and, integrating,
log r = θ cot α , or
r = εθ cot α .
As we have seen throughout our preliminary discussion, the two most important constants (or chief “specific characters,” as the naturalist would say) in any given logarithmic spiral, are (1) the magnitude of the angle of the spiral, or “constant angle,” α, and (2) the rate of increase of the radius vector for any given angle of revolution, θ. Of this latter, the simplest case is when θ = 2π, or 360°; that is to say when we compare the breadths, along the same radius vector, of two successive whorls. As our two magnitudes, that of the constant angle, and that of the ratio of the radii or breadths of whorl, are related to one another, we may determine either of them by actual measurement and proceed to calculate the other.
In any complete spiral, such as that of Nautilus, it is (as we have seen) easy to measure any two radii (r), or the breadths in {533} a radial direction of any two whorls (W). We have then merely to apply the formula
rn + 1 ⁄ rn = eθ cot α , or Wn + 1 ⁄ Wn = eθ cot α ,
which we may simply write r = eθ cot α , etc.; since our first radius or whorl is regarded, for the purpose of comparison, as being equal to unity.
Thus, in the diagram, OC ⁄ OE , or EF ⁄ BD , or DC ⁄ EF , being in each case radii, or diameters, at right angles to one another, are all equal to e(π ⁄ 2) cot α . While in like manner, EO ⁄ OF , EG ⁄ FH , or GO ⁄ HO , all equal eπ cot α ; and BC ⁄ BA , or CO ⁄ OB = e2π cot α .
