The Legacy of GREECE
Essays by Gilbert Murray, W. R. Inge, J. Burnet, Sir T. L. Heath, D’Arcy W. Thompson, Charles Singer, R. W. Livingston, A. Toynbee, A. E. Zimmern, Percy Gardner, Sir Reginald Blomfield
Edited by
R. W. LIVINGSTONE

OXFORD
AT THE CLARENDON PRESS

PRINTED IN ENGLAND AT THE
UNIVERSITY PRESS, OXFORD
BY JOHN JOHNSON
PRINTER TO THE UNIVERSITY

Transcriber’s Note

Short fragments of Greek text have a thin dotted blue underline. The transliterated version appears in a transient pop-up box when the mouse hovers over the words.

Longer Greek phrases and poems are followed by the transliterated version in braces.

In spite of many differences, no age has had closer affinities with Ancient Greece than our own; none has based its deeper life so largely on ideals which the Greeks brought into the world. History does not repeat itself. Yet, if the twentieth century searched through the past for its nearest spiritual kin, it is in the fifth and following centuries before Christ that they would be found. Again and again, as we study Greek thought and literature, behind the veil woven by time and distance, the face that meets us is our own, younger, with fewer lines and wrinkles on its features and with more definite and deliberate purpose in its eyes. For these reasons we are to-day in a position, as no other age has been, to understand Ancient Greece, to learn the lessons it teaches, and, in studying the ideals and fortunes of men with whom we have so much in common, to gain a fuller power of understanding and estimating our own. This book—the first of its kind in English—aims at giving some idea of what the world owes to Greece in various realms of the spirit and the intellect, and of what it can still learn from her.

The Editor.

October 1921.

CONTENTS

THE VALUE OF GREECE TO THE FUTURE OF THE WORLD

If the value of man’s life on earth is to be measured in dollars and miles and horse-power, ancient Greece must count as a poverty-stricken and a minute territory; its engines and implements were nearer to the spear and bow of the savage than to our own telegraph and aeroplane. Even if we neglect merely material things and take as our standard the actual achievements of the race in conduct and in knowledge, the average clerk who goes to town daily, idly glancing at his morning newspaper, is probably a better behaved and infinitely better informed person than the average Athenian who sat spellbound at the tragedies of Aeschylus. It is only by the standard of the spirit, to which the thing achieved is little and the quality of mind that achieved it much, which cares less for the sum of knowledge attained than for the love of knowledge, less for much good policing than for one free act of heroism, that the great age of Greece can be judged as something extraordinary and unique in value.

By this standard, if it is a legitimate and reasonable one to apply, we shall be able to understand why classical Greek literature was the basis of education throughout all later antiquity; why its re-discovery, however fragmentary and however imperfectly understood, was able to intoxicate the keenest minds of Europe and constitute a kind of spiritual ‘Re-birth’, and how its further and further exploration may be still a task worth men’s spending their lives upon and capable of giving mankind guidance as well as inspiration.

But is such a standard legitimate and reasonable? We shall gain nothing by unanalysed phrases. But I think surely it is merely the natural standard of any philosophical historian. Suppose it is argued that an average optician at the present day knows more optics than Roger Bacon, the inventor of spectacles; suppose it is argued that therefore he is, as far as optics go, a greater man, and that Roger Bacon has nothing to teach us; what is the answer? It is, I suppose, that Roger Bacon, receiving a certain amount of knowledge from his teachers, had that in him which turned it to unsuspected directions and made it immensely greater and more fruitful. The average optician has probably added a little to what he was taught, but not much, and has doubtless forgotten or confused a good deal. So that, if by studying Roger Bacon’s life or his books we could get into touch with his mind and acquire some of that special moving and inspiring quality of his, it would help us far more than would the mere knowledge of the optician.

This truth is no doubt hard to see in the case of purely technical science; in books of wider range, such as Darwin’s for instance, it is easy for any reader to feel the presence of a really great mind, producing inspiration of a different sort from that of the most excellent up-to-date examination text-book. In philosophy, religion, poetry, and the highest kinds of art, the greatness of the author’s mind seems as a rule to be all that matters; one almost ignores the date at which he worked. This is because in technical sciences the element of mere fact, or mere knowledge, is so enormous, the elements of imagination, character, and the like so very small. Hence, books on science, in a progressive age, very quickly become ‘out of date’, and each new edition usually supersedes the last. It is the rarest thing for a work of science to survive as a text-book more than ten years or so. Newton’s Principia is almost an isolated instance among modern writings.

Yet there are some few such books. Up till about the year 1900 the elements of geometry were regularly taught, throughout Europe, in a text-book written by a Greek called Eucleides in the fourth or third century B. C.[1] That text-book lasted over two thousand years. Now, of course, people have discovered a number of faults in Euclid, but it has taken them all that time to do it.

Again, I knew an old gentleman who told me that, at a good English school in the early nineteenth century, he had been taught the principles of grammar out of a writer called Dionysius Thrax, or Denis of Thrace. Denis was a Greek of the first century B. C., who made or carried out the remarkable discovery that there was such a thing as a science of grammar, i. e. that men in their daily speech were unconsciously obeying an extraordinarily subtle and intricate body of laws, which were capable of being studied and reduced to order. Denis did not make the whole discovery himself; he was led to it by his master Aristarchus and others. And his book had been re-edited several times in the nineteen-hundred odd years before this old gentleman was taught it.

To take a third case: all through later antiquity and the middle ages the science of medicine was based on the writings of two ancient doctors, Hippocrates and Galen. Galen was a Greek who lived at Rome in the early Empire, Hippocrates a Greek who lived at the island of Cos in the fifth century B. C. A great part of the history of modern medicine is a story of emancipation from the dead hand of these great ancients. But one little treatise attributed to Hippocrates was in active use in the training of medical students in my own day in Scotland and is still in use in some American Universities. It was the Oath taken by medical students in the classic age of Greece when they solemnly faced the duties of their profession. The disciple swore to honour and obey his teacher and care for his children if ever they were in need; always to help his patients to the best of his power; never to use or profess to use magic or charms or any supernatural means; never to supply poison or perform illegal operations; never to abuse the special position of intimacy which a doctor naturally obtains in a sick house, but always on entering to remember that he goes as a friend and helper to every individual in it.

We have given up that oath now: I suppose we do not believe so much in the value of oaths. But the man who first drew up that oath did a great deed. He realized and defined the meaning of his high calling in words which doctors of unknown tongues and undiscovered countries accepted from him and felt to express their aims for well over two thousand years.

Now what do I want to illustrate by these three instances? The rapidity with which we are now at last throwing off the last vestiges of the yoke of Greece? No, not that. I want to point out that even in the realm of science, where progress is so swift and books so short-lived, the Greeks of the great age had such genius and vitality that their books lived in a way that no others have lived. Let us get away from the thought of Euclid as an inky and imperfect English school-book, to that ancient Eucleides who, with exceedingly few books but a large table of sand let into the floor, planned and discovered and put together and re-shaped the first laws of geometry, till at last he had written one of the great simple books of the world, a book which should stand a pillar and beacon to mankind long after all the political world that Eucleides knew had been swept away and the kings he served were conquered by the Romans, and the Romans in course of time conquered by the barbarians, and the barbarians themselves, with much labour and reluctance, partly by means of Eucleides’ book, eventually educated; so that at last, in our own day, they can manage to learn their geometry without it. The time has come for Euclid to be superseded; let him go. He has surely held the torch for mankind long enough; and books of science are born to be superseded. What I want to suggest is that the same extraordinary vitality of mind which made Hippocrates and Euclid and even Denis of Thrace last their two thousand years, was also put by the Greeks of the great age into those activities which are, for the most part at any rate, not perishable or progressive but eternal.

This is a simple point, but it is so important that we must dwell on it for a moment. If we read an old treatise on medicine or mechanics, we may admire it and feel it a work of genius, but we also feel that it is obsolete: its work is over; we have got beyond it. But when we read Homer or Aeschylus, if once we have the power to admire and understand their writing, we do not for the most part have any feeling of having got beyond them. We have done so no doubt in all kinds of minor things, in general knowledge, in details of technique, in civilization and the like; but hardly any sensible person ever imagines that he has got beyond their essential quality, the quality that has made them great.

Doubtless there is in every art an element of mere knowledge or science, and that element is progressive. But there is another element, too, which does not depend on knowledge and which does not progress but has a kind of stationary and eternal value, like the beauty of the dawn, or the love of a mother for her child, or the joy of a young animal in being alive, or the courage of a martyr facing torment. We cannot for all our progress get beyond these things; there they stand, like light upon the mountains. The only question is whether we can rise to them. And it is the same with all the greatest births of human imagination. As far as we can speculate, there is not the faintest probability of any poet ever setting to work on, let us say, the essential effect aimed at by Aeschylus in the Cassandra-scene of the Agamemnon, and doing it better than Aeschylus. The only thing which the human race has to do with that scene is to understand it and get out of it all the joy and emotion and wonder that it contains.

This eternal quality is perhaps clearest in poetry: in poetry the mixture of knowledge matters less. In art there is a constant development of tools and media and technical processes. The modern artist can feel that, though he cannot, perhaps, make as good a statue as Pheidias, he could here and there have taught Pheidias something: and at any rate he can try his art on subjects far more varied and more stimulating to his imagination. In philosophy the mixture is more subtle and more profound. Philosophy always depends in some sense upon science, yet the best philosophy seems generally to have in it some eternal quality of creative imagination. Plato wrote a dialogue about the constitution of the world, the Timaeus, which was highly influential in later Greece, but seems to us, with our vastly superior scientific knowledge, almost nonsensical. Yet when Plato writes about the theory of knowledge or the ultimate meaning of Justice or of Love, no good philosopher can afford to leave him aside: the chief question is whether we can rise to the height and subtlety of his thought.

And here another point emerges, equally simple and equally important if we are to understand our relation to the past. Suppose a man says: ‘I quite understand that Plato or Aeschylus may have had fine ideas, but surely anything of value which they said must long before this have become common property. There is no need to go back to the Greeks for it. We do not go back and read Copernicus to learn that the earth goes round the sun.’ What is the answer? It is that such a view ignores exactly this difference between the progressive and the eternal, between knowledge and imagination. If Harvey discovers that the blood is not stationary but circulates, if Copernicus discovers that the earth goes round the sun and not the sun round the earth, those discoveries can easily be communicated in the most abbreviated form. If a mechanic invents an improvement on the telephone, or a social reformer puts some good usage in the place of a bad one, in a few years we shall probably all be using the improvement without even knowing what it is or saying Thank you. We may be as stupid as we like, we have in a sense got the good of it.

But can one apply the same process to Macbeth or Romeo and Juliet? Can any one tell us in a few words what they come to? Or can a person get the good of them in any way except one—the way of vivid and loving study, following and feeling the author’s meaning all through? To suppose, as I believe some people do, that you can get the value of a great poem by studying an abstract of it in an encyclopaedia or by reading cursorily an average translation of it, argues really a kind of mental deficiency, like deafness or colour-blindness. The things that we have called eternal, the things of the spirit and the imagination, always seem to lie more in a process than in a result, and can only be reached and enjoyed by somehow going through the process again. If the value of a particular walk lies in the scenery, you do not get that value by taking a short cut or using a fast motor-car.

In looking back, then, upon any vital and significant age of the past we shall find objects of two kinds. First, there will be things like the Venus of Milo or the Book of Job or Plato’s Republic, which are interesting or precious in themselves, because of their own inherent qualities; secondly, there will be things like the Roman code of the Twelve Tables or the invention of the printing-press or the record of certain great battles, which are interesting chiefly because they are causes of other and greater things or form knots in the great web of history—the first having artistic interest, the second only historical interest, though, of course, it is obvious that in any concrete case there is generally a mixture of both.

Now Ancient Greece is important in both ways. For the artist or poet it has in a quite extraordinary degree the quality of beauty. For instance, to take a contrast with Rome: if you dig about the Roman Wall in Cumberland you will find quantities of objects, altars, inscriptions, figurines, weapons, boots and shoes, which are full of historic interest but are not much more beautiful than the contents of a modern rubbish heap. And the same is true of most excavations all over the world. But if you dig at any classical or sub-classical site in the Greek world, however unimportant historically, practically every object you find will be beautiful. The wall itself will be beautiful; the inscriptions will be beautifully cut; the figurines, however cheap and simple, may have some intentional grotesques among them, but the rest will have a special truthfulness and grace; the vases will be of good shapes and the patterns will be beautiful patterns. If you happen to dig in a burying-place and come across some epitaphs on the dead, they will practically all—even when the verses do not quite scan and the words are wrongly spelt—have about them this inexplicable touch of beauty.

I am anxious not to write nonsense about this. One could prove the point in detail by taking any collection of Greek epitaphs, and that is the only way in which it can be proved. The beauty is a fact, and if we try to analyse the sources of it we shall perhaps in part understand how it has come to pass.

In the first place, it is not a beauty of ornament; it is a beauty of structure, a beauty of rightness and simplicity. Compare an athlete in flannels playing tennis and a stout dignitary smothered in gold robes. Or compare a good modern yacht, swift, lithe, and plain, with a lumbering heavily gilded sixteenth-century galleon, or even with a Chinese state junk: the yacht is far the more beautiful though she has not a hundredth part of the ornament. It is she herself that is beautiful, because her lines and structure are right. The others are essentially clumsy and, therefore, ugly things, dabbed over with gold and paint. Now ancient Greek things for the most part have the beauty of the yacht. The Greeks used paint a good deal, but apart from that a Greek temple is almost as plain as a shed: people accustomed to arabesques and stained glass and gargoyles can very often see nothing in it. A Greek statue has as a rule no ornament at all: a young man racing or praying, an old man thinking, there it stands expressed in a stately and simple convention, true or false, the anatomy and the surfaces right or wrong, aiming at no beauty except the truest. It would probably seem quite dull to the maker of a mediaeval wooden figure of a king which I remember seeing in a town in the east of Europe: a crown blazing with many-coloured glass, a long crimson robe covered with ornaments and beneath them an idiot face, no bones, no muscles, no attitude. That is not what a Greek meant by beauty. The same quality holds to a great extent of Greek poetry. Not, of course, that the artistic convention was the same, or at all similar, for treating stone and for treating language. Greek poetry is statuesque in the sense that it depends greatly on its organic structure; it is not in the least so in the sense of being cold or colourless or stiff. But Greek poetry on the whole has a bareness and severity which disappoints a modern reader, accustomed as he is to lavish ornament and exaggeration at every turn. It has the same simplicity and straightforwardness as Greek sculpture. The poet has something to say and he says it as well and truly as he can in the suitable style, and if you are not interested you are not. With some exceptions which explain themselves he does not play a thousand pretty tricks and antics on the way, so that you may forget the dullness of what he says in amusement at the draperies in which he wraps it.

But here comes an apparent difficulty. Greek poetry, we say, is very direct, very simple, very free from irrelevant ornament. And yet when we translate it into English and look at our translation, our main feeling, I think, is that somehow the glory has gone: a thing that was high and lordly has become poor and mean. Any decent Greek scholar when he opens one of his ancient poets feels at once the presence of something lofty and rare—something like the atmosphere of Paradise Lost. But the language of Paradise Lost is elaborately twisted and embellished into loftiness and rarity; the language of the Greek poem is simple and direct. What does this mean?

I can only suppose that the normal language of Greek poetry is in itself in some sense sublime. Most critics accept this as an obvious fact, yet, if true, it is a very strange fact and worth thinking about. It depends partly on mere euphony: Khaireis horôn fôs is probably more beautiful in sound than ‘You rejoice to see the light’, but euphony cannot be everything. The sound of a great deal of Greek poetry, either as we pronounce it, or as the ancients pronounced it, is to modern ears almost ugly. It depends partly, perhaps, on the actual structure of the Greek language: philologists tell us that, viewed as a specimen, it is in structure and growth and in power of expressing things, the most perfect language they know. And certainly one often finds that a thought can be expressed with ease and grace in Greek which becomes clumsy and involved in Latin, English, French or German. But neither of these causes goes, I think, to the root of the matter.

What is it that gives words their character and makes a style high or low? Obviously, their associations; the company they habitually keep in the minds of those who use them. A word which belongs to the language of bars and billiard saloons will become permeated by the normal standard of mind prevalent in such places; a word which suggests Milton or Carlyle will have the flavour of those men’s minds about it. I therefore cannot resist the conclusion that, if the language of Greek poetry has, to those who know it intimately, this special quality of keen austere beauty, it is because the minds of the poets who used that language were habitually toned to a higher level both of intensity and of nobility than ours. It is a finer language because it expresses the minds of finer men. By ‘finer men’ I do not necessarily mean men who behaved better, either by our standards or by their own; I mean men to whom the fine things of the world, sunrise and sea and stars and the love of man for man, and strife and the facing of evil for the sake of good, and even common things like meat and drink, and evil things like hate and terror, had, as it were, a keener edge than they have for us and roused a swifter and a nobler reaction.

Let us resume this argument before going further. We start from the indisputable fact that the Greeks of about the fifth century B. C. did for some reason or other produce various works of art, buildings and statues and books, especially books, which instead of decently dying or falling out of fashion in the lifetime of the men who made them, lasted on and can still cause high thoughts and intense emotions. In trying to explain this strange fact we notice that the Greeks had a great and pervading instinct for beauty, and for beauty of a particular kind. It is a beauty which never lies in irrelevant ornament, but always in the very essence and structure of the object made. In literature we found that the special beauty which we call Greek depends partly on the directness, truthfulness, and simplicity with which the Greeks say what they want to say, and partly on a special keenness and nobility in the language, which seems to be the natural expression of keen and noble minds. Can we in any way put all these things together so as to explain them—or at any rate to hold them together more clearly?

An extremely old and often misleading metaphor will help us. People have said: ‘The world was young then.’ Of course, strictly speaking, it was not. In the total age of the world or of man the two thousand odd years between us and Pericles do not count for much. Nor can we imagine that a man of sixty felt any more juvenile in the fifth century B. C. than he does now. It was just the other way, because at that time there were no spectacles or false teeth. Yet in a sense the world was young then, at any rate our western world, the world of progress and humanity. For the beginnings of nearly all the great things that progressive minds now care for were then being laid in Greece.

Youth, perhaps, is not exactly the right word. There are certain plants—some kinds of aloe, for instance—which continue for an indefinite number of years in a slow routine of ordinary life close to the ground, and then suddenly, when they have stored enough vital force, grow ten feet high and burst into flower, after which, no doubt, they die or show signs of exhaustion. Apart from the dying, it seems as if something like that happened from time to time to the human race, or to such parts of it as really bear flowers at all. For most races and nations during the most of their life are not progressive but simply stagnant, sometimes just managing to preserve their standard customs, sometimes slipping back to the slough. That is why history has nothing to say about them. The history of the world consists mostly in the memory of those ages, quite few in number, in which some part of the world has risen above itself and burst into flower or fruit.

We ourselves happen to live in the midst or possibly in the close of one such period. More change has probably taken place in daily life, in ideas, and in the general aspect of the earth during the last century than during any four other centuries since the Christian era: and this fact has tended to make us look on rapid progress as a normal condition of the human race, which it never has been. And another such period of bloom, a bloom comparatively short in time and narrow in area, but amazingly swift and intense, occurred in the lower parts of the Balkan peninsula from about the sixth to the fourth centuries before Christ.

Now it is this kind of bloom which fills the world with hope and therefore makes it young. Take a man who has just made a discovery or an invention, a man happily in love, a man who is starting some great and successful social movement, a man who is writing a book or painting a picture which he knows to be good; take men who have been fighting in some great cause which before they fought seemed to be hopeless and now is triumphant; think of England when the Armada was just defeated, France at the first dawn of the Revolution, America after Yorktown: such men and nations will be above themselves. Their powers will be stronger and keener; there will be exhilaration in the air, a sense of walking in new paths, of dawning hopes and untried possibilities, a confidence that all things can be won if only we try hard enough. In that sense the world will be young. In that sense I think it was young in the time of Themistocles and Aeschylus. And it is that youth which is half the secret of the Greek spirit.

And here I may meet an objection that has perhaps been lurking in the minds of many readers. ‘All this,’ they may say, ‘professes to be a simple analysis of known facts, but in reality is sheer idealization. These Greeks whom you call so “noble” have been long since exposed. Anthropology has turned its searchlights upon them. It is not only their ploughs, their weapons, their musical instruments, and their painted idols that resemble those of the savages; it is everything else about them. Many of them were sunk in the most degrading superstitions: many practised unnatural vices: in times of great fear some were apt to think that the best “medicine” was a human sacrifice. After that, it is hardly worth mentioning that their social structure was largely based on slavery; that they lived in petty little towns, like so many wasps’ nests, each at war with its next-door neighbour, and half of them at war with themselves!’

If our anti-Greek went further he would probably cease to speak the truth. We will stop him while we can still agree with him. These charges are on the whole true, and, if we are to understand what Greece means, we must realize and digest them. We must keep hold of two facts: first, that the Greeks of the fifth century produced some of the noblest poetry and art, the finest political thinking, the most vital philosophy, known to the world; second, that the people who heard and saw, nay perhaps, even the people who produced these wonders, were separated by a thin and precarious interval from the savage. Scratch a civilized Russian, they say, and you find a wild Tartar. Scratch an ancient Greek, and you hit, no doubt, on a very primitive and formidable being, somewhere between a Viking and a Polynesian.

That is just the magic and the wonder of it. The spiritual effort implied is so tremendous. We have read stories of savage chiefs converted by Christian or Buddhist missionaries, who within a year or so have turned from drunken corroborees and bloody witch-smellings to a life that is not only godly but even philanthropic and statesmanlike. We have seen the Japanese lately go through some centuries of normal growth in the space of a generation. But in all such examples men have only been following the teaching of a superior civilization, and after all, they have not ended by producing works of extraordinary and original genius. It seems quite clear that the Greeks owed exceedingly little to foreign influence. Even in their decay they were a race, as Professor Bury observes, accustomed ‘to take little and to give much’. They built up their civilization for themselves. We must listen with due attention to the critics who have pointed out all the remnants of savagery and superstition that they find in Greece: the slave-driver, the fetish-worshipper and the medicine-man, the trampler on women, the bloodthirsty hater of all outside his own town and party. But it is not those people that constitute Greece; those people can be found all over the historical world, commoner than blackberries. It is not anything fixed and stationary that constitutes Greece: what constitutes Greece is the movement which leads from all these to the Stoic or fifth-century ‘sophist’ who condemns and denies slavery, who has abolished all cruel superstitions and preaches some religion based on philosophy and humanity, who claims for women the same spiritual rights as for man, who looks on all human creatures as his brethren, and the world as ‘one great City of gods and men’. It is that movement which you will not find elsewhere, any more than the statues of Pheidias or the dialogues of Plato or the poems of Aeschylus and Euripides.

From all this two or three results follow. For one thing, being built up so swiftly, by such keen effort, and from so low a starting-point, Greek civilization was, amid all its glory, curiously unstable and full of flaws. Such flaws made it, of course, much worse for those who lived in it, but they hardly make it less interesting or instructive to those who study it. Rather the contrary. Again, the near neighbourhood of the savage gives to the Greek mind certain qualities which we of the safer and solider civilizations would give a great deal to possess. It springs swift and straight. It is never jaded. Its wonder and interest about the world are fresh. And lastly there is one curious and very important quality which, unless I am mistaken, belongs to Greek civilization more than to any other. To an extraordinary degree it starts clean from nature, with almost no entanglements of elaborate creeds and customs and traditions.

I am not, of course, forgetting the prehistoric Minoan civilization, nor yet the peculiar forms—mostly simple enough—into which the traditional Greek religion fell. It is possible that I may be a little misled by my own habit of living much among Greek things and so forgetting through long familiarity how odd some of them once seemed. But when all allowances are made, I think that this clean start from nature is, on the whole, a true claim. If a thoughtful European or American wants to study Chinese or Indian things, he has not only to learn certain data of history and mythology, he has to work his mind into a particular attitude; to put on, as it were, spectacles of a particular sort. If he wants to study mediaeval things, if he takes even so universal a poet as Dante, it is something the same. Curious views about the Pope and the emperor, a crabbed scholastic philosophy, a strange and to the modern mind rather horrible theology, floating upon the flames of Hell: all these have somehow to be taken into his imagination before he can understand his Dante. With Greek things this is very much less so. The historical and imaginative background of the various great poets and philosophers is, no doubt, highly important. A great part of the work of modern scholarship is now devoted to getting it clearer. But on the whole, putting aside for the moment the possible inadequacies of translation, Greek philosophy speaks straight to any human being who is willing to think simply, Greek art and poetry to any one who can use his imagination and enjoy beauty. He has not to put on the fetters or the blinkers of any new system in order to understand them; he has only to get rid of his own—a much more profitable and less troublesome task.

This particular conclusion will scarcely, I think, be disputed, but the point presents difficulties and must be dwelt upon.

In the first place, it does not mean that Greek art is what we call ‘naturalist’ or ‘realist’. It is markedly the reverse. Art to the Greek is always a form of Sophia, or Wisdom, a Technê with rules that have to be learnt. Its air of utter simplicity is deceptive. The pillar that looks merely straight is really a thing of subtle curves. The funeral bas-relief that seems to represent in the simplest possible manner a woman saying good-bye to her child is arranged, plane behind plane, with the most delicate skill and sometimes with deliberate falsification of perspective. There is always some convention, some idealization, some touch of the light that never was on sea or land. Yet all the time, I think, Greek art remains in a remarkable degree close to nature. The artist’s eye is always on the object, and, though he represents it in his own style, that style is always normal and temperate, free from affectation, free from exaggeration or morbidity and, in the earlier periods, free from conventionality. It is art without doubt; but it is natural and normal art, such as grew spontaneously when mankind first tried in freedom to express beauty. For example, the language of Greek poetry is markedly different from that of prose, and there are even clear differences of language between different styles of poetry. And further, the poetry is very seldom about the present. It is about the past, and that an ideal past. What we have to notice there is that this kind of rule, which has been usual in all great ages of poetry, is apparently not an artificial or arbitrary thing but a tendency that grew up naturally with the first great expressions of poetical feeling.

Furthermore, this closeness to nature, this absence of a unifying or hide-bound system of thought, acting together with other causes, has led to the extraordinary variety and many-sidedness which is one of the most puzzling charms of Ancient Greece as contrasted, say, with Israel or Assyria or early Rome. Geographically it is a small country with a highly indented coast-line and an interior cut into a great number of almost isolated valleys. Politically it was a confused unity made up of numerous independent states, one walled city of a few thousand inhabitants being quite enough to form a state. And the citizens of these states were, each of them, rather excessively capable of forming opinions of their own and fighting for them. Hence came in practice much isolation and faction and general weakness, to the detriment of the Greeks themselves; but the same cause led in thought and literature to immense variety and vitality, to the great gain of us who study the Greeks afterwards. There is hardly any type of thought or style of writing which cannot be paralleled in ancient Greece, only they will there be seen, as it were, in their earlier and simpler forms. Traces of all the things that seem most un-Greek can be found somewhere in Greek literature: voluptuousness, asceticism, the worship of knowledge, the contempt for knowledge, atheism, pietism, the religion of serving the world and the religion of turning away from the world: all these and almost all other points of view one can think of are represented somewhere in the records of that one small people. And there is hardly any single generalization in this chapter which the author himself could not controvert by examples to the contrary. You feel in general a great absence of all fetters: the human mind free, rather inexperienced, intensely interested in life and full of hope, trying in every direction for that excellence which the Greeks called aretê, and guided by some peculiar instinct toward Temperance and Beauty.

The variety is there and must not be forgotten; yet amid the variety there are certain general or central characteristics, mostly due to this same quality of freshness and closeness to nature.

If you look at a Greek statue or bas-relief, or if you read an average piece of Aristotle, you will very likely at first feel bored. Why? Because it is all so normal and truthful; so singularly free from exaggeration, paradox, violent emphasis; so destitute of those fascinating by-forms of insanity which appeal to some similar faint element of insanity in ourselves. ‘We are sick’, we may exclaim, ‘of the sight of these handsome, perfectly healthy men with grave faces and normal bones and muscles! We are sick of being told that Virtue is a mean between two extremes and tends to make men happy! We shall not be interested unless some one tells us that Virtue is the utter abnegation of self, or, it may be, the extreme and ruthless assertion of self; or again, that Virtue is all an infamous mistake! And for statues, give us a haggard man with starved body and cavernous eyes, cursing God—or give us something rolling in fat and colour....’

What is at the back of this sort of feeling? which I admit often takes more reasonable forms than these I have suggested. It is the same psychological cause that brings about the changes of fashion in art or dress: which loves ‘stunts’ and makes the fortunes of yellow newspapers. It is boredom or ennui. We have had too much of A; we are sick of it, we know how it is done and despise it; give us some B, or better still some Z. And after a strong dose of Z we shall crave for the beginning of the alphabet again. But now think of a person who is not bored at all; who is, on the contrary, immensely interested in the world, keen to choose good things and reject bad ones; full of the desire for knowledge and the excitement of discovery. The joy to him is to see things as they are and to judge them normally. He is not bored by the sight of normal, healthy muscles in a healthy, well-shaped body; he is delighted. If you distort the muscles for emotional effect, he would say with disappointment: ‘But that is ugly!’ or ‘But a man’s muscles do not go like that!’ He will have noted that tears are salt and rather warm; but if you say like a modern poet that your heroine’s tears are ‘more hot than fire, more salt than the salt sea’, he will probably think your statement απιθανον ‘unpersuasive’, and therefore ψυχρον ‘chilling’.

It is perhaps especially in the religious and moral sphere that we are accustomed to the habitual use of ecstatic language: expressions that are only true of exalted moments are used by us as the commonplaces of ordinary life. ‘It is a thousand times worse to see another suffer than to suffer oneself.’ ‘True love only desires the happiness of the beloved object.’ This kind of ‘high falutin’’ has become part of our regular mental habit, just as dead metaphors by the bushel are a part of our daily language. Consequently we are a little chilled and disappointed by a language in which people hardly ever use a metaphor except when they vividly realize it, and never utter heroic sentiments except when they are wrought up to the pitch of feeling them true. Does this mean that the Greek always remains, so to speak, at a normal temperature, that he never has intense or blinding emotions? Not in the least. It shows a lack of faith in the value of life to imagine such a conclusion. It implies that you can only reach great emotion by pretence, or by habitually exaggerating small emotions, whereas probably the exact reverse is the case. When the great thing comes, then the Greek will have the great word and the great thought ready. It is the habitual exaggerator who will perhaps be bankrupt. And after all—the great things are sure to come!

The power of seeing things straight and knowing what is beautiful or noble, quite undisturbed by momentary boredoms or changes of taste, is a very rare gift and never perhaps possessed in full by any one. But there is a profound rule of art, bidding a man in the midst of all his study of various styles or his pursuit of his own peculiar imaginations, from time to time se retremper dans la nature—‘to steep himself again in nature’. And in something the same way it seems as if the world ought from time to time to steep itself again in Hellenism: that is, it ought, amid all the varying affectations and extravagances and changes of convention in art and letters, to have some careful regard for those which arose when man first awoke to the meaning of truth and beauty and saw the world freely as a new thing.

Is this exaggeration? I think not. But no full defence of it can be attempted here. In this essay we have been concerned almost entirely with the artistic interest of Greece. It would be equally possible to dwell on the historical interest. Then we should find that, for that branch of mankind which is responsible for western civilization, the seeds of almost all that we count best in human progress were sown in Greece. The conception of beauty as a joy in itself and as a guide in life was first and most vividly expressed in Greece, and the very laws by which things are beautiful or ugly were to a great extent discovered there and laid down. The conception of Freedom and Justice, freedom in body, in speech and in mind, justice between the strong and the weak, the rich and the poor, penetrates the whole of Greek political thought, and was, amid obvious flaws, actually realized to a remarkable degree in the best Greek communities. The conception of Truth as an end to pursue for its own sake, a thing to discover and puzzle out by experiment and imagination and especially by Reason, a conception essentially allied with that of Freedom and opposed both to anarchy and to blind obedience, has perhaps never in the world been more clearly grasped than by the early Greek writers on science and philosophy. One stands amazed sometimes at the perfect freedom of their thought. Another conception came rather later, when the small City States with exclusive rights of citizenship had been merged in a larger whole: the conception of the universal fellowship between man and man. Greece realized soon after the Persian war that she had a mission to the world, that Hellenism stood for the higher life of man as against barbarism, for Aretê, or Excellence, as against the mere effortless average. First came the crude patriotism which regarded every Greek as superior to every barbarian; then came reflection, showing that not all Greeks were true bearers of the light, nor all barbarians its enemies; that Hellenism was a thing of the spirit and not dependent on the race to which a man belonged or the place where he was born: then came the new word and conception ανθρωποτης, humanitas, which to the Stoics made the world as one brotherhood. No people known to history clearly formulated these ideals before the Greeks, and those who have spoken the words afterwards seem for the most part to be merely echoing the thoughts of old Greek men.

These ideas, the pursuit of Truth, Freedom, Beauty, Excellence are not everything. They have been a leaven of unrest in the world; they have held up a light which was not always comforting to the eyes to see. There is another ideal which is generally stronger and may, for all we know, in the end stamp them out as evil things. There is Submission instead of Freedom, the deadening or brutalizing of the senses instead of Beauty, the acceptance of tradition instead of the pursuit of Truth, the belief in hallucination or passion instead of Reason and Temperate Thought, the obscuring of distinctions between good and bad and the acceptance of all human beings and all states of mind as equal in value. If something of this kind should prove in the end to be right for man, then Greece will have played the part of the great wrecker in human history. She will have held up false lights which have lured our ship to dangerous places. But at any rate, through calm and storm, she does hold her lights; she lit them first of the nations and held them during her short reign the clearest; and whether we believe in an individual life founded on Freedom, Reason, Beauty, Excellence and the pursuit of Truth, and an international life aiming at the fellowship between man and man, or whether we think these ideals the great snares of human politics, there is good cause for some of us in each generation at the cost of some time and trouble to study such important forces where they first appear consciously in the minds of our spiritual ancestors. In the thought and art of ancient Greece, more than any other, we shall find these forces, and also to some extent their great opposites, fresh, clean and comparatively uncomplicated, with every vast issue wrought out on a small material scale and every problem stated in its lowest terms.

Gilbert Murray.

RELIGION

Those who write about the Greeks must beware of a heresy which is very rife just now—the theory of racialism. Political ethnology, which is no genuine science, excused the ambition of the Germans to themselves, and helped them to wage war; it has suggested to the Allies a method of waging peace. The false and mischievous doctrine of superior and inferior races is used to justify oppression in Europe, and murder by torture in America. It will not help us to understand the Greeks. The Greeks were a nation of splendid mongrels, made up of the same elements, differently mixed, as ourselves. Their famous beauty, which had almost disappeared when Cicero visited Athens, was mainly the result of a healthy outdoor life and physical training, combined with a very becoming costume. They were probably not handsomer than Oxford rowing crews or Eton boys. Their flowering time of genius was due to the same causes which produced similar results in the Italian Renaissance. The city-state is a forcing-house of brilliant achievement, though it quickly uses up its human material. We cannot even regard the Greeks as a homogeneous mixed race. The Spartiates were almost pure Nordics; the Athenians almost pure Mediterraneans. The early colonists, from whom sprang so many of the greatest names in the Hellenic roll of honour, are not likely to have kept their blood pure. Nor was there ever a Greek culture shared by all the Greeks. The Spartan system, that of a small fighting tribe encamped in a subject country, recalls that of Chaka’s Zulus; Arcadia was bucolic, Aetolia barbarous, Boeotia stolid, Macedonia half outside the pale. The consciousness of race among the Greeks counted practically for about as much as the consciousness of being white men, or Christians, does in modern civilization.

Greece for our purposes means not a race, but a culture, a language and literature, and still more an attitude towards life, which for us begins with Homer, and persists, with many changes but no breaks, till the closing of the Athenian lecture-rooms by Justinian. The changes no doubt were great, when politically Greece was living Greece no more, and when the bearers of the tradition were no longer the lineal descendants of those who established it. But the tradition, enshrined in literature, in monuments, and in social customs, survived. The civilization of the Roman Empire was not Italian but Greek. After the sixth century, Hellenism—the language, the literature, and the attitude towards life—was practically lost to the West for nearly a thousand years. It was recovered at the Renaissance, and from that time to this has been a potent element in western civilization. The Dark Ages, and the early Middle Ages, are the period during which the West was cut off from Hellenism. Yet even then the severance was not complete. For these were the ages of the Catholic theocracy; and if we had to choose one man as the founder of Catholicism as a theocratic system, we should have to name neither Augustine nor St. Paul, still less Jesus Christ, but Plato, who in the Laws sketches out with wonderful prescience the conditions for such a polity, and the form which it would be compelled to take. Even in speculative thought we know that Augustine owed much to the Platonists, the Schoolmen to Aristotle, the mystics to the pupil of Proclus whom they called Dionysius. Only Greek science, and the scientific spirit, were almost completely lost, and a beginning de novo had to be made when the West shook off its fetters.

Hellenism then is not the mind of a particular ethnic type, nor of a particular period. It was not destroyed, though it was emasculated, by the loss of political freedom; it was neither killed nor died a natural death. Its philosophy was continuous from Thales to Proclus, and again from Ficino and Pico to Lotze and Bradley, after a long sleep which was not death. Its religion passes into Christian theology and cultus without any real break. The early Church spoke in Greek and thought in Greek. In the days of Greek freedom to be a Greek had meant to be a citizen of a Greek canton; after Alexander it meant to have Greek culture. None of the great Stoics were natives of Greece proper; Zeno himself was a Semite. Of the later Greek writers, Marcus Aurelius was a Romanized Spaniard, Plotinus possibly a Copt, Porphyry and Lucian Syrians, Philo, St. Paul, and probably the Fourth Evangelist were Jews. These men all belong to the history of Greek culture. And if these were Greeks how shall we deny the name to Raphael and Michael Angelo, to Spenser and Sidney, to Keats and Shelley? When Blake wrote—

The sun’s light when he unfolds it,
Depends on the organ that beholds it,

he was summing up, not only the philosophy of the Lake Poets but the fundamental dogma of the maturest Greek thought. Would not Plato have rejoiced in Michael Angelo’s confession of faith, which Wordsworth has translated for us?

Heaven-born, the soul a heavenward course must hold;
Beyond the visible world she soars to seek
(For what delights the sense is false and weak)
Ideal Form, the universal mould.
The wise man, I affirm, can find no rest
In that which perishes; nor will he lend
His heart to aught that doth on time depend.

Has the highest aspect of Greek religion ever been better expressed than by Wordsworth himself, to whom, as to Blake, it came by inspiration and not from books?

While yet a child, and long before his time
Had he perceived the presence and the power
Of greatness; and deep feelings had impressed
So vividly great objects that they lay
Upon his mind like substances, whose presence
Perplexed the bodily sense.

The spirit of man does not live only on tradition; it can draw direct from the fountain-head. We are dealing with a permanent type of human culture, which is rightly named after the Greeks, since it attained its chief glory in the literature and art of the Hellenic cities, but which cannot be separated from western civilization as an alien importation. Without what we call our debt to Greece we should have neither our religion nor our philosophy nor our science nor our literature nor our education nor our politics. We should be mere barbarians. We need not speculate how much we might ultimately have discovered for ourselves. Our civilization is a tree which has its roots in Greece, or, to borrow a more appropriate metaphor from Clement of Alexandria, it is a river which has received affluents from every side; but its head waters are Greek. The continuity of Greek thought and practice in religion and religious philosophy is especially important, and it is necessary to emphasize it because the accident of our educational curriculum leaves in the minds of most students a broad chasm between the Stoics and the Christians, ignores the later Greek philosophy of religion altogether, and traces Christian dogma back to Palestine, with which it has very little connexion.

Our sense of continuity is dulled in another way. There is a tendency to isolate certain aspects of Hellenic life and thought as characteristic, and to stamp others, which are equally found among the ancient Greeks, as untypical and exceptional. In the sphere of religion, with which we are concerned in this essay, we are bidden to regard Plato and Euripides as rebels against the national tradition, and not as normal products of their age and country. I do not feel at liberty to pick and choose in this fashion. A national character may be best exemplified in its rebels, a religion in its heretics. If Nietzsche was right in calling Plato a Christian before Christ, I do not therefore regard him as an unhellenic Greek. Rather, I trace back to him, and so to Greece, the religion and the political philosophy of the Christian Church, and the Christian type of mysticism. If Euripides anticipated to an extraordinary degree the devout agnosticism, the vague pantheism, the humanitarian sentiment of the nineteenth (rather than of the twentieth) century, I do not consider that he was a freak in fifth-century Athens, but that Greece showed us the way even in paths where we have not been used to look to her for guidance. I am equally reluctant to assume, without evidence, that the later Platonism, whether we call it religion or philosophy, is unhellenic. It is quite unnecessary to look for Asiatic influences in a school which clung close to the Attic tradition. It is more to the purpose to show how a religious philosophy of mystical revelation and introspection grew naturally out of the older nature-philosophies, just as in our own day metaphysics and science have both been driven back upon the theory of knowledge and psychology. It should not be necessary to remind Hellenists that ‘Know thyself’ passed for the supreme word of wisdom in the classical period, or that Heracleitus revealed his method in the words ‘I searched myself’.

We shall come presently to certain parts of our modern heritage which are not Greek either by origin or by affinity. These will not be found in Euripides or Plato any more than in Herodotus or Sophocles. But some developments of religion which our Hellenists particularly dislike, and are therefore anxious to disclaim as alien to Greek thought and practice, such as asceticism, sacramental magic, religious persecution, and timid reliance on authority, are maladies of the Greek spirit, and came into the Church from Hellenistic and not from Jewish sources. It was Cleanthes who wished to treat Aristarchus as the Church treated Galileo, for anticipating Galileo’s discovery. It was Plutarch, or rather his revered father, who said, ‘You seem to me to be handling a very great and dangerous subject, or rather to be raising questions which ought not to be raised at all, when you question the opinion we hold about the gods, and ask reasons and proofs for everything. The ancient and ancestral faith is enough; and if on one point its fixed and traditional character be disturbed, it will be undermined and no one will trust it’. It is true that Celsus accused the Christians of saying, ‘Do not inquire; only believe.’ But this was not the attitude of Clement and Origen, still less of that most courageous pioneer St. Paul; it was rather the attitude of the average devout pagan. At this time the defence of popular superstition was no longer a matter of mere policy but of heartfelt need. Marcus Aurelius was a great immolator of white cows. The Christians were disliked, not as superstitious, but as impious. Alexander of Abunoteichos expelled ‘Christians and Epicureans’ by name from his séances. Lucian is the Voltaire of a credulous age. As for sacerdotal magic, Ovid explicitly ascribed the ex opere operato doctrine to the Greeks.

Graecia principium moris fuit; illa nocentes
impia lustratos ponere facta putat,
a nimium faciles, qui tristia crimina caedis
fluminea tolli posse putatis aqua.

The Christian Church was the last great creative achievement of the classical culture. It is neither Asiatic nor mediaeval in its essential character. It is not Asiatic; Christianity is the least Oriental of all the great religions. The Semites either shook it off and reverted to a Judaism purged of its Hellenic elements, or enrolled themselves with fervour under the banner of Islam, which Westcott called ‘a petrified Judaism’. Christian missions have had no success in any Asiatic country. Nor is there anything specifically mediaeval about Catholicism. It preserved the idea of Roman imperialism, after the secular empire of the West had disappeared, and even kept the tradition of the secular empire alive. It modelled all its machinery on the Roman Empire, and consecrated the Roman claim to universal dominion, with the Roman law of maiestas against all who disputed its authority. Even its favourite penalty of the ‘avenging flames’ is borrowed from the later Roman codes. It maintained the official language of antiquity, and the imperial title of the autocrat who reigned on the Seven Hills. Nor were the early Christians so anxious as is often supposed to disclaim this continuity. At first, it is true, their apologetic was directed to proving their continuity with Judaism; but Judaism ceased to count for much after the destruction of the Holy City in A. D. 70, and the second-century apologists appeal for toleration on the ground that the best Greek philosophers taught very much the same as what Christians believe. ‘We teach the same as the Greeks’, says Justin Martyr, ‘though we alone are hated for what we teach.’ ‘Some among us’, says Tertullian, ‘who are versed in ancient literature, have written books to prove that we have embraced no tenets for which we have not the support of common and public literature.’ ‘The teachings of Plato’, says Justin again, ‘are not alien to those of Christ; and the same is true of the Stoics.’ ‘Heracleitus and Socrates lived in accordance with the divine Logos’, and should be reckoned as Christians. Clement says that Plato wrote ‘by inspiration of God’. Augustine, much later, finds that ‘only a few words and phrases’ need be changed to bring Platonism into complete accord with Christianity. The ethics of contemporary paganism, as Harnack shows, with special reference to Porphyry, are almost identical with those of the Christians of his day. They differ in many points from the standards of 500 years earlier and from those of 1,500 years later, but the divergences are neither racial nor credal. Catholic Christianity is historically continuous with the old civilization, which indeed continued to live in this region after its other traditions and customs had been shattered. There are few other examples in history of so great a difference between appearance and reality. Outwardly, the continuity with Judaism seems to be unbroken, that with paganism to be broken. In reality, the opposite is the fact.

This most important truth has been obscured from many causes. The gap in history made by our educational tradition has been already mentioned. And our histories of the early Church are too often warped by an unfortunate bias. Christianity has been judged at its best, paganism at its worst. The rhetorical denunciations of writers like Seneca, Juvenal, and Tacitus are taken at their face value, and few have remembered the convention which obliged a satirist to be scathing, or the political prejudice of the Stoics against the monarchy, or the non-representative character of fashionable life in the capital. The modern Church historian, as Mr. Benn says, has gathered his experience in a college quadrangle or a cathedral close, and knows little enough about his own country, next to nothing about what morality was in the Middle Ages, and nothing at all about what it still is in many parts of Europe. In the most recent books, however, there is a real desire to hold the scales fairly, and Christianity has nothing to fear from an impartial judgement.

There is also an assumption, which we find even in such learned writers as Harnack and Hatch, that the Hellenic element in Christianity is an accretion which transformed the new religion from its original purity and half-paganized Europe again. They would like to prove that underneath Catholicism was a primitive Protestantism, which owed nothing to Greece. The truth is that the Church was half Greek from the first, though, as I shall say presently, the original Gospel was not. St. Paul was a Jew of the Dispersion, not of Palestine, and the Christianity to which he was converted was the Christianity of Stephen, not of James the Lord’s brother. His later epistles are steeped in the phraseology of the Greek mysteries. The Epistle to the Hebrews and the Fourth Gospel are unintelligible without some knowledge of Philo, whose theology is more Greek than Jewish. In the conflict about the nature of the future life, it was the Greek eschatology which prevailed over the Jewish. St. Paul’s famous declaration, ‘We look not at the things which are seen, but at the things which are not seen; for the things which are seen are temporal, but the things which are not seen are eternal’, is pure Platonism and quite alien to Jewish thought. Judaic Christianity was a local affair, and had a very short life.

Further, too much is made of the conflict between the official cults of paganism and Christian public worship. It is forgotten how completely, in Hellenistic times, religion and philosophy were fused. Without under-estimating the simple piety which, especially in country districts, still attached itself to the temples and their ritual, we may say confidently that the vital religion of the empire was associated with the mystery-religions and with the discipline of the ‘philosophic life’. It is in this region that the continuity of Catholicism with Hellenism is mainly to be found. The philosophers at this time were preachers, confessors, chaplains, and missionaries. The clerical profession, in nearly all its activities, is directly descended from the Hellenistic philosophers.

This claim of continuity may seem paradoxical when we remember the savage persecutions of the Christians by the imperial government. Of these persecutions there were several causes. The empire, like all empires of the same type, rested partly on religious support. Augustus encouraged his court poets to advocate a revival of piety and sound morals. A government cannot inquire into religious conviction, but it can enforce conformity and outward respect for the forms of worship as ‘by law established’. The Christians and Epicureans were held guilty of the same political offence—‘atheism’. The State had no quarrel with the mystery-religions, which were a private matter, but open disrespect to the national deities was flat disloyalty. The pagans could not understand why the Church would make no terms with the fusion of religions (θεοκρασια) which seemed to them the natural result of the fusion of nationalities. Apuleius makes Isis say, when she reveals herself to Lucius, ‘cuius numen unicum multiformi specie, ritu vario, nomine multiiugo totus veneratur orbis’; and she then recounts her various names. This more than tolerant hospitality of the spirit seemed to the mixed population of the empire the logical recognition of the actual political situation, and those who deliberately stood outside it were at least potentially enemies of society. This was the real quarrel between the Church and the empire. It is the old State religion which Augustine attacks, ridiculing the innumerable Roman godlings whose names he perhaps found in Varro. It is true that Plato, Euripides, and Xenophanes had attacked the official mythology with hardly less asperity; but they did not escape censure, and the Christian alienation from the Olympians was far more fundamental.

The pagan revival under the empire was rather like Neo-Catholicism in France. It was patriotic, nationalistic, and conservative, rather than strictly religious. Celsus, in his lost book against the Christians, seems to have appealed to their patriotism, urging them to support their country and its government in dangerous times. As the Church grew in numbers and power, and the old traditions crumbled away, largely from the fall in the birth-rate among the upper and middle classes, the conservatives became more anxiously attached to their own culture, and saw in Christianity a ‘shapeless darkness’ which threatened to extinguish ‘all the beautiful things in the world’. We can partly sympathize with this alarm, though not with the foolish policy which it inspired. The early persecutions were like Russian ‘pogroms’, instigated or connived at by the government as a safety valve for popular discontent. For at this time the common people hated the Christians, and half believed the monstrous stories about them. The attacks were not continuous, and were half-hearted, very unlike the systematic extermination of Jews and Protestants in Spain. At Alexandria Hadrian found a money-loving population worshipping Christ and Sarapis almost indifferently. A wrong impression is formed if we picture to ourselves two sections of society engaged in constant war. The first real war was the last, under Diocletian; it was to decide whether paganism or Christianity was to be the state religion. However, there is no doubt that the persecutions helped to seal the fate of the old culture.

Harnack traces three stages in the Hellenization of Christianity. ‘In the earliest Christian writings, apart from Paul, Luke, and John’, he cannot find any considerable traces of Greek influence. ‘The real influx of Greek thought and life’ began about 130. The exception is so important as to make this statement of little or no value. After 130, he says, ‘the philosophy of Greece went straight to the core of the new religion’. A century or so later, ‘Greek mysteries and Greek civilization in the whole range of its development exercise their influence on the Church, but as yet not its mythology and polytheism; these were still to come’. ‘Another century had to elapse before Hellenism as a whole and in every phase of its development was established in the Church.’ The process which he describes began, in fact, as soon as Christian preachers used the Greek language, and was never so complete as he says. The Logos-Christology, to which he justly attributes the greatest importance, is already present in St. Paul’s epistles; the name only is wanting; and the sharp contradiction which he finds between the Christian idea of a revelation made through a person at a certain date, and the Greek idea of an apprehension of timeless and changeless truth, always open to individuals after the appropriate discipline, was faced and in part overcome by the Greek Fathers. Harnack also regards Gnosticism as an embodiment of the genuinely Greek view of revelation, forgetting that orthodox Platonism was as hostile to Gnosticism as the Church itself. In rejecting Gnosticism, the Church in fact decided for genuine Hellenism against a corrupted and barbarized development of it. On the other hand, there is no period at which we can speak of a complete conquest of Christianity by Greek ideas. There was a large part of the old tradition which perished with its defenders, who, obeying the melancholy law which directs human survival, died out to make way for immigrants and for the formerly submerged classes, the people with few wants, who were indifferent to a culture which they had never been allowed to share.

One more cause of misunderstanding may be illustrated from the writings of Matthew Arnold. He divides the human race into Hebraizers and Hellenizers, and classifies the modern English and Americans as Hebraizers. The fundamental maxim of Hebrew ethics, according to him, is ‘Walk by the light you have’; of Greek ethics, ‘Take heed that the light which is in thee is not darkness’. The Hebraizer is conscientious but unenlightened; the Hellenizer is clear-headed but unscrupulous. Professor Santayana has lately noted the same difference between the type of character developed by the Latin nations and by the Anglo-Saxons. The Mediterranean civilization, older and more sophisticated, is careful to get its values right; the northern man is bent on doing something big, no matter what, and follows Clough’s advice:

Go! say not in thine heart, And what then, were it accomplished,
Were the wild impulse allayed, what is the use and the good?

But Santayana does not make the mistake of regarding the Reformation as a return to Palestinian Christianity. This was, indeed, the opinion of the Reformers themselves; but all religious innovation seeks to base itself on some old tradition. Christianity at first sought for its credentials in Judaism, though the Jews saw very quickly that it ‘destroyed the Law’. The belief of the Reformers was plausible; for they rejected just those parts of Catholicism which had nothing to do with Palestine, but were taken over from the old Hellenic or Hellenistic culture. But the residuum was less Jewish than Teutonic. On one side, indeed, the Reformation was a return to Hellenism from Romanism. Early Christian philosophy was mainly Platonic; early Christian ethics (as exemplified especially in writers like Ambrose) were mainly Stoical. There had been a considerable fusion of Plato and the Stoa among the Neoplatonists, so that it was easy for the two to flourish together. Augustine banished Stoical ethics from the Church, and they were revived only at the Reformation. Calvinism is simply baptized Stoicism; it is logically pantheistic, since it acknowledges only one effective will in the universe. The creed of nineteenth-century science is very similar. Puritanism was not at all like Judaism, in spite of its fondness for the Old Testament; it was very like Stoicism. The Reformation was a revolt against Latin theocracy and the hereditary paganism of the Mediterranean peoples; it was not really a return to pre-Hellenic Christianity. It sheltered the humanism of Erasmus and the late-flowering English Renaissance, and Christian Platonism has nowhere had a more flourishing record than in Protestant Britain.

At the present time a more drastic revolt is in progress among the plebs urbana, which does in truth threaten with destruction ‘what we owe to Greece’. The industrial revolution has generated a new type of barbarism, with no roots in the past. For the second time in the history of Western Europe, continuity is in danger of being lost. A generation is growing up, not uneducated, but educated in a system which has little connexion with European culture in its historical development. The Classics are not taught; the Bible is not taught; history is not taught to any effect. What is even more serious, there are no social traditions. The modern townsman is déraciné: he has forgotten the habits and sentiments of the village from which his forefathers came. An unnatural and unhealthy mode of life, cut off from the sweet and humanizing influences of nature, has produced an unnatural and unhealthy mentality, to which we shall find no parallels in the past. Its chief characteristic is profound secularity or materialism. The typical town artisan has no religion and no superstitions; he has no ideals beyond the visible and tangible world of the senses. This of course opens an impassable gulf between him and Greek religion, and a still wider gulf between him and Christianity. The attempts which are occasionally made, especially in this country, to dress up the Labour movement as a return to the Palestinian Gospel, are little short of grotesque. The contrast is well summed up by Belfort Bax, in a passage quoted by Professor Gardner. ‘According to Christianity, regeneration must come from within. The ethics and religion of modern socialism on the contrary look for regeneration from without, from material conditions and a higher social life.’ Here the gauntlet is thrown down to Christ and Plato alike.

Quite logically the new spirit is in revolt against what it calls intellectualism, which means the application of the dry light of reason to the problems of human life. It wishes to substitute for reason what some of its philosophers call instinct, but which should rather be called sentiment and emotion. There is no reconciliation between this view of life and Hellenism. For science is the eldest and dearest child of the Greek spirit. One of the great battles of the future will be between science and its enemies. The misologists have numbers on their side; but ‘Nature’, whom all the Greeks honoured and trusted, will be justified in her children.

The new spirit is especially bitter against the Stoical ethics, which as we have seen were taken over, with the Platonic metaphysics, by Christianity. Stoicism teaches men to venerate and obey natural law; to accept with proud equanimity the misfortunes of life; to be beneficent, but to inhibit the emotion of pity; to be self-reliant and self-contained; to practise self-denial for the sake of self-conquest; and to regard this life as a stern school of moral discipline. All this is simply detestable to the new spirit, which is sentimental, undisciplined, and hedonistic. It remembers the hardness of Puritanism, and has no admiration for its virtues.

It is often said that the modern man has entirely lost the Greek love of beauty. This is, I think, untrue, and unjust to our present civilization, unlovely as it undoubtedly is in many ways. It is curious that modern critics of the Greeks have not called attention to the aesthetic obtuseness which showed itself in the defective reaction of the ancients against cruelty. It was not that they excluded beautiful actions from the sphere of aesthetics; they never thought of separating the beautiful from the good in this way. But they were not disgusted at the torture of slaves, the exposure of new-born children, or the massacre of the population of a revolted city. The same callousness appears in the Italian cities at the Renaissance; Ezzelino was a contemporary of the great architects and painters. I cannot avoid the conclusion that it is connected in some obscure way with the artistic creativeness of these two closely similar epochs. The extreme sensibility to physical suffering which characterizes modern civilization arose together with industrialism, and is most marked in the most highly industrialized countries. It has synchronized with the complete eclipse of spontaneous and unconscious artistic production, which we deplore in our time. Evelyn, in the seventeenth century, was still able to visit a prison in Paris to gratify his curiosity by seeing a prisoner tortured, and though he did not stay to the end of the exhibition he shows that his stomach was not easily turned. It is certain that our repugnance to such sights is aesthetic rather than moral, and probable that it is strongest in the lower social strata. Several years ago I went to the first night of a rather foolish play about ancient Rome, in which an early Christian is brought in to be very mildly tortured on the stage. At the first crack of the whip my neighbours sprang from their seats, crying, ‘Shame! Stop that!’; and the scene had to be removed in subsequent performances. The operatives in a certain factory stopped the engines for an hour because they heard a cat mewing among the machinery. Having with difficulty rescued the animal from being crushed they strangled it. The explanation of this extreme susceptibleness must be left to psychologists; but I am convinced that we have here a case of transferred aesthetic sensibility. We can walk unmoved down the streets of Plaistow, but we cannot bear to see a horse beaten. The Athenians set up no Albert Memorials, but they tortured slave-girls in their law-courts and sent their prisoners to work in the horrible galleries of the Laureion silver-mines.

This emergence of a new spirit, which seems to be almost independent of all traditions, makes it difficult to estimate our present indebtedness to Greece in matters of religion. It would be difficult even if the industrial revolution had not taken place. The northern Europeans have hardly yet attained to self-expression. Their religion is a mixture of Greek, Latin, and Hebrew elements which refuse to be harmonized, and which in this country sometimes clash with the ideal of a gentleman, that lay religion of the English-speaking peoples, which has no longer any connexion with heraldry or property in land. The English gentleman is not a Greek any more than he is a Jew. His code makes Odysseus an amusing rascal; Achilles a violent and sulky savage; and Aristotle’s μεγαλοψυχος (as has been said) is rather like a nobleman in a novel by Disraeli, but not like any other sort of gentleman. The Englishman is by nature religious; but Christianity in its developed form is a Mediterranean religion; in all external features it might have been very different if it had been first planted north of the Alps. There is, therefore, a chronic confusion in Protestantism which makes its conflicts with the Latin Church like the battles of undisciplined barbarians against well-drilled troops.

Nevertheless, though it is so difficult to separate out the various threads which make up the tangled skein of our modern religion, it may be worth while to make the attempt to distinguish, first, those parts of current Christianity which are not Greek, in the wide sense which I have chosen for the word, and then those which, in the same sense, are Greek by origin or affinity.

Among those elements which are not Greek, the first place must be given to the original Gospel, of which I have said nothing yet. Our records of the Galilean ministry, contained in the three synoptic Gospels, were not compiled till long after the events which they describe, and must not be used uncritically. But in my opinion, at any rate, the substance of the teaching of Christ comes out very clearly in these books. No Hellenic influence can be traced in it; there is not even any sign of the Hellenized Judaism which for us is represented by his contemporary Philo. But neither is it possible to call the Gospel Jewish, except with many qualifications. Christ came before his countrymen as a prophet; he deliberately placed himself in the line of the prophetic tradition. Like other prophets of his nation, he did not altogether eschew the framework of apocalyptic which was at that time the natural mould for prophecy. But he preached neither the popular nationalism, nor the popular ecclesiasticism, nor the popular ethics. His countrymen rejected him as soon as they understood him. The Gospel was, as St. Paul said, a new creation. It is most significant that it at once introduced a new ethical terminology. The Greek words which we translate love (or charity), joy, peace, hope, humility, are no part of the stock-in-trade of Greek moralists before Christ. Men do not coin new words for old ideas. Taken as a whole the Gospel is profoundly original; and a Christian can find strong evidence for his belief that in Christ a revelation was made to humanity at large, in which the religion of the Spirit, in its purest and most universal form, was for the first time presented to mankind. This revelation has to a considerable extent passed into the common consciousness of the civilized world; but its implications in matters of conduct, individual, social, and international, are still imperfectly understood and have never been acted upon, except feebly and sporadically. It is a reproach to us that the teaching of Christ must be regarded as only one of many elements which make up what we call Christianity. The Quakers, as a body, seem to me to come nearest to what a genuinely Christian society would be.

Secondly, the Greeks escaped the evils of priestly government. The Oriental type of theocracy, with which they were familiar in the Egypt of the Pharaohs, was alien to their civilization. Their sacrifices were for the most part of the genial type, a communion-meal with the god. But even in Greece we must remember the gloomy chthonian rites, and the degradations of Orphism mentioned by Plato in the Republic. ‘They persuade not only individuals but whole cities that expiations and atonements for sin may be made by sacrifices and amusements which fill a vacant hour, and are equally at the service of the living and of the dead; the latter sort they call mysteries, and they redeem us from the pains of hell, but if we neglect them no one knows what awaits us.’ This exploitation of sacramentalism was common enough in Greece; but the characteristic Caesaro-Papism of Byzantium and modern imperialism was wholly foreign to Hellenism. It was introduced by Constantine as part of the Orientalizing of the empire begun by Diocletian. As Seeley says, ‘Constantine purchased an indefeasible title by a charter. He gave certain liberties and received in return passive obedience. He gained a sanction for the Oriental theory of government; in return he accepted the law of the Church. He became irresponsible to his subjects on condition of becoming responsible to Christ.’

The Greeks never had a book-religion, in the sense in which Judaism became, and Islam always was, a book-religion. But they were in some danger of treating Homer and Hesiod as inspired scriptures. To us it is plain that a long religious history lies behind Homer, and that the treatment of the gods in Epic poetry proves that they had almost ceased to be the objects of religious feeling. Some of them are even comic characters, like the devil in Scottish folklore. To turn these poems into sacred literature was to court the ridicule of the Christians. But Homer was never supposed to contain ‘the faith once delivered to the saints’; no religion of authority could be built upon him, and Greek speculation remained far more unfettered than the thought of Christendom has been until our own day.

Those who have observed the actual state of Christianity in Mediterranean countries cannot lay much stress on the difference between Christian monotheism and pagan polytheism. The early Church fought against the tendency to interpose objects of worship between God and man; but Mariolatry came in through a loophole, and the worship of the masses in Roman Catholic countries is far more pagan than the service-books. In the imagination of many simple Catholics, Jesus, Mary, and Joseph are the chief potentates in their Olympus.

The doctrine of the creation of the world in time, which was denied by most pagan thinkers and affirmed by most Christian divines, belongs to philosophy rather than to religion. The disbelief in the pre-existence of the soul, a doctrine which for Greek thought stands or falls with the belief in survival after death, is more important, and may be partly attributable to Jewish influence. But pre-existence does not seem to have been believed by the majority of Greeks, and in fact almost disappears from Greek thought between Plato and the Neoplatonists. It is possible that the Pythagorean and Platonic doctrine may still have a future.

There are some who will insist that these differences are insignificant by the side of the fact that Christianity was the idealistic side of a revolt of the proletariat against the whole social order of the time. This notion, which made Christ ‘le bon sans-culotte’, has again become popular lately; some have even compared the early Christians with Bolsheviks. It is a fair question to ask at what period this was even approximately true. Christ and his apostles belonged to the prosperous peasantry of Galilee, a well-educated and comfortable middle class. The domestic slaves of wealthy Romans, who embraced the new faith in large numbers, were legally defenceless, but by no means miserable or degraded. After the second century the comparison of the Christians to modern revolutionists becomes too absurd for discussion. There is a good deal of rhetorical declamation about riches and poverty in the Christian Fathers; but unfortunately the Church seems to have done very little to protest against the crying economic injustices of the fourth and fifth centuries. From first to last there was nothing of the ‘Spartacus’ movement about the Catholic Church. As soon as the persecutions ceased, the bishops took their place naturally among the nobility.

When we turn to the obligations of modern religion to Greece, it is difficult to know where to begin.

The conception of philosophy as an ars vivendi is characteristically Greek. Nothing can be further from the truth than to call the Greeks ‘intellectualists’ in the disparaging sense in which the word is now often used. The object of philosophy was to teach a man to live well, and with that object to think rightly about God, the world, and himself. This close union between metaphysics, morals, and religion has remained as a permanent possession of the modern world. Every philosopher is now expected to show the bearing of his system on morality and religion, and the criticism is often justified that however bold the speculations of the thinker, he is careful, when he comes to conduct, to be conventional enough. The Hellenistic combination of Platonic metaphysics with Stoic ethics is still the dominant type of Christian religious philosophy. It is curious to observe how competing tendencies in these systems—the praise of isolated detachment and of active social sympathy—have continued to struggle against each other within the Christian Church.

The place of asceticism in religion is so important, and so much has been written rather unintelligently about the contrast between Hellenism and Christianity in this matter, that I propose to deal with it, briefly indeed, but with a little more detail than a strict attention to proportion would justify. It has often been assumed that a nation of athletes, who made heroes of Heracles and Theseus, Achilles and Hector, could have had nothing but contempt for the ascetic ideal. But in truth asceticism has a continuous history within Hellenism. Even Homer knows of the priests of chilly Dodona, the Selli, whose bare feet are unwashed, and who sleep on the ground. This is probably not, as Wilamowitz-Moellendorff thinks, a description of savage life, but of an ascetic school of prophets. For the fastdays which introduced the Thesmophoria were observed by the Athenian matrons in the same way; they went unshod and sat on the bare earth; and we may compare the Nudipedalia, ordered by the Romans in time of dearth and mentioned by Petronius and Tertullian. Prophets and prophetesses fasted at Miletus, Colophon, and other places. National fasts were ordered in times of calamity or danger, and Tarentum kept a yearly fast of thankfulness for deliverance from a siege. The flagellation of boys at Sparta hardly comes into account, being probably a substitute for human sacrifice; but the continuance of the cruel rite till nearly the end of antiquity causes surprise. The worship of Dionysus Zagreus in Thrace was accompanied by ascetic practices before Pythagoras. Vegetarianism, which has always played an important part in the ascetic life, was obligatory on all Pythagoreans; but in this school there was another motive besides the desire to mortify the flesh. Those who believe in the transmigration of souls into the bodies of animals must regard flesh-eating as little better than cannibalism. The Pythagorean and the Orphic rules of life were well known throughout antiquity, and were probably obeyed by large numbers. The rule of continence was far less strict than in the Catholic ‘religious’ life; but Empedocles, according to Hippolytus, advised abstinence from marriage and procreation, and the tendency to regard celibacy as part of the ‘philosophic life’ increased steadily. The Cynic Antisthenes is quoted by Clement of Alexandria as having expressed a wish to ‘shoot Aphrodite, who has ruined so many virtuous women’. But the asceticism of the early Cynics and of some Stoics was based not on self-devotion and spirituality but on the desire for independence, and often took repulsive forms. Of some among them it may be said that they did not object to sensual pleasure, they only objected to having to pay for it. Desire for self-sufficiency is always part of asceticism, but in the Christian saints it has been a small part. The Greeks who practised it were from first to last too anxious to be invulnerable; this was the main attraction of the philosophic life from the time of Antisthenes, and it remained the main attraction to the end. But Cynicism and Stoicism (which tend to run together) became gentler, more humane, and more spiritual under the Roman empire. Seneca, Epictetus, and Marcus Aurelius often seem to be half Christian. Direct influence of Christian ethics at this early period is perhaps unlikely; it is enough to suppose that the spirit of the age affected in a similar way all creeds and denominations. Self-mortification tended to assume more and more violent forms, till it culminated in the strange aberrations of Egyptian eremitism. It is impossible to regard these as either Greek or Christian; they indicate a pathological state of society, which can be partly but not entirely accounted for by the conditions of the time. After a few centuries a far more wholesome type of monachism supplanted the hermits; the anchorites of the Middle Ages retained the solitary life, but were very unlike the crazy savages of the Thebaid. In modern times, those who have been most under the Greek spirit have generally lived with austere simplicity, but without any of the violent self-discipline which is said to be still practised by some devout Catholics. The assiduous practice of self-mastery and the most sparing indulgence in the pleasures of sense are the ‘philosophic life’ which the Greek spirit recommends as the highest. The best Greeks would blame the life of an English clergyman, professor, or philosopher as too self-indulgent; we often forget how frugally and hardily the Greeks lived at all times. But here we have to consider the differences of climate, and the apparent necessity of a rather generous diet for the Nordic race.

The influence of the Greek mysteries upon Christianity is a keenly debated question, in which passion and prejudice play too large a part. The information necessary for forming a judgement has been much enlarged by recent discoveries in Egypt and elsewhere, and, as usually happens, the importance of the new facts has been sometimes exaggerated. Protestant theology has on the whole minimized the influence of the mysteries, and has post-dated it, from an unwillingness to allow that there was already a strong Catholic element in the Christianity of the first century. Orthodox Catholicism has ignored it from different but equally obvious motives. Modernist Catholicism has in my opinion antedated the irruption of crude sacramentalism into the Church, and has greatly overstated its importance in the religion of the first-century Christians. This school practically denies anything more than a half-accidental continuity between the preaching of the historical Christ, whom they strangely suppose to have been a mere apocalyptist, one of the many Messiahs or Mahdis who arose at this period in Palestine, and the Catholic Church, which according to them belonged to the same type of religion as the worship of Isis and Mithra. Another bone of contention is the value of the mystery-religions of Greece. The very able German scholars who have written on the subject, such as Reitzenstein and still more Rohde, seem to me much too unsympathetic in their treatment of the mystery-cults. Lastly, some competent critics have lately urged that this side of Christianity owed more to Judaism—Hellenized Judaism, of course—than has been hitherto supposed.

Plato in the Phaedo says that ‘those who established our mysteries declare that all who come to Hades uninitiated will lie in the mud; while he who has been purified and initiated will dwell with the gods’. For, as they say in the mysteries, ‘Many are the thyrsus-bearers, but few are the inspired’. This sacramentalism was not unchallenged, as we have already seen from Plato himself. Diogenes is said to have asked whether the robber Pataecion was better off in the other world than the hero Epaminondas, because the former had been initiated, and the latter had not. But Orphism, though liable to degradation, purified and elevated the old Bacchic rites. As Miss Harrison says, the Bacchanals hoped to attain unity with God by intoxication, the Orphics by abstinence. The way to salvation was now through ‘holiness’ (ὁσιοτης). To the initiated the assurance was given, ‘Happy and blessed one! Thou shalt be a god instead of a mortal.’ To be a god meant for a Greek simply to be immortal; the Orphic saint was delivered from the painful cycle of recurring births and deaths. And Orphic purity was mainly, though not entirely, the result of moral discipline. Cumont says that the mystery-cults brought with them two new things—mysterious means of purification by which they proposed to cleanse away the defilements of the soul, and the assurance that an immortality of bliss would be the reward of piety. The truth, says Mr. H. A. Kennedy, was presented to them in the guise of divine revelations, esoteric doctrines to be carefully concealed from the gaze of the profane, doctrines which placed in their hands a powerful apparatus for gaining deliverance from the assaults of malicious demonic influences, and above all for overcoming the relentless tyranny of fate. This demonology was believed everywhere under the Roman empire, the period of which Mr. Kennedy is thinking in this sentence, and it has unfortunately left more traces in St. Paul’s epistles than we like to allow. The formation of brotherhoods for mystic worship was also an important step in the development of Greek religion. These brotherhoods were cosmopolitan, and seem to have flourished especially at great seaports. They were thoroughly popular, drawing most of their support from the lower classes, and within them national and social distinctions were ignored. Their ultimate aim cannot be summed up better than in Mr. Kennedy’s words—‘to raise the soul above the transiency of perishable matter through actual union with the Divine’. It has been usual to distinguish between the dignified and officially recognized mysteries, like those of Eleusis, and the independent voluntary associations, some of which became important. But there was probably no essential difference between them. In neither case was there much definite teaching; the aim, as Aristotle says, was to produce a certain emotional state (ου μαθειν τι δειν αλλα παθειν). A passion-play was enacted amid the most impressive surroundings, and we need not doubt that the moral effect was beneficial and sometimes profound. When the Egyptian mysteries of Isis and Osiris were fused with the Hellenic, a type of worship was evolved which was startlingly like Christianity. A famous Egyptian text contains the promise: ‘As truly as Osiris lives, shall he [the worshipper] live; as truly as Osiris is not dead, shall he not die.’ The thanksgiving to Isis at the end of the Metamorphoses of Apuleius is very beautiful in itself, though it is an odd termination of a licentious novel. The Hermetic literature also contains doctrine of a markedly Johannine type, as notably in a prayer to Isis: ‘Glorify me, as I have glorified the name of thy son Horus.’ I agree with those critics (Cumont, Zielinski, and others) who attach the ‘higher’ Hermetic teaching to genuinely Hellenic sources. But it is not necessary to ascribe all the higher teaching to Greece and the lower to Egypt.

Much of St. Paul’s theology belongs to the same circle of ideas as these mysteries. Especially important is the psychology which divides human nature into spirit, soul, and body, spirit being the divine element into which those who are saved are transformed by the ‘knowledge of God’. This knowledge is a supernatural gift, which (in the Poimandres) confers ‘deification’. St. Paul usually prefers ‘Pneuma’ as the name of this highest part of human nature; in the Hermetic literature it is not easy to distinguish between Pneuma and Nous, which holds exactly the same place in Neoplatonism. The notion of salvation as consisting in the knowledge of God is not infrequent in St. Paul; compare, for example, 1 Cor. xiii. 12 and a still more important passage, Phil. ii. 8-10. This knowledge was partly communicated by visions and revelations, to which St. Paul attributed some importance; but on the whole he is consistent in treating knowledge as the crown and consummation of faith. The pneumatic transformation of the personality is the centre of St. Paul’s eschatology. ‘Though our outward man perish, our inward man is renewed day by day.’ The ‘spiritual body’ is the vehicle of the transformed personality; for ‘flesh and blood cannot inherit the kingdom of God’. The expression ‘to be born again’ is common in the mystery literature.

It would be easy to find many other parallels in St. Paul’s epistles, in the Johannine books which are the best commentary upon them, and in the theology of the Greek Fathers, which prove the close connexion of early Christianity with the mystery-religions of the empire. Twenty years ago it might have been worth while to draw out these resemblances in greater detail, even in so summary a survey as this. But at present the tendency is, if not to over-estimate the debt of the Christian religion to Hellenistic thought and worship, at any rate to ignore the great difference between the higher elements in the mystery-religions, which the new faith could gladly and readily assimilate, and the lower type, the theosophy, magic, and theurgy, which was not in the line of Hellenic development, and is not to be found in the New Testament. Wendland, always a judicious critic, has said very truly that St. Paul stands to the mystery-religions as Plato to Orphism; they are not the centre of his religious life, but they gave him effective forms of expression for his religious experience. Or, as Weinel says, ‘St. Paul’s doctrine of the Spirit and of Christ is not an imitation of mystery-doctrine, but inmost personal experience metaphysically interpreted after the manner of his time.’ Writers like Loisy, who say that for St. Paul Jesus was ‘a Saviour God, after the manner of Osiris, Attis, or Mithra’, and who proceed to draw out obvious parallels between the sufferings, death, and resurrection of these mythological personages and the gospels of the Christian Church, surely forget that St. Paul was a Jew, and that there are some transformations of which the religious mind is incapable. He never speaks of Christ as a ‘Saviour God’. Even more perverse are the arguments which are used to prove that the centre of St. Paul’s religion was a gross and materialistic sacramental magic. The apostle, whose antipathy to ritual in every shape is stamped upon all his writings, who thanks God that he baptized very few of the Corinthians, who declares that ‘Christ sent him not to baptize but to preach the Gospel’, is accused of regarding baptism as ‘an opus operatum which secures a man’s admission into the kingdom apart from the character of his future conduct’. And yet in the Epistle to the Romans, as Weinel says, ‘baptism only once enters his mind, and the Lord’s Supper not even once’. Baptism for him is no opus operatum, but a ceremony of social significance, a symbol conditioning a deeper experience of divine grace, already embraced by faith. These same critics proceed to illustrate St. Paul’s doctrine of the Lord’s Supper by references to the religion of the Aztecs and other barbarians. But it is hardly worth while to argue with those who suppose that a man with St. Paul’s upbringing and culture could have dallied with the notion of ‘eating a god’. The ‘table of the Lord’ is the table at which the Lord is the spiritual host, not the table on which his flesh is placed. Does any one suppose that ‘the table of demons’ which is contrasted with the ‘table of the Lord’ is the table at which demons are eaten? Demons had no bodies, as we learn from the ουκ ειμι δαιμονιον ασωματον of a well-known passage in a New Testament manuscript.

Crude sacramentalism certainly came in later. Its parentage may be traced, if we will, to those mystery-mongers whom Plato mentions with disapproval. If Hellenism is the name of a way of thinking, this form of religion is not healthy Hellenism; that it was held by many Hellenes cannot be denied.

The biblical doctrine of the Fall of Man, which the Hebrews would never have evolved for themselves, remained an otiose dogma in Jewish religion. It was revivified in Christianity under Greek influence. Man, as Empedocles and others had taught, was ‘an exile and vagabond from God’; his body was his tomb; he is clothed in ‘an alien garment of flesh’. He is in a fallen state and needs redemption. Hellenism had become a religion of redemption; the empire was quite ready to accept this part of Christian doctrine. The sin of Adam became the first scene in the great drama of humanity, which led up to the Atonement. At the same time the whole process was never mere history; its deepest meaning was enacted in the life-story of each individual. Greek thought gave this turn to dogmas which for a Jew would have been a flat historical recital. In modern times the earlier scenes in the story, at any rate, are looked upon as little more than the dramatization of the normal experience of a human soul. But Greek thought, while it remained true to type, never took sin so tragically as Christianity has done. The struggle against evil has become sterner than it ever was for the Greeks. It must, however, be remembered that the large majority of professing Christians do not trouble themselves much about their sins, and that the best of the Greeks were thoroughly in earnest in seeking to amend their lives.

Redemption was brought to earth by a Redeemer who was both God and Man. This again was in accordance with Greek ideas. The Mediator between God and Man must be fully divine, since an intermediate Being would be in touch with neither side. The victory of Athanasius was in no sense a defeat for Hellenism. The only difficulty for a Greek thinker was that an Incarnate God ought to be impassible. This was a puzzle only for philosophers; popular religion saw no difficulty in a Christus patiens. The doctrine of the Logos brought Christianity into direct affinity with both Platonism and Stoicism, and the Second Person of the Trinity was invested with the same attributes as the Nous of the Neoplatonists. But the attempts to equate the Trinity with the three divine hypostases of Plotinus was no more successful than the later attempt of Hegel to set the Trinity in the framework of his philosophy.

The subject of eschatology is so vast that it is hopeless to deal with it, even in the most summary fashion, in one paragraph. It is usually said that the resurrection of the body is a Jewish doctrine, the immortality of the soul a Greek doctrine. But the Jews were very slow to bring the idea of a future life into their living faith; to this day it does not seem to be of much importance in Judaism. Some form of Millenarianism—a reign of the saints on earth—would seem to be the natural form for Jewish hopes to take. This belief, which was the earliest mould into which the treasure of the new revelation was poured, has never quite disappeared from the Church, and in times of excitement and upheaval it tends to reassert itself. The maturest Greek philosophy regards eternity as the divine mode of existence, while mortals are born, live, and die in time. Man is a microcosm, in touch with every rung of the ladder of existence; and he is potentially a ‘participator’ in the divine mode of existence, which he can make his own by living, so far as may be, in detachment from the vain shadows and perishable goods of earth. That this conception of immortality has had a great influence upon Christian thought and practice needs no demonstration. It is and always has been the religion of the mystic. But the Orphic tradition, with its pictures of purgatory and of eternal bliss and torment, has on the whole dominated the other two in popular Christian belief. It has been stripped of its accessories—the belief in reincarnation and the transmigration of souls, doctrines which maintain a somewhat uneasy existence within the scheme of the Neoplatonists. The picture of future retribution is even more terrifying without them. Both the philosophical and the popular beliefs about the other world are far more Greek than Jewish; but the attempt to hold these very discrepant beliefs together has reduced Christian eschatology to extreme confusion, and many Christians have given up the attempt to formulate any theories about what are called the four last things. On such a mysterious subject, definiteness is neither to be expected nor desired. The original Gospel does not encourage the natural curiosity of man to know his future fate; and the three types of eschatology which we have described have all their value as representing different aspects of religious faith and hope. We must after all confess the truth of St. Paul’s words, that ‘eye hath not seen, nor ear heard, neither hath it entered into the heart of man to conceive, the things that God hath prepared for them that love him’. The same apostle reminds us that ‘now we see through a mirror, in riddles, and know only in part’; the face to face vision, and the knowledge which unites the knower and the known, may be ours when we have finished our course. In these words, which recall Plato’s famous myth of the Cave, St. Paul is fundamentally at one with the Platonists; and it may well be that it is by this path that our contemporaries may recover that belief in eternal life which is at present burning very dimly among us.

In conclusion, what has the religion of the Greeks to teach us that we are most in danger of forgetting? In a word, it is the faith that Truth is our friend, and that the knowledge of Truth is not beyond our reach. Faith in honest seeking (ζητησις) is at the heart of the Greek view of life. ‘Those who would rightly judge of truth’, says Aristotle, ‘must be arbitrators, not litigants’. ‘Happy is he who has learnt the value of research’ (ἱστορια), says Euripides in a fragment. Curiosity, as the Greeks knew and the Middle Ages knew not, is a virtue, not a vice. Nature, for Plato, is God’s vicegerent and revealer, the Soul of the universe. Human nature is the same nature as the divine; no one has proclaimed this more strongly. Nature is for us; chaos and ‘necessity’ are the enemy. The divorce between religion and humanism began, it must be admitted, under Plato’s successors, who unhappily were indifferent to natural science, and did not even follow the best light that was to be had in physical knowledge. In the Dark Ages, when the link with Greece was broken, the separation became absolute. The luxuriant mythology of the early Greeks was not unscientific. In the absence of knowledge gaps were filled up by the imagination, and the ‘method of trial and error’. The dramatic fancy which creates myths is the raw material of both poetry and science. Of course religious myths may come to be a bar to progress in science; they do so when, in a rationalizing age, the question comes to be one of fact or fiction. It is a mistake to suppose that the faith of a ‘post-rational’ age, to use a phrase of Santayana, can be the same as that of an unscientific age, even when it uses the same formulas. The Greek spirit itself is now calling us away from some of the vestments of Greek tradition. The choice before us is between a ‘post-rational’ traditionalism, fundamentally sceptical, pragmatistic, and intellectually dishonest, and a trust in reason which rests really on faith in the divine Logos, the self-revealing soul of the universe. It is the belief of the present writer that the unflinching eye and the open mind will bring us again to the feet of Christ, to whom Greece, with her long tradition of free and fearless inquiry, became a speedy and willing captive, bringing her manifold treasures to Him, in the well-grounded confidence that He was not come to destroy but to fulfil.

W. R. Inge.

PHILOSOPHY

If we consider the philosophical tendencies of the day, we shall probably observe first of all that the artificial wall of partition between philosophy and science—and especially mathematical science—is beginning to wear very thin. On the other hand, we cannot fail to notice a reaction against what is called intellectualism. This reaction takes many forms, the most characteristic perhaps of which is a renewed interest in Mysticism. It leads also to a strong insistence on the practical aspect of philosophic thought, and to a view of its bearing on what had been regarded as primarily theoretical issues, which is known by the rather unfortunate name of Pragmatism. Now it is just on these points that we have most to learn from the Greeks, and Greek philosophy is therefore of special importance for us at the present time. At its best, it was never divorced from science, while it found a way of reconciling itself both with the interests of the practical life and with mysticism without in any way abating the claims of the intellect. It is solely from these points of view that it is proposed to regard Greek philosophy here. It would be futile to attempt a summary of the whole subject in the space available, and such a summary would have no value. Many things will therefore be passed over in silence which are important in themselves and would have to be fully treated in a complete account. All that can be done now is to indicate the points at which Greek philosophy seems to touch our actual problems. It will be seen that here, as elsewhere, ‘all history is contemporary history’, and that the present can only be understood in the light of the past.

The word ‘philosophy’ is Greek and so is the thing it denotes. Unless we are to use the term in so wide a sense as to empty it of all special meaning, there is no evidence that philosophy has ever come into existence anywhere except under Greek influences. In particular, mystical speculation based on religious experience is not itself philosophy, though it has often influenced philosophy profoundly, and for this reason the pantheism of the Upanishads cannot be called philosophical. It is true that there is an Indian philosophy, and indeed the Hindus are the only ancient people besides the Greeks who ever had one, but Indian science was demonstrably borrowed from Greece after the conquest of Alexander, and there is every reason to believe that those Indian systems which can be regarded as genuinely philosophical are a good deal more recent still. On the other hand, the earliest authenticated instance of a Greek thinker coming under Indian influence is that of Pyrrho (326 B. C.), and what he brought back from the East was rather the ideal of quietism than any definite philosophical doctrine. The barrier of language was sufficient to prevent any intercourse on important subjects, for neither the Greeks nor the Indians cared to learn any language but their own. Of course philosophy may culminate in theology, and the best Greek philosophy certainly does so, but it begins with science and not with religion.

By philosophy the Greeks meant a serious endeavour to understand the world and man, having for its chief aim the discovery of the right way of life and the conversion of people to it. It would not, however, be true to say that the word had always borne this special sense. At any rate the corresponding verb (φιλοσοφειν) had at first a far wider range. For instance, Herodotus (i. 30) makes Croesus say that Solon had travelled far and wide ‘as a philosopher’ (φιλοσοφεων), and it is clear from the context that this refers to that love of travel for the sake of the ‘wonders’ to be seen in strange lands which was so characteristic of the Ionian Greeks in the fifth century B. C. That is made quite plain by the phrase ‘for the sake of sightseeing’ (θεωριης ἑινεκεν) with which the word is coupled. Again, when Thucydides (ii. 40) makes Pericles say of his fellow citizens ‘we follow philosophy without loss of manliness’ (φιλοσοφουμεν ανευ μαλακιας), it is certainly not of philosophy in the special sense he is thinking. He is only contrasting the culture of Athens with the somewhat effeminate civilization of the Ionians in Asia Minor. Even in the next century, Isocrates tried to revert to this wider sense of the word, and he regularly uses it of the art of political journalism which he imparted to his pupils.

Tradition ascribes the first use of the term ‘philosophy’ in the more restricted sense indicated above to Pythagoras of Samos, an Ionian who founded a society for its cultivation in southern Italy in the latter half of the sixth century B. C. It is notoriously difficult to make any positive statements about Pythagoras, seeing that he wrote nothing; but it is safer on general grounds to ascribe the leading ideas of the system to the master rather than to his followers. Moreover, this particular tradition is confirmed by the fact, for which there is sufficient evidence, that the name ‘philosophers’ originally designated the Pythagoreans in a special way. For instance, we know that Zeno of Elea (c. 450 B. C.) wrote a book ‘Against the Philosophers’, and in his mouth that can only mean ‘Against the Pythagoreans’. Now the Pythagorean use of the term depends on a certain way of regarding man, which there is good reason for ascribing to Pythagoras himself. It has become more or less of a commonplace now, but we must try to seize it in its original freshness if we wish to understand the associations the word ‘philosophy’ came to have for the Greeks. To state it briefly, it is the view that man is something intermediate between God and ‘the other animals’ (ταλλα ζωα). As compared with God, he is ‘mere man’, liable to error and death (both of which are spoken of as specially human, ανθρωπινα); as compared with ‘the other animals’, he is kindly and capable of civilization. The Latin word humanus took over this double meaning, which is somewhat arbitrarily marked in English by the spellings human and humane. Now it is clear that, for a being subject to error and death, wisdom (σοφια) in the full sense is impossible; that is for God alone. On the other hand, man cannot be content, like ‘the other animals’ to remain in ignorance. If he cannot be wise, he can at least be ‘a lover of wisdom’, and it follows that his chief end will be ‘assimilation to God so far as possible’ (ὁμοιωσις τω θεω κατα το δυνατον), as Plato put it in the Theaetetus. The mathematical studies of the Pythagoreans soon brought them face to face with the idea of a constant approximation which never reaches its goal. There is, then, sufficient ground for accepting the tradition which makes Pythagoras the author of this special sense of the word ‘philosophy’ and for connecting it with the division of living creatures into God, men and ‘the other animals’. If the later Pythagoreans went a step further and classified rational animals into gods, men and ‘such as Pythagoras’, that was due to the enthusiasm of discipleship, and is really a further indication of the genuinely Pythagorean character of this whole range of ideas. We may take it, then, that the word ‘philosophy’ had acquired its special sense in southern Italy before the beginning of the fifth century B. C.

It is even more certain that this sense was well known at Athens, at least in certain circles, not long after the middle of the fifth century. To all appearance, this was the work of Socrates (470-399 B.C.). Whatever view may be taken of the philosophy of Socrates or of its relation to that expounded in Plato’s earlier dialogues (a point which need not be discussed here), it is at least not open to question that he was personally intimate with the leading Pythagoreans who had taken refuge at Thebes and at Phlius in the Peloponnesus when their society came to be regarded as a danger to the state at Croton and elsewhere in southern Italy. That happened about the middle of the fifth century, and Socrates must have made the acquaintance of these men not long after. At that time it would be quite natural for them to visit Athens; but, after the beginning of the Peloponnesian War (431 B. C.), all intercourse with them must have ceased. They were resident in enemy states, and Socrates was fighting for his country. With the exception of the brief interval of the Peace of Nicias (421 B. C.), he can have seen nothing of them for years. Nevertheless it is clear that they did not forget him; for we must accept Plato’s statement in the Phaedo that many of the most distinguished philosophers of the time came to Athens to be with Socrates when he was put to death, and that those of them who could not come were eager to hear a full account of what happened. It is highly significant that, even before this, two young disciples of the Pythagorean Philolaus, Simmias and Cebes, had come from Thebes and attached themselves to Socrates. For that we have the evidence of Xenophon as well as of Plato, and Xenophon’s statement is of real value here; for it was just during these few years that he himself associated with Socrates, though he saw him for the last time a year or two before his trial and death. Whatever other inferences may be drawn from these facts, they are sufficient to prove that Socrates had become acquainted with some of the leading philosophers of the Greek world before he was forty, and to make it highly probable that it was he who introduced the word ‘philosophy’ in its Pythagorean sense to the Athenians.

So much for the word; we have next to ask how there came to be such a thing as philosophy at all. It has been mentioned that Pythagoras was an Ionian, and we should naturally expect to find that he brought at least the beginnings of what he called philosophy from eastern Hellas. Now it has been pointed out that Greek philosophy was based on science, and science originated at Miletus on the mainland of Asia Minor nearly opposite the island of Samos, which was the original home of Pythagoras. The early Milesians were, in fact, men of science rather than philosophers in the strict sense. The two things were not differentiated yet, however, and the traditional account of the matter, according to which Greek philosophy begins with Thales (c. 585 B. C.), is after all quite justified. The rudimentary mathematical science of which, as explained elsewhere in this volume, he was the originator in fact led him and his successors to ask certain questions about the ultimate nature of reality, and these questions were the beginning of philosophy on its theoretical side. It is true that the Milesians were unable to give any but the crudest answers to these questions, and very likely they did not realise their full importance. These early inquirers only wanted to know what the world was made of and how it worked, but the complete break with mythology and traditional views which they effected cleared the way for everything that followed. It was no small thing that they were able to discard the old doctrine of what were afterwards known as the ‘elements’—Fire, Air, Earth, and Water—and to regard all these as states of a single substance, which presented different appearances according as it was more or less rarefied or condensed. Moreover, Anaximander at least (c. 546 B. C.), the successor of Thales, shook himself free of the idea that the earth required support of some kind to keep in its place. He held that it swung free in space and that it remained where it was because there was no reason for it to fall in one direction rather than another. In general these early cosmologists saw that weight was not an inherent quality of bodies and that it could not be used to explain anything. On the contrary, weight was itself the thing to be explained. Anaximander also noted the importance of rotary or vortex motion in the cosmical scheme, and he inferred that there might be an indefinite number of rotating systems in addition to that with which we are immediately acquainted. He also made some very important observations of a biological character, and he announced that man must be descended from an animal of a different species. The young of most animals, he said, can find their food at once, while that of the human species requires a prolonged period of nursing. If, then, man had been originally such as he is now, he could never have survived. All this, no doubt, is rudimentary science rather than philosophy, but it was the beginning of philosophy in this sense, that it completely transformed the traditional view of the world, and made the raising of more ultimate problems inevitable.

This transformation was effected in complete independence of religion. What we may call secularism was, in fact, characteristic of all eastern Ionian science to the end. We must not be misled by the fact that Anaximander called his innumerable worlds ‘gods’ and that his successor Anaximenes spoke of Air as a ‘god’. These were never the gods of any city and were never worshipped by any one, and they did not therefore answer at all to what the ordinary Greek meant by a god. The use of the term by the Milesians means rather that the place once occupied by the gods of religion was now being taken by the great fundamental phenomena of nature, and the later Greeks were quite right, from their own point of view, in calling that atheism. Aristophanes characterizes this way of speaking very accurately indeed in the Clouds when he makes Strepsiades sum up the teaching he has received in the words ‘Vortex has driven out Zeus and reigns in his stead’, and when he makes Socrates swear by ‘Chaos, Respiration and Air’. So too the Milesians spoke of the primary substance as ‘ageless and deathless’, which is a Homeric phrase used to mark the difference between gods and men, but this only means that the emotion formerly attached to the divine was now being transferred to the natural.

The Milesians, then, had formed the conception of an eternal matter out of which all things are produced and into which all things return, and the conception of Matter belongs to philosophy rather than to science. But besides this they had laid the foundations of geometry, and that led in other hands to the formulation of the correlative conception of Limit or Form. It is needless to enumerate here the Milesian and Pythagorean contributions to plane geometry; it will be sufficient to remind the reader that they covered most of the ground of Euclid, Books I, II, IV, and VI, and probably also of Book III. In addition, Pythagoras founded Arithmetic, that is, the scientific theory of numbers (αριθμητικη), as opposed to the practical art of calculation (λογιστικη). We also know that he discovered the sphericity of the earth, and the numerical ratios of the intervals between the concordant notes of the octave. It is obvious that he was a scientific genius of the first order, and it is also clear that his methods included those of observation and experiment. The discovery of the earth’s spherical shape was due to observation of eclipses, and that of the intervals of the octave can only have been based on experiments with a stretched string, though the actual experiments attributed by tradition to Pythagoras are absurd. It was no doubt this last discovery that led him to formulate his doctrine in the striking saying ‘Things are numbers’, thus definitely giving the priority to the element of form or limit instead of to the indeterminate matter of his predecessors.

Pythagoras further differed from his predecessors in one respect which proved of vital moment. So far was he from ignoring religion, that he founded a society in southern Italy which was primarily a religious community. It is quite possible that he was influenced by the growth of the Orphic societies which had begun to spread everywhere in the course of the sixth century, but his religion differed from the Orphic in many ways. In particular, Apollo and not Dionysus was the chief god of the Pythagoreans, and all our evidence points to the conclusion that Pythagoras brought his religion, as he had brought his science, from eastern Hellas, though rather from the islands of the Aegean than from mainland Ionia. He was much influenced, we can still see, by certain traditions of the temple of Delos, which had become the religious centre of the Ionic world. There had, of course, been plenty of religious speculation among the Greeks before Pythagoras, and it was of a type not unlike that we find in India, though there are insuperable difficulties in the way of assuming any Aegean influence on India or any Indian influence on the Aegean at this date. It may be that the beginnings of such ideas go back to the time when the Greeks and the Hindus were living together, though it is still more likely that both the Greeks and the Indians were affected by a movement originating in the north, which brought to both of them a new view of the soul. The Delian legend of the Hyperboreans may be thought to point in this direction. However that may be, the main purpose of the religious observances practised by the Orphics and Pythagoreans alike was to secure by means of ‘purifications’ (καθαρμοι) the ransom (λυσις) of the soul, which was regarded as a fallen god, from the punishment of imprisonment in successive bodies. There is no reason to suppose that Pythagoras displayed any particular originality in this part of his teaching. It all depends on the doctrine of transmigration or rebirth (παλιγγενεσια), which is often incorrectly designated by the late and inaccurate term ‘metempsychosis’. There is no doubt that Pythagoras taught this, and also the rule of abstinence from animal flesh which is its natural corollary, but such ideas had been well known in many parts of Greece before his time. The real difficulty is to see the connexion between all this and his scientific work. Here we are of course confined to inferences from what we are told by later writers; but, if the doctrine which Plato makes Socrates expound in the early part of the Phaedo is Pythagorean, as it is generally supposed to be, we may say that what Pythagoras did was to teach that, while the ordinary methods of purification were well enough in their way, the best and truest purification for the soul was just scientific study. It is only in some such way as this that we can explain the religious note which is characteristic of all the best Greek science. It involves the doctrine that the Theoretic Life is the highest way of life for man, a belief still held by Plato and Aristotle, and to which we shall have to return. We may note at once, however, that it is not an ‘intellectualist’ ideal. There is no question of idle contemplation; it is a strenuous way of life, the aim of which is the soul’s salvation, and it gives rise to an eager desire to convert other men. Just for that reason, the Pythagorean philosopher will take part in practical life when the opportunity offers, and he will even rule the state if called upon to do so. The Pythagorean society was a proselytizing body from the first, and it tried to bring in all it could reach, without distinction of nationality, social position, or sex (for women played a great part in it from the first). It was precisely its zeal for the reform of human life, and its attempt to set up a Rule of the Saints in the cities of southern Italy that led to its unpopularity. If the Pythagoreans had contented themselves with idle speculation, they would not have been massacred or forced to take refuge in flight, a fate which overtook them before the middle of the fifth century.

It soon proved, however, that the Pythagorean doctrine in its entirety was too high a one for its adherents, and a rift between Pythagorean religion and Pythagorean science was inevitable. Those who were capable of appreciating the scientific side of the movement would tend more and more to neglect the religious rule which it prescribed, and we find accordingly that before the end of the fifth century the leading Pythagoreans, the men whose names we know, are first of all men of science, and more and more inclined to drop what they doubtless regarded as the superstitious side of the doctrine. In the end they were absorbed in the new philosophical schools which arose at Athens. The mass of the faithful, on the other hand, took no interest in arithmetic, geometry, music, and astronomy, and with them to follow Pythagoras meant to go barefoot and to abstain from animal flesh and beans. These continued the tradition even after scientific Pythagoreanism had become extinct as such, and they were a favourite subject of ridicule with the comic poets of the fourth century B. C.

It is easy for us to see now that all this indicates a real weakness in Pythagoreanism. Science and religion are not to be brought into union by a simple process of juxtaposition. We do not know how far Pythagoras himself was conscious of the ambiguity of his position; it would not be surprising if he came to feel it towards the end of his life, and we know for certain that he lived long enough to witness the beginnings of the revolt against his society in Croton and elsewhere. It is for this reason that he removed to Metapontum where he died, and where Cicero was able to visit his tomb long afterwards. We shall see later what the weak point in his system was, and we shall have to consider how the discord he had left unresolved was ultimately overcome. For the present, it is more important to note that he was the real founder both of science, and of philosophy as we understand them now. It is specially true of science that it is the first steps which are the most difficult, and Pythagoras left a sufficient achievement in mathematics behind him for others to elaborate. The Greeks took less than three centuries to complete the edifice, and that was chiefly due to Pythagoras, who had laid the foundations truly and well.

We have now seen how the two great conceptions of Matter and Form were reached; the next problem Greek philosophy had to face was that of Motion. At first the fact of movement had simply been taken for granted. The Ionian tendency was to see motion everywhere; it was rest that had to be explained, or rather the appearance of it. However, when the new conception of an eternal matter began to be taken seriously, difficulties made themselves felt at once. If reality was regarded as continuous, it appeared that there was no room for anything else, not even for empty space, which could only be identified with the unreal, and it was easy to show that the unreal could not exist. But, if there is no empty space, it seems impossible that there should be any motion, and the world of which we suppose ourselves to be aware must be an illusion. Such, briefly stated, was the position taken up by another Ionian of southern Italy, Parmenides of Elea (c. 475 B. C.), who had begun as a Pythagorean, but had been led to apply the rigorous method of reasoning introduced into geometry with such success by the Pythagoreans to the old question of the nature of the world which had occupied the Milesians. The remarkable thing about the earliest geometers is, in fact, that they did not formulate the conception of Space, which seems to us at the present day fundamental. They were able to avoid it because they possessed the conception of Matter, and regarded Air as the normal state of the material substratum. The confusion of air with empty space is, of course, a natural one, though it may be considered surprising that it should not have been detected by the founders of geometrical science. Such failures to draw all the consequences from a new discovery are common enough, however, in the history of scientific thought.

Parmenides cleared up this ambiguity, not by affirming the existence of empty space, but by denying the possibility of such a thing, even before it had been asserted by any one. He saw that the Pythagoreans really implied it, though they were quite unconscious of the fact. He is interesting to us as the first philosopher who thought of expounding his system in verse. It was not a very happy thought, as the arguments in which he deals do not readily lend themselves to this mode of expression, and we may be thankful that none of his successors except Empedocles followed his example. It has the very great inconvenience of making it necessary to use different words for the same thing to suit the exigencies of metre. And if there ever was an argument that demanded precise statement, it was that of Parmenides. As it is, his poem has the faults we should look for in a metrical version of Euclid. On the other hand, Parmenides is the first philosopher of whom we have sufficient remains to enable us to follow a continuous argument; for we have nothing of Pythagoras at all, and only detached fragments of the rest. We can see that he was ready to follow the argument wherever it might lead. He took the conception of matter which had been elaborated by his predecessors and he showed that, if it is to be taken seriously, it must lead to the conclusion that reality is continuous, finite, and spherical, with nothing outside it and no empty space within it. For such a reality motion is impossible, and the world of the senses is therefore an illusion. Of course that was not a result in which it was possible for men to acquiesce for long, and historically speaking, the Eleatic doctrine must be regarded as a reductio ad absurdum of earlier speculation. There is no reason to believe, however, that Parmenides himself meant it to be understood in this way. He believed firmly that he had found the truth.

Several attempts were made to escape the conclusions of Parmenides, and they all start by abandoning the assumption of the homogeneity and continuity of matter which had been implicit in the earlier systems, though it was first brought to the light of day by Parmenides. Here again the influence of contemporary science on philosophic thought is clearly marked. Empedocles of Agrigentum (c. 460 B. C.), the only citizen of a Dorian state who finds a place in the early history of science and philosophy, was the founder of the Sicilian school of medicine, and it was probably his pre-occupation with that science that led him to revive the old doctrine of Fire, Air, Earth, and Water, which the Milesians had cast aside, but which lent itself readily to the physiological theories of the day. He did not use the word afterwards translated ‘elements’ (στοιχεια) for these. It means literally ‘letters of the alphabet’, and appears to have been first employed in this connexion by the Pythagoreans at a later date, when they found it necessary to take account of the new theory. Empedocles spoke of the ‘four roots’ of things, and by this he meant to imply that these four forms of matter were equally original and altogether disparate. That furnished at least a partial answer to the arguments of Parmenides, which depended on the assumption that matter was homogeneous. He also found it necessary to assume two sources of motion or forces, as we might call them, though Empedocles thought of them as substances, one of which tended to separate the ‘four roots’ and the other to combine them. These he called Love and Strife, and he supposed the life of the world to take the form of alternate cycles, in which one or the other prevailed in turn. In all this he was plainly influenced by his physiological studies. He thinks of the world as an animal organism subject to what are now called anabolism and catabolism. The details of the theory make this quite clear. A similar doctrine was taught by Anaxagoras (c. 460 B. C.), who came from Clazomenae in Asia Minor to Athens after the Persian Wars, and was one of the teachers of Pericles. His doctrine of ‘seeds’, in which the traditional ‘opposites’—wet and dry, cold and hot—were combined in different proportions, is rather more subtle than that of Empedocles, and it is possible to see in it a curious anticipation of certain features in modern chemistry. Anaxagoras too felt it necessary to assume a force or source of motion, but he thought that one would suffice to account for the rotation (περιχωρησις) to which he attributed the formation of the world. He called that force Mind (νους), but his own description of it shows that he regarded it as corporeal, though he thought it was something more tenuous and unmixed than other bodies. There is little doubt that he selected the term in order to mark the identity of the source of motion in the world with that in the animal organism. That again is in accordance with the scientific interests of the time. In his astronomical theories, however, Anaxagoras showed himself a true eastern Ionian, and lagged far behind the Pythagoreans. For him, as for the Ionians of the Aegean down to and including Democritus, the earth was flat, and the eddy or vortex which gave rise to the world was still rotation in a plane. A more satisfying answer to Parmenides was the doctrine of Atomism, which frankly accepted the existence of space, and asserted that it was just as real as body. The first hint of such a solution was given by Melissus (c. 444 B. C.), who was a Samian but a member of the Eleatic school. He said, ‘If things are a many, then each of them must be such as I have shown the One to be.’ That was meant as a reductio ad absurdum; but, when Leucippus of Miletus (c. 440 B. C.), who had also studied in the school of Elea, ventured to assert the existence of the Void, there was no longer any reason for shirking the conclusion which Melissus had stated only to show its impossibility. The atoms are, in fact, just the continuous indivisible One of Parmenides multiplied ad infinitum in an infinite empty space. On that side at least, the theory of body was now complete, and the question asked by Thales was answered, and it is of great interest to observe that this was brought about by the renewal of intercourse between the Ionians of Italy and those of the Aegean, a renewal which was made possible by the establishment of the Athenian Empire. Nothing makes us feel the historical connexion more vividly than the re-emergence of the names of Miletus and Samos after all these years. There were, however, certain more fundamental problems which Atomism could not solve, and which were first attacked at Athens itself. So far, it will be noted, Athens has played no part at all in our story, and in fact no more than two Athenians ever became philosophers of the first rank. It is true that they were called Socrates and Plato, so the exception is a considerable one. It was the foundation of the Athenian Empire that made Athens the natural meeting-place of the most diverse philosophical and scientific views. It was here that the east and west of Hellas came together, and that the two streams of tradition became one, with the result that a new tradition was started which, though often interrupted for a time, continues to the present day.

If we wish to understand the development of Greek philosophy, it is of the first importance that we should realize the intellectual ferment which existed at Athens in the great days of the Periclean age. It has been mentioned already that Anaxagoras of Clazomenae had settled there, and it was not long before his example was followed by others. In particular, Zeno of Elea (c. 450 B. C.), the favourite disciple of Parmenides, had a considerable following at Athens. He made it his business to champion the doctrine of his master by showing that those who refused to accept it were obliged to give their assent to views which were at least as repugnant to common sense, and in this way he incidentally did much for mathematics and philosophy by raising the difficulties of infinite divisibility and continuity in an acute form. All that is something quite apart from the influence of the ‘sophists’ at a rather later date, though they too came both from the east and from the west, and though they had been influenced by the more strictly philosophical schools of these regions. It was into this Athens that Socrates was born (470 B. C.) about ten years after the battle of Salamis, and he was naturally exposed to all these conflicting influences, of which Plato has given us a vivid description in the Phaedo, from his earliest youth. He cannot, in fact, be understood at all unless this historical background is kept constantly in view. There can be no reasonable doubt that at a very early age he attached himself to Archelaus, an Athenian who had succeeded Anaxagoras, when that philosopher had to leave Athens for Lampsacus. Ion of Chios, a contemporary witness, said that Socrates had visited Asia Minor with Archelaus, and that appears to refer to the siege of Samos, when Socrates was under thirty. There is no reason whatever to doubt the statement, which Plato makes more than once, that he had met Parmenides and Zeno at a still earlier date. At any rate, the influence of Zeno on the dialectic of Socrates is unmistakable. We may also take it that he was familiar with all sorts of Orphic and Pythagorean sectaries. Aeschines of Sphettos wrote a dialogue entitled Telauges, in which he represented Socrates as rallying the extreme asceticism of the strict followers of Pythagoras. So far, however, as we can form a picture of him for ourselves, he was not the sort of man to become the disciple of any one. He was a genuine Athenian in respect of what is called his ‘irony’, which implies a certain humorous reserve which kept him from all extravagances, however interested he might be in the extravagances of others. Nevertheless, while still quite a young man, he had somehow acquired a reputation for ‘wisdom’, though he himself disclaimed anything of the sort. He had also, it appears, gathered round him a circle of ‘associates’ (ἑταιροι). The only direct evidence we have for these early days is the Clouds of Aristophanes (423 B. C.), which is of course a comedy and must not be taken too literally. On the other hand, a comic poet who knew his business (and surely Aristophanes did) could hardly present a well-known man to the Athenian public in a manner which had no relation to fact at all. It is fortunate that there is a passage in Xenophon’s Memorabilia (i. 6) which seems to supply us with the very background we need to make the Clouds intelligible. It represents Socrates in an entirely different light from that in which he appears in the rest of the work, and it can hardly be Xenophon’s own invention. It seems to refer to a time when Plato and Xenophon were babies, if not to a time before they were born, and it is probable that it comes from some literary source which we can no longer trace. We are told, then, that Antiphon the sophist was trying to detach his companions (συνουσιασται) from Socrates, and a conversation followed in which he charged him with teaching his followers to be miserable rather than happy, and added that he was right not to charge a fee for his teaching, since in fact it was of no value. It will be seen that this implies a regular relation between Socrates and his followers which was sufficiently well known to arouse professional jealousy. Socrates does not attempt to deny the fact. He says that what he and his companions do is to spend their time together in studying the wisdom of the men of old which they have left behind them in books, and that, if they come upon anything which they think is good, they extract it for their own use, and count it great gain if, in doing this, they become friends to one another. It is obvious that this suggests something quite different from the current view of Socrates as a talker at street corners, something much more like a regular school, and that, so far as it goes, it explains the burlesque of Aristophanes.

The Socrates of whom we know most is, however, quite differently engaged. He has devoted his life to a mission to his fellow men, and especially to his fellow citizens. If we may so far trust Plato’s Apology, the occasion of that was the answer received from the Delphic oracle by Chaerephon, whom we know from Aristophanes as one of the leading disciples of Socrates in the earlier part of his life. Chaerephon asked the god of Delphi whether there was any one wiser than Socrates, and this of course implies that Socrates had a reputation for ‘wisdom’ before his mission began. The oracle declared that there was no one wiser, and Plato makes Socrates say in the Apology that this was the real beginning of that mission. He set out at first to prove that the oracle was wrong, and for that purpose he tried to discover some one wiser than himself, a search in which he was disappointed, since he could only find people who thought they were wise, and no one who really was so. He therefore concluded that what the oracle really meant was that Socrates was wiser than other people in one respect only. Neither he nor any one else was really ‘wise’, but Socrates was wiser than the rest because he knew he was not wise and they thought they were. It ought to be clear that this is mostly ‘irony’, and it is not to be supposed that Socrates attached undue importance to the oracle, which he speaks of quite lightly, but he could hardly have told the story at all unless it was generally known that his mission did in fact date roughly from that period of his life. Historically it would probably be truer to say that the outbreak of the Peloponnesian War, in which Socrates served with great distinction as a hoplite, marked the decisive turning-point. It was in the camp at Potidaea that he once stood in a trance for twenty-four hours (431 B. C.), and that seems to point to some great psychological change, which may very well have been occasioned or accelerated by his experiences in the war. At any rate we now find him entirely devoted to the conversion of his fellow citizens, and we must try to understand what the message he had for them was.

In the Apology Socrates declares that his mission was divinely imposed upon him, so that he dare not neglect it, even if it should lead to his death, as in fact it did. The tone here is quite different from the half-humorous style in which he deals with the Delphic oracle, and even the ‘divine sign’. That only warned him not to do things, mostly quite trivial things, which he was about to do, and never told him to do anything; this, on the contrary, was a positive command, laid upon him by God, and there can be no doubt that Plato means us to understand this to have been the innermost conviction of Socrates. It is hard to believe that Plato could have misrepresented his master’s attitude on such a point. He was present at the trial, and the Apology must have been written not very long afterwards, when the memory of it was still fresh in people’s minds. Now Plato tells us quite clearly that what Socrates tried to get the Athenians to understand was the duty of ‘caring for their souls’ (ψυχης επιμελειο). That is confirmed from other sources, and indeed it is generally admitted. The phrase has, however, become so familiar that it does not at once strike us as anything very new or important. To an Athenian of the fifth century B. C., on the other hand, it must have seemed very strange indeed. The word translated ‘soul’ (ψυχη) occurs often enough, no doubt, in the literature of the period, but it is never used of anything for which we could be called upon to ‘care’ in the sense evidently intended by Socrates. Its normal use is to denote the breath of life, the ‘ghost’ a man ‘gives up’ at the moment of death. It can therefore be rendered by ‘life’ in all cases where there is a question of risking or losing life or of clinging to it when we ought to be prepared to sacrifice it, but it is not used for the seat of conscious life at all. It is sometimes employed to signify the seat of the dream-consciousness or of what is now called the subconscious or subliminal self, but never of the ordinary waking consciousness which is the seat of knowledge and ignorance, goodness and badness.[2] On the other hand, that use of the word is quite common in the fourth century, and it may be inferred that this change was due to Socrates. More than once Aristophanes ridicules him for holding some strange view of the ‘soul’, and these jests were made at a time when Plato was only a child. We cannot, of course, expect to get any very definite idea from them as to the real teaching of Socrates on this subject, but it is not impossible to see what it was, if we take into account the views of the soul which had been held by the philosophical schools of eastern and western Ionia.

The Ionians of Asia Minor had certainly identified the soul with that in us which is conscious, and which is the seat of goodness and badness, wisdom and folly; but they did not regard it as what we call the self or treat it as an individual. Anaximenes and his school held that the soul was what they called Air, but that was just because they regarded Air as the primary substance of which all things are made. The soul was something, in fact, that comes to us from outside (θυραθεν) by means of respiration. As Diogenes of Apollonia expresses it, it is ‘a small portion of the god’, that is, of the primary substance, enclosed in a human body for a time, and returning at death to the larger mass of the same substance outside. The formula ‘Earth to earth and air to air’ was accepted as an adequate description of what takes place at death. The western Ionians, and especially the Pythagoreans, held a very different view. For them, the soul was something divine. It was, in fact, a fallen god, imprisoned in the body as a punishment for antenatal sin, and it deserved our care in this sense, that it was our chief business in life to purify it so as to secure its release from the necessity of reincarnation in another body. But, during this present life, they held that this divine element slumbers, except in prophetic dreams. As Pindar puts it, ‘It sleeps when the limbs are active.’ Neither of these views was familiar to the ordinary Athenian, but Socrates of course knew both well, and felt satisfied with neither. When he spoke of the soul he did not mean any mysterious fallen god which was the temporary tenant of the body, but the conscious self which it lies with us to try to make wise and good. On the other hand, his insistence on our duty to ‘care for’ it is quite inconsistent with the view that it is merely something extrinsic, as all the eastern Ionians down to Anaxagoras had taught. It is, on the contrary, our very self, the thing in us which is of more importance to us than anything else whatever. It was to this doctrine of the soul and our duty to it that Socrates felt he must convert mankind and especially his fellow-citizens. It was a strange and novel doctrine then; and, if it has become a commonplace since, that only shows that he was successful, if not in persuading his fellowmen to act on this knowledge, at least in making them aware of it. It was in this way that Socrates healed the rift between science and religion which had proved fatal to the Pythagorean society, and it may be suggested that the significance of his teaching is not exhausted yet. As has been indicated above it is to be found clearly stated in Plato’s Apology of Socrates, and it furnishes the only clue to a right understanding of the great series of Platonic dialogues down to and including the Republic in which Socrates is represented as the chief speaker. Whether Plato added much or little of his own to the doctrine of his master in these dialogues is an interesting historical problem, but it need not concern the ordinary reader, at least in the first instance. We know from the allusions of Aristophanes that Socrates himself taught a new doctrine of the soul when Plato was a child, and no sympathetic reader can fail to see that the passage of the Apology to which we have referred is intended to be a faithful account of that doctrine. All the rest is simply its legitimate development, and it is not of very great importance for us to determine whether that development is due to Socrates or to Plato. The inspiration which has been derived from these writings by many generations will not be lessened by any decision we may come to on this point, so long as we keep clearly in mind that the new doctrine of soul is their principal theme, and that this must be understood in the light of the doctrines which had prepared the way for it. What Socrates did was really this. He deepened the meaning of the Eastern Ionian doctrine by informing it with some of the feeling and emotion which had characterized the Pythagorean teaching on the subject, while on the other hand he rationalized the Pythagorean theory by identifying the soul with our conscious personality.

Now if this is a correct account of what Socrates taught, he must be regarded as inaugurating an entirely new period in the history of philosophy. That is implied in the common term ‘Presocratics’ generally applied to his predecessors, though the ordinary textbooks are by no means clear as to the grounds for assigning this pre-eminent position to Socrates. We can also see how natural it was for him to lay such emphasis on the conversion of souls as he certainly did. That purpose continued to dominate Greek philosophy to the very end. No doubt successive schools varied in their conception of what conversion meant, but that is the link which binds them all together. In fact, it gave rise to a new literary form, the ‘hortatory discourse’ (προτρεπτικος λογος), which was more and more cultivated as time went on, and was at last taken over by the fathers of the Christian church along with much else of a more fundamental character.

It has been noted already that Socrates had followers among all the leading philosophical schools of the time, and the possibility is not to be excluded that we may still learn more of him from the discovery of new sources. For the present, the recovery of some new and fairly extensive fragments of the Alcibiades of Aeschines of Sphettos is the chief addition to our sources of information. We know that Aeschines was a disciple of Socrates, and the tradition of antiquity was that his dialogues gave the most faithful picture of the man as he really was. If so, that was probably because Aeschines had no philosophy of his own. For us the chief importance of the new fragments is that, if we read them along with those already known (and it is unfortunate that the old and the new have not yet been printed together), they strongly confirm the impression we get from Plato of the manner of Socrates and his method of argument, and that helps to reassure us as to the essentially historical character of the Platonic Socrates. The fragments of Aeschines also corroborate Plato by showing that the conversion of Alcibiades (whose life he had saved when a young man) was one of the things that lay nearest his heart.

But the real successor of Socrates was, of course, Plato himself (427-347 B. C.). It is not possible to give even an outline of Plato’s philosophy here. Indeed the time has hardly come for that yet, though much admirable work is now being done, especially by a French professor, M. Robin, which promises more certain conclusions than have yet been possible. All that can be attempted here is to indicate the attitude of Plato to some of the problems we have been discussing. His very great contributions to the theory of knowledge will be passed over, as they are beginning to be well understood, and the Theaetetus in particular, with its sequel the Sophist, is more and more coming to occupy its rightful place as the best introduction to philosophy in general. It is necessary, however, just to notice in passing a fundamental question of method which the Platonic dialogues themselves suggest. It is this. While Socrates is present in every one of them except the Laws, he takes practically no part in some of them, and the dialogues in which this is the case are known on other grounds to belong to the later years of Plato’s life. There must be some reason for this, and it is obviously prudent to treat these later dialogues in the first instance as our primary evidence for Plato’s own views. Indeed, it is only after his philosophy has been reconstructed from these sources and from the sometimes obscure references to it in Aristotle, that it will be safe to attempt an answer to the question of how much there may be in the dialogues of his early life which is properly to be assigned to Plato himself rather than to Socrates. That is a historical question of great interest; but, as has been said, the solution of it, if that should ever prove possible, would not greatly affect the impression that Athenian philosophy leaves upon us as a whole.

Now, if we consider Plato’s later, and presumably therefore most independent writings, we find, just as we should expect from a disciple of Socrates, that the doctrine of soul holds the first place, but that it has certain features of its own which there is no sufficient ground for attributing to Socrates. We are too apt to think of Plato as mainly occupied with what is called the ‘theory of Ideas’, a theory which is discussed once or twice in his earlier dialogues, and which is there ascribed to Socrates, but which plays no part at all in his mature works. There the chief place is undoubtedly taken by the doctrine of the soul, and we can see that it is of the first importance for Plato. Soul is regarded as the source of all motion in the world, because it is the only thing in the world that moves without being itself moved by anything else. It is this and this alone that enables Plato to account for the existence of the world and of mankind, and to avoid the theory of ‘two worlds’ into which, as he points out in the Sophist, ‘the friends of the Ideas’, whoever they may have been, were only too apt to fall. In Plato this view of the soul culminates in theology of a kind which he nowhere attributes to Socrates. He represents him, indeed, as a man of a deeply religious nature, but we do not gather that he had felt the need of a formal doctrine of God. Plato, on the other hand, has left us the first systematic defence of Theism we know of, and it is based entirely on his doctrine of soul as the self-moved mover. But the highest soul, or God, is not only the ultimate source of motion, but also supremely good. Now, since there are many things in the world which are not good, and since it would be blasphemy to attribute these to God, there must be other souls in the world which are relatively at least independent. God is not, directly at least, the cause of all things, but it is not easy to discover the relation in which these other souls are thought of as standing to God. In the Timaeus, the matter is put in this way. The soul of the world, and all other souls human and divine, are the work of the Creator, who is identified with God, and they are not inherently indestructible, since anything that has been made can be unmade. They are, however, practically indestructible, since God made all things because He was good and wished them also to be as good as possible. His goodness, therefore, will not suffer Him to destroy what He has once made. That of course is mythically expressed, and Plato is not committed to it as a statement of his own belief, since it is only the account which Timaeus puts into the mouth of the Creator. We can see, however, what was the problem with which he was occupied, and it is not perhaps illegitimate to infer that he approached the question which still baffles speculation from the point of view that God’s omnipotence, as we should call it, is limited by his goodness. This is a much more important limitation than that imposed by the existence of matter, to which Timaeus also refers. In that, he is simply following the tradition of the Pythagorean society to which he belonged, as is shown by his identification of matter with space, or rather with ‘room’. So far as can be seen at present, we are not entitled to ascribe this view to Plato without more ado, but that is a point on which the last word has not yet been said.

The description of the creation given by Timaeus is of course to be regarded as mythical in its details, but it has features from which we may learn a good deal as to the direction taken by Plato’s thoughts about the world. In particular, while the important part played by geometry is quite intelligible in the mouth of a Pythagorean, he makes use of certain theories which we know to belong to the most recent mathematics of the day, in particular the complete doctrine of the five regular solids, which was due to Theaetetus, who was one of the earliest members of the Academy, and whom Plato represents as having made the acquaintance of Socrates just before the master’s death. Theaetetus died young, but we know enough of him to feel sure that he was one of the few great original mathematicians who have appeared in history. In the Timaeus the theory of the regular solids is used to get rid once more of the doctrine of four ultimate ‘elements’. These, Timaeus says, are so far from being elements or letters of the alphabet, that they are not even syllables. The way in which the so-called elements are built up out of molecules corresponding in their configuration to the regular solids, and the explanations of their transmutation into one another based on the geometrical construction of these figures, is apt to strike the average reader as fantastic, but one of the most distinguished living mathematicians and physicists has stated that he is struck most of all by their resemblance to the scientific theories of the twentieth century. It will be well, therefore, to avoid hasty judgements on this point. It is at any rate easy to understand how the study of mathematics came to hold the preponderating place it did in the Platonic Academy.

In accordance with the plan of this paper, something must now be said of Plato’s attitude to the practical life, a point on which it is very easy to make mistakes. No one has insisted more strongly than he has on the primacy of the Theoretic Life. The philosopher is the man who is in love with the spectacle of all time and all existence and that is what delivers him from petty ambitions and low desires. He has made the toilsome ascent out of the Cave in which the mass of men dwell, and in which they only behold the shadows of reality. But, even in this enthusiastic description of the philosophic life, an equal stress is laid on the duty of philosopher to descend into the Cave in turn and to rescue as many of their former fellow-prisoners as may be, even against their will, by turning them to the light and dragging them up into the world of truth and reality. It is quite easy to understand, in view of this, that Plato devoted some of the best years of his life to practical affairs and that he relinquished the studies of the Academy for a time in order to direct the education of Dionysius II. The thing appeared well worth doing; for Greek civilization in Sicily, and consequently, as we can now see, the civilization of western Europe, was seriously threatened by the Carthaginians. They had been held at bay by Dionysius I, but after his death everything depended on his successor. Now the education of Dionysius II had been completely neglected, but he had good natural abilities, and his uncle Dion, who was Plato’s friend, was ready to answer for his good intentions. Plato could not turn a deaf ear to such a call. Unfortunately Dionysius was vain and obstinate, and he soon became impatient of the serious studies which Plato rightly regarded as necessary to prepare him for his task. The result was a growing estrangement between Plato and his pupil, which made it impossible to hope for a successful issue to the plans of Dion. It is unnecessary to tell the whole story here, but it is right to say that there was nothing at all impracticable in what Plato undertook, and that he was certainly justified in holding that the education of Dionysius must be completed before it would be safe to entrust him with the championship of the cause of Hellenism in the west.

His failure to make anything of Dionysius did not lead Plato to abandon his efforts to heal the wounds of Hellenism. One of the studies most ardently pursued in the Academy was Jurisprudence, of which he is the real founder. It was not uncommon for Greek states to apply to the Academy for legislators to codify existing law or to frame a new code for colonies which had just been founded. That is the real explanation of the remarkable work entitled the Laws, which must have occupied Plato for many years, and which was probably begun while he was still directing the studies of Dionysius. It appears to have been left unfinished; for, while some parts of it are highly elaborated, there are others which make upon us the impression of being a first draft. Even so, it is a great work if we regard it from the proper point of view. It is, in the first place, a codification of Greek, and especially Athenian law, of course with those reforms and improvements which suggest themselves when the subject is systematically treated, and it formed the basis of Hellenistic, and through that of Roman law, to which the world owes so much. There is no more useful corrective of the popular notion of Plato as an unpractical visionary than the careful study of the dullest and most technical parts of the Laws in the light of the Institutes.

No attempt has been made here to describe the system of Plato as a whole, and indeed the time has not yet come when such an attempt can profitably be made. We have no direct knowledge of his teaching in the Academy; for we only possess the works which he wrote with a wider public in view. In the case of Aristotle (384-322 B. C.), a similar reservation must be made, though for just the opposite reason. We have only fragments of his published works and what we possess is mainly the groundwork of his lectures in the Lyceum. It will be seen that there is still very much to be done here too. From the nature of the case, notes for lectures take a great deal for granted that would be more fully explained when the lectures were delivered, and some of the most important points are hardly developed at all. Nevertheless there are certain things which come out clearly enough, and it so happens that they are points of great importance from which we can learn something with regard to the philosophical problems of the present day.

In the first place, it is desirable to point out that Aristotle was not an Athenian, but an Ionian from the northern Aegean, and that he was strongly influenced by eastern Ionian science, especially by the system of Democritus (which Plato does not appear to have known) and by the medical theories of the time. That is why he is so unsympathetic to the western schools of philosophy, and especially to the Pythagoreans and the Eleatics. Empedocles alone, who was a biologist like himself, and the founder of a medical school, finds favour in his eyes. He is not, therefore, at home in mathematical matters and his system of Physics can only be regarded as retrograde when we compare it with that of the Academy. He did indeed accept the doctrine of the earth’s sphericity, but with that exception his cosmological views must be called reactionary. Where he is really great is in biology, a field of research which was not entirely neglected by the Academy, but which had been treated as secondary in comparison with mathematics and astronomy. The contrast between Plato and Aristotle in this respect seems to repeat on a higher plane that between Pythagoras and Empedocles, and this suggests something like a law of philosophical development which may perhaps throw light on the present situation. It seems as if this alternation of the mathematical and the biological interest was fundamental in the development of scientific thought and that the philosophy of different periods takes its colour from it. The philosophy of the nineteenth century was dominated in the main by biological conceptions, while it seems as if that of the twentieth was to be chiefly mathematical in its outlook on the world. We must not, of course, make too much of such formulas, but it is instructive to study such alternations in the philosophy of the Greeks, where everything is simpler and more easily apprehended.

On the other hand, Aristotle had been a member of the Academy for twenty years, and that could not fail to leave its mark upon him. This no doubt explains the fact, which has often been noted, that there are two opposite and inconsistent strains in all Aristotle’s thinking. On the one hand, he is determined to avoid everything ‘transcendental’, and his dislike of Pythagorean and Platonist mathematics is mainly due to that. On the other hand, despite his captious and sometimes unfair criticisms of Plato, he evidently admired him greatly and had been much influenced by him. It may be suggested that the tone of his criticisms is partly due to his annoyance at finding that he could not shake off his Platonism, do what he would. This is borne out by the fact that, when he has come to the furthest point to which his own system will take him, he is apt to take refuge in metaphors of a mythical or ‘transcendental’ character, for which we are not prepared in any way and of which no explanation is vouchsafed us. That is particularly the case when he is dealing with the soul and the first mover. On the whole his account of the soul is simply a development of eastern Ionian theories, and we feel that we are far removed indeed from the Platonist conception of the soul’s priority to everything else. But, when he has told us that the highest and most developed form of soul is Mind, we are suddenly surprised by the statement that Mind in this sense is merely passive, while there is another form of it which is separable from matter, and that alone is immortal and everlasting. This has given rise to endless controversy which does not concern us here, but it seems best to interpret it as an involuntary outburst of the Platonism Aristotle could not wholly renounce. Very similar is the passage where he tries to explain how the first mover, though itself unmoved, communicates motion to the world. ‘It moves it like a thing beloved,’ he tells us, and leaves us to make what we can of that. And yet we cannot help feeling that, in passages like this, we come far nearer to the beliefs Aristotle really cared about than we do anywhere else. At heart he is a Platonist in spite of himself.

Aristotle’s attitude to the practical life is also dependent on Plato’s. In the Tenth Book of the Ethics he puts the claims of the Contemplative Life even higher than Plato ever did, so that the practical life appears to be only ancillary to it. He does not feel in the same degree as Plato the call for the philosopher to descend once more into the Cave for the sake of the prisoners there, and altogether he seems far more indifferent to the practical interests of life. Nevertheless he followed Plato’s lead in giving much of his time to the study of Politics and that too with the distinctly practical aim of training legislators. He has often been criticized for his failure to see that the days of the city-state were numbered, and for the way in which he ignores the rise of an imperial monarchy in the person of his own pupil Alexander the Great. That, however, is not quite fair. Aristotle had a healthy dislike of princes and courts, and the city-state still appealed to him as the normal form of political organization. He could not believe that it would ever be superseded, and he wished to contribute to its better administration. He had, in fact, a much more conservative outlook than Plato, who was inclined to think with Isocrates, that the revival of monarchy was the only thing that could preserve Hellenism as things were then. We must remember that Aristotle was not himself a citizen of any free state, and that he could hardly be expected to have the same political instincts as Plato, who belonged by birth to the governing classes of Athens and had inherited the liberal traditions of the Periclean Age. This comes out best of all perhaps, in the attitude of the two philosophers to the question of slavery. In the Laws, which deals with existing conditions, Plato of course recognizes the de facto existence of slavery, though he is very sensible of its dangers and makes many legislative proposals with a view to their mitigation. In the Republic, on the other hand, where there is no need to trouble about existing conditions, he makes Socrates picture for us a community in which there are apparently no slaves at all. Aristotle is also anxious to mitigate the worst abuses of slavery, but he justifies the institution as a permanent one by the consideration that barbarians are ‘slaves by nature’ and that it is for their own interest to be ‘living tools’. This insistence upon the fundamental distinction between Greeks and barbarians must have seemed an anachronism to many of Aristotle’s contemporaries and it had been expressly denounced by Plato as unscientific.

The immediate effect of Aristotle’s rejection of Platonist mathematics was one he certainly neither foresaw nor intended. It was to make a breach between philosophy and science. Mathematical science, whether Aristotle realized it or not, was still in the vigour of its first youth, and mathematicians were stirred by the achievements of the last generation to attempt the solution of still higher problems. If the Lyceum turned away from them, they were quite prepared to carry on the Academic tradition by themselves, and they succeeded for a time beyond all expectation. The third century B. C. was, in fact, the Golden Age of Greek mathematics, and it has been suggested that this was due to the emancipation of mathematics from philosophy. If that were true, it would be very important for us to know it; but it can, I think, be shown that it is not true. The great mathematicians of the third century were certainly carrying on the tradition of their predecessors who had been philosophers as well as mathematicians, and it is not to be wondered at that they were able to do so for a time. But the really striking fact is surely that Greek mathematics became sterile in a comparatively short time, and that no further advance was made till the days of Descartes and Leibniz, with whom philosophy and mathematics once more went hand in hand.

Nor was the effect of this divorce on philosophy itself less disastrous. Theophrastus continued Aristotle’s work on Aristotle’s lines, and founded the science of Botany as his predecessor had founded that of Zoology, but the Peripatetic School practically died out with him and had very little influence till the study of Aristotle was revived long afterwards by the Neoplatonists.

For the present, the divorce of science and philosophy was complete. The Stoics and the Epicureans had both, indeed, a scientific system, but their philosophy was in no sense based upon it. The attitude of Epicurus to science is particularly well marked. He took no interest in it whatever as such, but he used it as an instrument to free men from the religious fear to which he attributed human unhappiness. For that purpose, the science of the Academy, which had led up to a theology, was obviously unsuitable, and, like a true eastern Ionian as he was, Epicurus harked back to the atomic theory of Democritus, adding to it, however, certain things which really made nonsense of it, such, for instance, as the theory of absolute weight and lightness, which Aristotle had unfortunately taught. The Stoics too were corporealists, and found such science as they required in the system of Heraclitus, though they also adopted for polemical purposes much of Aristotle’s Logic, taking pains, however, to alter his terminology. Both these schools, in fact, while remaining faithful to the idea of philosophy as conversion, forgot that it had always been based on science in its best days. It was this, no doubt, which chiefly commended Stoicism and Epicureanism to the Romans, who were never really interested in science. Both Stoicism and Epicureanism made a practical appeal, though of a different kind, and that served to gain credit for them at Rome.

The Academy which Plato had founded still continued to exist, though it was diverted from its original purpose not more than a generation after Plato’s death. Mathematics, we have seen, had made itself independent, and the most pressing necessity of the time was certainly the criticism of the new dogmatism which the Stoics had introduced. That was really carrying on one side of Platonism and not the least important. It is true indeed that the Academy appears to us at this distance of time mainly as a school of scepticism, but we must remember that its scepticism was directed entirely to the sensible world, as to which the attitude of Plato himself was not fundamentally different. The real sceptics always refused to admit that the Academics were sceptics in the proper sense of the word, and it is possible that the tradition of Platonism proper was never wholly broken. At any rate, by the first century B. C., we begin to notice that Stoicism tends to become more and more Platonic. The study of Plato’s Timaeus came into favour again, and the commentary which Posidonius (c. 100 B. C.) wrote upon it had great influence on the development of philosophy down to the end of the Middle Ages. It is this period of eclecticism which is reflected for us in the philosophical writings of Cicero. It had great importance for the history of civilization, but it is far removed from the spirit of genuine Greek philosophy. That was dead for the present, and it did not come to life again till the third century of our era, when Platonism was revived at Rome by Plotinus.

It is only quite recently that historians of Greek philosophy have begun to do justice to ‘Neoplatonism’. That is partly due to the contemporary philosophical tendencies noted at the beginning of this paper, and partly to historical investigations into the philosophy of the Middle Ages, which is more and more seen to be dependent mainly on Neoplatonism down to and including the system of St. Thomas Aquinas. It was in fact the most decisive fact in the history of Western European civilization that Plotinus founded his school at Rome rather than at Athens or Alexandria; for that is how Western Europe became the real heir to the philosophy of Greece. Every one knows, of course, that Plotinus was a ‘mystic’, but the term is apt to suggest quite wrong ideas about him. He is often spoken of still as a man who introduced oriental ideas into Greek philosophy, and he is popularly supposed to have been an Egyptian. That is most improbable; and, if it were true, it would only make it the more remarkable that, though he certainly studied at Alexandria for eleven years, he never even mentions the religion of Isis, which was so fashionable at Rome in his day, and which had fascinated so genuine a Greek as Plutarch some generations before. There is no doubt that what Plotinus believed himself to be teaching was genuine Platonism, and that he had prepared himself for the task by a careful study of Aristotle and even of Stoicism, so far as that served his purpose. No doubt he was too great a man to make himself the mere mouthpiece of another’s thought; but, for all that, he was the legitimate successor of Plato, and it may be added that M. Robin, who has taken upon himself the arduous task of extracting Plato’s real philosophy from the writings of Aristotle, has come to the conclusion that there is a great deal more ‘Neoplatonism’ in Plato than is sometimes supposed.

Plotinus is a mystic, then, though not at all in the sense in which the term is often misused. He sets before his disciples a ‘way of life’ which leads by stages to the highest life of all, but that is just what Pythagoras and Plato had done, and it is only the continuation of a tradition which goes back among the Greeks to the sixth century B. C., nearly a thousand years before the time of Plotinus. His aim, like that of his predecessors, is the conversion of souls to this way of life, and he differs from such thinkers as the Stoics and the Epicureans in holding that the ‘way of life’ to which he calls them must be based once more on a systematic doctrine of God, the World and Man. The result was that the divorce which had existed for centuries between science and philosophy was once more annulled. We cannot say, indeed, that Plotinus himself made any special study of Mathematics, but there is no doubt at all that his followers did, and it is due to them, and especially to Proclus, that we know as much of Greek Mathematics as we do. Proclus was indeed the systematizer of the doctrine of Plotinus, though he differs from him on certain points, and his influence on later philosophy cannot be overestimated. It can be distinctly traced even in Descartes, whom it reached through a number of channels, the study of which has recently been undertaken by a French scholar, Professor Gilson, of the University of Strasbourg. When his researches are complete, the continuity of Greek and modern philosophy will be plainly seen, and the part played by Platonism in the making of the modern European mind will be made manifest. We shall then understand better than ever why Greek philosophy is a subject of perennial interest.

The history of Greek philosophy is, in fact, the history of our own spiritual past, and it is impossible to understand the present without taking it into account. In particular, the Platonist tradition underlies the whole of western civilization. It was at Rome, as has been pointed out, that Plotinus taught, and it was in certain Latin translations of the writings of his school that St. Augustine found the basis for a Christian philosophy he was seeking. It was Augustine’s great authority in the Latin Church that made Platonism its official philosophy for centuries. It is a complete mistake to suppose that the thinking of the Middle Ages was dominated by the authority of Aristotle. It was not till the thirteenth century that Aristotle was known at all, and even then he was studied in the light of Platonism, just as he had been by Plotinus and his followers. It was only at the very close of the Middle Ages that he acquired the predominance which has made so strong an impression on the centuries that followed. It was from the Platonist tradition, too, that the science of the earlier Middle Ages came. A considerable portion of Plato’s Timaeus had been translated into Latin in the fourth century by Chalcidius with a very elaborate commentary based on ancient sources, while the Consolation of Philosophy, written in prison by the Roman Platonist Boethius in A. D. 525, was easily the most popular book of the Middle Ages. It was translated into English by Alfred the Great and by Chaucer, and into many other European languages. It was on these foundations that the French Platonism of the twelfth century, and especially that of the School of Chartres, was built up, and the influence of that school in England was very great indeed. The names of Grosseteste and Roger Bacon may just be mentioned in this connexion, and it would not be hard to show that the special character of the contribution which English writers have been able to make to science and philosophy is in large measure attributable to this influence.

But the interest of Greek philosophy is not only historical; it is full of instruction for the future too. Since the time of Locke, philosophy has been apt to limit itself to discussions about the nature of knowledge, and to leave questions about the nature of the world to specialists. The history of Greek philosophy shows the danger of this unnatural division of the province of thought, and the more we study it, the more we shall feel the need of a more comprehensive view. The ‘philosophy of things human’, as the Greeks called it, is only one department among others, and the theory of knowledge is only one department of that. If studied in isolation from the whole, it must inevitably become one-sided. From Greek philosophy we can also learn that it is fatal to divorce speculation from the service of mankind. The notion that philosophy could be so isolated would have been wholly unintelligible to any of the great Greek thinkers, and most of all perhaps to the Platonists who are often charged with this very heresy. Above all, we can learn from Greek philosophy the paramount importance of what we call the personality and they called the soul. It was just because the Greeks realized this that the genuinely Hellenic idea of conversion played so great a part in their thinking and in their lives. That, above all, is the lesson they have to teach, and that is why the writings of their great philosophers have still the power to convert the souls of all that will receive their teaching with humility.

J. Burnet.

MATHEMATICS AND ASTRONOMY

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 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.

The methods of solution of problems were no doubt first applied in particular cases and then gradually systematized; the technical terms for them were probably invented later, after the methods themselves had become established.

One method of solution was the reduction of one problem to another. This was called απαγωγη, a term which seems to occur first in Aristotle. But instances of such reduction occurred long before. Hippocrates of Chios reduced the problem of duplicating the cube to that of finding two mean proportionals in continued proportion between two straight lines, that is, he showed that, if the latter problem could be solved, the former was thereby solved also; and it is probable that there were still earlier cases in the Pythagorean geometry.

Next there is the method of mathematical analysis. This method is said to have been ‘communicated’ or ‘explained’ by Plato to Leodamas of Thasos; but, like reduction (to which it is closely akin), analysis in the mathematical sense must have been in use much earlier. Analysis and its correlative synthesis are defined by Pappus: ‘in analysis we assume that which is sought as if it were already done, and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of principles. But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis, and, by arranging in their natural order as consequences what were before antecedents and successively connecting them one with another, we arrive finally at the construction of that which was sought.’

The method of reductio ad absurdum is a variety of analysis. Starting from a hypothesis, namely the contradictory of what we desire to prove, we use the same process of analysis, carrying it back until we arrive at something admittedly false or absurd. Aristotle describes this method in various ways as reductio ad absurdum, proof per impossibile, or proof leading to the impossible. But here again, though the term was new, the method was not. The paradoxes of Zeno are classical instances.

Lastly, the Greeks established the form of exposition which still governs geometrical work, simply because it is dictated by strict logic. It is seen in Euclid’s propositions, with their separate formal divisions, to which specific names were afterwards assigned, (1) the enunciation (προτασις), (2) the setting-out (εκθεσις), (3) the διορισμος, being a re-statement of what we are required to do or prove, not in general terms (as in the enunciation), but with reference to the particular data contained in the setting-out, (4) the construction (κατασκευη), (5) the proof (αποδειξις), (6) the conclusion (συμπερασμα). In the case of a problem it often happens that a solution is not possible unless the particular data are such as to satisfy certain conditions; in this case there is yet another constituent part in the proposition, namely the statement of the conditions or limits of possibility, which was called by the same name διορισμος, definition or delimitation, as that applied to the third constituent part of a theorem.

We have so far endeavoured to indicate generally the finality and the abiding value of the work done by the creators of mathematical science. It remains to summarize, as briefly as possible, the history of Greek mathematics according to periods and subjects.

The Greeks of course took what they could in the shape of elementary facts in geometry and astronomy from the Egyptians and Babylonians. But some of the essential characteristics of the Greek genius assert themselves even in their borrowings from these or other sources. Here, as everywhere else, we see their directness and concentration; they always knew what they wanted, and they had an unerring instinct for taking only what was worth having and rejecting the rest. This is illustrated by the story of Pythagoras’s travels. He consorted with priests and prophets and was initiated into the religious rites practised in different places, not out of religious enthusiasm ‘as you might think’ (says our informant), but in order that he might not overlook any fragment of knowledge worth acquiring that might lie hidden in the mysteries of divine worship.

This story also illustrates an important advantage which the Greeks had over the Egyptians and Babylonians. In those countries science, such as it was, was the monopoly of the priests; and, where this is the case, the first steps in science are apt to prove the last also, because the scientific results attained tend to become involved in religious prescriptions and routine observances, and so to end in a collection of lifeless formulae. Fortunately for the Greeks, they had no organized priesthood; untrammelled by prescription, traditional dogmas or superstition, they could give their reasoning faculties free play. Thus they were able to create science as a living thing susceptible of development without limit.

Greek geometry, as also Greek astronomy, begins with Thales (about 624-547 B. C.), who travelled in Egypt and is said to have brought geometry from thence. Such geometry as there was in Egypt arose out of practical needs. Revenue was raised by the taxation of landed property, and its assessment depended on the accurate fixing of the boundaries of the various holdings. When these were removed by the periodical flooding due to the rising of the Nile, it was necessary to replace them, or to determine the taxable area independently of them, by an art of land-surveying. We conclude from the Papyrus Rhind (say 1700 B. C.) and other documents that Egyptian geometry consisted mainly of practical rules for measuring, with more or less accuracy, (1) such areas as squares, triangles, trapezia, and circles, (2) the solid content of measures of corn, &c., of different shapes. The Egyptians also constructed pyramids of a certain slope by means of arithmetical calculations based on a certain ratio, se-qeṭ, namely the ratio of half the side of the base to the height, which is in fact equivalent to the co-tangent of the angle of slope. The use of this ratio implies the notion of similarity of figures, especially triangles. The Egyptians knew, too, that a triangle with its sides in the ratio of the numbers 3, 4, 5 is right-angled, and used the fact as a means of drawing right angles. But there is no sign that they knew the general property of a right-angled triangle (= Eucl. I. 47), of which this is a particular case, or that they proved any general theorem in geometry.

No doubt Thales, when he was in Egypt, would see diagrams drawn to illustrate the rules for the measurement of circles and other plane figures, and these diagrams would suggest to him certain similarities and congruences which would set him thinking whether there were not some elementary general principles underlying the construction and relations of different figures and parts of figures. This would be in accord with the Greek instinct for generalization and their wish to be able to account for everything on rational principles.

The following theorems are attributed to Thales: (1) that a circle is bisected by any diameter (Eucl. I, Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I. 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I. 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I. 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle, which must mean that he was the first to discover that the angle in a semicircle is a right angle (cf. Eucl. III. 31).

Elementary as these things are, they represent a new departure of a momentous kind, being the first steps towards a theory of geometry. On this point we cannot do better than quote some remarks from Kant’s preface to the second edition of his Kritik der reinen Vernunft.

‘Mathematics has, from the earliest times to which the history of human reason goes back, (that is to say) with that wonderful people the Greeks, travelled the safe road of a science. But it must not be supposed that it was as easy for mathematics as it was for logic, where reason is concerned with itself alone, to find, or rather to build for itself, that royal road. I believe on the contrary that with mathematics it remained for long a case of groping about—the Egyptians in particular were still at that stage—and that this transformation must be ascribed to a revolution brought about by the happy inspiration of one man in trying an experiment, from which point onward the road that must be taken could no longer be missed, and the safe way of a science was struck and traced out for all time and to distances illimitable.... A light broke on the first man who demonstrated the property of the isosceles triangle (whether his name was Thales or what you will)....’

Thales also solved two problems of a practical kind: (1) he showed how to measure the distance of a ship at sea, and (2) he found the heights of pyramids by means of the shadows thrown on the ground by the pyramid and by a stick of known length at the same moment; one account says that he chose the time when the lengths of the stick and of its shadow were equal, but in either case he argued by similarity of triangles.

In astronomy Thales predicted a solar eclipse which was probably that of the 28th May 585 B. C. Now the Babylonians, as the result of observations continued through centuries, had discovered the period of 223 lunations after which eclipses recur. It is most likely therefore that Thales had heard of this period, and that his prediction was based upon it. He is further said to have used the Little Bear for finding the pole, to have discovered the inequality of the four astronomical seasons, and to have written works On the Equinox and On the Solstice.

After Thales come the Pythagoreans. Of the Pythagoreans Aristotle says that they applied themselves to the study of mathematics and were the first to advance that science, going so far as to find in the principles of mathematics the principles of all existing things. Of Pythagoras himself we are told that he attached supreme importance to the study of arithmetic, advancing it and taking it out of the region of practical utility, and again that he transformed the study of geometry into a liberal education, examining the principles of the science from the beginning.

The very word μαθηματα, which originally meant ‘subjects of instruction’ generally, is said to have been first appropriated to mathematics by the Pythagoreans.

In saying that arithmetic began with Pythagoras we have to distinguish between the uses of that word then and now. Αριθμητικη with the Greeks was distinguished from λογιστικη, the science of calculation. It is the latter word which would cover arithmetic in our sense, or practical calculation; the term αριθμητικη was restricted to the science of numbers considered in themselves, or, as we should say, the Theory of Numbers. Another way of putting the distinction was to say that αριθμητικη dealt with absolute numbers or numbers in the abstract, and λογιστικη with numbered things or concrete numbers; thus λογιστικη included simple problems about numbers of apples, bowls, or objects generally, such as are found in the Greek Anthology and sometimes involve simple algebraical equations.

The Theory of Numbers then began with Pythagoras (about 572-497 B. C.). It included definitions of the unit and of number, and the classification and definitions of the various classes of numbers, odd, even, prime, composite, and sub-divisions of these such as odd-even, even-times-even, &c. Again there were figured numbers, namely, triangular numbers, squares, oblong numbers, polygonal numbers (pentagons, hexagons, &c.) corresponding respectively to plane figures, and pyramidal numbers, cubes, parallelepipeds, &c., corresponding to solid figures in geometry. The treatment was mostly geometrical, the numbers being represented by dots filling up geometrical figures of the various kinds. The laws of formation of the various figured numbers were established. In this investigation the gnomon played an important part. Originally meaning the upright needle of a sun-dial, the term was next used for a figure like a carpenter’s square, and then was applied to a figure of that shape put round two sides of a square and making up a larger square. The arithmetical application of the term was similar. If we represent a unit by one dot and put round it three dots in such a way that the four form the corners of a square, three is the first gnomon. Five dots put at equal distances round two sides of the square containing four dots make up the next square (3²), and five is the second gnomon. Generally, if we have n² dots so arranged as to fill up a square with n for its side, the gnomon to be put round it to make up the next square, (n+1)², has 2n+1 dots. In the formation of squares, therefore, the successive gnomons are the series of odd numbers following 1 (the first square), namely 3, 5, 7, ... In the formation of oblong numbers (numbers of the form n(n+1)), the first of which is 1. 2, the successive gnomons are the terms after 2 in the series of even numbers 2, 4, 6.... Triangular numbers are formed by adding to 1 (the first triangle) the terms after 1 in the series of natural numbers 1, 2, 3 ...; these are therefore the gnomons (by analogy) for triangles. The gnomons for pentagonal numbers are the terms after 1 in the arithmetical progression 1, 4, 7, 10 ... (with 3, or 5-2, as the common difference) and so on; the common difference of the successive gnomons for an a-gonal number is a-2.

From the series of gnomons for squares we easily deduce a formula for finding square numbers which are the sum of two squares. For, the gnomon 2n+1 being the difference between the successive squares n² and (n+1)², we have only to make 2n+1 a square. Suppose that 2n+1=m²; therefore n=½(m²-1), and {½(m²-1)}²+m²={½(m²+1)}², where m is any odd number. This is the formula actually attributed to Pythagoras.

Pythagoras is said to have discovered the theory of proportionals or proportion. This was a numerical theory and therefore was applicable to commensurable magnitudes only; it was no doubt somewhat on the lines of Euclid, Book VII. Connected with the theory of proportion was that of means, and Pythagoras was acquainted with three of these, the arithmetic, geometric, and sub-contrary (afterwards called harmonic). In particular Pythagoras is said to have introduced from Babylon into Greece the ‘most perfect’ proportion, namely:

a:(a+b)/2=2ab/(a+b):b,

where the second and third terms are respectively the arithmetic and harmonic mean between a and b. A particular case is 12:9=8:6.

This bears upon what was probably Pythagoras’s greatest discovery, namely that the musical intervals correspond to certain arithmetical ratios between lengths of string at the same tension, the octave corresponding to the ratio 2:1, the fifth to 3:2 and the fourth to 4:3. These ratios being the same as those of 12 to 6, 8, 9 respectively, we can understand how the third term, 8, in the above proportion came to be called the ‘harmonic’ mean between 12 and 6.

The Pythagorean arithmetic as a whole, with the developments made after the time of Pythagoras himself, is mainly known to us through Nicomachus’s Introductio arithmetica, Iamblichus’s commentary on the same, and Theon of Smyrna’s work Expositio rerum mathematicarum ad legendum Platonem utilium. The things in these books most deserving of notice are the following.

First, there is the description of a ‘perfect’ number (a number which is equal to the sum of all its parts, i.e. all its integral divisors including 1 but excluding the number itself), with a statement of the property that all such numbers end in 6 or 8. Four such numbers, namely 6, 28, 496, 8128, were known to Nicomachus. The law of formation for such numbers is first found in Eucl. IX. 36 proving that, if the sum (Sn) of n terms of the series 1, 2, 2², 2³ ... is prime, then Sn.2n-1 is a perfect number.

Secondly, Theon of Smyrna gives the law of formation of the series of ‘side-’ and ‘diameter-’ numbers which satisfy the equations 2x²-y²=±1. The law depends on the proposition proved in Eucl. II. 10 to the effect that (2x+y)²-2(x+y)²=2x²-y², whence it follows that, if x, y satisfy either of the above equations, then 2x+y, x+y is a solution in higher numbers of the other equation. The successive solutions give values for y/x, namely 1/1, 3/2, 7/5, 17/12, 41/29, ..., which are successive approximations to the value of √2 (the ratio of the diagonal of a square to its side). The occasion for this method of approximation to √2 (which can be carried as far as we please) was the discovery by the Pythagoreans of the incommensurable or irrational in this particular case.

Thirdly, Iamblichus mentions a discovery by Thymaridas, a Pythagorean not later than Plato’s time, called the επανθημα (‘bloom’) of Thymaridas, and amounting to the solution of any number of simultaneous equations of the following form:

x+x1 + x2 + ... + xn-1 = s,
x + x1 = a1,
x + x2 = a2,
....
x+xn-1 = an-1,

the solution being x=((a1+a2+...+an-1)-s)/(n-2).

The rule is stated in general terms, but the above representation of its effect shows that it is a piece of pure algebra.

The Pythagorean contributions to geometry were even more remarkable. The most famous proposition attributed to Pythagoras himself is of course the theorem of Eucl. I. 47 that the square on the hypotenuse of any right-angled triangle is equal to the sum of the squares on the other two sides. But Proclus also attributes to him, besides the theory of proportionals, the construction of the ‘cosmic figures’, the five regular solids.

One of the said solids, the dodecahedron, has twelve regular pentagons for faces, and the construction of a regular pentagon involves the cutting of a straight line ‘in extreme and mean ratio’ (Eucl. II. 11 and VI. 30), which is a particular case of the method known as the application of areas. This method was fully worked out by the Pythagoreans and proved one of the most powerful in all Greek geometry. The most elementary case appears in Eucl. I. 44, 45, where it is shown how to apply to a given straight line as base a parallelogram with one angle equal to a given angle and equal in area to any given rectilineal figure; this construction is the geometrical equivalent of arithmetical division. The general case is that in which the parallelogram, though applied to the straight line, overlaps it or falls short of it in such a way that the part of the parallelogram which extends beyond or falls short of the parallelogram of the same angle and breadth on the given straight line itself (exactly) as base is similar to any given parallelogram (Eucl. VI. 28, 29). This is the geometrical equivalent of the solution of the most general form of quadratic equation ax±mx²=C, so far as it has real roots; the condition that the roots may be real was also worked out (=Eucl. VI. 27). It is in the form of ‘application of areas’ that Apollonius obtains the fundamental property of each of the conic sections, and, as we shall see, it is from the terminology of application of areas that Apollonius took the three names parabola, hyperbola, and ellipse which he was the first to give to the three curves.

Another problem solved by the Pythagoreans was that of drawing a rectilineal figure which shall be equal in area to one given rectilineal figure and similar to another. Plutarch mentions a doubt whether it was this problem or the theorem of Eucl. I. 47 on the strength of which Pythagoras was said to have sacrificed an ox.

The main particular applications of the theorem of the square on the hypotenuse, e. g. those in Euclid, Book II, were also Pythagorean; the construction of a square equal to a given rectangle (Eucl. II. 14) is one of them, and corresponds to the solution of the pure quadratic equation x²=ab.

The Pythagoreans knew the properties of parallels and proved the theorem that the sum of the three angles of any triangle is equal to two right angles.

As we have seen, the Pythagorean theory of proportion, being numerical, was inadequate in that it did not apply to incommensurable magnitudes; but, with this qualification, we may say that the Pythagorean geometry covered the bulk of the subject-matter of Books I, II, IV and VI of Euclid’s Elements. The case is less clear with regard to Book III of the Elements; but, as the main propositions of that Book were known to Hippocrates of Chios in the second half of the fifth century B. C., we conclude that they, too, were part of the Pythagorean geometry.

Lastly, the Pythagoreans discovered the existence of the incommensurable or irrational in the particular case of the diagonal of a square in relation to its side. Aristotle mentions an ancient proof of the incommensurability of the diagonal with the side by a reductio ad absurdum showing that, if the diagonal were commensurable with the side, it would follow that one and the same number is both odd and even. This proof was doubtless Pythagorean.

A word should be added about the Pythagorean astronomy. Pythagoras was the first to hold that the earth (and no doubt each of the other heavenly bodies also) is spherical in shape, and he was aware that the sun, moon and planets have independent movements of their own in a sense opposite to that of the daily rotation; but he seems to have kept the earth in the centre. His successors in the school (one Hicetas of Syracuse and Philolaus are alternatively credited with this innovation) actually abandoned the geocentric idea and made the earth, like the sun, the moon, and the other planets, revolve in a circle round the ‘central fire’, in which resided the governing principle ordering and directing the movement of the universe.

The geometry of which we have so far spoken belongs to the Elements. But, before the body of the Elements was complete, the Greeks had advanced beyond the Elements. By the second half of the fifth century B. C. they had investigated three famous problems in higher geometry, (1) the squaring of the circle, (2) the trisection of any angle, (3) the duplication of the cube. The great names belonging to this period are Hippias of Elis, Hippocrates of Chios, and Democritus.

Hippias of Elis invented a certain curve described by combining two uniform movements (one angular and the other rectilinear) taking the same time to complete. Hippias himself used his curve for the trisection of any angle or the division of it in any ratio; but it was afterwards employed by Dinostratus, a brother of Eudoxus’s pupil Menaechmus, and by Nicomedes for squaring the circle, whence it got the name τετραγωνιζουσα, quadratrix.

Hippocrates of Chios is mentioned by Aristotle as an instance to prove that a man may be a distinguished geometer and, at the same time, a fool in the ordinary affairs of life. He occupies an important place both in elementary geometry and in relation to two of the higher problems above mentioned. He was, so far as is known, the first compiler of a book of Elements; and he was the first to prove the important theorem of Eucl. XII. 2 that circles are to one another as the squares on their diameters, from which he further deduced that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of lunes, which was intended to lead up to the squaring of the circle. The essential portions of the tract are preserved in a passage of Simplicius’s commentary on Aristotle’s Physics, which contains substantial extracts from Eudemus’s lost History of Geometry. Hippocrates showed how to square three particular lunes of different kinds and then, lastly, he squared the sum of a circle and a certain lune. Unfortunately the last-mentioned lune was not one of those which can be squared, so that the attempt to square the circle in this way failed after all.

Hippocrates also attacked the problem of doubling the cube. There are two versions of the origin of this famous problem. According to one story an old tragic poet had represented Minos as having been dissatisfied with the size of a cubical tomb erected for his son Glaucus and having told the architect to make it double the size while retaining the cubical form. The other story says that the Delians, suffering from a pestilence, consulted the oracle and were told to double a certain altar as a means of staying the plague. Hippocrates did not indeed solve the problem of duplication, but reduced it to another, namely that of finding two mean proportionals in continued proportion between two given straight lines; and the problem was ever afterwards attacked in this form. If x, y be the two required mean proportionals between two straight lines a, b, then a:x=x:y=y:b, whence b/a=(x/a)³, and, as a particular case, if b=2a, x³=2a³, so that, when x is found, the cube is doubled.

Democritus wrote a large number of mathematical treatises, the titles only of which are preserved. We gather from one of these titles, ‘On irrational lines and solids’, that he wrote on irrationals. Democritus realized as fully as Zeno, and expressed with no less piquancy, the difficulty connected with the continuous and the infinitesimal. This appears from his dilemma about the circular base of a cone and a parallel section; the section which he means is a section ‘indefinitely near’ (as the phrase is) to the base, i. e. the very next section, as we might say (if there were one). Is it, said Democritus, equal or not equal to the base? If it is equal, so will the very next section to it be, and so on, so that the cone will really be, not a cone, but a cylinder. If it is unequal to the base and in fact less, the surface of the cone will be jagged, like steps, which is very absurd. We may be sure that Democritus’s work on ‘The contact of a circle or a sphere’ discussed a like difficulty.

Lastly, Archimedes tells us that Democritus was the first to state, though he could not give a rigorous proof, that the volume of a cone or a pyramid is one-third of that of the cylinder or prism respectively on the same base and having equal height, theorems first proved by Eudoxus.

We come now to the time of Plato, and here the great names are Archytas, Theodoras of Cyrene, Theaetetus, and Eudoxus.

Archytas (about 430-360 B. C.) wrote on music and the numerical ratios corresponding to the intervals of the tetrachord. He is said to have been the first to write a treatise on mechanics based on mathematical principles; on the practical side he invented a mechanical dove which would fly. In geometry he gave the first solution of the problem of the two mean proportionals, using a wonderful construction in three dimensions which determined a certain point as the intersection of three surfaces, (1) a certain cone, (2) a half-cylinder, (3) an anchor-ring or tore with inner diameter nil.

Theodorus, Plato’s teacher in mathematics, extended the theory of the irrational by proving incommensurability in certain particular cases other than that of the diagonal of a square in relation to its side, which was already known. He proved that the side of a square containing 3 square feet, or 5 square feet, or any non-square number of square feet up to 17 is incommensurable with one foot, in other words that √3, √5 ... √17 are all incommensurable with 1. Theodorus’s proof was evidently not general; and it was reserved for Theaetetus to comprehend all these irrationals in one definition, and to prove the property generally as it is proved in Eucl. X. 9. Much of the content of the rest of Euclid’s Book X (dealing with compound irrationals), as also of Book XIII on the five regular solids, was due to Theaetetus, who is even said to have discovered two of those solids (the octahedron and icosahedron).

Plato (427-347 B. C.) was probably not an original mathematician, but he ‘caused mathematics in general and geometry in particular to make a great advance by reason of his enthusiasm for them’. He encouraged the members of his school to specialize in mathematics and astronomy; e. g. we are told that in astronomy he set it as a problem to all earnest students to find ‘what are the uniform and ordered movements by the assumption of which the apparent motions of the planets may be accounted for’. In Plato’s own writings are found certain definitions, e. g. that of a straight line as ‘that of which the middle covers the ends’, and some interesting mathematical illustrations, especially that in the second geometrical passage in the Meno (86E-87C). To Plato himself are attributed (1) a formula (n²-1)²+(2n)²=(n²+1)² for finding two square numbers the sum of which is a square number, (2) the invention of the method of analysis, which he is said to have explained to Leodamas of Thasos (mathematical analysis was, however, certainly, in practice, employed long before). The solution, attributed to Plato, of the problem of the two mean proportionals by means of a frame resembling that which a shoemaker uses to measure a foot, can hardly be his.

Eudoxus (408-355 B. C.), an original genius second to none (unless it be Archimedes) in the history of our subject, made two discoveries of supreme importance for the further development of Greek geometry.

(1) As we have seen, the discovery of the incommensurable rendered inadequate the Pythagorean theory of proportion, which applied to commensurable magnitudes only. It would no doubt be possible, in most cases, to replace proofs depending on proportions by others; but this involved great inconvenience, and a slur was cast on geometry generally. The trouble was remedied once for all by Eudoxus’s discovery of the great theory of proportion, applicable to commensurable and incommensurable magnitudes alike, which is expounded in Euclid’s Book V. Well might Barrow say of this theory that ‘there is nothing in the whole body of the elements of a more subtile invention, nothing more solidly established’. The keystone of the structure is the definition of equal ratios (Eucl. V, Def. 5); and twenty-three centuries have not abated a jot from its value, as is plain from the facts that Weierstrass repeats it word for word as his definition of equal numbers, and it corresponds almost to the point of coincidence with the modern treatment of irrationals due to Dedekind.

(2) Eudoxus discovered the method of exhaustion for measuring curvilinear areas and solids, to which, with the extensions given to it by Archimedes, Greek geometry owes its greatest triumphs. Antiphon the Sophist, in connexion with attempts to square the circle, had asserted that, if we inscribe successive regular polygons in a circle, continually doubling the number of sides, we shall sometime arrive at a polygon the sides of which will coincide with the circumference of the circle. Warned by the unanswerable arguments of Zeno against infinitesimals, mathematicians substituted for this the statement that, by continuing the construction, we can inscribe a polygon approaching equality with the circle as nearly as we please. The method of exhaustion used, for the purpose of proof by reductio ad absurdum, the lemma proved in Eucl. X. 1 (to the effect that, if from any magnitude we subtract not less than half, and then from the remainder not less than half, and so on continually, there will sometime be left a magnitude less than any assigned magnitude of the same kind, however small): and this again depends on an assumption which is practically contained in Eucl. V, Def. 4, but is generally known as the Axiom of Archimedes, stating that, if we have two unequal magnitudes, their difference (however small) can, if continually added to itself, be made to exceed any magnitude of the same kind (however great).

The method of exhaustion is seen in operation in Eucl. XII. 1-2, 3-7 Cor., 10, 16-18. Props. 3-7 Cor. and Prop. 10 prove that the volumes of a pyramid and a cone are one-third of the prism and cylinder respectively on the same base and of equal height; and Archimedes expressly says that these facts were first proved by Eudoxus.

In astronomy Eudoxus is famous for the beautiful theory of concentric spheres which he invented to explain the apparent motions of the planets and, particularly, their apparent stationary points and retrogradations. The theory applied also to the sun and moon, for each of which Eudoxus employed three spheres. He represented the motion of each planet as produced by the rotations of four spheres concentric with the earth, one within the other, and connected in the following way. Each of the inner spheres revolves about a diameter the ends of which (poles) are fixed on the next sphere enclosing it. The outermost sphere represents the daily rotation, the second a motion along the zodiac circle; the poles of the third sphere are fixed on the latter circle; the poles of the fourth sphere (carrying the planet fixed on its equator) are so fixed on the third sphere, and the speeds and directions of rotation so arranged, that the planet describes on the second sphere a curve called the hippopede (horse-fetter), or a figure of eight, lying along and longitudinally bisected by the zodiac circle. The whole arrangement is a marvel of geometrical ingenuity.

Heraclides of Pontus (about 388-315 B. C.), a pupil of Plato, made a great step forward in astronomy by his declaration that the earth rotates on its own axis once in 24 hours, and by his discovery that Mercury and Venus revolve about the sun like satellites.

Menaechmus, a pupil of Eudoxus, was the discoverer of the conic sections, two of which, the parabola and the hyperbola, he used for solving the problem of the two mean proportionals. If a:x=x:y=y:b, then x²=ay, y²=bx and xy=ab. These equations represent, in Cartesian co-ordinates, and with rectangular axes, the conics by the intersection of which two and two Menaechmus solved the problem; in the case of the rectangular hyperbola it was the asymptote-property which he used.

We pass to Euclid’s times. A little older than Euclid, Autolycus of Pitane wrote two books, On the Moving Sphere, a work on Sphaeric for use in astronomy, and On Risings and Settings. The former work is the earliest Greek textbook which has reached us intact. It was before Euclid when he wrote his Phaenomena, and there are many points of contact between the two books.

Euclid flourished about 300 B. C. or a little earlier. His great work, the Elements in thirteen Books, is too well known to need description. No work presumably, except the Bible, has had such a reign; and future generations will come back to it again and again as they tire of the variegated substitutes for it and the confusion resulting from their bewildering multiplicity. After what has been said above of the growth of the Elements, we can appreciate the remark of Proclus about Euclid, ‘who put together the Elements, collecting many of Eudoxus’s theorems, perfecting many of Theaetetus’s and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors’. Though a large portion of the subject-matter had been investigated by those predecessors, everything goes to show that the whole arrangement was Euclid’s own; it is certain that he made great changes in the order of propositions and in the proofs, and that his innovations began at the very beginning of Book I.

Euclid wrote other books on both elementary and higher geometry, and on the other mathematical subjects known in his day. The elementary geometrical works include the Data and On Divisions (of figures), the first of which survives in Greek and the second in Arabic only; also the Pseudaria, now lost, which was a sort of guide to fallacies in geometrical reasoning. The treatises on higher geometry are all lost; they include (1) the Conics in four Books, which covered almost the same ground as the first three Books of Apollonius’s Conics, although no doubt, for Euclid, the conics were still, as with his predecessors, sections of a right-angled, an obtuse-angled, and an acute-angled cone respectively made by a plane perpendiular to a generator in each case; (2) the Porisms in three Books, the importance and difficulty of which can be inferred from Pappus’s account of it and the lemmas which he gives for use with it; (3) the Surface-Loci, to which again Pappus furnishes lemmas; one of these implies that Euclid assumed as known the focus-directrix property of the three conics, which is absent from Apollonius’s Conics.

In applied mathematics Euclid wrote (1) the Phaenomena, a work on spherical astronomy in which ὁ ὁριζων (without κυκλος or any qualifying words) appears for the first time in the sense of horizon; (2) the Optics, a kind of elementary treatise on perspective: these two treatises are extant in Greek; (3) a work on the Elements of Music. The Sectio Canonis, which has come down under the name of Euclid, can, however, hardly be his in its present form.

In the period between Euclid and Archimedes comes Aristarchus of Samos (about 310-230 B. C.), famous for having anticipated Copernicus. Accepting Heraclides’s view that the earth rotates about its own axis, Aristarchus went further and put forward the hypothesis that the sun itself is at rest, and that the earth, as well as Mercury, Venus, and the other planets, revolve in circles about the sun. We have this on the unquestionable authority of Archimedes, who was only some twenty-five years later, and who must have seen the book containing the hypothesis in question. We are told too that Cleanthes the Stoic thought that Aristarchus ought to be indicted on the charge of impiety for setting the Hearth of the Universe in motion.

One work of Aristarchus, On the sizes and distances of the Sun and Moon, which is extant in Greek, is highly interesting in itself, though it contains no word of the heliocentric hypothesis. Thoroughly classical in form and style, it lays down certain hypotheses and then deduces therefrom, by rigorous geometry, the sizes and distances of the sun and moon. If the hypotheses had been exact, the results would have been correct too; but Aristarchus in fact assumed a certain angle to be 87° which is really 89° 50', and the angle subtended at the centre of the earth by the diameter of either the sun or the moon to be 2°, whereas we know from Archimedes that Aristarchus himself discovered that the latter angle is only ½°. The effect of Aristarchus’s geometry is to find arithmetical limits to the values of what are really trigonometrical ratios of certain small angles, namely

1/18 > sin 3° > 1/20, 1/45 > sin 1° > 1/60, 1 > cos 1° > 89/90.

The main results obtained are (1) that the diameter of the sun is between 18 and 20 times the diameter of the moon, (2) that the diameter of the moon is between 2/45ths and 1/30th of the distance of the centre of the moon from our eye, and (3) that the diameter of the sun is between 19/3rds and 43/6ths of the diameter of the earth. The book contains a good deal of arithmetical calculation.

Archimedes was born about 287 B. C. and was killed at the sack of Syracuse by Marcellus’s army in 212 B. C. The stories about him are well known, how he said ‘Give me a place to stand on, and I will move the earth’ (πα βω και κινω ταν γαν; how, having thought of the solution of the problem of the crown when in the bath, he ran home naked shouting ἑυρηκα, ἑυρηκα; and how, the capture of Syracuse having found him intent on a figure drawn on the ground, he said to a Roman soldier who came up, ‘Stand away, fellow, from my diagram.’ Of his work few people know more than that he invented a tubular screw which is still used for pumping water, and that for a long time he foiled the attacks of the Romans on Syracuse by the mechanical devices and engines which he used against them. But he thought meanly of these things, and his real interest was in pure mathematical speculation; he caused to be engraved on his tomb a representation of a cylinder circumscribing a sphere, with the ratio 3/2 which the cylinder bears to the sphere: from which we infer that he regarded this as his greatest discovery.

Archimedes’s works are all original, and are perfect models of mathematical exposition; their wide range will be seen from the list of those which survive: On the Sphere and Cylinder I, II, Measurement of a Circle, On Conoids and Spheroids, On Spirals, On Plane Equilibriums I, II, the Sandreckoner, Quadrature of the Parabola, On Floating Bodies I, II, and lastly the Method (only discovered in 1906). The difficult Cattle-Problem is also attributed to him, and a Liber Assumptorum which has reached us through the Arabic, but which cannot be his in its present form, although some of the propositions in it (notably that about the ‘Salinon’, salt-cellar, and others about circles inscribed in the αρβηλος, shoemaker’s knife) are quite likely to be of Archimedean origin. Among lost works were the Catoptrica, On Sphere-making, and investigations into polyhedra, including thirteen semi-regular solids, the discovery of which is attributed by Pappus to Archimedes.

Speaking generally, the geometrical works are directed to the measurement of curvilinear areas and volumes; and Archimedes employs a method which is a development of Eudoxus’s method of exhaustion. Eudoxus apparently approached the figure to be measured from below only, i. e. by means of figures successively inscribed to it. Archimedes approaches it from both sides by successively inscribing figures and circumscribing others also, thereby compressing them, as it were, until they coincide as nearly as we please with the figure to be measured. In many cases his procedure is, when the analytical equivalents are set down, seen to amount to real integration; this is so with his investigation of the areas of a parabolic segment and a spiral, the surface and volume of a sphere, and the volume of any segments of the conoids and spheroids.

The newly-discovered Method is especially interesting as showing how Archimedes originally obtained his results; this was by a clever mechanical method of (theoretically) weighing infinitesimal elements of the figure to be measured against elements of another figure the area or content of which (as the case may be) is known; it amounts to an avoidance of integration. Archimedes, however, would only admit that the mechanical method is useful for finding results; he did not consider them proved until they were established geometrically.

In the Measurement of a Circle, after proving by exhaustion that the area of a circle is equal to a right-angled triangle with the perpendicular sides equal respectively to the radius and the circumference of the circle, Archimedes finds, by sheer calculation, upper and lower limits to the ratio of the circumference of a circle to its diameter (what we call π). This he does by inscribing and circumscribing regular polygons of 96 sides and calculating approximately their respective perimeters. He begins by assuming as known certain approximate values for √3, namely 1351/780 > √3 > 265/153, and his calculations involve approximating to the square roots of several large numbers (up to seven digits). The text only gives the results, but it is evident that the extraction of square roots presented no difficulty, notwithstanding the comparative inconvenience of the alphabetic system of numerals. The result obtained is well known, namely 3-1/7 > π > 3-10/71.

The Plane Equilibriums is the first scientific treatise on the first principles of mechanics, which are established by pure geometry. The most important result established in Book I is the principle of the lever. This was known to Plato and Aristotle, but they had no real proof. The Aristotelian Mechanics merely ‘refers’ the lever ‘to the circle’, asserting that the force which acts at the greater distance from the fulcrum moves the system more easily because it describes a greater circle. Archimedes also finds the centre of gravity of a parallelogram, a triangle, a trapezium and finally (in Book II) of a parabolic segment and of a portion of it cut off by a straight line parallel to the base.

The Sandreckoner is remarkable for the development in it of a system for expressing very large numbers by orders and periods based on powers of myriad-myriads (10,000²). It also contains the important reference to the heliocentric theory of the universe put forward by Aristarchus of Samos in a book of ‘hypotheses’, as well as historical details of previous attempts to measure the size of the earth and to give the sizes and distances of the sun and moon.

Lastly, Archimedes invented the whole science of hydrostatics. Beginning the treatise On Floating Bodies with an assumption about uniform pressure in a fluid, he first proves that the surface of a fluid at rest is a sphere with its centre at the centre of the earth. Other propositions show that, if a solid floats in a fluid, the weight of the solid is equal to that of the fluid displaced, and, if a solid heavier than a fluid is weighed in it, it will be lighter than its true weight by the weight of the fluid displaced. Then, after a second assumption that bodies which are forced upwards in a fluid are forced upwards along the perpendiculars to the surface which pass through their centres of gravity, Archimedes deals with the position of rest and stability of a segment of a sphere floating in a fluid with its base entirely above or entirely below the surface. Book II is an extraordinary tour de force, investigating fully all the positions of rest and stability of a right segment of a paraboloid floating in a fluid according (1) to the relation between the axis of the solid and the parameter of the generating parabola, and (2) to the specific gravity of the solid in relation to the fluid; the term ‘specific gravity’ is not used, but the idea is fully expressed in other words.

Almost contemporary with Archimedes was Eratosthenes of Cyrene, to whom Archimedes dedicated the Method; the preface to this work shows that Archimedes thought highly of his mathematical ability. He was indeed recognized by his contemporaries as a man of great distinction in all branches, though the names Beta and Pentathlos[4] applied to him indicate that he just fell below the first rank in each subject. Ptolemy Euergetes appointed him to be tutor to his son (Philopator), and he became librarian at Alexandria; he recognized his obligation to Ptolemy by erecting a column with a graceful epigram. In this epigram he referred to the earlier solutions of the problem of duplicating the cube or finding the two mean proportionals, and advocated his own in preference, because it would give any number of means; on the column was fixed a bronze representation of his appliance, a frame with right-angled triangles (or rectangles) movable along two parallel grooves and over one another, together with a condensed proof. The Platonicus of Eratosthenes evidently dealt with the fundamental notions of mathematics in connexion with Plato’s philosophy, and seems to have begun with the story of the origin of the duplication problem.

The most famous achievement of Eratosthenes was his measurement of the earth. Archimedes quotes an earlier measurement which made the circumference of the earth 300,000 stades. Eratosthenes improved upon this. He observed that at the summer solstice at Syene, at noon, the sun cast no shadow, while at the same moment the upright gnomon at Alexandria cast a shadow corresponding to an angle between the gnomon and the sun’s rays of 1/50th of four right angles. The distance between Syene and Alexandria being known to be 5,000 stades, this gave for the circumference of the earth 250,000 stades, which Eratosthenes seems later, for some reason, to have changed to 252,000 stades. On the most probable assumption as to the length of the stade used, the 252,000 stades give about 7,850 miles, only 50 miles less than the true polar diameter.

In the work On the Measurement of the Earth Eratosthenes is said to have discussed other astronomical matters, the distance of the tropic and polar circles, the sizes and distances of the sun and moon, total and partial eclipses, &c. Besides other works on astronomy and chronology, Eratosthenes wrote a Geographica in three books, in which he first gave a history of geography up to date and then passed on to mathematical geography, the spherical shape of the earth, &c., &c.

Apollonius of Perga was with justice called by his contemporaries the ‘Great Geometer’, on the strength of his great treatise, the Conics. He is mentioned as a famous astronomer of the reign of Ptolemy Euergetes (247-222 B. C.); and he dedicated the fourth and later Books of the Conics to King Attalus I of Pergamum (241-197 B. C.).

The Conics, a colossal work, originally in eight Books, survives as to the first four Books in Greek and as to three more in Arabic, the eighth being lost. From Apollonius’s prefaces we can judge of the relation of his work to Euclid’s Conics, the content of which answered to the first three Books of Apollonius. Although Euclid knew that an ellipse could be otherwise produced, e. g. as an oblique section of a right cylinder, there is no doubt that he produced all three conics from right cones like his predecessors. Apollonius, however, obtains them in the most general way by cutting any oblique cone, and his original axes of reference, a diameter and the tangent at its extremity, are in general oblique; the fundamental properties are found with reference to these axes by ‘application of areas’, the three varieties of which, application (παραβολη), application with an excess (ὑπερβολη) and application with a deficiency (ελλειψις), give the properties of the three curves respectively and account for the names parabola, hyperbola, and ellipse, by which Apollonius called them for the first time. The principal axes only appear, as a particular case, after it has been shown that the curves have a like property when referred to any other diameter and the tangent at its extremity, instead of those arising out of the original construction. The first four Books constitute what Apollonius calls an elementary introduction; the remaining Books are specialized investigations, the most important being Book V (on normals) and Book VII (mainly on conjugate diameters). Normals are treated, not in connexion with tangents, but as minimum or maximum straight lines drawn to the curves from different points or classes of points. Apollonius discusses such questions as the number of normals that can be drawn from one point (according to its position) and the construction of all such normals. Certain propositions of great difficulty enable us to deduce quite easily the Cartesian equations to the evolutes of the three conics.

Several other works of Apollonius are described by Pappus as forming part of the ‘Treasury of Analysis’. All are lost except the Sectio Rationis in two Books, which survives in Arabic and was published in a Latin translation by Halley in 1706. It deals with all possible cases of the general problem ‘given two straight lines either parallel or intersecting, and a fixed point on each, to draw through any given point a straight line which shall cut off intercepts from the two lines (measured from the fixed points) bearing a given ratio to one another’. The lost treatise Sectio Spatii dealt similarly with the like problem in which the intercepts cut off have to contain a given rectangle.

The other treatises included in Pappus’s account are (1) On Determinate Section; (2) Contacts or Tangencies, Book II of which is entirely devoted to the problem of drawing a circle to touch three given circles (Apollonius’s solution can, with the aid of Pappus’s auxiliary propositions, be satisfactorily restored); (3) Plane Loci, i. e. loci which are straight lines or circles; (4) Νευσεις, Inclinationes (the general problem called a νευσις being to insert between two lines, straight or curved, a straight line of given length verging to a given point, i. e. so that, if produced, it passes through the point, Apollonius restricted himself to cases which could be solved by ‘plane’ methods, i. e. by the straight line and circle only).

Apollonius is also said to have written (5) a Comparison of the dodecahedron with the icosahedron (inscribed in the same sphere), in which he proved that their surfaces are in the same ratio as their volumes; (6) On the cochlias or cylindrical helix; (7) a ‘General Treatise’, which apparently dealt with the fundamental assumptions, &c., of elementary geometry; (8) a work on unordered irrationals, i. e. irrationals of more complicated form than those of Eucl. Book X; (9) On the burning-mirror, dealing with spherical mirrors and probably with mirrors of parabolic section also; (10) ωκυτοκιον (‘quick delivery’). In the last-named work Apollonius found an approximation to π closer than that in Archimedes’s Measurement of a Circle; and possibly the book also contained Apollonius’s exposition of his notation for large numbers according to ‘tetrads’ (successive powers of the myriad).

In astronomy Apollonius is said to have made special researches regarding the moon, and to have been called ε (Epsilon) because the form of that letter is associated with the moon. He was also a master of the theory of epicycles and eccentrics.

With Archimedes and Apollonius Greek geometry reached its culminating point; indeed, without some more elastic notation and machinery such as algebra provides, geometry was practically at the end of its resources. For some time, however, there were capable geometers who kept up the tradition, filling in details, devising alternative solutions of problems, or discovering new curves for use or investigation.

Nicomedes, probably intermediate in date between Eratosthenes and Apollonius, was the inventor of the conchoid or cochloid, of which, according to Pappus, there were three varieties. Diocles (about the end of the second century B. C.) is known as the discoverer of the cissoid which was used for duplicating the cube. He also wrote a book περι πυρειων, On burning-mirrors, which probably discussed, among other forms of mirror, surfaces of parabolic or elliptic section, and used the focal properties of the two conics; it was in this work that Diocles gave an independent and clever solution (by means of an ellipse and a rectangular hyperbola) of Archimedes’s problem of cutting a sphere into two segments in a given ratio. Dionysodorus gave a solution by means of conics of the auxiliary cubic equation to which Archimedes reduced this problem; he also found the solid content of a tore or anchor-ring.

Perseus is known as the discoverer and investigator of the spiric sections, i. e. certain sections of the σπειρα, one variety of which is the tore. The spire is generated by the revolution of a circle about a straight line in its plane, which straight line may either be external to the circle (in which case the figure produced is the tore), or may cut or touch the circle.

Zenodorus was the author of a treatise on Isometric figures, the problem in which was to compare the content of different figures, plane or solid, having equal contours or surfaces respectively.

Hypsicles (second half of second century B. C.) wrote what became known as ‘Book XIV’ of the Elements containing supplementary propositions on the regular solids (partly drawn from Aristaeus and Apollonius); he seems also to have written on polygonal numbers. A mediocre astronomical work (Αναφορικος) attributed to him is the first Greek book in which we find the division of the zodiac circle into 360 parts or degrees.

Posidonius the Stoic (about 135-51 B. C.) wrote on geography and astronomy under the titles On the Ocean and περι μετεωρων. He made a new but faulty calculation of the circumference of the earth (240,000 stades). Per contra, in a separate tract on the size of the sun (in refutation of the Epicurean view that it is as big as it looks), he made assumptions (partly guesswork) which give for the diameter of the sun a figure of 3,000,000 stades (39-1/4 times the diameter of the earth), a result much nearer the truth than those obtained by Aristarchus, Hipparchus, and Ptolemy. In elementary geometry Posidonius gave certain definitions (notably of parallels, based on the idea of equidistance).

Geminus of Rhodes, a pupil of Posidonius, wrote (about 70 B. C.) an encyclopaedic work on the classification and content of mathematics, including the history of each subject, from which Proclus and others have preserved notable extracts. An-Nairīzī (an Arabian commentator on Euclid) reproduces an attempt by one ‘Aganis’, who appears to be Geminus, to prove the parallel-postulate.

But from this time onwards the study of higher geometry (except sphaeric) seems to have languished, until that admirable mathematician, Pappus, arose (towards the end of the third century A. D.) to revive interest in the subject. From the way in which, in his great Collection, Pappus thinks it necessary to describe in detail the contents of the classical works belonging to the ‘Treasury of Analysis’ we gather that by his time many of them had been lost or forgotten, and that he aimed at nothing less than re-establishing geometry at its former level. No one could have been better qualified for the task. Presumably such interest as Pappus was able to arouse soon flickered out; but his Collection remains, after the original works of the great mathematicians, the most comprehensive and valuable of all our sources, being a handbook or guide to Greek geometry and covering practically the whole field. Among the original things in Pappus’s Collection is an enunciation which amounts to an anticipation of what is known as Guldin’s Theorem.

It remains to speak of three subjects, trigonometry (represented by Hipparchus, Menelaus, and Ptolemy), mensuration (in Heron of Alexandria), and algebra (Diophantus).

Although, in a sense, the beginnings of trigonometry go back to Archimedes (Measurement of a Circle), Hipparchus was the first person who can be proved to have used trigonometry systematically. Hipparchus, the greatest astronomer of antiquity, whose observations were made between 161 and 126 B. C., discovered the precession of the equinoxes, calculated the mean lunar month at 29 days, 12 hours, 44 minutes, 2½ seconds (which differs by less than a second from the present accepted figure!), made more correct estimates of the sizes and distances of the sun and moon, introduced great improvements in the instruments used for observations, and compiled a catalogue of some 850 stars; he seems to have been the first to state the position of these stars in terms of latitude and longitude (in relation to the ecliptic). He wrote a treatise in twelve Books on Chords in a Circle, equivalent to a table of trigonometrical sines. For calculating arcs in astronomy from other arcs given by means of tables he used propositions in spherical trigonometry.

The Sphaerica of Theodosius of Bithynia (written, say, 20 B. C.) contains no trigonometry. It is otherwise with the Sphaerica of Menelaus (fl. A. D. 100) extant in Arabic; Book I of this work contains propositions about spherical triangles corresponding to the main propositions of Euclid about plane triangles (e.g. congruence theorems and the proposition that in a spherical triangle the three angles are together greater than two right angles), while Book III contains genuine spherical trigonometry, consisting of ‘Menelaus’s Theorem’ with reference to the sphere and deductions therefrom.

Ptolemy’s great work, the Syntaxis, written about A. D. 150 and originally called Μαθηματικη συνταξις, came to be known as Μεγαλη συνταξις; the Arabs made up from the superlative μεγιστος the word al-Majisti which became Almagest.

Book I, containing the necessary preliminaries to the study of the Ptolemaic system, gives a Table of Chords in a circle subtended by angles at the centre of ½° increasing by half-degrees to 180°. The circle is divided into 360 μοιραι, parts or degrees, and the diameter into 120 parts (τμηματα); the chords are given in terms of the latter with sexagesimal fractions (e. g. the chord subtended by an angle of 120° is 103p 53′ 23″). The Table of Chords is equivalent to a table of the sines of the halves of the angles in the table, for, if (crd. 2 α) represents the chord subtended by an angle of 2 α (crd. 2 α)/120 = sin α. Ptolemy first gives the minimum number of geometrical propositions required for the calculation of the chords. The first of these finds (crd. 36°) and (crd. 72°) from the geometry of the inscribed pentagon and decagon; the second (‘Ptolemy’s Theorem’ about a quadrilateral in a circle) is equivalent to the formula for sin (θ-φ), the third to that for sin ½ θ. From (crd. 72°) and (crd. 60°) Ptolemy, by using these propositions successively, deduces (crd. 1½°) and (crd. ¾°), from which he obtains (crd. 1°) by a clever interpolation. To complete the table he only needs his fourth proposition, which is equivalent to the formula for cos (θ+φ).

Ptolemy wrote other minor astronomical works, most of which survive in Greek or Arabic, an Optics in five Books (four Books almost complete were translated into Latin in the twelfth century), and an attempted proof of the parallel-postulate which is reproduced by Proclus.

Heron of Alexandria (date uncertain; he may have lived as late as the third century A. D.) was an almost encyclopaedic writer on mathematical and physical subjects. He aimed at practical utility rather than theoretical completeness; hence, apart from the interesting collection of Definitions which has come down under his name, and his commentary on Euclid which is represented only by extracts in Proclus and an-Nairīzī, his geometry is mostly mensuration in the shape of numerical examples worked out. As these could be indefinitely multiplied, there was a temptation to add to them and to use Heron’s name. However much of the separate works edited by Hultsch (the Geometrica, Geodaesia, Stereometrica, Mensurae, Liber geëponicus) is genuine, we must now regard as more authoritative the genuine Metrica discovered at Constantinople in 1896 and edited by H. Schöne in 1903 (Teubner). Book I on the measurement of areas is specially interesting for (1) its statement of the formula used by Heron for finding approximations to surds, (2) the elegant geometrical proof of the formula for the area of a triangle Δ = √{s (s-a) (s-b) (s-c)}, a formula now known to be due to Archimedes, (3) an allusion to limits to the value of π found by Archimedes and more exact than the 3-1/7 and 3-10/71 obtained in the Measurement of a Circle.

Book I of the Metrica calculates the areas of triangles, quadrilaterals, the regular polygons up to the dodecagon (the areas even of the heptagon, enneagon, and hendecagon are approximately evaluated), the circle and a segment of it, the ellipse, a parabolic segment, and the surfaces of a cylinder, a right cone, a sphere and a segment thereof. Book II deals with the measurement of solids, the cylinder, prisms, pyramids and cones and frusta thereof, the sphere and a segment of it, the anchor-ring or tore, the five regular solids, and finally the two special solids of Archimedes’s Method; full use is made of all Archimedes’s results. Book III is on the division of figures. The plane portion is much on the lines of Euclid’s Divisions (of figures). The solids divided in given ratios are the sphere, the pyramid, the cone and a frustum thereof. Incidentally Heron shows how he obtained an approximation to the cube root of a non-cube number (100). Quadratic equations are solved by Heron by a regular rule not unlike our method, and the Geometrica contains two interesting indeterminate problems.

Heron also wrote Pneumatica (where the reader will find such things as siphons, Heron’s Fountain, penny-in-the-slot machines, a fire-engine, a water-organ, and many arrangements employing the force of steam), Automaton-making, Belopoeïca (on engines of war), Catoptrica, and Mechanics. The Mechanics has been edited from the Arabic; it is (except for considerable fragments) lost in Greek. It deals with the puzzle of ‘Aristotle’s Wheel’, the parallelogram of velocities, definitions of, and problems on, the centre of gravity, the distribution of weights between several supports, the five mechanical powers, mechanics in daily life (queries and answers). Pappus covers much the same ground in Book VIII of his Collection.

We come, lastly, to Algebra. Problems involving simple equations are found in the Papyrus Rhind, in the Epanthema of Thymaridas already referred to, and in the arithmetical epigrams in the Greek Anthology (Plato alludes to this class of problem in the Laws, 819 B, C); the Anthology even includes two cases of indeterminate equations of the first degree. The Pythagoreans gave general solutions in rational numbers of the equations x²+y²=z² and 2x²-y²=±1, which are indeterminate equations of the second degree.

The first to make systematic use of symbols in algebraical work was Diophantus of Alexandria (fl. about A. D. 250). He used (1) a sign for the unknown quantity, which he calls αριθμος, and compendia for its powers up to the sixth; (2) a sign (

) with the effect of our minus. The latter sign probably represents ΛΙ, an abbreviation for the root of the word λειπειν (to be wanting); the sign for αριθμος (

) is most likely an abbreviation for the letters αρ; the compendia for the powers of the unknown are ΔΥ for δυναμις, the square, ΚΥ for κυβος, the cube, and so on. Diophantus shows that he solved quadratic equations by rule, like Heron. His Arithmetica, of which six books only (out of thirteen) survive, contains a certain number of problems leading to simple equations, but is mostly devoted to indeterminate or semi-determinate analysis, mainly of the second degree. The collection is extraordinarily varied, and the devices resorted to are highly ingenious. The problems solved are such as the following (fractional as well as integral solutions being admitted): ‘Given a number, to find three others such that the sum of the three, or of any pair of them, together with the given number is a square’, ‘To find four numbers such that the square of the sum plus or minus any one of the numbers is a square’, ‘To find three numbers such that the product of any two plus or minus the sum of the three is a square’. Diophantus assumes as known certain theorems about numbers which are the sums of two and three squares respectively, and other propositions in the Theory of Numbers. He also wrote a book On Polygonal Numbers of which only a fragment survives.

With Pappus and Diophantus the list of original writers on mathematics comes to an end. After them came the commentators whose names only can be mentioned here. Theon of Alexandria, the editor of Euclid, lived towards the end of the fourth century A. D. To the fifth and sixth centuries belong Proclus, Simplicius, and Eutocius, to whom we can never be grateful enough for the precious fragments which they have preserved from works now lost, and particularly the History of Geometry and the History of Astronomy by Aristotle’s pupil Eudemus.

Such is the story of Greek mathematical science. If anything could enhance the marvel of it, it would be the consideration of the shortness of the time (about 350 years) within which the Greeks, starting from the very beginning, brought geometry to the point of performing operations equivalent to the integral calculus and, in the realm of astronomy, actually anticipated Copernicus.

T. L. Heath.

NATURAL SCIENCE

Aristotle

There is a little essay of Goethe’s called, simply, Die Natur. It comes among those tracts on Natural Science in which the poet and philosopher turned his restless mind to problems of light and colour, of leaf and flower, of bony skull and kindred vertebra; and it sounds like a prose-poem, a noble paean, eulogizing the love and glorifying the study of Nature. Some twenty-five hundred years before, Anaximander had written a book with the same title, Concerning Nature, περι φυσεως: but its subject was not the same. It was a variant of the old traditional cosmogonies. It told of how in the beginning the earth was without form and void. It sought to trace all things back to the Infinite, το απειρον—to That which knows no bounds of space or time but is before all worlds, and to whose bosom again all things, all worlds, return. For Goethe Nature meant the beauty, the all but sensuous beauty of the world; for the older philosopher it was the mystery of the Creative Spirit.

Than Nature, in Goethe’s sense, no theme is more familiar to us, for whom many a poet tells the story and many a lesser poet echoes the conceit; but if there be anywhere in Greek such overt praise and worship of Nature’s beauty, I cannot call it to mind. Yet in Latin the divini gloria ruris is praised and Natura daedala rerum worshipped, as we are wont to praise and worship them, for their own sweet sakes. It is one of the ways, one of the simpler ways, in which the Roman world seems nearer to us than the Greek: and not only seems, but is so. For compared with the great early civilizations, Rome is modern and of the West; while, draw her close as we may to our hearts, Greece brings along with her a breath of the East and a whisper of remote antiquity. A Tuscan gentleman of to-day, like a Roman gentleman of yesterday, is at heart a husbandman, like Cato; he is ruris amator, like Horace; he gets him to his little farm or vineyard (O rus, quando te aspiciam!), like Atticus or the younger Pliny. As Bacon praised his garden, so does Pliny praise his farm, with its cornfields and meadowland, vineyard and woodland, orchard and pasture, bee-hives and flowers. That God made the country and man made the town was (long before Cowper) a saying of Varro’s; but in Greek I can think of no such apophthegm.

As Schiller puts it, the Greeks looked on Nature with their minds more than with their hearts, nor ever clung to her with outspoken admiration and affection. And Humboldt, asserting (as I would do) that the portrayal of nature, for her own sake and in all her manifold diversity, was foreign to the Greek idea, declares that the landscape is always the mere background of their picture, while their foreground is filled with the affairs and actions and thoughts of men. But all the while, as in some old Italian picture—of Domenichino or Albani or Leonardo himself—the subordinated background is delicately traced and exquisitely beautiful; and sometimes we come to value it in the end more than all the rest of the composition.

Deep down in the love of Nature, whether it be of the sensual or intellectual kind, and in the art of observation which is its outcome and first expression, lie the roots of all our Natural Science. All the world over these are the heritage of all men, though the inheritance be richer or poorer here and there: they are shown forth in the lore and wisdom of hunter and fisherman, of shepherd and husbandman, of artist and poet. The natural history of the ancients is not enshrined in Aristotle and Pliny. It pervades the vast literature of classical antiquity. For all we may say of the reticence with which, the Greeks proclaim it, it greets us nobly in Homer, it sings to us in Anacreon, Sicilian shepherds tune their pipes to it in Theocritus: and anon in Virgil we dream of it to the coo of doves and the sound of bees’ industrious murmur.

Not only from such great names as these do we reach the letter and the spirit of ancient Natural History. We must go a-wandering into the by-ways of literature. We must eke out the scientific treatises of Aristotle and Pliny by help of the fragments which remain of the works of such naturalists as Speusippus or Alexander the Myndian; add to the familiar stories of Herodotus the Indian tales of Ctesias and Megasthenes; sit with Athenaeus and his friends at the supper table, gleaning from cook and epicure, listening to the merry idle troop of convivial gentlemen capping verses and spinning yarns; read Xenophon’s treatise on Hunting, study the didactic poems, the Cynegetica and Halieutica, of Oppian and of Ovid. And then again we may hark back to the greater world of letters, wherein poet and scholar, from petty fabulist to the great dramatists, from Homer’s majesty to Lucian’s wit, share in the love of Nature and enliven the delicate background of their story with allusions to beast and bird.

Such allusions, refined at first by art and hallowed at last by familiar memory, lie treasured in men’s hearts and enshrine themselves in our noblest literature. Take, of a thousand crowding instances, that great passage in the Iliad where the Greek host, disembarking on the plains of the Scamander, is likened to a migrating flock of cranes or geese or long-necked swans, as they fly proudly over the Asian meadows and alight screaming by Cayster’s stream—and Virgil echoes more than once the familiar lines. The crane was a well-known bird. Its lofty flight brings it, again in Homer, to the very gates of heaven. Hesiod and Pindar speak of its far-off cry, heard from above the clouds: and that it ‘observed the time of its coming’, ‘intelligent of seasons’, was a proverb old in Hesiod’s day—when the crane signalled the approach of winter, and when it bade the husbandman make ready to plough. It follows the plough, in Theocritus, as persistently as the wolf the kid and the peasant-lad his sweetheart. The discipline of the migrating cranes, the serried wedge of their ranks in flight, the good order of the resting flock, are often, and often fancifully, described. Aristotle records how they have an appointed leader, who keeps watch by night and in flight keeps calling to the laggards; and all this old story Euripides, the most naturalistic of the great tragedians, puts into verse:

The ordered host of Libyan birds avoids
The wintry storm, obedient to the call
Of their old leader, piping to his flock.

Lastly, Milton gathers up the spirit and the letter of these and many another ancient allusion to the migrating cranes:

Part loosely wing the region; part more wise,
In common ranged in figure, wedge their way
Intelligent of seasons, and set forth
Their aery caravan, high over seas
Flying, and over lands; with mutual wing
Easing their flight; so steers the prudent crane.

But the natural history of the poets is a story without an end, and in our estimation, however brief it be, of ancient knowledge, there are other matters to be considered, and other points of view where we must take our stand.

When we consider the science of the Greeks, and come quickly to love it and slowly to see how great it was, we likewise see that it was restricted as compared with our own, curiously partial or particular in its limitations. The practical and ‘useful’ sciences of chemistry, mechanics, and engineering, which in our modern world crowd the others to the wall, are absent altogether, or so concealed that we forget and pass them by. Mathematics is enthroned high over all, as it is meet she should be; and of uncontested right she occupies her throne century after century, from Pythagoras to Proclus, from the scattered schools of early Hellenic civilization to the rise and fall of the great Alexandrine University. Near beside her sits, from of old, the daughter-science of Astronomy; and these twain were worshipped by the greatest scientific intellects of the Greeks. But though we do not hear of them nor read of them, we must not suppose for a moment that the practical or technical sciences were lacking in so rich and complex a civilization. China, that most glorious of all living monuments of Antiquity, tells us nothing of her own chemistry, but we know that it is there. Peep into a Chinese town, walk through its narrow streets, thronged but quiet, wherein there is neither rumbling of coaches nor rattling of wheels, and you shall see the nearest thing on earth to what we hear of Sybaris. To the production of those glowing silks and delicate porcelains and fine metal-work has gone a vast store of chemical knowledge, traditional and empirical. So was it, precisely, in ancient Greece; and Plato knew that it was so—that the dyer, the perfumer, and the apothecary had subtle arts, a subtle science of their own, a science not to be belittled nor despised. We may pass here and there by diligent search from conjecture to assurance; analyse a pigment, an alloy or a slag; discover from an older record than the Greeks’, the chemical prescription wherewith an Egyptian princess darkened her eyes, or study the pictured hearth, bellows, oven, crucibles with which the followers of Tubal-Cain smelted their ore. Once in a way, but seldom, do we meet with ancient chemistry even in Greek literature. There is a curious passage (its text is faulty and the translation hard) in the story of the Argonauts, where Medea concocts a magic brew. She put divers herbs in it, herbs yielding coloured juices such as safflower and alkanet, and soapwort and fleawort to give consistency or ‘body’ to the lye; she put in alum and blue vitriol (or sulphate of copper), and she put in blood. The magic brew was no more and no less than a dye, a red or purple dye, and a prodigious deal of chemistry had gone to the making of it. For the copper was there to produce a ‘lake’ or copper-salt of the vegetable alkaloids, which copper-lakes are among the most brilliant and most permanent of colouring matters; the alum was there as a ‘mordant’; and even the blood was doubtless there incorporated for better reasons than superstitious ones, in all probability for the purpose of clarifying (by means of its coagulating albumen) the seething and turbid brew.

The ‘Orphic’ version of the story, in which this passage occurs, is probably an Alexandrine compilation, and whether the ingredients of the brew had been part of the ancient legend or were merely suggested to the poet by the knowledge of his own day we cannot tell; in either case the prescription is old enough, and is at least pre-Byzantine by a few centuries. Such as it is, it does not stand alone. Other fragments of ancient chemistry, more or less akin to it, have been gathered together; in Galen’s book on The making of Simples, in Pliny, in Paulus Aegineta, and for that matter in certain Egyptian papyri (especially a certain very famous one, still extant, of which Clement of Alexandria speaks as a secret or ‘hermetic’ book), we can trace the broken and scattered stones of a great edifice of ancient chemistry.

Nevertheless, all this weight of chemical learning figures scantily in literature, and is conspicuously absent from our conception of the natural genius of the Greeks. We have no reason to suppose that ancient chemistry, or any part of it, was ever peculiarly Greek, or that this science was the especial property of any nation whatsoever; moreover it was a trade, or a bundle of trades, whose trade-secrets were too precious to be revealed, and so constituted not a science but a mystery. So has it always been with chemistry, the most cosmopolitan of sciences, the most secret of arts. Quietly and stealthily it crept through the world; the tinker brought it with his solder and his flux; the African tribes who were the first workers in iron passed it on to the great metallurgists who forged Damascan and Toledan steel.

This ‘trade’ of Chemistry was never a science for a Gentleman, as philosophy and mathematics were; and Plato, greatest of philosophers, was one of the greatest of gentlemen. Long, long afterwards, Oxford said the same thing to Robert Boyle—that Chemistry was no proper avocation for a gentleman; but he thought otherwise, and the ‘brother of the Earl of Cork’ became the Father of scientific Chemistry.

Now I take it that in regard to biology Aristotle did much the same thing as Boyle, breaking through a similar tradition; and herein one of the greatest of his great services is to be found. There was a wealth of natural history before his time; but it belonged to the farmer, the huntsman, and the fisherman—with something over (doubtless) for the schoolboy, the idler, and the poet. But Aristotle made it a science, and won a place for it in Philosophy. He did for it just what Pythagoras had done (as Proclus tells us) for mathematics in an earlier age, when he discerned the philosophy underlying the old empirical art of ‘geometry’, and made it the basis of ‘a liberal education’.[5]

The Mediterranean fisherman, like the Chinese fisherman or the Japanese, has still, and always has had, a wide knowledge of all that pertains to and accompanies his craft. Our Scottish fishermen have a limited vocabulary, which scarce extends beyond the names of the few common fishes with which the market is supplied. But at Marseilles or Genoa or in the Levant they have names for many hundreds of species, of fish and shell-fish and cuttle-fish and worms and corallines, and all manner of swimming and creeping things; they know a vast deal about the habits of their lives, far more, sometimes, than do we ‘scientific men’; they are naturalists by tradition and by trade. Neither, by the way, must we forget the ancient medical and anatomical learning of the great Aesculapian guild, nor the still more recondite knowledge possessed by various priesthoods (again like their brethren of to-day in China and Japan) of the several creatures, sacred fish, pigeons, guinea-fowl, snakes, cuttlefish, and what not, which time out of mind they had reared, tended, and venerated.

Of what new facts Aristotle actually discovered it is impossible to be sure. Could it ever be proved that he discovered many, or could it even be shown that of his own hand he discovered nothing at all, it would affect but little our estimate of his greatness and our admiration of his learning. He was the first of Greek philosophers and gentlemen to see that all these things were good to know and worthy to be told. This was his great discovery.

I have sought elsewhere to show that Aristotle spent two years, the happiest years perhaps of all his life—a long honeymoon—by the sea-side in the island of Mytilene, after he had married the little Princess, and before he began the hard work of his life: before he taught Alexander in Macedon, and long before he spoke urbi et orbi in the Lyceum. Here it was that he learned the great bulk of his natural history, in which, wide and general as it is, the things of the sea have from first to last a notable predominance.

I have tried to illustrate elsewhere (as many another writer has done) something of the variety and the depth of Aristotle’s knowledge of animals—choosing an example here and there, but only drawing a little water from an inexhaustible well.

A famous case is that of the ‘molluscs’, where either Aristotle’s knowledge was exceptionally minute, or where it has come down to us with unusual completeness.

These are the cuttle fish, which have now surrendered their Aristotelian name of ‘molluscs’ to that greater group which is seen to include them, together with the shell-fish or ‘ostracoderma’ of Aristotle. These cuttle-fishes are creatures that we seldom see, but in the Mediterranean they are an article of food and many kinds are known to the fishermen. All or wellnigh all of these many kinds were known to Aristotle. He described their form and their anatomy, their habits, their development, all with such faithful accuracy that what we can add to-day seems of secondary importance. He begins with a methodical description of the general form, tells us of the body and fins, of the eight arms with their rows of suckers, of the abnormal position of the head. He points out the two long arms of Sepia and of the calamaries, and their absence in the octopus; and he tells us, what was only confirmed of late, that with these two long arms the creature clings to the rock and sways about like a ship at anchor. He describes the great eyes, the two big teeth forming the beak; and he dissects the whole structure of the gut, with its long gullet, its round crop, its stomach and the little coiled coecal diverticulum: dissecting not only one but several species, and noting differences that were not observed again till Cuvier re-dissected them. He describes the funnel and its relation to the mantle-sac, and the ink-bag, which he shows to be largest in Sepia of all others. And here, by the way, he seems to make one of those apparent errors that, as it happens, turn out to be justified: for he tells us that in Octopus, unlike the rest, the funnel is on the upper side; the fact being that when the creature lies prone upon the ground, with all its arms outspread, the funnel-tube (instead of being flattened out beneath the creature’s prostrate body) is long enough to protrude upwards between arms and head, and to appear on one side or other thereof, in a position apparently the reverse of its natural one. He describes the character of the cuttle-bone in Sepia, and of the horny pen which takes its place in the various calamaries, and notes the lack of any similar structure in Octopus. He dissects in both sexes the reproductive organs, noting without exception all their essential and complicated parts; and he had figured these in his lost volume of anatomical diagrams. He describes the various kinds of eggs, and, with still more surprising knowledge, shows us the little embryo cuttle-fish, with its great yolk-sac attached, in apparent contrast to the chick’s, to the little creature’s developing head.

But there is one other remarkable feature that he knew ages before it was rediscovered, almost in our own time. In certain male cuttle-fishes, in the breeding season, one of the arms develops in a curious fashion into a long coiled whip-lash, and in the act of breeding may then be transferred to the mantle-cavity of the female. Cuvier himself knew nothing of the nature or the function of this separated arm, and indeed, if I am not mistaken, it was he who mistook it for a parasitic worm. But Aristotle tells us of its use and its temporary development, and of its structure in detail, and his description tallies closely with the accounts of the most recent writers.