THE CURVES OF LIFE: A CRITICISM 397 



The equiangular or logarithmic spiral, upon which hangs so 

 much of Mr. Cook's theories, is itself connected with an older 

 proportion, to which the names of Sectio aurea, Sectio divina, 

 were given in the Middle Ages. The question of proportion in 

 relation to beauty of form had then been long in men's minds. 

 Apparently Polyclitus, as mentioned by Vitruvius, Galen, and 

 Pliny, was the first who recognised that there were certain pro- 

 portions in the beautiful human form, to which he gave expression 

 in his statues ; so that the people of the time said " Polyclitus 

 has created Art." He determined for each part of the body a 

 certain size, but the human body refused to allow itself to be 

 expressed in lifeless numbers, and others after him set up new 

 systems to show that his normal was not absolute. The Sectio 

 aurea is only a name for the extreme and mean proportion of 

 the geometricians, which may be stated thus : " The first part 

 is to the second part as the second is to the whole or sum of the 

 two parts ": i.e., 



A 1 B 



C 



BC : AC : : AC : AB. 



The Greeks knew this ratio, for we find it in Euclid (300 b.c) ; 

 and in 1509 Fra Luca Pacioli, a friend of Leonardo's, gave the 

 name of Divina proportione to his book on these proportions, 

 since which time a number of books have been written on the 

 subject without producing any definite result. 



Up to the present time it had not been found possible to 

 express this ratio arithmetically, with any accuracy. A famous 

 series, called the Fibonacci series from its inventor, which is 

 well known in connection with work on phyllotaxis, comes near 

 to it, but, as will be seen, it is not accurate. This additive series 

 is that obtained by the summation of 1+2 = 3, 2 + 3 =5, 

 3 + 5 =8, etc., giving the following series : 1, 2, 3, 5, 8, 13, 21, 34, 

 55, etc. If we place any of these numbers in the above proportion 

 the result is not perfect ; there is a difference. For instance, 



3 : 5 : : 5 : § 



8 x 3 = 24 - 5 x 5 = 25. 



Now Mr. Cook has been so fortunate as to have Mr. Mark 

 Barr and Mr. William Schooling as his collaborators on the 

 mathematical side of his work, and they discovered that if, in a 

 geometrical progression, the sum of any two terms is to equal 



