512 



THE LOGARITHMIC SPIRAL 



[CH. 



If we take any one of these figures, for instance the isosceles 

 triangle which we have just described, 

 and add to it (or subtract from it) in 

 succession a series of gnomons, so con- 

 verting it into larger and larger (or smaller 

 and smaller) triangles all similar to the 

 first, we find that the apices (or other 

 corresponding points) of all these triangles 

 have their locus upon a logarithmic spiral : 

 a result which follows directly from that 

 alternative definition of the logarithmic 

 spiral which I have quoted from Whit- 

 worth (p. 507). 



Again, we may build up a series of 

 right-angled triangles, each of which is a 

 gnomon to the preceding figure; and here again, a logarithmic 

 spiral is the locus of corresponding points in these successive 

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



Fig. 255. 



Fig. 256. Logarithmic spiral derived from corresponding points in 

 a system of squares. 



