At the beginning of his Elements, Euclid places his three Postulates: "Let it be granted
and all the constructions used in the first six books are built up from these three operations only. The first two tell us what Euclid could do with his ruler or straight edge. It can have had no graduation, for he does not use it to carry a distance from one position to another, but only to draw straight lines and produce them. The first postulate gives us that part of the straight line AB which lies between the given points A, B; and the second gives us the parts lying beyond A and beyond B; so that together they give the power to draw the whole of the straight line which is determined by the two given points, or rather as much of it as may be required for any problem in hand.
The last postulate tells us what Euclid could do with his compasses. Again, he does not use them to carry distance, except from one radius to another of the same circle; his instrument, whatever it was, must have collapsed in some way as soon as the centre was shifted, or either point left the plane. The three postulates then amount to granting the use of ruler and compasses, in order to draw a straight line through two given points, and to describe a circle with a given centre to pass through a given point; and these two operations carry us through all the plane constructions of the Elements. The term Euclidean construction is used of any construction, whether contained in his works or not, which can be carried out with Euclid's two operations repeated any finite number of times.
In fact, Euclid gives only very few of the constructions which can be carried out with ruler and compasses, and probably every student of geometry has at some time or other constructed a figure which no one else had ever made before. But from very early times there were certain figures which everyone tried to make with rule and compasses, and no one succeeded. The most famous of these baffling figures are the square equal in area to a given circle, and the angle equal to the third part of a given angle; and it has at last been proved that neither of these can possibly be constructed by a finite series of Euclidean operations.
The first question treated in this book is the one which naturally arises here: what constructions can be built upon Euclid's postulates, and what cannot? or, in other words, what problems can be solved by ruler and compasses only?...
In the next chapter we shall show how each step of a rule and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics....Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider what these algebraic processes can lead to, when combined in every possible way, and that enables us to answer the question before us and say (p. 19) that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and (pp. 23, 30) that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only.
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