580 THE LOGAKITHMIC SPIRAL [ch. 



ellipsoid. If the attachment of the septum be not in one 

 plane, but form a sinuous line of contact with the cone, then 

 the septum will be a saddle-shaped surface, of great complexity 

 and beauty. In all cases, provided only that the membrane be 

 isotropic, the form assumed will be precisely that of a soap-bubble 

 under similar conditions of attachment : that is to say, it will be 

 (with the usual limitations or conditions) a surface of minimal 

 area. 



If our cone be no longer straight, but curved, then the septa 

 will be symmetrically deformed in consequence. A beautiful and 

 interesting case is afforded us by Nautilus itself. Here the 

 outline of the septum, referred to a plane, is approximately 

 bounded by two elhptic curves, similar and similarly situated, 

 whose areas are to one another in a definite ratio, namelv as 



A.2, r^r' 



- in cot a 



and a similar ratio exists in Ammonites and all other close-whorled 

 spirals, in which however we cannot always make the simple 

 assumption of elhptical form. In a median section of Nautilus, 

 we see each septum forming a tangent to the inner and to the 

 outer wall, just as it did in a section of the straight Orthoceras ; 

 but the curvatures in the neighbourhood of these two points of 

 contact are not identical, for they now vary inversely as the radii, 

 drawn from the pole of the spiral shell. The contour of the septum 

 in this median plane is a spiral curve identical with the original 

 logarithmic spiral. Of this it is the "invert," and the fact that 

 the original curve and its invert are .both identical is one of the 

 most beautiful properties of the logarithmic spiral*. 



But while the outline of the septum in median section is simple 

 and easy to determine, the curved surface of the septum in its 

 entirety is a very complicated matter, even in Nautilus which is 

 one of the simplest of actual cases. For, in the first place, since 

 the form of the septum, as seen in median section, is that of a 

 logarithmic spiral, and as therefore its curvature is constantly 

 altering, it follows that, in successive transverse sections, the 



* It was this that led James Bernoulli, in imitation of Archimedes, to have 

 the logarithmic spiral graven on his tomb, with the pious motto, Eadem muhita 

 resurgam. On Goodsir's grave the same symbol is reinscribed. 



