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The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5) | A. S. Smogorzhevskii

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The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5)
Original Title The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5)
Author A. S. Smogorzhevskii
Publication date

Topics mathematics, soviet, geometry, geometrical constructions, constructions with ruler only, inverse with a circle, cross ratio, projective geometry, rectilinear figures, circle, triangles, polygons, brianchon theorem, pascal’s theorem, proofs
Collection mir-titles, additional_collections
Language English
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The study of the constructive power of a ruler and compas­ ses, 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 centu­ ry. 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 histoiy 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, dupli­ cation 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.
 Foreword
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 syn­ thetic 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 origin­ ality, 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 impossibility1 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 Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5)
      
 | A. S. Smogorzhevskii
The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5) | A. S. Smogorzhevskii
The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5) Original T
The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5)
      
 | A. S. Smogorzhevskii
The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5) | A. S. Smogorzhevskii
The Ruler In Geometrical Constructions (Popular Lectures In Mathematics Vol. 5) Original T
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