SPIRAL (ML. spiralis, from Lat. spire, coil, spire, from Gk. arapa, speira, coil, spire, twist). A curve which during its gradual regression from a point winds repeatedly around it. A plane spiral is ge.nerated by a point moving along a line according to a fixed law, while the line revolves uniformly about a fixed point in the plane. A spiral which is not plane is generated by a point moving on a given surface other than a plane, about a fixed point according to a given law, e.g. the loxodroine (q.v.). A great many spirals have been studied. Of these the most Common are the spiral of Archimedes, the hyper bolic, parabolic, Cotes's, logarithmic or equi angular spirals, and the /ituus.
If p denotes the perpendicular distance from the pole to the tangent at any point P of the hr curve, and if p = , various spirals may be formed representing this equation. These are known as Cotes's spirals, and are of scientific The spiral of Archimedes, probably discovered by Conon, has the equation r = a0. In this curve the point moves with a constant velocity along the radius vector, and the length of the radius is proportional to the angle described. The curve may he constructed by points as fol lows: Draw a circle of radius a about 0 as a interest, especially in their relation to trajecto ries (q.v.). If b = a the equation is that of the logarithmic or equiangular spiral. The charac teristic property of this spiral is that the angle between any radius vector and the corresponding tangent is constant. The equation of the loga rithmic spiral is logy = a0. It is evident from the equation that the curve has an infinite num ber of spires. The evolute (q.v.) of the logarith mic spiral is a similar logarithmic spiral.
centre; draw radii dividing the circumference into n equal parts, and lay off from 0 on these a 2a 3a 4a radii the distaneesri. 'it. St etc. The cir
St ele nsed in this construction is called the measur ing circle of the spiral. 1 f the point so moves that the radius vector varies inversely as the angle described, the curve is called a hyperbolic k, or reciprocal spiral. Its equation is k being the circumference of the measuring circle. It follows from the equation that an infinite number of spires are necessary for the curve to reach the origin. The curve received its name from the fact that it can also be con structed by means of an auxiliary equilateral hyperbola. If r varies directly as the square root of 0 the equation becomes a0, and we have the parabolic spiral. The figure represents the curve for both positive and negative values of r.
A spiral in which the square of the radius vector varies inversely as the angle described is called the lituus or trumpet, a carve described by Cotes (1682-1716). Its equation is The curve begins at infinity and winds round the origin, but cannot reach it by a finite number of spires.
One of the best, although not recent. mono graphs on the general subject is to be found in the Jlemoires de PAcademie des Sciences (1704, pp. 47, 69). For the general and special bibliog raphies and for a list of various spirals studied, consult Brown], Notes de bibliographic dcs courbcs gt'uaicuigucs (Bar-lc-Due, 1897, p. 252; pantie compliment:lire, 1899, p. 160). On the spiral of Archimedes, consult ['fait, lietrach tuuyen iiber die Npirale (Slunich, 1830). On the parabolic spiral and that of Cotes, consult Sacclu in the Smurcites Annales dc .Nathf3 s+atilues 0860). On the logarithmic spiral, Turquan in the same journal (1546) and Whit worth (1669). On the hyperbolic spiral, Fouret and Lebon in the same journal (1880).