SPIRAL, a name belonging properly to curves which wind round a point in successive convolutions. The easiest mode of representing such curves algebraically is by means of polar COORDINATES : hence, in many of the older English works, any curve referred to such coordi nates is said to be considered as a spiral. Thus we have the circle considered as a spiral; the ellipse considered as a spiral, and so on. The rest of this article is intended only for those who have some knowledge of the mathematical part of the subject.
If r be the radius vector of a curve, 0 the angle which it makes with a given line, and r= (0) the equation of the curve, it is obvious that if IA be a common trigonometrical function of sin 0, cos 0, &c., the curve will not have an unlimited number of convolutions. The whole of the curve from 0=27 to will be merely a repetition of that from 0=0 to Thus, r=ain 0 is tho equation of a circle of a unit diameter, tangent at the origin to the line from which r sets out ; the fifteenth half-revolution of the radius vector is only the fifteenth description of this circle. It is then only when the angle 0 occurs independently of trigonometrical quantities, that any curve is retire Rented which can properly be called a spiral. Thus, the spiral of Archimedes, or Conon, of which the equation is r=a0, has a convolu tion in which r changes from 0 to 2wa, while 0 changes from 0 to 2w; another, in which r changes from 2Ta to Ora, while 0 changes from 2w to 4w and so on. The principal spirals to which distinct names have given, are— Equation.
I. Spiral of Archimedes . . . r a0 2. Reciprocal Spiral . . . . r0 ea a 3. Lituus . . . . . . ss a 0 4. Logarithmic or Equiangular Spiral r = all with some others of less note. The figures of these spirals are given in all books on the application of algebra to geometry.
It has hitherto been universal to consider spirals in a manner which has deprived these curves of half their convolutions ; this has been done by refusing to entertain negative values of the radius. For example, in the spiral of Archimedes r=a0, a being a positive quantity, the curve is supposed to have no convolutions when 0 is negative, or when the radius revolves negatively. The consequence is, that the curve begins abruptly at the origin. It would be a matter of little importance to insist on the existence of the additional branches which belong to the negative radii, if it were not that the other mode of representing curves, by means of rectangular coordinates, always gives the additional branches : so that, Have refuse to receive the latter as coming from the polar equation, we have only the alternative of sup posing that the mere transformation of coordinates destroys a part of the curve. In the spiral of Archimedes, for example, the rectangular
and polar equations are— + g = a, tan a . r = a0.
The first, treated in the usual way, gives a curve of which there is one succession of convolutions beginning with onee n, and another beginning with oebcd. But the second equation, which is only the first in a different form, does not yield any of the second set of con volutions, unless by means of the negative values of the radius vector answering to negative values of 0.
The manner in which the negative value of r is to be treated, is as follows :—Every line passing through the origin, as P 0 fa, makes two angles with the positive side of the axis of x, PO D, less than a right angle in the diagram, and Q o n, between two and three right angles: the second of which may be considered as the common angle fa o n, taken nega tively. The bounding directions of these angles are different, o r and o g : the rule is, whichever angle the straight line Q o r is supposed to make with o n, let the bounding direction of that angle be the positive direction, and the other direction negative. Thus, when r o D is the angle, o r is positive and o o negative : when g o o is the angle, o Q is positive and 0 P negative. In this manner it will be found that the first three of the four spirals above enumerated have never been com pletely drawn. Thera is little need to insist much on the necessity of the extension here described : one more instance may suffice. Let the reader trace the curve whose equation is— derived from r=1-2 cos 0. The rectangular equation gives a curve of two loops, of which the polar equation will only yield one, un less negative values of r be employed, in the manlier above described. Nevertheless, if the process had been inverted, and the polar equation deduced from the rectangular, we should have found r= +1-2 cos 0 for the former ; and the effect of the double sign is that tire positive values of r only, in the two equations r=1-2 cos 0, and r= —1-2 cos 0, will give the complete curve deduced from the rectaugdlar equation. As far as this instance goes, it might seem as if the complete polar equation, as reduced from the 'rectangulae, would give the whole curve by means of positive radii ; though at the same time a single instance hardly proves anything. But even granting that the passage from the rectangular to the polar equation will always give forms enough to the latter to trace the whole curve from positive radii, it remains indisputable that the other transition, from the polar to the rectangular, requires the negative radii to be taken into account