XI] OR LOGARITHMIC SPIRAL 755 



The many specific properties of the equiangular spiral are so 

 interrelated to one another that we may choose pretty well any 

 one of them as the basis of our definition, and deduce the others 

 from it either by analytical methods or by elementary geometry. 

 In algebra, when m^ = n, x is called the logarithm of n to the base 

 m. Hence, in this instance, the equation r = a^ may be written in 

 the form log r = 6 log a, or 6 = log r/log a, or (since a is a constant) 

 6 = A; log /•*. Which is as much as to say that (as Descartes dis- 

 covered) the vector angles about the pole are proportional to the 

 logarithms of the successive radii; from which circumstance the 

 alternative name of the "logarithmic spiral" is derivedf. 



Moreover, for as many properties as the curve exhibits, so many 

 names may it more or less appropriately receive. ^James Bernoulli 

 called it the logarithmic spiral, as we still often do ; P. Nicolas called 

 it the geometrical spiral, because radii at equal polar angles are in 

 geometrical progression; Halley, the proportional spiral, because 

 the parts of a radius cut off by successive whorls are in continued 

 proportion; and lastly, Roger Cotes, going back to Descartes' first 

 description or first definition of all, called it the equiangular spiral J. 

 We may also recall Newton's remarkable demonstration that, had 

 the force of gravity varied inversely as the cube instead of the 

 square of the distance, the planets, instead of being bound to their 



* Instead of r=a^, we might write r = rQa^; in which case r^ is the value of r 

 for zero value of 6. 



f Of the two names for this spiral, equiangular and logarithmic, I used the 

 latter in my first edition, but equiangular spiral seems to be the better name; 

 for the constant angle is its most distinguishing characteristic, and that which 

 leads to its remarkable property of continuous self-similarity. Equiangular spiral 

 is its name in geometry; it is the analyst who derives from its geometrical pro- 

 perties its relation to the logarithm. The mechanical as well^as the mathematical 

 properties of this curve are very numerous. A Swedish admiral, in the eighteenth 

 century, shewed an equiangular spiral (of a certain angle) to be the best form for 

 an anchor-fluke {Sv. Vet. Akad. Hdl. xv, pp. 1-24, 1796), and in a parrot's 

 beak it has the same efl&ciency. Macquorn Rankine shewed its advantages in 

 the pitch of a cam or non-circular wheel {^Manual of Mechanics, 1859, pp. 99-102; 

 cf. R. C. Archibald, Scripta Mathem. m (4), p. 366, 1935). 



X James Bernoulli, in Acta Eruditorum, 1691, p. 282; P. Nicolas, De novis 

 spiralibus, Tolosae, 1693, p. 27; E. Halley, Phil. Trans, xix, p. 58, 1696; Roger 

 Cotes, ibid. 1714, and Harmonia Mensurarum, 1722, p. 19. For the further history 

 of the curve see (e.g.) Gomes de Teixeira, Traite des courbes remarqicables, Coimbre, 

 1909, pp. 76-86; Gino Loria, Spezielle algebrdische Kurven, ii, p. 60 scq., 1911; 

 R. C. Archibald (to whom I am much indebted) in Amer. Mathem. Monthly, xxv, 

 pp. 189-193, 1918, and in Jay Hambidge's Dynamic Symmetry, 1920, pp. 146-157. 



