learn how, by the properties of the curve, we may further extend our knowledge or simplify our descriptions of the shell; and (3) to understand the factors by which the charôÙacôÙterôÙisôÙtic form of any particular logarithmic spiral is determined, and so to comprehend the nature of the specific or generic characters by which one spiral shell is found to differ from another.
Of the elementary properties of the logarithmic spiral, so far as we have now enumerated them, the following are those which we may most easily inôÙvesôÙtiôÙgate in the concrete case, such as we have to do with in the molluscan shell: (1) that the polar radii of points whose vectorial angles are in arithmetical progression, are themselves in geometrical progression; and (2) that the tangent at any point of a logarithmic spiral makes a constant angle (called the angle of the spiral) with the polar radius vector.
The former of these two propositions may be written in what is, perhaps, a simpler form, as follows: radii which form equal angles about the pole of the logarithmic spiral, are themselves continued proportionals. That is to say, in Fig. 261, when the angle ROQ is equal to the angle QOP, then ORã₤:ã₤OQã₤::ã₤OQã₤:ã₤OP.
A particular case of this proposition is when the equal angles are each angles of 360ô¯: that is to say when in each case the radius vector makes a complete revolution, and when, therefore P, Q and R all lie upon the same radius. {518}
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