Fig. 270.
As soon, then, as we have prepared tables for these values, the determination of the constant angle α in a particular shell becomes a very simple matter.
A complete table would be cumbrous, and it will be sufficient to deal with the simple case of the ratio between the breadths of adjacent, or immediately succeeding, whorls.
Here we have r = e2π cot α , or log r = log e × 2π × cot α , from which we obtain the following figures[513]: {534}
| Ratio of breadth of each whorl to the next preceding r ⁄ 1 | Constant angle α |
|---|---|
| 1·1 | 89° 8′ |
| 1·25 | 87 58 |
| 1·5 | 86 18 |
| 2·0 | 83 42 |
| 2·5 | 81 42 |
| 3·0 | 80 5 |
| 3·5 | 78 43 |
| 4·0 | 77 34 |
| 4·5 | 76 32 |
| 5·0 | 75 38 |
| 10·0 | 69 53 |
| 20·0 | 64 31 |
| 50·0 | 58 5 |
| 100·0 | 53 46 |
| 1,000·0 | 42 17 |
| 10,000 | 34 19 |
| 100,000 | 28 37 |
| 1,000,000 | 24 28 |
| 10,000,000 | 21 18 |
| 100,000,000 | 18 50 |
| 1,000,000,000 | 16 52 |
We learn several interesting things from this short table. We see, in the first place, that where each whorl is about three times the breadth of its neighbour and predecessor, as is the case in Nautilus,
Fig. 271.
the constant angle is in the neighbourhood of 80°; and hence also that, in all the ordinary Ammonitoid shells, and in all the typically spiral shells of the Gastropods[514], the constant angle is also a large one, being very seldom less than 80°, and usually between 80° and 85°. In the next place, we see that with smaller angles the apparent form of the spiral is greatly altered, and the very fact of its being a spiral soon ceases to be apparent (Figs. [271], 272). Suppose one whorl to be an inch in breadth, then, if the angle of the spiral were 80°, the {535} next whorl would (as we have just seen) be about three inches broad; if it were 70°, the next whorl would be nearly ten inches, and if it were 60°, the next whorl would be nearly four feet broad. If the angle were 28°, the next whorl would be a mile and a half in breadth; and if it were 17°, the next would be some 15,000 miles broad.
