Fig. 281. An ammonitoid shell (Macroscaphites) to shew change of curvature.
of Oekotraustes, a subgenus of Ammonites. If on the other hand, the angle α gradually diminishes, and even falls away to zero, we shall have the spiral curve opening out, as it does in Scaphites, Ancyloceras and Lituites, until the spiral coil is replaced by a spiral curve so gentle as to seem all but straight. Lastly, there are a few cases, such as Bellerophon expansus and some Goniatites, where the outer spiral does not perceptibly change, but the whorls become more “embracing” or the whole shell more involute. Here it is the angle of retardation, the ratio of growth between the outer and inner parts of the whorl, which undergoes a gradual change.
In order to understand the relation of a close-coiled shell to one of its straighter congeners, to compare (for example) an {551} Ammonite with an Orthoceras, it is necessary to estimate the length of the right cone which has, so to speak, been coiled up into the spiral shell. Our problem then is, To find the length of a plane logarithmic spiral, in terms of the radius and the constant angle α. In the annexed diagram, if OP be a radius vector, OQ a line of reference perpendicular to OP, and PQ a tangent to the curve, PQ, or sec α, is equal in length to the spiral arc OP. And this is practically obvious: for PP′ ⁄ PR′ = ds ⁄ dr = sec α, and therefore sec α = s ⁄ r, or the ratio of arc to radius vector.
Fig. 282.
Accordingly, the ratio of l, the total length, to r, the radius vector up to which the total length is to be measured, is expressed by a simple table of secants; as follows:
| α | l ⁄ r |
|---|---|
| 5° | 1·004 |
| 10 | 1·015 |
| 20 | 1·064 |
| 30 | 1·165 |
| 40 | 1·305 |
| 50 | 1·56 |
| 60 | 2·0 |
| 70 | 2·9 |
| 75 | 3·9 |
| 80 | 5·8 |
| 85 | 11·5 |
| 86 | 14·3 |
| 87 | 19·1 |
| 88 | 28·7 |
| 89 | 57·3 |
| 89° 10′ | 68·8 |
| 20 | 85·9 |
| 30 | 114·6 |
| 40 | 171·9 |
| 50 | 343·8 |
| 55 | 687·5 |
| 59 | 3437·7 |
| 90 | Infinite |
Putting the same table inversely, so as to shew the total {552} length in whole numbers, in terms of the radius, we have as follows:
|
Total length (in terms of the radius) | Constant angle |
|---|---|
| 2 | 60° |
| 3 | 70 31′ |
| 4 | 75 32 |
| 5 | 78 28 |
| 10 | 84 16 |
| 20 | 87 8 |
| 30 | 88 6 |
| 40 | 88 34 |
| 50 | 88 51 |
| 100 | 89 26 |
| 1000 | 89 56′ 36″ |
| 10,000 | 89 59 30 |