SPIRAL. 454 SPIRAL. SPIRAL (ML. spiralis, from Lat. spira, coil, spire, from Gk. airnpa, speira, coil, spire, twist). A curve which during its gradual regression from a point winds repeatedly around it. A plane spiral is gcjierated by a point moving along a line according to a fixed law, Avhile 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 loxodrome (q.v.). A great many spirals have been studied. Of these the most common are the spiral of Archimedes, the hyper- bolic. parahoUc, Cotes's, lognrithmic or equi- angular spirals, and the lituus. If p denotes the perpendicular distance from the pole to the tangent at any point P of the br curve, and if p =: — , various siiirals may l/a' + H be formed representing this equation. These are known as Cotes's spirals, and are of scientific ARCHIMEDES SPIRAL. The spiral of Archimedes, probably discovered by Conon, has the equation r = a6. 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 be constructed by points as fol- lows: Draw a circle of radius a about O as a /l r / x;
( j/ ^ A ^ '• ^ y M TARABOLIC SPIRAL. interest, especially in their relation to trajecto- ries (q.v.). If i) = 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 log;- = aS. 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. C HYPERBOLIC SPIRAL. centre; draw radii dividing the circumference into n equal parts, and lay off from O on these a 2a. 3a 4a radii the distances -. — . — . — • etc. The cir- n n n n cle used in this construction is called the measur- ing circle of the spiral. If the point so moves that the radius vector varies inversely as the angle described, the curve is called a hyperbolia k or reciprocal spiral. Its equation is r —-^• 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 d the equation becomes r" = a 8, and we have the parabolic spiral. The figure represents the curve for both positive and negative values of r. LOGARITHMIC SPIRAL. A spiral in which the square of the radius vector varies inversely as the angle described is called the lituus or trumpet, a curve described by Cotes (1682-171G). 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 lI4moires de VAcademie des Sciences (1704, pp. 47, 69). For the general and special bibliog- raphies and for a list of various spirals studied,
