QUADRATRIX
, or Quadratix, in Geometry, is a mechanical line, by means of which, right lines are found equal to the circumference of circles, or other curves, and of the parts of the same. Or, more accurately, the Quadratrix of a curve, is a transcendental curve described on the same axis, the ordinates of which being given, the quadrature of the correspondent parts in the other curve is likewise given. See Curve.—Thus, for example, the curve AND may be called the Quadratrix of the parabola AMC, when the area APMA bears some such relation as the following to the absciss AP or ordinate PN, viz,
when | APM = PN^{2}, | |
or | APM = AP | X PN, |
or | APM = a | X PN, |
The most distinguished of these Quadratices are, those of Dinostrates and of Tschirnhausen for the circle, and that of Mr. Perks for the hyperbola.
Quadratrix of Dinostrates, is a curve AMD, by which the quadrature of the circle is effected, though not geometrically, but only mechanically. It is so called from its inventor Dinostrates; and the genesis or description of it is as follows: Divide the quadrantal arc ANB into any number of equal parts, in the points N, n, n, &c; and also the radius AC into the same number of parts at the points P, p, p, &c. To the points of N, n, n, &c, draw the radii CN, Cn, &c; and from the points P, p, &c, the parallels to CB, as PM, Pm, &c: then through all the points of intersection draw the curve AMmD, and it will be the Quadratrix of Dinostrates.
Or the same curve may be conceived as described by a continued motion, thus: Conceive a radius CN to revolve with a uniform motion about the centre C, from the position AC to the position BC; and at the same time a ruler PM always moving uniformly parallel towards CB; the two uniform motions being so regulated that the radius and the ruler shall arrive at the position BC at the same time. For thus the continual intersection M, m, &c. of the revolving radius, and moving ruler, will describe the Quadratrix AMm &c. Hence,
1. For the Equation of the Quadratrix: Since, from the relation of the uniform motions, it is always, AB : AN :: AC : AP; therefore if AB = a, AC = r, AP = x, and AN = z, it will be a : z :: r : x, or ax = rz, which is the equation of the curve.
Or, if s denote the sine NE of the arc AN, and y = PM the ordinate of the curve AM, its absciss AP being x; then, by similar triangles, CE : CP :: EN : PM, that is, , and hence , the equation of the curve. And when the relation between AB and AN is given, in terms of that between AC and AP, hence will be expressed the relation between the sine EN and the radius CB, or s will be expressed in terms of r and x; and consequently the equation of the curve will be expressed in terms of r, x, and y only.
2. The base of the Quadratrix CD is a third proportional to the quadrant AB and the radius AC or CB; i. e. CD : CB :: CB : AB. Hence the rectification and quadrature of the circle.
3. A quadrantal arc DF de- scribed with the centre C and radius CD, will be equal in length to the radius CA or CB.
4. CDF being a quadrant inscribed in the Quadratrix AMD, if the base CD be = 1, and the circular arc DG = x; then in the area . See Quadrature. |
Quadratrix of Tscbirnhau- sen, is a transcendental curve AMmB by which the quadrature of the circle is likewise effected. This was invented by M. Tschirnhausen, and its genesis, in imitation of that of Dinostrates, is as follows: Divide the quadrant ANC, and the radius AC, each into equal parts, as before; and from the points P, p, &c, draw the lines PM, pm, &c, parallel to CB; also from the points N, n, &c, the lines NM, nm, &c, parallel to the other radius AC; so shall all the intersections M, m, &c, be in the curve of the Quadratrix AMmB.
Now for the Equation of this Quadratrix; it is, as before, .
Or, because here y = PM = EN = s; therefore s, as before, expressed in terms of r and x, gives the equation of this Quadratrix in terms of r, x, and y, and that in a simpler form than the other. Thus, from the nature of the circle and the construction of the Quadratrix, it is , where A, B, C, &c, are the preceding terms; which is the equation of the curve or Quadratrix of Tschirnhausen.
By either Quadratix, it is evident that an arc or angle is easily divided into three, or any other number of equal parts; viz, by dividing the corresponding radius, or part of it, into the same number of equal parts: for AN is always the same part of AB, that AP is of AC.