What Have the Greeks Done for Modern Civilisation?

by J. P. Mahaffy

But I must not attempt to give you a lecture on the Elements of Euclid, of which some of you may have evil recollections. For it is the misfortune, as well as the glory, of a great work not only to be repeated for centuries, but to be parroted and travestied by those who merely accept its greatness from the voice of ages, and who come to think that the words of inspiration only require blind repetition to instruct men. So if Euclid has become in many classical schools a sort of amulet or fetish (which must for common decency be put in the programme but which may be learned by committing the proofs to memory without any intelligence) such a misfortune is not the fault of Euclid, but the most pathetic tribute to his genius. Let me also add, for the benefit of those of you who have never seen more than six books of the Elements, and who probably thought six more than enough, that these are but the introduction to the discussion of higher and more complex questions, which show the large advance made by the Greeks in this science, and which explain also how in other arts, such as architecture, there is no defect for want of scientific accuracy. Books VII-X are not on geometry, but on higher arithmetic, and even treat, as in Book X, of incommensurable or irrational quantities. With XI he begins to teach solid geometry, the measurement of pyramids, cones, spheres, and the like, ending (XIII) with the discussion of the five regular polyhedra, of which Plato had long since spoken.

From the great sequence of discoverers and teachers of pure mathematics, I need only here pick out three immortal names: Apollonius of Perga, living about 200 B.C., whose geometrical treatment of conic sections is, I am informed, a splendid monument of genius, which would still be the basis of modern study had not the treatment of these figures by analysis entirely superseded the geometrical method. Then there is Pappus in the second century A.D., who gives us in eight books a review of all the previous masters, with important additions of his own. The third name is Diophantus, who lived much later, perhaps in the fourth century, and whose work is considered the first great step toward the science of Algebra.

All these speculations were developed in the direction of mathematical physics by Archimedes, Heron, and other great men of the Alexandrian school. The triumphs of Archimedes in mechanics astonished the Romans, who, in the defence of Syracuse against their attack, found him equal to a host. But how little Archimedes confined himself to practical problems is shown by his famous method of determining the area of a circle by approximation, by inscribing and circumscribing polygons of a great number of sides, which can of course be treated and measured as a complex of triangles. This is still, I am told, the proof admitted by modern mathematicians as the best.

The works of Heron show not only an excellent practical knowledge of mechanics, but of hydrostatics, from which he deduces a number of most ingenious inventions, such as our penny in the slot, and even the construction of a whole scene acted by marionettes moving by a most elaborate hidden machinery. It[35] is a fine specimen of his ingenuity in using the ordinary mechanical contrivances. He postulates a tall hollow basis, adorned with pilasters, and having an architrave, with boards covering its upper surface. Over this stands a little round temple, visible from all sides, with six pillars. It is covered with a conical roof, and on the apex is a figure of Victory with outspread wings and holding in her right hand a garland. Under the centre of the roof stands a figure of Bacchus, holding a thyrsus in his left hand, and a cup in his right. At his feet lies a little stuffed panther. Before and behind Bacchus, and outside the temple, stands an altar with dry shavings of wood. Also on each side, outside the temple, a Bacchante, in a proper costume and attitude. The whole concern being set up at some suitable spot, the exhibitor will retire, and the automatic machine will presently move forward to a fixed spot. The moment it stops, the altar fire in front of Bacchus will light up, and from his thyrsus will flow milk or water, and from his cup wine will be poured out on the panther beneath him, the pilasters beneath will be adorned with garlands, the Bacchantes will dance round the temple; drums and cymbals will be heard. When this noise ceases, the figure of Bacchus will turn round to the other altar and all the movements be repeated in the other direction. As soon as this has happened the second time, the show is over, and the whole machine will return to its original place. We have felt bound, he adds, to make the measurements (which he gives) small, for if made large, the suspicion naturally arises in the audience that there is a man inside the machine producing all the movements. This precaution, then, should be observed in making any automatic machine.

He then proceeds to give in great detail the construction of this machine. It is as ingenious as any construction of the present day, but cannot be presented to you without a series of figures, which are given in his book. Any of you may read it in the Greek (Teubner text), to which is added an excellent German translation. It will be enough to mention that the lighting of the altar fires is done by concealing a lamp inside the altar immediately under the wood, and by withdrawing a metal plate which separates them. The flowing of milk and wine is produced by concealing two little reservoirs in the summit of the building, and leading the liquor by pipes down the inside of the pillars, and up the inside of the figure of Bacchus, so that, when the cocks are turned by machinery, the milk and wine flow and rise to the level of the thyrsus and the cup, which are set underneath the level of the cisterns. It is evident enough that people who could do these things were capable of inventing the sakia now in use throughout Egypt, where a horizontal wheel worked round a capstan by oxen moves another set perpendicularly, at right angles to it, furnished with jars, which get filled below and, when they pass over the highest point of their revolution, are emptied into a water course, and so irrigate a higher level. This is well known to have been the invention of these Alexandrian mechanicians, whose theory had long preceded their practice, and whose applications of science they never valued so highly as their pure speculations.

Perhaps before leaving the subject I should tell you what was the moving force in the automatic machinery. It was a weight suspended in the air by a rope over a pulley, which, as soon as it was allowed to sink from its support, made the rope, wrapped round the axle of a large wheel, move the wheel, that was in its turn connected with other wheels. With very great and ingenious contrivance, as the machinery was all carefully concealed, the exhibitor could take his seat among the spectators, and make the ignorant believe that the whole effect was produced by some magic.

Nor were the laws of optics and the correction of the illusions of sight neglected. Euclid wrote a work on the subject which is now lost; but the praise of it by competent men of the Alexandrian school shows that it was on a level with his other scientific productions. To our educated public, the work of the Greeks in most fields is known at least by hearsay; the great library of Greek mathematics, scores of volumes, some of which are only quite recently published, is, except for Euclid, absolutely unknown. Yet from it is derived not only the scanty knowledge of science that filtered through the Romans into Western Europe, but also that adopted by the Arabs, and which in translations from Arabic versions came from them into awakening Italy and Germany and France. But let me add that now, when their discoveries in pure mathematics are being weighed by the light of expert knowledge, we are assured by all those really competent to judge that in no field of learning have the old Greeks shown their amazing originality and acuteness more signally than in higher arithmetic and in higher geometry.

The great fathers of the exact sciences are therefore in arithmetic the Pythagoreans, whose history is too obscure to mention from it any single name before Archytas, Euclid, and Theon of Smyrna; in geometry, Euclid; in mechanics, Archimedes; in conic sections, Apollonius of Perga; in hydrostatics, Heron; in astronomy, Eudoxus and Hipparchus; last, but not least, in higher arithmetic and algebra, Diophantus; all of these were, moreover, men who did not confine themselves to any single department, but promoted accurate thinking in many. These, and others hardly less great, have left a record and a legacy to posterity second to none in its mighty consequences.

But among them all Aristotle stands out as the “master of those that knew”—the man who attained in the Middle Ages such celebrity and authority that he narrowly escaped being canonised as a saint in the Roman calendar. If that distinction really belonged to the benefactors of mankind, I know not that any man ever lived who had a better claim to it. For his life and activity mark an epoch not only in the progress of many sciences, but in the general culture of the human mind, to which I know no parallel. He was brought up under the influence of the Socratic method of inquiry as perfected by Plato, but, though in some popular works (now lost) he adopted the dialogue as the correct method of teaching, there can be no doubt that the sober and practical tone of his mind made him despise all the delays and delights of character-drawing, and of spinning out the subject, for what we have from him is pre-eminently plain and scientific in form. There is seldom an unnecessary sentence; if there be a metaphor, it is a mere flash of colour across the cold severity of his argument. He writes like a man who had no time to waste and a vast world of subjects to teach. If it was still an age when the sciences had not entered upon the path of observation and experiment, but were philosophical speculations, Aristotle did more than any man to establish a separation between philosophy and science, while fully recognising, what in our day most scientists ignore, that positive science without a sound knowledge of philosophy is apt to run into fatal mistakes.

Of course this immense programme which Aristotle set before him could not be carried out without large collaboration, and so we know that, as Plato seems to have underrated such collaboration, and thus have failed in fruitfulness among his pupils, Aristotle, who was not chosen as his successor by the school (I suppose as usual there were jealousies among the commonplace and docile pupils toward the great original thinker), formed and stimulated a band of helpers, who gathered special observations in botany, mineralogy, zoölogy, physics as the science of nature, and others who put into shape his views on rhetoric and on poetry, on ethics and on theology. We have, in my opinion, a new specimen of such delegated work in the now famous Constitution of Athens, which was known and quoted as Aristotle’s through later antiquity, but which is rather the work of a pupil and not a brilliant one. But then we know that Aristotle either wrote or brought out 158 of these tracts on Greek constitutions. To this I shall return in a subsequent lecture.