SPIRAL

In Geometry, the name given to a cla of curves distinguished by this general property, that they continually recede from a centre or pole, while they continue to revolve about it. Spirals receive different names from the properties by which they are characterized, or from their inventors; thus, t\ie equable spiral, the hyperbolic spiral, the logarithmic spiral, the spirals of Cotes, Sic. Equable spiral, or spiral qf Arch i me des. This curve may be supposed to be generated as follows: — If a straight line P A B C turn uniformly about its extremity P, a point in the straight line which advances from P with a uniform motion and arrives succe ively at a. A, b, B, c, C, d, will describe the spiral. The point P is called the pole, and the portion of the revolving line inter SPIRITS. cepted between the pole and any point of the curve is the radiant, or radius vector at that point. Let Q be a point in the spiral, make the radiant P Q = «, and let 6 be the angle described by the revolving line while the travelling point advances from P to Q; then, since from the definition the radiant u and the angle 6 are both proportional to the time, they are proportional to each other; and the polar equation of the equable spiral isu = aB, where a is a constant. If we suppose M = r when the revolving line has made a complete revolution, or when (i = 2vr; then a = r-7- 2sr, and the equation becomes u=: — B. 2 sr Some of the principal properties of this spiral are the following: — I. The area of the spiral P a Ab Q is equal to — - = or the area generated while the revolving line makes one revolution, is equal to a third of a circle whose radius is r. If M = 2 r, the expre ion for the area becomes § JT (2 r)2; and the spiral area is therefore the third part of a space that is double of a circle described with a radius = 2 r. And generally the whole area generated by the radiant, from the beginning of the motion till after any number of revolutions, is equal to the third part of a space which is the same multiple of the circle whose radius is equal to the greatest radiant as the number of revolutions is of unit. 2. The arc of the spiral between its origin and the radiant « is equal to that of a parabola whose lat us rectum is 2 o included between the vertex and an ordinate = u, and the corresponding area of the spiral is equal to one half the corresponding area of the parabola. 3. The subtangent to any point Q of the spiral is equal to the arc of a circle described by the radiant P Q; and hence at the termination of the first, second, third, fourth, . revolutions, the corresponding subtangents are as the series of square numbers, 1, 4, 9, 16, .; for the second, third, fourth, . the subtangents are equal to 2, 3, 4, . circumferences, whose radii are at the sarne time doubled, tripled, quadrupled, . The equable spiral was proposed by Conon to Arch i me des, who, in his treatise IXe^/ EX/;^;e», has investigated its quadrature, and some of its chief properties. The hyperbolic spiral belongs to a cla of spirals of which the general equation is « =a 0 ", If we suppose n = 1, we have u = aB~,orud = a, which is the equation of the hyperbolic spiral; and from which it follows, that if from a point in a straight line, given by position, arcs be described of a given length, and having each one extremity in the given straight line, the locus of their other extremities will be in the spiral. This curve was proposed by James Bernoulli, and has been called the hyperbolic spiral, from the analogy between its polar equation and the equation to rectangular co-ordinates of the common hyperbola, when referred to its asymptotes; namely, x y = A, a. constant area. The logarithmic spiral differs from the equable spiral in this respect, that the point which describes the curve, instead of advancing equably along the uniformly revolving straight line, is carrieci along it with a velocity increasing in proportion to the distance from the centre. This curve, which was noticed by Des Cartes, has many remarkable properties, of which the more abstruse were investigated by James Bernoulli. See Logarithmic Spiral. For the investigation of the properties of spirals, see Leslie's Geometrical Analysis; Peacock's Examples of the Differential and Integral Calculus; Maclaurin's Fluxions, . SPl'RAL VESSELS, in Plants, are membranous tubes with conical extremities, lined in the inside by a fibre twisted spirally, and capable of unrolling with elasticity; their function is that of the conveyance of air. They are found in almost any part of plants except the bark; but are most abundant in leaves and flowers, and least common in the stem and root, except in the medullary sheath of the former. [s. 1152]