...duplication of the cube, we are given the length s of a side and seek a number x such that x 3 = 2s 3 . But usually it is convenient to employ analytic geometry;...located by means of points either initially given or obtained as the intersections of two straight lines, a straight line and a circle, or two circles....

...numbers which define analytically the desired geometric elements can be derived from those defining Die given elements by a finite number of rational operations...located by means of points either initially given or obtained as the intersections of two straight lines, a straight line and a circle, or two circles....

...the construction, and therefore also the length ± Xi of the segment joining two of them, are found by a finite number of rational operations and extractions of real square roots, performed upon rational numbers and numbers obtained by earlier ones of these operations. If xi is...

...the construction, and therefore also the length ± xi of the segment joining two of them, are found by a finite number of rational operations and extractions of real square roots, performed upon rational numbers and numbers obtained by earlier ones of these operations. If xi is...

...volumes. These numbers may be said to define analytically the various geometric elements involved. CRITERION. A proposed construction is possible by...rational operations and extractions of real square roots. In § 29 we were given the numbers a and 6, and constructed lines of lengths Proof. First, we grant...

...the Greek mathematicians were able to construct any desired geometric element that could be derived by a finite number of rational operations and extractions of real square roots from the given elements. To illustrate : Suppose we are given the elements^, b, and the unit element....