uniform thickness, so in the whole spiral coil is each whorl of the same breadth as that which precedes and as that which follows it. Using its ancient definition, we may define it by saying, that ãIf a straight line revolve uniformly about its extremity, a point which likewise travels uniformly along it will describe the equable spiral3.ã Or, putting the same thing into our more modern words, ãIf, while the radius vector revolve uniformly about the pole, a point (P) travel with uniform velocity along it, the curve described will be that called the equable spiral, or spiral of Archimedes.ã {504}
It is plain that the spiral of Archimedes may be compared to a cylinder coiled up. And it is plain also that a radius (r =ã₤OP), made up of the successive and equal whorls, will increase in arithmetical progression: and will equal a certain constant quantity (a) multiplied by the whole number of whorls, or (more strictly speaking) multiplied by the whole angle (ö¡) through which it has revolved: so that r =ã₤aö¡.
But, in contrast to this, in the logarithmic spiral of the Nautilus or the snail-shell, the whorls gradually increase in breadth, and do so in a steady and unchanging ratio. Our definition is as follows: ãIf, instead of travelling with a uniform velocity, our point move along the radius vector with a velocity increasing as its distance from the pole , then the path described is called a logarithmic spiral.ã Each whorl which the radius vector intersects will be broader than its predecessor in a definite ratio; the radius vector will increase in length in geometrical progression, as it sweeps through successive equal angles; and the equation to the spiral will be r =ã₤ a ÿ£¢ ö¡ ã₤. As the spiral of Archimedes, in our example of the coiled rope,