Beckman, Peter. A History of Pi.
A million years or so have passed since the tool-wielding animal called man made its appearance on this planet. During this time it learned to recognize shapes and directions; to grasp the concepts of magnitude and number; to measure; and to realize that there exist relationships between certain magnitudes. (9)
By 2,000 B.C., men had grasped the significance of the constant that is today denoted by П, and that they had found a rough approximation of its value. (9)
Long before the invention of the wheel, man must have learned to identify the peculiarly regular shape of the circle…and perhaps he was pleased by its infinite symmetry as he drew its shape in the sand with a stick. (9)
Then, one might speculate, men began to grasp the concept of magnitude – there were large circles and small circles, tall trees and small trees, heavy stones, heavier stones, very heavy stones. The transition fro, these qualitative statements to quantitative measure was the dawn of mathematics. It must have been a long and arduous road, but it is a safe guess that it was first taken for quantities that assume only integral values – people, animals, trees, stones, sticks. For counting is a quantitative measurement: The measurement of the amount of a multitude of items. (9-10)
Man first learned to count to two, and a long time elapsed before he learned to count to higher numbers. There is a fair amount of evidence for this, perhaps none of it more fascinating than that preserved in man’s launguages: In Czech, until the Middle Ages, there used to be two kinds of plural – one for two items, another for many (more than two) items, and apparently in Finnish this is so to this day. There is evidently no connection between the (Germanic) words two and half; there is none in the Romance languages (French: deux and moitié)nor in the Slavic languages (Russian: dva and pol) … This suggests that men grasped the concept of a ratio, and the idea of a relation between a number and its reciprocal, only after they had learned to count beyond two. (10)
The next step was to discover relations between various magnitudes. Again, it seems certain that such relations were first expressed qualitatively. It must have been noticed that bigger stones are heavier, or to put it into more complicated words, that there is a relation between the volume and the weight of a stone. It must have been observed than an older tree is taller, that a faster runner covers a longer distance, that more prey gives more food, that larger fields yield bigger crops. Among all these kinds of relationships, there was one which could hardly have escaped notice, and which, moreover, had not exceptions:
The wider a circle is “across,” the longer it is “around.” (11)
And again, this line of qualitative reasoning must have been followed by quantitative considerations. If the volume of a stone is doubled, the weight is doubled; if you run twice as fast, you cover double the distance; if you treble the fields, you treble the crop; if you double the diameter of a circle, you double its circumference. (11)
Neolithic man was hardly concerned with monotonic functions; but it is certain that men learned to recognize, consciously or unconsciously, by experience, instinct, reasoning, or all of these, the concept of proportionality; that is, they learned to recognize pairs of magnitude such that if the one was doubled, trebled, quadrupled, halved or left alone, then the other would also double, treble, quadruple, halve or show no change. (11)
And then came the great discovery. By recognizing certain specific properties, and by defining them, little is accomplished. But a great scientific discovery has been made when the observations are generalized in such a way that a generally valid rule can be stated. The greater its range of validity, the greater its significance. To say that one field will feed half the tribe, two fields will field the whole tribe, three fields will feed one and a half tribes, all this applies only to certain fields and tribes. To say that one bee has six legs, three bees have eighteen legs, etc., is a statement that applies, at best, to the class of insects. But somewhere along the line some inquisitive and smart individuals must have seen something in common in the behavior of the magnitudes in these and similar statements:
No matter how the two proportional quantities are varied, their ratio remains constant.
For the fields, this constant is 1 : ½ = 2 : 1 = 3 : 1½ = 2. For the bees, this constant is 1 : 6 = 3 : 18 = 1/6. And thus, man had discovered a general, not a specific, truth. (11)
This constant ratio was not obtained by numerical division (and certainly not by the use of Arabic numerals, as above); more likely, the ratio was expressed geometrically, for geometry was the first mathematical discipline to make substantial progress. (11)