CHAPTER VIII. ON THE GENERATION OF CURVES AND THE INVESTIGATION OF THEIR EQUATIONS FROM THEIR GEOMETRICAL PROPERTies. As in the following chapters frequent reference will be made to certain curves which have acquired historical importance, and have in consequence been distinguished by particular names, I shall here describe the mode of their generation and deduce their equations from their definitions, adding some notice of the principal properties which possess interest. (1) The Cissoid of Diocles. This curve, named after Diocles, a Greek mathematician, who is supposed to have lived about the sixth century of our era, was invented by him for the purpose of constructing the solution of the problem of finding two mean proportionals. The curve is generated in the following manner: In the diameter ACB (fig. 12) of the circle ADBE take BM = AN, erect the ordinates QM, RN and, join AQ; the locus of the point P where the line AQ cuts the ordinate RN is the cissoid of Diocles. To find its equation, put AN = X, PN = Y, AC = a: then as PN QM y (2a x – x*) or y* (2a – x) = x, which is the equation to the curve. The curve has an equal and similar branch on the other side of AB; the two branches meet in a cusp at the point A, and have the line HK as a common asymptote. The area included between the curve and the asymptote is three times the area of the generating circle. The application of this curve to the solution of the problem of two mean proportionals is very simple. Pappus has shown that if BC, Cs be the two quantities between which the two mean proportionals are to be inserted, and if the line APQ be drawn so that QT = PT, the line CT is the first of the two mean proportionals : it is obvious from this that P is a point in the cissoid. If therefore we wish to find two mean proportionals between BC and CS, we construct the cissoid AND and produce BS till it meet the curve in a point P. Joining AP, and producing it to meet CS produced, we determine the line CT which is the first of the two mean proportionals required. According to the geometrical ideas of the ancients a problem was not thought to be completely solved unless a mechanical construction was given. To complete therefore the theory of the cissoid, Newton * invented the following means of describing it by continuous motion. At the centre Cof the circle ADB (fig. 13) erect the perpendicular CDE, of indefinite length. Take a point 0 in CA produced such that AO = AC; then if the reetangular ruler NLM, of which the leg LM is equal to the diameter of the circle, be moved so that the leg NL always slides along 0, while the end M slides along CDE, the middle point P of LM will trace out the cissoid. (2) The Conchoid of Nicomedes. This curve, the invention of Nicomedes, who lived about the second century of our era, was, like the preceding, first formed for the purpose of constructing the solution of the problem of finding two mean proportionals, or the duplication of the cube, but it is more readily applicable to another problem not less celebrated among the ancients, that of the trisection of an angle. The curve is generated in the following manner : take the indefinite straight line HK, (fig. 14) and from a fixed point o draw a line OMP cutting the line HK in M; take the point P, such that PM shall be always of a constant length: the locus of the point P is the conchoid. The point P may be taken * Append. ad Arith. Univ. between 0 and M, in which case it will trace out another branch of the curve which is called the inferior conchoid. To determine the equation, let AN = x, PN = y, PM (which is of constant length) = a, 0A = b. Then as PM – PN + MN?, and MN = AN – AM = AN - AMN, we have w* y* = (a’ – y')(b + y)', which is the equation to the curve, including both the superior and the inferior conchoid. It is evident from the construction of the curve that the line KH is an asymptote to both branches. When a>b there is a loop in the inferior conchoid at O as in the figure ; when a = b the loop degenerates into a cusp; and when a<b there are two points of contrary flexure, one on each side of the line 0A. The application of this curve to the construction of the problem of the trisection of an angle is as follows. It may be readily shown, that if AOB (fig. 15) be the angle to be trisected, and if the line OMP be so drawn that the part MP, intercepted between AB and BC at right angles to each other, is double of OB, the angle AOM is the third part of AOB. Now if we describe a conchoid with O as pole and the line AB as directrix, the constant parameter being equal to twice OB, its intersection with BC will determine the point P. Nicomedes appears to have been led to the invention of this curve as a means of solving the celebrated problems mentioned above, by the facility with which it could be constructed mechanically. For if we take a grooved rule HK (fig. 16) and another grooved rule PQ, having a fixed pin at a point M, and bearing a pencil at P, and if we cause the pin at M to slide along the groove HK while the groove MQ slides along a pin fixed at 0, the point P will trace out the conchoid. (3) The Witch of Agnesi. take a point P, such that PN : AB = MN : AN; the locus of the point P is the curve called the Witch. Putting AC = a, AN = ln PN = y, we find as the equation to the curve Xy' = 4 a’(2a – x'). This curve is given by Donna Maria Agnesi in her Instituzioni Analitiche, Art. 238, and is called by her the “ Versiera.” The line KAH is an asymptote to the curve, which has 3 a two points of contrary flexure corresponding to x = *. (4) The Lemniscate of Bernoulli. If a point be taken such that the product of the lines drawn from it to two fixed points is constant, it will trace out the curve called the lemniscate* If 2a be the distance between the fixed points, and if the origin be taken at the middle point between them, the equation to the curve is {y2 + (a + x)}} {y' + (a – x)?} = c. When c = a, the equation is reduced to (x2 + y)2 = 2a’ (23 – y'). This was the curve used by James Bernoullit in the construction of the curve along which a body under the action of gravity will advance or recede uniformly from a fixed point. It is the locus of the intersections of tangents to a rectangular hyperbola with perpendiculars drawn to them from the centre, and its form is that of the figure so. Of the properties of the arcs of this curve, which have been investigated by Fagnani and Euler, we shall treat in the chapter on the comparison of Transcendents in the Integral Calculus. If we assume x = r cos 0, y = r sind, we find + Opera, p. 609. portional to the logarithm of the ordinate. Hence its equation is y = beč, or, as it is generally written, y = a*. The subtangent is constant, and the axis of « is an asymptote. The whole area included between the curve, the axis of X and any ordinate is equal to twice the triangle formed by the ordinate, the tangent at its extremity and the axis of w; and the solid formed by the revolution of the curve round its asymptote is equal to a cylinder, the radius of whose base is the bounding ordinate, and whose height is the tangent at its extremity. This curve was invented by James Gregorie*, who investigated some of its properties : others were discovered by Huyghens. Eulert, and more recently Vincent, in the Annales de Gergonne, Vol. xv. p. 1, have conceived that the equation y = a* expresses, besides the continuous curve, a series of discontinuous points, forming what the latter calls a “courbe pointillée.” This conclusion appears to me to be founded on an erroneous conception of the principles of the interpretation of algebraical expressions, and I have elsewhere stated my reasons for believing that these discontinuous points belong each to a separate continuous curve which does not lie in the plane of reference, and that they cannot be properly included in the equation to one curve. As however the question is more closely connected with the analytical Theory of Logarithms than with the subject of which we here treat, I shall not now enter into the argument, but shall content myself with referring the reader, who is curious in such matters, to the papers quoted above, and to De Morgan's Differential Calculus, p. 383, where he will find the views of Vincent supported and illustrated %. • Geometriæ Pars Universalis, Pref. Camb. Math. Journal, Vol. 1. p. 231, and p. 264. & Professor De Morgan says, “that those who object to the pointed branch as introducing discontinuity, must choose between its discontinuity and that of an abrupt termination." It appears to me that if we interpret our analytical symbols with proper generality so as to introduce those branches of curves which do not lie in the plane of reference, we avoid the second horn of his dilemma. |