The duplication of the cube requires the de termination of the edge of a cube x, such that its cube shall be twice the volume of a Oven unit. That is xi'=.2. This equation is irre ducible, since otherwise would be rational. Moreover, the equation is a cubic and its de gree is not of the form 2". Hence the solution is in general impossible.
The problem of the trisection of an arbi trary angle corresponds to the solution of P+ V-1 sin and it follows that in general this is impossible by Euclidean methods.
History.— The quadrature of the circle is attempted in the Rhind Papyrus (2000 ac.), the oldest known mathematical work, in which Ahmes, an Egyptian priest, lays down the em pirical rule: 'Cut off one ninth of the diameter; the square on the remainder will equal the area of the circle.' This rule affords 9 3.16 . . . . , a value reasonably accurate as compared with3, the value assumed in the Bible (1 Kings vii, 23; Chronicles iv, 2). Archimedes (200 ac.) invented the method still used by students of plane geometry that depends on inscribing and circumscribing regular poly gons, which, save for an improvement by Huy gens (1654), remained in use until the invention of the infinitesimal calculus by Newton and Leibnitz in the last of the 17th Hippocrates of Chios (470 B.c.) was the first to investigate areas bounded by curves and to bring into prominence the problems of squar ing the circle and duplication of the cube. Ac cording to Philoponus, the Athenians were suffering from a severe plague of typhoid fever in 430 a.c., and were told by the oracle at Delos that Apollo required an altar in the form of a cube twice the size of the one existing. A new one was, therefore, constructed, having each edge twice the length of the old one, but the plague was worse than ever. Suspecting some mystery, confirmed by the insistence of the oracle, the Greeks applied to Plato, the most illustrious of their geometricians, and were re ferred by him to Hippocrates. He succeeded in reducing the problem to the determination of two mean proportionals: for, if a: x xft.x: y: 2a, then x'•c.ay and ex..2ax, the equations of two parabolas, which intersect in a point whose abscissa is .1' 620'. This is equivalent to a graphical solution by means of conics, but to draw the curves, a different instrument would be required than those prescribed by Euclid. The problem was thereafter known as the Delian problem. The Cissoid of Diodes (150 a.c.) and the Conchoid of Nicomedes (150 e.c.) are curves of the third and fourth degree, respectively, invented for the special purpose of duplicating the cube and trisecting any angle. The curve known as the quadratrix "of Dinostratus (350 a.c.) which, however, had previously been constructed by Hippias of Elis (420 a.c.) for the trisection of the angle, suf ficed also to determine the length of a circular arc and for the quadrature of the circle. It be longs to the class known in modern times as in tegral curves, since the ordinate can be expressed as .an integral, such a curve being known to the ancients as a quadratrix. If OA and OB are two perpendicular radii of a circle and two points M and L move with constant veloc ity one upon the radius OB and the other upon arc AB, such that starting from 0 and A at the same time they both arrivesimultaneously at B; then the intersection of OL and MP drawn parallel to OA is a point on the quadratrix.
The ordinate, y, is proportional to 0, the angle between OL and OA, and since rim 1, when 8 = -' 2 x y. But = tan-12 ' the equation of the quadratrix tan 2 - y.
It cuts the axes of X at the point whose scissa is x.z_ 2 goo -.• y= 0 IT IT tan- y 2 .
It follows that the radius of the circle is the mean proportional between the length of the quadrant and the abscissa of the intersection of the curve with the axis of X. Hence any apparatus that will describe the quadratrix will enable us to determine r graphically. Evidently the curve y = sin n- Lx is much more convenient for this purpose, since w is one of the ordinates of this curve when x equals zero; but this curve does not appear to have been used by the Greeks. It is called the sinusoid, the axis being vertical, and any transcendental apparatus which will trace the sinusoid by continuous mo tion would afford a geometrical construction of w. Such an apparatus has been• invented recently by a Russian engineer, Abdank-Aba kanowicz, and constructed by Coradi of Zurich. It is called the integraph, and with its aid it is possible not only to lay off w but to trace the integral nerve.
Y =F (x)= f f(x) dx, when the differen tial f(x) is given.
se of modern analysis after Newton and Leibnitz developed many new methods for the evolution of w, the best known being the so-called series of Leibnitz, ((=1— 1+ • • • After the invention of logarithms by Napier (1614), Euler, by the introduction of complex quantities, developed the celebrated relation be tween the Napenan base e and w, ax which for x.r-ir reduces to the most remark able relation in mathematics, , 8 s — 1 The modern proofs of the transcendency of IT are based upon this relation, since they all de pend upon that of e. In 1873 Hermite ((Sur la Function exponentielle,' Comptes rendus, 1873) first proved the transcendence of e, and this was followed, in 1882, by an analogous demonstration for given by Lindemann ((Vier die Zahl Mathematische Annalen, XX 1882). This demonstration is equivalent to proving that the Euclidean problem of the quadrature of the circle is impossible, and closes in this generation a question that has occupied mathematicians for 4,000 years. The long and difficult proofs of Hermite and Lindemann have been much simplified, first by Weierstrass (Ber liner Berichte, 1885), and in particular by Hil bert, Hurwitz and Gordon (Mathematische Annalen, Vol. XLIII). The questions involved treated without requiring a knowledge of the calculus have become familiar to English mathe maticians through the translation by ilcman and Smith of Klein's (Famous Problems of Ele mentary Geometry' (1897). Consult also the chapter on (The History and Transcendence of a, in (Monographs on Topics in Modern Mathe matics) (New York 1911) by D. E. Smith.