THE BEGINNINGS OF CERTITUDE 55 



less of the same length. A stadium was simply a " stage " 

 that is, about the distance that one man could readily shout 

 to another and be understood. When they had need of news 

 more swiftly than it might be carried by horse or man, a line 

 of shouters were stationed at intervals, sometimes over long 

 distances. It was thus that word of the disaster at Thermopylae 

 was brought to the Persian king. 



When these units of measure had been grouped up into a 

 rude system, it is obvious that through a growing proclivity 

 to barter must have come a growing sense of the fact of pro- 

 portion. When for so many horses or slaves or wives a man 

 traded so many skins or sacks of wheat, or weights of meat 

 or roods of land, there came insensibly a sense of abstract 

 values that is, the relations of units of one kind to units of 

 another. There would be an inevitable comparison of signs 

 and symbols representing like quantities of different things. 

 From this would have come an intimate sense of the fixity of 

 numbers, and likewise of their inter-relations. 



It was a far higher step when he had come to perceive that 

 one number might bear the same relation to a second as a 

 third to a fourth our familiar rule of three. The utilisation 

 of this method must have been among the earliest instances of 

 that method of prevision and subsequent verification which is 

 in a sense the foundation of exact knowledge. 



The application of the method must have been very slow. 

 There is a legend preserved by Diogenes of how Thales, the 

 wise man, when he went among the Egyptians, showed them 

 a simple means of measuring the height of the Pyramids. This 

 was by measuring the length of the pyramid's shadow at the 

 moment that the shadow cast by a staff equalled in length the 

 height of the staff. Doubtless the story was a fable ; it may 

 have been merely Thales' own boastfulness ; the boasting trait 

 is strong in primitive man. The Pyramids are still a marvel 

 of constructive accuracy. The base of the Great Pyramid 

 is 756 feet on each side, and so near to a perfect square that 

 the mean error of the four sides is only six-tenths of an 

 inch. The height is to the total length of the sides as the 

 quantity symbolised by ?r to the circumference of a circle ; and 

 there are other indications in the interior that the ratio of if 

 was accurately known. The Pyramids were built several 

 thousand years before Thales was born. 



