282 Mr. H. Norton on Fractional Values 



tion, I doubt whether the former would not be found fully as 

 accurate as the latter. 



I have constructed Rober's figure from the quantities given 

 by Sir William, and also 



8.r 8 + 4r 9 -4vr-l = 0, and z 3 + z' 2 -2z-l=0 



in the forms 



a? 



and 



the latter by placing the chord of — - in two intersecting circles. 



I think I can understand the whole train of thought which led 

 to Roberts plan, and I certainly find no evidence of any attempt 

 to do more than to discover a coincidence. I have not seen 

 Roberts book. 



Supposing that it were proved that the heptagon did enter 

 into the plans of ancient architects, it must have been attribu- 

 table to some superstitious reverence for the figure itself : reve- 

 rence for the number 7 would in all probability lead to a 

 thorough investigation of the polygon, not to mention that it 

 was the first polygonal figure which resisted all their attempts 

 to find a formal method of construction. How were they likely 

 to have attempted such an investigation in such early times ? 

 The earliest astronomy consisted in a long and patient obser- 

 vation of the periods of the heavenly bodies, and in a careful 

 examination, of the ratios of those periods one to another in order 

 to obtain some simple fraction. This would appear the more 

 mysterious in proportion as the numbers were low : continued 

 fractions probably owed their origin to such inquiries. They 

 would also reduce those fractions to their prime factors, in order 

 to see at a glance whether they were capable of simple relations 

 with other quantities or not. 



If this was their practice with regard to time, nothing is more 

 likely, than that they would apply the same method to space : 

 any two lines whose ratio was incapable of rigorous demonstra- 

 tion would be subjected to actual measurement. A very large 

 figure, constructed either tentatively or by approximation, would 

 give them the measures with considerable accuracy ; and having 

 them, they had a chance (by no means despicable) of finding the 

 above fraction, although not equal to my own chance, with Hut- 

 ton's Tables and Goodwyn's ' First Centenary of Decimal Quo- 

 tients '*. I think they are more likely to have done this than 



* The actual fraction is not there; the Table goes only from toW to iV • 



