752 THE EQUIANGULAR SPIRAL [ch. 



material; and in all alike the parts once formed remain in being, 

 and are thenceforward incapable of change. 



In a shghtly different, but closely cognate way, the same is true 

 of the spirally arranged florets of the sunflower. For here again 

 we are regarding serially arranged portions of a composite structure, 

 which portions, similar to one another in form, differ in age; and 

 differ also in magnitude in the strict ratio of their age. Somehow 

 or other, in the equiangular spiral the time-element always enters 

 in; and to this important fact, full of curious biological as well as 

 mathematical significance, we shall afterwards return. 



In the elementary mathematics of a spiral, we speak of the point 

 of origin as the pole (0) ; a straight hne having its extremity in the 

 pole, and revolving about it, is called the radius vector; and a 

 point (P), travelling 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 equiangular or 

 logarithmic spiral. 



The former may be roughly illustrated by the way a sailor 

 coils a rope upon the deck; as the rope is of 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 ''If a straight fine 

 revolve uniformly about its extremity, a point which Hkewise 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." It is plain that the 

 spiral of Archimedes may be compared, but again roughly, to a 

 cylinder coiled up. It is plain also that a radius {r = OP), made 

 up of the successive and equal whorls, will increase in arithmetical 

 progression : and will equal a certain constant quantity (a) multiplied 



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

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

 Mathematik, i, p. 262, 1880. 



