• "Plato believed that the 'Truths' of geometry, as formulated by the Greeks, were 'eternal', laid up in the Mind of God, and independent of human activity. Even Kant, some two thousand years later, still thought that Euclid's work was 'True', a priori, for the whole of humanity. Their familiarity with smooth 'geometrical' shapes blinded them to the fact that such shapes, or even close approximations to them, occur very infrequently in 'nature', and that their multiple replication by human agency was a necessary preliminary to their conceptualisation.
• Now many of the 'less civilised' peoples of the world have made no effort whatsoever to schematize or standardise their geometrical 'habits', and their languages include very few names for geometrical concepts. It was easy enough for Plato, in the Meno, to devise Socrates' demonstration that his slave had an 'inborn' knowledge of geometrical 'Truth' in spite of never having studied the subject; he conveniently ignored the circumstance that the slave had been living among a people who habitually used linear measurements, constructed rectangular buildings, and compared the size of 'areas' by dividing them as nearly as possible into 'equal squares'.
• If the slave had recently arrived, say, from the Congo, he would not have begun to understand what Socrates was talking about. It would have been almost impossible for him to grasp the concepts involved until he had lived in Athens for long enough to become familiar with the visual methods of construction and measurement. In Equatorial Africa, the (up)right angle never left the 'vertical' plane until the arrival of Arabs and Europeans, and linear measurement, except for rough estimates of short distances in terms of paces or foot-lengths, was unknown. Even nowadays, distances across country are very often given in 'hours', based on the informant's estimate of his hearer's likely walking pace.
• If we wish to arrive at the empirical foundations of Greek geometry, we have first to diagnose the reasons why large sections of the human race have found it important to engage in the multiple replication of geometrical shapes. This in turn requires us to seek their pre-geometrical prototypes, and to suggest reasons for the importance attributed to them.
• As far as the circle is concerned, there need be no mystery whatsoever. This shape is bound to function as a 'frame of reference', in the most literal sense of that expression, for all visual shape-perception, since it is the shape of the aperture of the eye - which presumably looks much the same from the inside as it does from the outside. Nowhere else on the surface of the human body is there anything approximating in this fashion to a smooth 'geometrical' shape, and furthermore, it is the only bodily shape which shows virtually no variation from one human body to another.
• As far as we can judge, the circle was in fact the first geometrical shape to be intentionally replicated. 'Hut-circles' go back to Mesolithic times, and there is no reason to suppose that the method of 'drawing' them differed from that in use in Africa to-day. A strip of rawhide is attached to a central stake, and the circle is generated at the other end of the strip, which is held by someone who walks round the central stake while keeping the strip tightly stretched.
• This practice gives a likely clue to the reason for the importance of the 'straight line', which, like the circle. is seldom found in nature with anything like the smoothness which characterises 'geometrical' shapes. Empirical instances of this shape can, however, be readily produced, in the form of tightly stretched threads. The Old-English equivalent 'streccat lin' in fact meant just that. It refers to the process of spinning, during which the raw material is both 'straightened' and 'straitened'. The filament by which a spider hangs on a windless day is an excellent 'natural' example; it is noteworthy that such filaments have been employed as 'cross-wires' in the eye-pieces of microscopes on account of their exemplary -straightness'. The discovery that the shape of a tightly stretched thread coincided with that of the most rapid transfer of attention between two discrete areas of the visual field must have been 'empirical'; it led to the definition of a straight line as 'the shortest route between two points'. This does not involve us in any circularity, since we maintain that the concept of the 'point' is derived from touch, and has nothing to do with vision. A visual 'quasi-point' is only a sign for the possibility of a tactile one.
• The empirical importance of the 'upright angle' is obvious. 'Flat' surfaces are clearly desirable as dwelling-places, their important characteristic being that bodies placed on them, particularly the human variety, show little or no tendency to lateral movement. As soon as it became general practice to surround the dwelling-places with walls^*, in order to provide protection from the weather and perhaps from animals, it must also have been discovered that the walls had to be built at a certain visible angle with the ground if they were to remain standing. Furthermore, a 'plumb-line' could be used during the building, so as to ensure the correctness of the angle."
• ^{* Editor's note: Joseph S. Ellul, who mentions the 'Prehistoric Yard' in the context of his discussion of the famous Mnajdra isoceles triangle inside one of Malta's ancient temples said to date from before Noah's Flood, and the apex of which is, in any case, undoubtedly oriented towards the Solar solstice, provocatively asks: "Which idea originated first, the directional or the measurement?" - or, in the words of Michæl Weber, his German translator: "Was war zuerst, die Maßeinheit oder die astronomische Ausrichtung?"}

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