Of all these methods by which the mathematical constants, or specific characters, of a given spiral shell may be determined, the only one of which much use has been made is that which Moseley first employed, namely, the simple method of determining {539} the relative breadths of the whorl at distances separated by some convenient vectorial angle (such as 90°, 180°, or 360°).
Very elaborate measurements of a number of Ammonites have been made by Naumann[517], by Sandberger[518], and by Grabau[519], among which we may choose a couple of cases for consideration. In the following table I have taken a portion of Grabau’s determinations of the breadth of the whorls in Ammonites (Arcestes)
| Breadth of whorls (180° apart) mm. | Ratio of breadth of successive whorls (360° apart) | The angle (α) as calculated |
|---|---|---|
| 0·30 | — | — — |
| 0·30 | 1·333 | 87° 23′ |
| 0·40 | 1·500 | 86 19 |
| 0·45 | 1·500 | 86 19 |
| 0·60 | 1·444 | 86 39 |
| 0·65 | 1·417 | 86 49 |
| 0·85 | 1·692 | 85 13 |
| 1·10 | 1·588 | 85 47 |
| 1·35 | 1·545 | 86 2 |
| 1·70 | 1·630 | 85 33 |
| 2·20 | 1·441 | 86 40 |
| 2·45 | 1·432 | 86 43 |
| 3·15 | 1·735 | 85 0 |
| 4·25 | 1·683 | 85 16 |
| 5·30 | 1·482 | 86 25 |
| 6·30 | 1·519 | 86 12 |
| 8·05 | 1·635 | 85 32 |
| 10·30 | 1·416 | 86 50 |
| 11·40 | 1·252 | 87 57 |
| 12·90 | — | — — |
| Mean | 86° 15′ | |
{540}
intuslabiatus; these measurements Grabau gives for every 45° of arc, but I have only set forth one quarter of these measurements, that is to say, the breadths of successive whorls measured along one diameter on both sides of the pole. The ratio between alternate measurements is therefore the same ratio as Moseley adopted, namely the ratio of breadth between contiguous whorls along a radius vector. I have then added to these observed values the corresponding calculated values of the angle α, as obtained from our usual formula.
There is considerable irregularity in the ratios derived from these measurements, but it will be seen that this irregularity only implies a variation of the angle of the spiral between about 85° and 87°; and the values fluctuate pretty regularly about the mean, which is 86° 15′. Considering the difficulty of measuring the whorls, especially towards the centre, and in particular the difficulty of determining with precise accuracy the position of the pole, it is clear that in such a case as this we are scarcely justified in asserting that the law of the logarithmic spiral is departed from.
In some cases, however, it is undoubtedly departed from. Here for instance is another table from Grabau, shewing the corresponding ratios in an Ammonite of the group of Arcestes tornatus. In this case we see a distinct tendency of the ratios to
| Breadth of whorls (180° apart) mm. | Ratio of breadth of successive whorls (360° apart) | The spiral angle (α) as calculated |
|---|---|---|
| 0·25 | — | — — |
| 0·30 | 1·400 | 86° 56′ |
| 0·35 | 1·667 | 85 21 |
| 0·50 | 2·000 | 83 42 |
| 0·70 | 2·000 | 83 42 |
| 1·00 | 2·000 | 83 42 |
| 1·40 | 2·100 | 83 16 |
| 2·10 | 2·179 | 82 56 |
| 3·05 | 2·238 | 82 42 |
| 4·70 | 2·492 | 81 44 |
| 7·60 | 2·574 | 81 27 |
| 12·10 | 2·546 | 81 33 |
| 19·35 | — | — — |
| Mean | 83° 22′ |
{541}
increase as we pass from the centre of the coil outwards, and consequently for the values of the angle α to diminish. The case is precisely comparable to that of a cone with slightly curving sides: in which, that is to say, there is a slight acceleration of growth in a transverse as compared with the longitudinal direction.