768 THE EQUIANGULAR SPIRAL [ch. 



diverge by successive equal angles from that plane of reference; and by 

 this means, there will be superposed upon the equiangular spiral a twist or 

 screw. And, in the particular case where this latter angle of divergence is 

 just equal to 180°, or two right angles, the successive shoots will once more 

 come to lie in a plane, but they will appear to come off from one another on 

 alternate sides, as in Fig. 365 A. This is the Schrauhel or Bostryx of Schimper, 

 the cyme unipare helicoide of Bravais. The equiangular spiral is still latent 

 in it, as in the other; but is concealed from view by the deformation resulting 

 from the helicoid. Many botanists did not recognise (as the brothers Bravais 

 did) the mathematical significance of the latter case, but were led by the 

 snail-like spiral of the scorpoid cyme to transfer the name "helicoid" to it*. 



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 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 equiangular 

 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 an 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. 



prohibitum) ipsius aeternae generationis Filii, qui Patris veluti Imago, et ab illo ut 

 Lumen a Lumine emanans, eidem bixoLovaLo% existit, qualiscunque adumbratio. Aut, 

 si mavis, quia Curva nostra mirabilis in ipsa mutatione semper sibi constantissime 

 manet similis at numero eadem, poterit esse vel fortitudinis et constantiae in 

 adversitatibus, vel etiam Carnis nostrae post varias alterationes et tandem ipsam 

 quoque mortem, ejusdem numero resurrecturae symbolum: adeo quidem, ut si 

 Archimedem imitandi hodiernum consuetudo obtineret, libenter Spiram hanc tumulo 

 meo juberem incidi, cum Epigraphe, Eadem numero mutata resurgeV; Acta Erudi- 

 torum, M. Mali, 1692, p. 213. Cf. L. Isely, Epigraphes tumulaires de mathe- 

 maticiens, Bull. Soc. Set. nat. Neuchdtel. xxvii, p. 171, 1899. 



* The names of these structures have been often confused and misunderstood ; 

 cf. S. H. Vines, The history of the scorpioid cyme, Journ. Bot. (n.s.), x, pp. 3-9, 

 1881. 



