The Legacy of Greece. Various
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It has been well said that, if we would study any subject properly, we must study it as something that is alive and growing and consider it with reference to its growth in the past. As most of the vital forces and movements in modern civilization had their origin in Greece, this means that, to study them properly, we must get back to Greece. So it is with the literature of modern countries, or their philosophy, or their art; we cannot study them with the determination to get to the bottom and understand them without the way pointing eventually back to Greece.
When we think of the debt which mankind owes to the Greeks, we are apt to think too exclusively of the masterpieces in literature and art which they have left us. But the Greek genius was many-sided; the Greek, with his insatiable love of knowledge, his determination to see things as they are and to see them whole, his burning desire to be able to give a rational explanation of everything in heaven and earth, was just as irresistibly driven to natural science, mathematics, and exact reasoning in general, or logic.
To quote from a brilliant review of a well-known work: ‘To be a Greek was to seek to know, to know the primordial substance of matter, to know the meaning of number, to know the world as a rational whole. In no spirit of paradox one may say that Euclid is the most typical Greek: he would know to the bottom, and know as a rational system, the laws of the measurement of the earth. Plato, too, loved geometry and the wonders of numbers; he was essentially Greek because he was essentially mathematical. … And if one thus finds the Greek genius in Euclid and the Posterior Analytics, one will understand the motto written over the Academy, μηδεις αγεωμετρητος εισιτω. To know what the Greek genius meant you must (if one may speak εν αινιγματι) begin with geometry.’
Mathematics, indeed, plays an important part in Greek philosophy: there are, for example, many passages in Plato and Aristotle for the interpretation of which some knowledge of the technique of Greek mathematics is the first essential. Hence it should be part of the equipment of every classical student that he should have read substantial portions of the works of the Greek mathematicians in the original, say, some of the early books of Euclid in full and the definitions (at least) of the other books, as well as selections from other writers. Von Wilamowitz-Moellendorff has included in his Griechisches Lesebuch extracts from Euclid, Archimedes and Heron of Alexandria; and the example should be followed in this country.
Acquaintance with the original works of the Greek mathematicians is no less necessary for any mathematician worthy of the name. Mathematics is a Greek science. So far as pure geometry is concerned, the mathematician’s technical equipment is almost wholly Greek. The Greeks laid down the principles, fixed the terminology and invented the methods ab initio; moreover, they did this with such certainty that in the centuries which have since elapsed there has been no need to reconstruct, still less to reject as unsound, any essential part of their doctrine.
Consider first the terminology of mathematics. Almost all the standard terms are Greek or Latin translations from the Greek, and, although the mathematician may be taught their meaning without knowing Greek, he will certainly grasp their significance better if he knows them as they arise and as part of the living language of the men who invented them. Take the word isosceles; a schoolboy can be shown what an isosceles triangle is, but, if he knows nothing of the derivation, he will wonder why such an apparently outlandish term should be necessary to express so simple an idea. But if the mere appearance of the word shows him that it means a thing with equal legs, being compounded of ισος, equal, and σκελος, a leg, he will understand its appropriateness and will have no difficulty in remembering it. Equilateral, on the other hand, is borrowed from the Latin, but it is merely the Latin translation of the Greek ισοπλευρος, equal-sided. Parallelogram again can be explained to a Greekless person, but it will be far better understood by one who sees in it the two words παραλληλος and γραμμη and realizes that it is a short way of expressing that the figure in question is contained by parallel lines; and we shall best understand the word parallel itself if we see in it the statement of the fact that the two straight lines so described go alongside one another, παρ’ αλληλας, all the way. Similarly a mathematician should know that a rhombus is so called from its resemblance to a form of spinning-top (ῥομβος from ῥεμβω, to spin) and that, just as a parallelogram is a figure formed by two pairs of parallel straight lines, so a parallelepiped is a solid figure bounded by three pairs of parallel planes (παραλληλος, parallel, and επιπεδος, plane); incidentally, in the latter case, he will be saved from writing ‘parallelopiped’, a monstrosity which has disfigured not a few textbooks of geometry. Another good example is the word hypotenuse; it comes from the verb ὑποτεινειν (c. ὑπο and acc. or simple acc.), to stretch under, or, in its Latin 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