## The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5) – Smogorzhevskii

In this post, we will see the book The Ruler In Geometrical Constructions by A. S. Smogorzhevskii. This book is Volume 5 of the Popular Lectures In Mathematics series. The study of the constructive power of a ruler and compasses, that is of the set of problems soluble by means of these classical tools of geometric constructions (both together or each separately), was carried out fully only in the 19th century. Until then some mathematicians regarded the ruler and compasses as universal instruments, which, if used together, were capable of solving any construction problem. This point of view played a negative role in the history of geometry. It prompted a premeditated attempt to regard each problem on construction as soluble by means of a ruler and compasses and led to the misuse of enormous effort on the futile search for non-existing solutions; this happened, for instance, with problems on squaring the circle, trisecting an angle, duplication of the cube.
The investigation of constructions carried out by means of a ruler alone was given a stimulus by the development of the theory of perspective and by the necessity of performing constructions over large portions of the earth’s surface, where the application of compasses with a large opening is technically impossible, while the construction of straight lines is easily achieved by the use of surveying instruments. In the present book the most typical construction problems, soluble by means of ruler alone, are considered.

The cases when the effectiveness of the use of the ruler is enhanced by the use of a previously drawn definite auxiliary figure in the plane of construction (for example two parallel straight lines or two intersecting circles) are worthy of attention. Many of these cases are also considered by us. In our presentation, we shall keep to the methods of synthetic geometry, i.e. we avoid the application of methods characteristic of arithmetic and algebra. We only permitted some minor deviations from this principle in some of the initial sections, motivated by the desire to simplify the presentation.
We should observe that the proofs of theorems and solutions of problems based on the application of methods of synthetic geometry are often distinguished by great elegance and originality; we hope that the reader will find in this book many examples confirming these words.

We draw the attention of the reader to Section 18, where it is shown that, using the ruler alone, it is impossible to construct the centres of two given non-concentric circles if these circles have no common point. It is well known, that “proofs of impossibility” belong, mostly, to the class of difficult mathematical problems and are usually based on profound and difficult reasoning. We think, that the reader will be interested in the contents of the section mentioned above, where one such proof is to be found.

The book was translated from Russian by Halina Moss and was edited by Ian N. Sneddon. The book was published in 1961.

You can get the book here.

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# Contents

Foreword vii

## PART 1 SOME THEOREMS OF SYNTHETIC AND PROJECTIVE GEOMETRY

1. Infinitely distant elements of a plane 3
2. Inverse with respect to a circle
3. The power of a point with respect to a circle. The radical axis of two circles. The radical centre of three circles 10
4. Pencils of straight lines and coaxial circles 15
5. Cross ratio 18
6. The harmonic distribution of four points on a straight line, and four straight lines of a pencil 21
7. ‘The harmonic properties of a complete quadrangle 23
8. Conic sections 25
9. Polar properties of conic sections 27
10. The theorems of Brianchon and Pascal 33

## PART 2 GEOMETRICAL CONSTRUCTIONS WITH THE AID OF A RULER

11. The construction of certain rectilinear figures by means of a ruler 41
12. Ruler constructions connected with conic sections 44
13. Ruler constructions, given two parallel straight lines 51
14. Ruler constructions, given a parallelogram or a square 56
15. Ruler constructions, given a circle and its centre 59
16. Ruler constructions, given the centre of a circle and its arc 67
17. The construction, by means of a ruler, of circles belonging to a given coaxial system 71
18. On the impossibility of constructing the centre of a circle by means of a ruler 75
19. Cases, when it is possible to construct the centres of two drawn circles by means of a ruler 78
20. On the construction by means of a ruler, of the
centres of several circles 83 