XI] AND THE SPIRAL OF ARCHIMEDES 503 



and a point (P) which is conceived as travelhng along the radius 

 vector under definite conditions of velocity, will then describe our 

 spiral curve. 



Of several mathematical curves whose form and development 

 may be so conceived, the two most important (and the only two 

 with which we need deal), are those which are known as (1) the 

 equable spiral, or spiral of Archimedes, and (2) the logarithmic, 

 or equiangular spiral. 



The former may be illustrated by the spiral coil in which a 

 sailor coils a rope upon the deck ; as the rope is of uniform thick- 

 ness, so in the whole spiral coil is each whorl of the same breadth 



Fig. 244. 



as that which precedes and as that which follows it. Using 

 its ancient definition, we may define it by saying, that "If a 

 straight line revolve uniformly about its extremity, a point which 

 likewise travels uniformly along it will describe the equable 

 spiral*." Or, putting the same thing into our more modern 

 words, "If, 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." 



* Leslie's Geometry of Curved Lines, p. 417, 1821. This is practically identical 

 with Archimedes' own definition (ed. Torelli, p. 219); cf. Cantor, Geschichte der 

 Mathematik, i, p. 262, 1880. 



