It was by observing, with the help of very careful measurement, this continued proportionality, that Moseley was enabled to verify his first assumption, based on the general appearance of the shell, that the shell of Nautilus was actually a logarithmic spiral, and this demonstration he was immediately afterwards in a position to generalise by extending it to all the spiral Ammonitoid and Gastropod mollusca[505].
For, taking a median transverse section of a Nautilus pompilius, and carefully measuring the successive breadths of the whorls (from the dark line which marks what was originally the outer surface, before it was covered up by fresh deposits on the part of the growing and advancing shell), Moseley found that “the distance of any two of its whorls measured upon a radius vector is one-third that of the two next whorls measured upon the same radius vector[506]. Thus (in Fig. [262]), ab is one-third of bc, de of ef, gh of hi, and kl of lm. The curve is therefore a logarithmic spiral.”
The numerical ratio in the case of the Nautilus happens to be one of unusual simplicity. Let us take, with Moseley, a somewhat more complicated example.
From the apex of a large specimen of Turbo duplicatus[507] a {519} line was drawn across its whorls, and their widths were measured upon it in succession, beginning with the last but one. The measurements were, as before, made with a fine pair of compasses and a diagonal scale. The sight was assisted by a magnifying glass. In a parallel column to the following admeasurements are the terms of a geometric progression, whose first term is the width of the widest whorl measured, and whose common ratio is 1·1804.
Fig. 262.
|
Widths of successive whorls measured in inches and parts of an inch |
Terms of a geometrical progression, whose first term is the width of the widest whorl, and whose common ratio is 1·1804 |
|---|---|
| 1·31 | 1·31 |
| 1·12 | 1·1098 |
| ·94 | ·94018 |
| ·80 | ·79651 |
| ·67 | ·67476 |
| ·57 | ·57164 |
| ·48 | ·48427 |
| ·41 | ·41026 |
The close coincidence between the observed and the calculated figures is very remarkable, and is amply sufficient to justify the conclusion that we are here dealing with a true logarithmic spiral.
Nevertheless, in order to verify his conclusion still further, and to get partially rid of the inaccuracies due to successive small {520} measurements, Moseley proceeded to investigate the same shell, measuring not single whorls, but groups of whorls, taken several at a time: making use of the following property of a geometrical progression, that “if µ represent the ratio of the sum of every even number (m) of its terms to the sum of half that number of terms, then the common ratio (r) of the series is represented by the formula
r = (µ − 1)2 ⁄ m .”