consists of further explanatory remarks and hints.

Overall, I find myself in total agreement with Robin Hartshorne, who describes the book as consisting of ‘a series of vignettes, each crafted with elegance and economy’. It needn’t be read all at once, but can be dipped into for the purpose of giving meaning to those empty moments between life’s succession of minor and major worries.

Peter Ruane was introduced to geometry via C. V. Durrell’s New Geometry for Schools (1958), which made Euclid accessible without distorting its message.


Preface.- The Pythagorean theorem.- Ceva's theorem.- Perpendicular bisectors; concurrence.- The Nine-point circle and euler line.- The Taylor circle.- Coordinate systems with respect to a triangle.- The Area of a triangle as a function of the barycentric coordinates of its vertices.- The Distances from a point to the vertices of a triangle.- The Simson line.- Morley's triangle.- Inequalities in a triangle.- The Mixed area of two parallel polygons.- The Isoperimetric inequaltiy.- Poncelet polygons.- A Closure problem for triangles.- A Class of special triangles.- Two unusual conditions for a triangle.- A Counterpart for the euler line.- Menelaus's theorem; cross-ratios and reciprocation.- The Theorems of desargues, pappus, and pascal.- Inversion.- The Theorems of ptolemy and casey.- Pedal triangles; brocard points.- Isogonal conjugation; the symmedian point.- Isotomic conjugation.- Triangles with two equal angle bisectors.- The Inscribed triangle with the smallest perimeter; the fermat point.- Appendix: remarks and hints.- References.- Index.-