The Legacy of Greece. Various

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СКАЧАТЬ form, to subtend, which term is used quite generally for ‘to be opposite to’; in our phraseology the word hypotenuse is restricted to that side of a right-angled triangle which is opposite to the right angle, being short for the expression used in Eucl. i. 47, ἡ την ορθην γωνιαν ὑποτεινουσα πλευρα, ‘the side subtending the right angle’, which accounts for the feminine participial form ὑποτεινουσα, hypotenuse. If mathematicians had had more Greek, perhaps the misspelt form ‘hypothenuse’ would not have survived so long.

      To take an example outside the Elements, how can a mathematician properly understand the term latus rectum used in conic sections unless he has seen it in Apollonius as the erect side (ορθια πλευρα) of a certain rectangle in the case of each of the three conics?[3] The word ordinate can hardly convey anything to one who does not know that it is what Apollonius describes as ‘the straight line drawn down (from a point on the curve) in the prescribed or ordained manner (τεταγμενως κατηγμενη)’. Asymptote again comes from ασυμπτωτος, non-meeting, non-secant, and had with the Greeks a more general signification as well as the narrower one which it has for us: it was sometimes used of parallel lines, which also ‘do not meet’.

      Again, if we take up a textbook of geometry written in accordance with the most modern Education Board circular or University syllabus, we shall find that the phraseology used (except where made more colloquial and less scientific) is almost all pure Greek. The Greek tongue was extraordinarily well adapted as a vehicle of scientific thought. One of the characteristics of Euclid’s language which his commentator Proclus is most fond of emphasizing is its marvellous exactness (ακριβεια). The language of the Greek geometers is also wonderfully concise, notwithstanding all appearances to the contrary. One of the complaints often made against Euclid is that he is ‘diffuse’. Yet (apart from abbreviations in writing) it will be found that the exposition of corresponding matters in modern elementary textbooks generally takes up, not less, but more space. And, to say nothing of the perfect finish of Archimedes’s treatises, we shall find in Heron, Ptolemy and Pappus veritable models of concise statement. The purely geometrical proof by Heron of the formula for the area of a triangle, Δ=√{s(s-a) (s-b) (s-c)}, and the geometrical propositions in Book I of Ptolemy’s Syntaxis (including ‘Ptolemy’s Theorem’) are cases in point.

      The principles of geometry and arithmetic (in the sense of the theory of numbers) are stated in the preliminary matter of Books I and VII of Euclid. But Euclid was not their discoverer; they were gradually evolved from the time of Pythagoras onwards. Aristotle is clear about the nature of the principles and their classification. Every demonstrative science, he says, has to do with three things, the subject-matter, the things proved, and the things from which the proof starts (εξ ὡν). It is not everything that can be proved, otherwise the chain of proof would be endless; you must begin somewhere, and you must start with things admitted but indemonstrable. These are, first, principles common to all sciences which are called axioms or common opinions, as that ‘of two contradictories one must be true’, or ‘if equals be subtracted from equals, the remainders are equal’; secondly, principles peculiar to the subject-matter of the particular science, say geometry. First among the latter principles are definitions; there must be agreement as to what we mean by certain terms. But a definition asserts nothing about the existence or non-existence of the thing defined. The existence of the various things defined has to be proved except in the case of a few primary things in each science the existence of which is indemonstrable and must be assumed among the first principles of the science; thus in geometry we must assume the existence of points and lines, and in arithmetic of the unit. Lastly, we must assume certain other things which are less obvious and cannot be proved but yet have to be accepted; these are called postulates, because they make a demand on the faith of the learner. Euclid’s Postulates are of this kind, especially that known as the parallel-postulate.

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