The ratio of the circumference to the diam eter is the same for all circles: the constant thus arising has been generally denoted by the symbol r since the time of Euler. It was proved quite simply by Legendre that it is not rational (i.e., cannot be represented ex actly by the ratio of any two integers, and hence, in particular, cannot be represented by a terminating decimal). The difficulty consists in showing that r is transcendental, that is, is not the root of any algebratic equation atom . . . where n is a positive integer, and the coeffi cients are any integers. This was finally proved by Lindemann in 1882, after Hermite in 1873 had shown that e, the base of the Napierian system of logarithms, is transcen dental. The two numbers are connected by the remarkable relation 1, where i is the imaginary unit number \/ 1. Since It cannot satisfy any algebraic equation, it cer tainly cannot be expressed by square roots. Hence Theorem II proves the impossibility.
Approximate The prob lems considered cannot be solved exactly by ruler and compass, but they can be solved to any required degree of approximation. Thus a simple approximate solution of the rectifica tion problem is the following: Let 0 be the centre and AB any diameter of the given circle. At the middle point E of AO construct a perpendicular cutting the circumference in C and D. On AB prolonged lay off EF CD. Draw FD, and on this line lay off Then the segment HD is approximately one fourth the circumference. The error is less than one part in 5,000.
Other The problems con sidered may be solved exactly if other instru ments in addition to ruler and compass are allowed. Thus the trisection and duplication problems (like all problems depending on cubic and biquadratic equations) can be solved by the instruments for drawing parabolas or other conics, or by appropriate linkages. The quad rature of the circle, being a transcendental problem, cannot be effected by any instrument which draws algebraic curves. It can be solved by various transcendental curves (quad ratrix, sinusoid, cycloid); or by the integraph (an instrument which draws the curve y =ff (x) dx, where y=f (x) is a given curve).
We consider now various restrictions which may be imposed on Euclidean constructions.
(1) Ruler Here only the straight-edge is allowed. For the possibility of such a construction it is necessary but not sufficient that the corresponding algebraic ex pression should be rational. If two parallel lines are given, then through a given point a line may be drawn parallel to given lines by a ruler construction. But this is not the case when a line is to be drawn through a given point parallel to a given line. The impossibility proof, based upon projection, may be carried out by pure geometry.
(2) Mascheroni Constructions. Here only the compass is allowed. A straight line is con sidered as known when two of its points are determined. Mascheroni, in 1797, showed that all problems which can be solved by the ruler and compass can be solved by the compass alone.
(3) Poncelet and Steiner have shown that if a single fixed circle with its centre is given, all elementary constructions may be carried out by means of the straight-edge. Again, if
a ruler with parallel edges may be employed (it is then, for instance, possible to place the instrument so that each edge goes through an assigned point), all elementary problems may be solved without the compass.
(4) Hilbert considers constructions with the straight-edge and sect-carrier. The latter de notes a compass used not to draw circles, but merely to lay off a given segment on a given line. All such constructions can be carried out by the straight-edge and a movable unit sect. The test for deciding the possibility or impossibility of a problem in this sense is exceedingly complicated, depending on the higher theory of algebraic numbers.
Geometrography. A problem of ele mentary geometry can usually be solved in a variety of ways by the rule and compass. Thus for the Apollonian problem (to construct a circle touching three given circles) over 100 distinct solutions have been worked out (Apol lonius, Poncelet, Steiner, Lemoine, Study, etc.). How can we compare these as regards sim plicity? It is necessary to adopt some stand ard or measure of simplicity. One method, for instance, would be to take the number of lines and circles as the measure of simplicity.
A more complete (but still somewhat arti ficial) discussion has been elaborated by E. Lemoine in his 'Geometrography.' Construc tions are analyzed into the following element ary operations: Operation C. consists in plac ing one point of the compass on a given point in the plane of construction; including a given length between the points of the compass is then denoted by 2Cs; placing a point of the compass on an undetermined point of a line is operation C2; drawing a circle is C.; malting the edge of the ruler pass through an assigned point is operation and through two as signed points is finally, drawing a straight line is operation RL Any construction may then be represented by a symbol 12Ri m2C2+ ni.C., where the coefficients represent the numbers of elementary opera tions involved. The simplicity is measured by 11+12+ tn.+ tn.+ nis, and the exactness by 11+ m. m, (the preparatory operations). The number of lines employed is given by 1. and of circles by m.. In case of construction for the bisection of an angle explained above the symbol is The construction which leads to the smallest possi ble value for the simplicity is termed the geo metrographic solution. There may be more than one solution of this kind.
Bibliography. Casey, 'Sequel to Euclid' (Dublin 1888) ; Euclid, (Elements' (Editions by Hall and Stevens, Harper, Nixon, Todhun ter, etc.) ; Hadamard, 'Lecons de geometric elementaire' (2 vols., Paris 1898, 1901) ; Hil bert, 'Grundlagender Geometric> (Leipzig, 1899, 2d ed. 1901); translation by Townsend (Chicago 1900) Halsted, 'Rational Geometry' (New York 1964); Kempe, 'How to draw a straight line' (London 1877); Klein, 'Elemen targeometrie (Leipzig 1895); translation by Beman and Smith (Boston 1897); Phillips and Fisher, 'Elements of Geometric' (New York 1896); Legendre, 'Elements de geom &de' (Paris 1794); Lindemann, 'Ueber die Zahl ir' (Mathematische Annalen, Vol. XX, 1882) Vahlen, 'Abstracte Geometric' (Leip zig 1905).