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The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science.

Aspects of Science, 1925.

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# An Investigation of Historical Geometric Constructions

## Introduction

From ancient Greek times, mathematicians have considered three famous geometric construction problems. These problems are: (1) the duplication of the cube – construct the edge of a cube whose volume is twice that of a given cube; (2) the trisection of any angle – construct an angle that equals one-third of a given angle; and (3) the squaring of a circle – construct a square whose area equals the area of a given circle. Given the tools at the time, a compass and an unmarked ruler, these problems challenged mathematicians for centuries.

The best Greek intellect was bent upon them; Arabic learning was applied to them; some of the best mathematicians of the Western Renaissance wrestled with them.  Trained minds and untrained minds, wise men and cranks, all endeavoured to conquer these problems which the best brains of preceding ages had tried but failed to solve. (Cajori, 1924, p. 54)

Although these three problems are closely linked, we choose to examine the latter two problems. We must divulge that the given problems do not have solutions if we limit ourselves to constructions using only compass and straightedge. However, using curves created by other means the Greeks resolved all three problems.