516 THE LOGARITHMIC SPIRAL [ch. 



successively, and permanently, laid down. In the main, the 

 logarithmic spiral is characteristic, not of the Uving tissues, but 

 of the dead. And for the same reason, it will always or nearly 

 always be accompanied, and adorned, by a pattern formed of 

 "hues of growth," the lasting record of earlier and successive 

 stages of form and magnitude. 



It is evident that the spiral curve of the shell is, in a sense, 

 a vector diagram of its own growth ; for it shews at each instant 

 of time, the direction, radial and tangential, of growth, and the 

 unchanging ratio of velocities in these directions. Regarding the 

 actual velocity of growth in the shell, we know very little (or 

 practically nothing), by way of experimental measurement; but 

 if we make a certain simple assumption, then we may go a good 

 deal further in our description of the logarithmic spiral as it appears 

 in this concrete case. 



Let us make the assumption that similar increments are added 

 to the shell in equal times ; that is to say, that the amount of 

 growth in unit time is measured by the areas subtended by equal 

 angles. Thus, in the outer whorl of a spiral shell a definite area 

 marked out by ridges, tubercles, etc., has very different linear 

 dimensions to the corresponding areas of the inner whorl, but the 

 symmetry of the figure imphes that it subtends an equal angle 

 with these; and it is reasonable to suppose that the successive 

 regions, marked out in this way by successive natural boundaries 

 or patterns, are produced in equal intervals of time. 



If this be so, the radii measured from the pole to the boundary 

 of the shell will in each case be proportional to the velocity of 

 growth at this point upon the circumference, and at the time when 

 it corresponded with the outer hp, or region of active growth; 

 and while the direction of the radius vector corresponds with the 

 direction of growth in thickness of the animal, so does the tangent 

 to the curve correspond with the direction, for the time being, of 

 the animal's growth in length. The successive radii are a measure 

 of the acceleration of growth, and the spiral curve of the shell 

 itself is no other than the hodograph of the growth of the contained 

 organism. 



