770 THE EQUIANGULAR SPIRAL [ch. 



The first of these propositions may be written in a simpler form, 

 as follows: radii which form equal angles about the pole of the 

 equiangular spira] are themselves continued proportionals. That 

 is to say, in Fig. 366, when the angle ROQ is equal 

 to the angle QOP, then OP:OQ::OQ: OR. 



A particular case of this proposition is when the 

 equal angles are each angles of 360° : that is to say 

 when in each case the radius vector makes a complete 

 revolution, and when, therefore, P, Q and R all he 

 upon tjie same radius. 



It was by observing with the help of very careftil 

 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 an equiangular 

 spiral, and this demonstration he was soon after- 



Via ^ftfi 



wards in a position to generalise by extending it to 

 all spiral Ammonitoid and Gastropod mollusca*. 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 Tjy 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 vectorf. Thus (in 



* The Rev. H. Moseley, On the geometrical forms of turbinated and discoid 

 shells, Phil. Trans. 1838, Pt. i, pp. 351-370. Reaumur, in describing the snail-shell 

 (Mem. Acad, des Sci. 1709, p. 378), had a glimpse of the same geometrical law: 

 " Le diametre de chaque tour des spirale, ou sa plus grande longueur, est a peu pres 

 double de celui qui la precede et la moitie de celui qui la suit." Leslie (in his 

 Geometry of Curved Lines, 1822, p. 438) compared the "general form and the 

 elegant septa of the Nautilus'" to an equiangular spiral and a series of its involutes. 



f It will be observed that here Moseley, speaking as a mathematician and 

 considering the linear spiral, speaks of whorls when he means the linear boundaries, 

 or lines traced by the revolving radius vector; while the conchologist usually 

 applies the term whorl to the whole space between the two boundaries. As con- 

 chologists, therefore, we call the breadth of a whorl what Moseley looked updn as 

 the distance between two consecutive whorls. But this latter nomenclature Moseley 

 himself often uses. Observe also that Moseley gets a very good approximate result 

 by his measurements "upon a radius vector," although he has to be content with 

 a very rough determination of the pole. 



