XI] CONCERNING GNOMONS 763 



triangles have their locus upon a equiangular spiral: a result which 

 follows directly froni that alternative definition of the equiangular 

 spiral which I have quoted from Whit worth (p. 757). 



If in this, or any other isosceles triangle, we take corresponding 

 median lines of the successive triangles, by joining C to the mid-' 

 point (M) of AB, and D to the mid-point (N) of BC, then the pole 

 of the spiral, or centre of similitude of ABC and BCD, is the point 

 of intersection of CM and DN*. 



Again, ,we may build up a series of right-angled triangles, each 

 of which is a gnomon to the preceding figure; and here again, an 

 equiangular spiral is the locus of corresponding points in these suc- 

 cessive triangles. And lastly, whensoever we fill up space with a 

 collection of equal and similar figures, as in Figs. 360, 361, there 

 we can always discover a series of equiangular spirals in their 

 successive multiples*)*. 



Once more, then, we may modify our definition, and say that: 

 "Any plane curve proceeding from a fixed point (or pole), and such 

 that the vectorial area of any sector is always a gnomon to the 

 whole preceding figure, is called an equiangular, or logarithmic, 

 spiral." And we may now introduce this new concept and nomen- 

 clature into our description of the Nautilus shell and other related 

 organic forms, by saying that: (1) if a growing structure be built 

 up of successive parts, similar in form, magnified in geometrical 

 progression, and similarly situated with respect' to a centre of 

 similitude, we can always trace through corresponding points a 

 series of equiangular spirals; and (2) it is characteristic of the 



* I owe this simple but novel construction, like so much else, to Dr G. T. Bennett. 



t In each and all of these gnomonic figures we may now recognise a never- 

 ending polygon, with equal angles at its corners, and with its successive sides in 

 geometrical progression; and such a polygon we may look upon as the natural 

 precursor of the equiangular spiral. If we call the exterior or "bending" angle 

 of th'^ polygon j3, and the ratio of its sides A, then the vertices lie on an equiangular 

 spiral of angle a, given by logg A = j3 cot a. In the spiral of Fig. 359 the constant 

 angle is thus found to be about 75° 40', in "that of Fig. 355, 77° 40', and in that of 

 Fig. 356, 72° 50'. 



The calculation is as follows. Taking, for example, the successive triangles of 

 Fig. 359, the ratio (A) of the sides, as BC : AC, is that of the golden section, 1 : 1-618. 

 The external angle (^), as ADB, is 108°, or in radians 1-885. Then 

 log 1-618=0-209, from which log, 1-618=0-481 



and cota = !^ = ^^=0-255 = cot75°4y. 



