On Growth and Form - D'Arcy Wentworth Thompson

or,

1 ⁄ k = θ ⁄ log(r ⁄ a).

Then, if θ increase by 2π, while r increases to r₁ ,

1 ⁄ k = (θ + 2π) ⁄ log(r₁ ⁄ a),

which leads, by subtraction to

1 ⁄ k · log(r₁ ⁄ r) = 2π.

Now, as α tends to 0, k (i.e. cot α) tends to ∞, and therefore, as k → ∞, log(r₁ ⁄ r) → ∞ and also r₁ ⁄ r → ∞.

Therefore if one whorl exists, the radius vector of the other is infinite; in other words, there is nowhere, even in the near neighbourhood of the pole, a complete revolution of the spire. Our spiral shells of small constant angle, such as Dentalium, may accordingly be considered to represent sufficiently well the true commencement of their respective spirals.

Let us return to the problem of how to ascertain, by direct measurement, the spiral angle of any particular shell. The method already employed is only applicable to complete spirals, that is to say to those in which the angle of the spiral is large, and furthermore it is inapplicable to portions, or broken fragments, of a shell. In the case of the broken fragment, it is plain that the determination of the angle is not merely of theoretic interest, but may be of great practical use to the conchologist as being the one and only way by which he may restore the outline of the missing portions. We have a considerable choice of methods, which have been summarised by, and are partly due to, a very careful student of the Cephalopoda, the late Rev. J. F. Blake[515]. {537}

Fig. 274.
(1) The following method is useful and easy when we have a portion of a single whorl, such as to shew both its inner and its outer edge. A broken whorl of an Ammonite, a curved shell such as Dentalium, or a horn of similar form to the latter, will fall under this head. We have merely to draw a tangent, GEH, to the outer whorl at any point E; then draw to the inner whorl a tangent parallel to GEH, touching the curve in some point F. The straight line joining the points of contact, EF, must evidently pass through the pole: and, accordingly, the angle GEF is the angle required. In shells which bear longitudinal striae or other ornaments, any pair of these will suffice for our purpose, instead of the actual boundaries of the whorl. But it is obvious that this method will be apt to fail us when the angle α is very small; and when, consequently, the points E and F are very remote.
Fig. 275. An Ammonite, to shew corrugated surface-pattern.
Fig. 276.
(2) In shells (or horns) shewing rings, or other transverse ornamentation, we may take it that these ornaments are set at a constant angle to the spire, and therefore to the radii. The angle (θ) between two of them, as AC, BD, is therefore equal to the angle θ between the polar radii from A and B, or from C and D; and therefore BD ⁄ AC = e^{θ cot α} , which gives us the angle α in terms of known quantities. {538}
(3) If only the outer edge be available, we have the ordinary geometrical problem,—given an arc of an equiangular spiral, to find its pole and spiral angle. The methods we may employ depend (1) on determining directly the position of the pole, and (2) on determining the radius of curvature.