Chapter VIII: Higher Thinking, Philosophy, Speculative and Practical Theology 213 (4)

When we pass from the monumental arts of architecture and sculpture to those of a more subjective character, which use more fleeting vehicles for their expression, we have in modern life painting and music, which we may expect to be more independent of Greek models than the rest. For, _ex hypothesi_, pictures so far as they are on panels of wood or canvas can hardly survive the lapse of ages of neglect,[29] and as for music, the notation is so small and poor a clue to its real meaning, that even if we understood it perfectly, we should still be a long way from grasping the full meaning as felt by the Greek public. I will give you an illustration of this from my own experience. There is in our English and Irish cathedrals a tradition of the way in which certain anthems are to be sung--a tradition generally derived from those who sang them in the composer’s presence, or under his influence. The older editions of these anthems seldom give any expression marks, the performance being entrusted to the taste of the choir or its knowledge of the composer’s intentions. A signal example is the finest of Blow’s anthems, “I beheld and lo! a great multitude,” composed in the reign of Charles II. (1680), and sung ever since by sundry cathedral choirs, amongst others those of Dublin, where there has from long since been a great school of church music. In Dublin, this is one of the most moving and dramatic anthems, owing to the great liberties taken with the time by the Vicars Choral, who have kept the tradition unbroken. I chanced to hear it sung by the very excellent choir of Magdalen College, Oxford, on their high day--All Saints’ Day, when it is annually performed. I was astonished to find that they merely sang it from the text without any of the traditional liberties. The effect was so poor and unmeaning as to be almost ridiculous to one who had been taught to understand the inner sense of the work.

We are not quite so destitute as to Greek painting, for we have at least a good many fresco pictures, by more or less obscure and incompetent workmen of the Hellenistic age, to show us what the Greeks aimed at; we have on the many beautiful examples of pottery preserved to us the representations of mythical or other scenes which must have had some analogy with the paintings of the same or similar scenes. Lastly, we have many descriptions and epigrams from those who admired the masterpieces of this art, and although these are inadequate, and are often the observations of incompetent rhetorical critics, they still give us far more definite ideas than any description of a musical work could possibly supply. As to the Golden Age of painting, we have nothing but these, for our specimens of frescoes on the walls are all either from pre-historic palaces, or from Græco-Roman houses. If we wish, therefore, to obtain any understanding of this side of Greek art, we must not be content with our poor and sporadic examples, but must enter upon some general considerations which will afford a larger and deeper basis for our judgment. For our inferences from the Pompeiian frescoes to the lost masterpieces are just as hazardous as if we had lost all the masterpieces of sculpture, and endeavoured to judge of their quality by reasoning from the terra cotta figurines of Tanagra and other places, which are often graceful, but almost always faulty in their modelling. Should we indeed have inferred that the modelling of statues in marble and in bronze was absolutely perfect?

The two æsthetic qualities requisite for success in painting are obviously a sense of form, and a sense of colour; without a natural appreciation of the beauty inhering in each of these, the highest technical skill, however valuable, does not suffice. After what you have heard about Greek architecture and sculpture, I need not say another word to show that in the sense of form the Greeks were supreme and unapproachable. But what about their sense of colour? On this the evidence is not so clear and has given rise to divers interpretations. First of all, the Homeric poets, in their vivid pictures of old Greek life, are singularly vague and confused in their words for colour, so much so that people used to imagine that the poet, because he was blind, or the poets, because they were primitive, had no distinct colour-sense. I remember this latter view being pressed upon me by Mr. Gladstone in conversation, together with the reply he had from Charles Darwin, which he gave me to read, that as even insects are guided by a very clear sense of colour, it was absurd to say that the most primitive men should not possess it. This argument seamed both to him and to me hardly conclusive, for the faculties which are now human need not have developed at the same rate from lower forms, or kept abreast of one another in acuteness. Thus human development might not require an acute sense of colour, while that of the insect made it essential, and so lower forms of life might be infinitely more developed in some respects than those far higher in the general condition of their senses and their intelligence. I therefore took another line in my objection: that we know the Egyptians, centuries before the oldest date allowed for Homer, had at least ten distinct names for colour. And this was not because they felt the difference more distinctly, but because in their arts and crafts they produced the varied shades, and therefore found names for them. Even nowadays, it is not the poet, or even the artist, that invents names for subtle shades of colours, but the milliner or the modiste. When I was young, there were two shades of grey known in the phraseology of these people--one as _gris de souris_, the other as _gris de souris poursuivie_. This is but a more minute subdivision of our sensations of colour invented by those that produce it for trade purposes. The want of names for colours is therefore not confined to the Greeks. More important is the fact that their early painters are known to have used but a few and primary colours, and also the further fact that their temples, which they always coloured (and, to my mind, rightly) for effect, were adorned on a simple and primitive plan,--red, blue, white, yellow, being, so far as I know, the colours generally used.

Now these facts seem to me to harmonise with the small development of a sense of the picturesque in landscape, which is characteristic of the Greeks. The principles of reproducing perspective with lines and colours on a flat surface were indeed discovered in the fifth century B.C., by a certain Agatharchus, whose book on shade painting seems, however, to have been a work on scene-painting, as an aid to producing illusions on the stage. Nor does the idea of representing external nature seem to have been a want felt by Greek artists, seeing that they had adopted the very peculiar device of representing mountains and rivers by figures of the gods and nymphs which inhabited them and in which they were personified. The heads of the horses rising from the sea represented on the Parthenon the advent of the day. The graceful figures of nymphs on the pediments of the great temple at Olympia represented the scenery in which the action was laid. Looking down the whole history of Greek painting, from the rude frescoes at Tiryns to the decorations of houses at Pompeii, I cannot find that landscape as such ever occupied Greek artists, and here therefore we have one of the very few departments in which the modern world may boast itself independent of its almost universal teacher.

It is not so in the case of portrait-painting, and painting of scenes in mythical or in real life, for here even the faint echoes of Greek genius affected powerfully the artists of the Renaissance. In this field, however, the influence of Hellenistic sculpture and of relief work was so combined with that of the few specimens of actual painting from Herculaneum, Pompeii, and other sites, that the separate effect of Greek painting on the modern artists is not so easily appreciated. The mythical subjects at all events were told and glorified in countless epigrams of the Anthology, and as soon as this collection became known and popular, it was sure to dominate the fancy of sentimental artists like Botticelli. But if the direct influence of Greek on modern painting was baulked for want of models, the indirect effect of Greek art on the best of modern painters is very great. Consider for a moment the two most refined of modern English painters, the late Lord Leighton, and the still living and working Sir Edward Alma-Tadema. The latter generally calls his subjects Roman, but anyone that knows what the elegances of Roman life owe to the Greeks, sees at once that the whole spirit of the artist, and of the subjects he delights in, is Greek. The case is still more undisguised with Leighton. All his most striking pictures are from Greek life or from Greek legend; his whole conception of beauty is derived from the same models, and I well remember, when I used to visit him in his delightful studio in Kensington, seeing it all set round with copies of Greek sculpture, and his fervid utterance that to these unapproachable models he owed all his art.

In the absence of the actual paintings, great use has been made of the scenes painted on Greek vases of the best period, some of which attain to quite a high level, and we cannot but feel with Sir Alma-Tadema that from this source he has drawn not a few of his ideas. Surely the products that inspired Keats with his exquisite Ode make clear to us how the fruitfulness of Greek genius is not dead or even exhausted, but still kindles a pure light in modern minds sensitive enough to catch the flame.

It is to be observed, before we pass on to another subject, that the art of painting among the Greeks began, and long remained, a branch of decoration, and was therefore subsidiary to architecture, or stately furniture, or fine pottery. The fashion of producing easel pictures painted for their own sake, readily movable and therefore displayed in galleries, as well as upon the walls of palaces, only came in with the decadence or at least the full ripeness of other arts. The products of the painter were akin to those of the epigrammatist, whose elegance may well be called by a poorer word--_finish_--and is to us rather the exhibition of great cleverness than the outcome of genius. It was the day also of social decadence, when the mere artist became the idol of society, and could parade his conceit and his vulgarity without fear of censure from patrons who only valued him as the ephemeral fashion. The gossip we hear about the old painters often exhibits this painfully modern triviality.

I now turn to the topic of music, in interest second to none, but one in which I must endeavour to make my discussion intelligible to those who have only a practical knowledge of this subject. In most histories of Greek art, music is simply omitted; in the special works upon it, there is much that is not only so difficult, but so dry and technical that the average student of Greek life can hardly be expected to approach it.

As regards existing specimens, we are just as miserably provided as we are in the case of painting. We have recovered a few scraps of the musical notation accompanying poetical words; and as we understand this notation, it is an easy task to reproduce the so-called melody. We have also a scrap or two in the notation of instrumental music (apparently an accompaniment), a notation, strange to say, differing from the oral. But here the melody is missing. And let me tell you at once that no living musician could attempt to supply it with the smallest verisimilitude. The same is the case with our texts of melody. There was a much lauded hymn found a few years ago on the wall of one of the houses uncovered at Delphi. In some places the surface of the stone was broken; so that there were gaps here and there of a bar or two in the music. No living musician who knows his business would undertake to supply any one of these gaps.[30] Were it a modern composition, we could with certainty offer two or three alternatives, and we could exclude a vast number of restorations as absolutely impossible. Such is not the case with the Greek specimens we know, neither do they appeal to our modern taste. To say that these specimens, when played for us, are hideous, is merely the expression of that violated taste. There are many, perhaps even some in this audience, who would say the same thing of the plain song which the present Pope has ordered to be used in Roman Catholic churches to the exclusion of more modern music.

The real conclusion is that so far Greek music is to us unintelligible; and yet in all the other arts nothing is more intelligible to modern minds than the products of Greek taste which are our best and clearest models. Is it that a highly artistic nation may be wanting in one particular department? We have before us the case of the modern Japanese, whose artistic work in most directions is of great excellence and fully appreciated by the world, but who confess (at least I have heard one most intelligent native confess) that their music is far below the level of European compositions. But here we probably start from a difference of scale, whereas the Greek scales (or at least the diatonic) are the parents of all modern European scales.

And now that you have before you the actual problem raised by the extant remnants of Greek music, let us turn to the Greeks themselves, and see what light their writings throw upon the matter. In the first place music was not only popular but universal among the Greeks. Those who did not cultivate it were worse than Shakspere’s “man that has not music in his soul.” All Greek poetry, even the epic of Homer, was recited musically; the lyric poets were as much musicians as poets; great tragedians composed the music for their choral odes, and indeed a Greek tragedy when performed must have far more resembled an Italian opera than a play in our sense. This is the combination which Richard Wagner strove to realise. But to be gifted in two directions of art is indeed very rare. The music of Æschylus and Sophocles was probably as inferior to their text as Wagner’s text is inferior to his music. All Greek educators imply that every boy can learn music; we never hear a word about want of ear, a want of musical faculty. This was to me in former years a great puzzle, for, like all of you, I was brought up in a society where a few had gifts for music, and the remainder were incapable of singing in time or in tune, or of learning to play an instrument with intelligence--and so we drifted away from the older fashion of making at least every girl play or sing as an inevitable infliction on society, and now only those who show a keen desire for it spend their time at music. But in the new schools, where choirs are taught on the tonic _sol-fa_ system, I am informed by the most competent teachers that an inability to appreciate music, or to sing in tune, is quite rare, and that the great body of our children can be taught to make and to appreciate good music. If this be so, the Greeks were again right, and we in our older generation less wise than they.

In their opinion, this general possibility of learning music was a necessary condition of another settled conviction among educators, which is foreign to us--I mean the conviction that the practice of music has a direct and powerful effect upon the morals of average men. On this point the Greek educators were very explicit, and it is of great practical importance to us nowadays to consider what they say. It was not at all identical with a very widespread belief among modern parents that the pursuit of music generally is a refined pleasure, and will save the young from some lower or more mischievous recreation. That view was quite familiar to the Greeks. But their distinctive theory was this: that the performing or hearing of certain kinds of music had a direct effect, either moral or immoral, upon the mind, and that therefore wise educators must encourage the right sort of music only, and banish the rest from their pupils. I know very well that there were stray voices, especially from the Epicurean philosophers, saying that this is all nonsense, that music can have no such effect, and that the only moral or immoral part of the performance lies in the words.[31] But this only shows that the opinion of the vast majority and of the wisest men was not adopted without criticism, and without the other side of the question being clearly before them. The modern world is under mental conditions such as the Epicureans. We have generally assumed that music as such had no influence in moulding morals. We feel that it may be so in the accessories--that the constant singing of love duets and the associating with theatrical company may do harm and the associating with serious musicians may do good--but modern people seem hardly to dream that music such as Wagner’s, apart from the words, may have a direct effect upon morals. And yet it is here that we might have incurred a great and honourable debt to the Greeks, and have used their wisdom to save our youth from serious danger. This is a conviction of mine, not of to-day or yesterday, but of forty years’ standing, derived indeed from the suggestions of Plato, but verified by frequent contact with music and musicians. I will here give you one striking illustration. Anyone at first hearing of Wagner’s _Tristan und Isolde_ would perceive that it was a most immoral subject, expressed in highly emotional music. It is an artistic glorification of adultery, palliated by the old and vulgar excuse of a magical love-potion. All this is so obvious that I wonder sober people would not keep their children from witnessing the work just as they endeavour to keep them from reading immoral novels. To me it seemed even worse, for I could not but perceive, and had often and long since asserted, that the composer himself wrote the music under the influence of some such moral aberration, and that, apart from the words, it was intended to express his criminal longings and disappointments. It is only a year or two since the correspondence of a lady, published after her death, showed that this anticipation was literally true, that these phrases of love-sickness were actually composed and sent to her because she had awakened in him a passion which she was not wicked enough to satisfy.

I know there are people who think transcendent genius such as that of Napoleon, or, in his way, of Wagner, affords a justification, or at least an excuse, for such lawlessness. And you have heard much talk about the _Superman_, whose main attribute seems to me _infra_ human, when the rights of others are concerned. To me the veritable Superman is not the slave of his own passions, who satisfies them at the expense of others, but the master of himself, who, because he is pure, feels and helps the weakness of his neighbours. Not Sir Lancelot but Sir Galahad is the ideal of chivalry. Of the one,

“His honour rooted in dishonour stood,
And faith unfaithful kept him falsely true”;

but of the other,

“My strength is as the strength of ten, because my heart is pure!”

Perhaps, during this digression, the objection may have been rising in your minds that if Greek music was so universally believed to have a moral or immoral influence, this was because it differed wholly in quality from that which we pursue, and that therefore an inference from one to the other is very hazardous. This is supported by the fact already adduced, that the actual remains of Greek music, though legible and intelligible in the literal sense, have no power whatever to speak to our musical emotions. We must therefore turn back from practice to theory and prove to you that, in spite of these difficulties, Greek music was distinctly the source and forerunner of our own. And I may say by way of preface to this part of my discourse that the simplicity of music, far from being a cause of its lesser emotional effect, may be the very reason why the great mass of people feel it more deeply. The intricacy and difficulty of our modern music tend to estrange it from the feelings of the larger public and to confine its influence to the special class of trained musicians. The Greeks left us no practical work on music, no criticism of existing compositions, no comparison of the effects produced on audiences by this or that artist, by this or that kind of instrument. We find only obvious generalities, such as the flute being more exciting than the harp. There is indeed one passage where Plato goes deeper and inveighs against purely instrumental music as more exciting and therefore possibly more mischievous than vocal music with an accompaniment, showing that he did not lay the stress of the emotion upon the words. Those who have gone deeply into modern music will agree with him; they feel that the emotions produced by a symphony of Beethoven are more subtle, and, because more subtle, deeper and more lasting than those produced by any vocal music, unless it be eight-part music, which approaches the richness of an orchestra.

But this suggestive remark is quite an exception. The extant musical tracts are wholly theoretical, and are concerned with the scientific basis of music, not its application to practice. And the first problem to which they applied themselves, which they solved, and have handed down to us, their heirs in art, is the determination of the proper scale or scales in which music should be composed. This was no easy thing to do, and if you take the trouble to hear the music of any people who have not adopted the Greek solution, or one like it, you will at once perceive the difference. I well remember persuading, with great difficulty, a band of gipsies, in Hungary, to play for me not the music of the Hungarians, for which they are so celebrated, but some of their own Oriental stuff, which they play among themselves in private. I found it wholly unintelligible on account of the scale, which seemed to have thirteen or fourteen notes within the octave. All this the Greeks had contemplated, and in some of their early scales they used quarter-tones and intervals strange and disagreeable to us. But, after much hesitation, they fixed upon the diatonic scale, which became the basis of their music, and in due time of ours. The varieties of this scale which they used were far greater than ours. We are contented with the variation of major and minor, and repeat the same intervals in the same order with a mere difference of pitch, very slightly modified by the temperament of our tuning. The Greeks thought the position of the two semitones far more important, and considered that the quality of the scale, quite apart from pitch, was produced by the variety in the placing of these intervals. But I must repeat that our extant treatises are so absolutely scientific and not practical that it would be impossible to attempt an analysis of them in a popular lecture. The discovery of the scientific basis of concord or harmony and its difference from discord had been made very early by the Pythagoreans, and I have often thought that their famous theory that numerical relations were the key of the universe was much stimulated and fortified by finding that octaves, fifths, and fourths, which are recognised by the ear as concords, can be produced by stopping a vibrating string at the points dividing it into portions represented by 1:2, 2:3, and 3:4. They did not acknowledge our favorite major ⅓ as a concord, the proportion being more complex, _viz._ 4:5 or 5:6; and indeed if the major ⅓ on our instruments be tuned to its full height of two full tones, it sounds sharp and very disagreeable. In this as in most detail we can follow and understand the Greek theory. When Aristotle tells us that the middle note of the scale is that to which the melody always returns, he is evidently speaking of the unaccompanied melody, and there are scores of our melodies that move up and down round this keynote, which may in these cases well represent the central note of the scale.

It is not possible for me to delay longer on this topic. I therefore sum up the result thus: the Greeks had a music to some extent homogeneous with ours; they attributed to its varieties great and direct effects on the morals of men. Seeing that in all their other arts they were so singularly modern and reasonable, it is surely well worth the careful consideration of educators whether similar effects be not latent in our music, _e.g._ whether the study of Handel, Corelli, Palestrina, may not have a strengthening effect on the mind, whereas the study of Chopin, of Verdi, even of Beethoven, with all the vague _Weltschmerz_ which they contain, the unsatisfied longings, the unreasoning discontent, the suspended harmony, may not contribute directly to the vices of modern society, vices not unknown in the fashionable cities of this Commonwealth.

We now turn to the subject of household furniture and decoration, in which you will find that there are many and the best of our ideas borrowed from the Greeks.

We have not had the good fortune to unearth a Greek town of the best epoch from under lava or from beneath the débris of an earthquake. But it is likely that even if the ruins of Antioch were cleared of the great rocks that tumbled down upon it, in the many earthquakes of the early centuries of our era, some splendid houses might be discovered. So far, however, I do not know that, except at Delos, we have been able to find clear evidences that the wall decorations and the furniture of a Greek house were the same in kind as those which a century and a half of excavation has brought up from the dead in Pompeii and Herculaneum. These towns, as well as Naples, which was well known to Cicero as an essentially Greek town, were in close proximity to Puteoli, which again was for several centuries the great port for all Alexandrian luxuries since the second Ptolemy had made friends with the Romans. Through Puteoli, then, Greek artists and Greek designs made their way to that coast, and even the worship of Isis, and the frequent use of the ibis and the crocodile in their designs, show that the Hellenistic artists had felt the influence of native Egyptian work, just as the workman of the French “Empire” felt the breath of old Egypt, when Napoleon’s Commission brought out its splendid work on that mysterious country.

Although, therefore, all the little texts scrawled upon the walls by children are in Latin, I take it the furniture and decoration of the smart houses or villas uncovered are in Greek style, and may thus give us some suggestion of the inside of a Greek house. And let me add at once, that the discoveries of such ruins and remains at Rome in the time of the Renaissance moulded all the taste of that age, and produced house decoration, in direct imitation of the antique, which has been copied down to the present day.

NOTE.--I thought that I could not bring before the audience the
character of this decoration adequately, except by showing some of
the designs, and some of the furniture, on a screen. Some of the
pictures were taken from Niccolini’s magnificent _Art of Pompeii_
(Naples, 1876-92), the Curator having allowed me to use the
expensive process of photographing in colours, in order to show not
only the design, but the rich colours of the Pompeian walls.

VI

SCIENCE: GRAMMAR--LOGIC--MATHEMATICS--MEDICINE

When I speak to you of Greek Science, of course I use the word in the old and proper sense to include all strict reasoning, especially of the deductive kind, particularly therefore pure Mathematics, and not merely the inferences from observation and experiment which now commonly assume and even monopolise the title of Science. I often see in educational programmes Science and Mathematics contrasted as distinct things, which indeed in this case they are, only because the Science so-called is often unworthy of the name. Sciences of observation were, I think, not formulated by the Greeks except in the case of Medicine, in which their results are still quoted with respect; in the case of Hydrostatics, as Heron’s great book shows; and in the case of Natural History, in which they made the first collection of facts that modern men of science can use; but we have lost what they said on their artistic observations, namely their minute observations of the anatomy of the human body, which, as I have told you, their sculptors learned to represent with such accuracy that no modern anatomist can find a flaw in their work. This was done by careful external observation, for the practice of dissecting the human body would have seemed to them impious and horrible. But, whenever it was possible, the Greeks went back to first principles and framed a theory from which they deduced the facts; and this it is which has made their science so valuable. It will not be hard to show you how in Logic--the Science of Reasoning,--in Arithmetic, and in Geometry--the science of the laws of lines, of figures, and of solid bodies in space--they are our teachers to the present day.

It is well to approach the subject of Logic through the avenue by which the Greeks approached it, through the analysis of ordinary language and as the natural expression of thinking. The early poets and great prose writers had so far perfected the use of language that the Greeks in the catalogue of human acquisitions came to put their speech on a very high pedestal. Delighted with it, and despising all other tongues as barbarous, they convinced themselves that the Greek word adequately expressed the nature of the thing it signified, and therefore that to understand their language properly was to understand the nature of things. Λὁγος meant not only speech (_oratio_), but reason (_ratio_), and so, after first seeking to obtain clear conceptions of abstract ideas, they advanced to the structure of sentences and analysed speech in so accurate a way that their technical terms are our technical terms of to-day. When you talk of _infinitives_, or _genitives_, or _participles_, you are only using words borrowed from Latin translations, often mistranslations, of the Greek. You find these logical studies in their beginning, but by no means in their infancy, in the _Dialogues_ of Plato. Whole conversations are employed in trying to fix the connotation of important moral terms, such as _holiness_, or _valour_, or _temperance_. And we also find in some of the dialogues an appreciation of the difficulties contained in the form of simple propositions, the meaning of affirmation or negation, and the nature of the deduction of one proposition from another.

But I need not detain you with particulars about these early preparations for science, when we have before us in Aristotle various treatises on the analysis of speech from its logical side, and the laying down of the laws of formal thinking with such accuracy and completeness that nothing of importance has ever been added to it. We hear it often said that a single man apprehended and systematised these laws. That is not true; there were plenty of tentative essays before his time. But if there be one achievement which has made his name and fame everlasting, it is his treatment of the theory of Reasoning.

The mediæval universities knew this well, and so do the modern universities of Europe which are worthy of the name. I need not bear witness to the vast importance of common Logic by telling you that in my own youth nothing ever woke me up like having a good Logic put into my hands at the age of fourteen. For since that time I have been often teaching it and have watched its effects on hundreds of intelligent youths. Among all the subjects that we teach, not for the purpose of supplying mere facts, but for the purpose of training youth to judge facts and co-ordinate their knowledge, I know nothing that benefits the average student like the study of Aristotelian Logic. May I add that, so far as I know American education, the most serious defect I have observed in it is the small attention paid to this subject, and hence the vast number of your men and women who are unable to distinguish a sound from an unsound argument, still less to point out where the fallacy lies.

There are here present, I have no doubt, a large number of people, otherwise highly educated, who, were I to propose a stock example for their criticism, would feel at a loss how to deal with it. Let me give an illustration. “Every hen comes from an egg; every egg comes from a hen; therefore every egg comes from an egg.” Is this a correct argument? If not, where is it at fault? If you had all been trained in Whately’s Logic, or any other Logic of the kind, as we were in our youth, such a question would present no difficulty whatever.

But if you have failed to derive this lesson from the old Greeks, your English ancestors were better advised. All the subtlety of the mediæval schools, all the disputations of their universities, were based on Greek Logic; and, if they often wasted their time on idle problems, it must always be remembered that by this means Europe was trained to accuracy and subtlety in argument, and hence to weigh vague and random theorising and to make men competent critics of any new dogma. We often remark from our side of the Atlantic how many wild theories in religion, how many sham theories in science, blossom and flourish in this country, inhabited though it may be by a most shrewd and intelligent population. The simplest answer is to point to their ignorance of common Logic, and hence their liability to be deceived by the most vulgar fallacies. It would be easy to mention a book popular in this country, the pages of which any logically trained people would only use to wrap sardines or to heat a stove.

The Greeks do not parade their logic in their writings, though we know they were fond of subtleties; there are indeed examples of it in the _Sophist_ of Plato, where this sort of thing is ridiculed in his travesty of two professional educators. But there are two great and solid proofs of the power which strict Logic had upon their minds. The first comes out in their literature. Wherever they undertake to argue an issue, whether political, social, or religious, their reasoning is clear and easily followed. They of course often start from traditional beliefs, which may not now command assent, but they always reason from these with clear and sober thinking. There was no more important cause for the permanence of that great literature. Its sound thinking has kept it from all extravagance and made it acceptable to educated men of all ages and nations. The second proof is my chief subject to-day: it is the peculiarly logical character of Greek mathematics which has made this too the model of the scientific thinking of the world.

Let me go back to the infancy of Greek science and give you evidence for this statement. Setting aside for the present the metaphysical thinkers, who will occupy us in another chapter, we may safely say that the earliest mathematicians were the school of Pythagoras, and also that their work started (so far as they did not start from the highest of all--pure thinking) from Arithmetic. To this science Pythagoras and his school attached such importance that they were supposed to hold that numbers were the essence of the universe. If you think that such a theory is mere nonsense, I may tell you that I have often heard my colleagues, distinguished in modern science, discuss a theory, alive at the present day, that the so-called material universe consists of mere motion, without anything to be moved! At the root of these speculations lies the fundamental distinction of form and matter, of the definite and the indefinite; and the Pythagoreans had got a glimpse of the eternal truth that it is only through our intuitions of space and time, and through abstract concepts explicating these, that we can bring the myriad phenomena of nature under intelligible law. It was an early anticipation, so far as we can explain it, of the great theory of Descartes, that all the universe could be reduced to mathematical relations, and these handled by algebra, which is in its essence but a very abstract and generalised arithmetic. If therefore all parts of the world stand in mutual arithmetical relations, of which the chemical law of definite proportions is the most signal example, the science of numbers must be the capital of every scientific man.

And remember that in Greek parlance this was the strict meaning of their _arithmetic_--a pure science, while they used the term _logistic_ (or computation) for the working of practical rules. At the basis of their theory of numbers lay of course the one great assumption which makes the science possible--I mean the absolute equality of the units of any number used for the purpose of calculation.

This is not merely the abstraction from all their differences, as when I say that the present audience consists of five hundred people, regardless of the countless variations existing between the units of this crowd. It is the assumption of an ideal and accurate identity between each of the units, as to magnitude, which makes the expression of geometrical truths arithmetically possible.

The truth that 3² + 4² = 5² applies not only to numbers but to lines, and probably suggested the geometrical proof to Euclid (1, 47). But it is only true if the units in the measurement of each line are exactly equal.

Starting from this first assumption, the Pythagoreans began to speculate on the peculiarities of the natural series of units in use among men, and to deduce from these general considerations various theorems, which they believed might solve the secrets of nature. At the very outset they were struck with the obvious contrast between odd and even, which Plato, following them, regarded as a fundamental distinction in nature. Had they been told that, thousands of years later, men of science would find that a most primitive and fundamental distinction among animals is founded on this difference, I mean that of _artio-dactyle_, and _perisso-dactyle_, actually called by the Greek words, they would have said that this caused them no surprise, as their arithmetic had long since laid down the distinction as a law of nature. As simple specimens of the sort of treatment that the science of numbers received from them, I may cite the following: The successive additions of the odd numbers produce the squares of the series of even and odd.[32] The series of even numbers when added give us no such result, but rather this--that the addition of even numbers gives us figures which are the products of successive numbers differing by only one, _e.g._ 2 + 4 = 3 × 2; 2 + 4 + 6 = 4 × 3, and so on. These latter numbers were regarded as rectangles, when expressed in lines. It was by the discovery of the relation of the sides to the base of a right-angled triangle that they, so to speak, stumbled upon irrational numbers. If the two sides are each equal to 1, the hypothenuse is equal to √2, which is no integral number, but a problem in itself.[33]

All the results of this Pythagorean research lived through into the days of Plato and Aristotle and then, as we know from Euclid and Theon, into the learning of Alexandria. The importance recognised by them in the numbers ten and twelve was shown by the general adoption of a decimal system of notation, and of the division of time on a duodecimal system.

You will ask me what symbols the Greeks had which could enable them to treat arithmetical figures of any complexity, and on this I could give you now a very definite reply, but the details would lead us away from our subject, seeing that this notation was lost in the Dark Ages and was ultimately replaced by the Arabic numerals. But we now know that they had a very practical system of decimal notation based on the use of the letters of the alphabet; and the fact that several letters obsolete in the alphabet of the fifth century B.C. appear as symbols, proves that it was current as early as Pythagorean days. The sign for 6 is the _digamma_, that for 90 is the _koph_ of the Phœnician alphabet, which is still found in Locrian inscriptions; the Phœnician letter known as _sampi_ is used for 900. We know the practical management of this easy notation perfectly from the mass of accounts both private and public found on Egyptian papyri. It can express large numbers far more compendiously than the Roman system, often more compendiously even than ours. Suppose you desire to express any large number, say 20,050, here it is β/ΜΝ; say 47,678, it is δ/ΜΖΧΟΗ, and if there be small gain in simplicity here, I will give you 800,000 = 10,000 × 80 = π/Μ. But these are practical matters, though without an easy notation even the most scientific thinkers could not make large progress.[34]

The next great step was to pass from arithmetic to geometry as the science of space and to show how far the same laws governed both.

If we are not well informed upon the beginnings of arithmetic, we are more fortunate in the case of geometry, and here, if anywhere, the old Greeks have been the acknowledged teachers of modern Europe. For we have in the so-called _Elements_ of Euclid, composed most probably at Alexandria about 300 B.C., a summary of all that had been discovered up to his day, doubtless with many new things of his own. He had distinctly built upon his predecessors; he has before him all through his book a problem discussed in Plato, that of the possible number of regular polyhedra, and its solution forms the climax of his work. But he begins from the very beginning and builds up his whole doctrine with such accuracy that a flaw in the demonstration is hard to be found.

How did this great master attain to such perfection? The form of his demonstrations does not suggest an intimacy with the logic of his immediate predecessor Aristotle; but from him he might easily have obtained the whole notion of a strictly deductive science, which, starting from the smallest possible number of primary data, proceeds to derive from these by strict demonstration proposition after proposition. Philosophers of our own time have often expressed wonder at the clearness with which these data are laid down. They are three in kind: first the _common notions_, which apply to all science and all practical life, such as “the whole is greater than its part”; secondly, the _axioms_ peculiar to our intuition of space, such as “two right lines cannot enclose a space”; and thirdly the very simple _postulates_, which amount to the use of a ruler and compass with a pencil. There are besides very careful definitions, so careful that they are at first obscure, because they apply to the ideal construction of the mind in its intuition of pure space and do not concern themselves about the flaws of actual figures. Thus his “point which has no parts” is not nothing at all, but the minimum of definite place; his right line, “which lies in the same way (όμαλῶς) between any two points taken upon its length,” is simply unity of direction. Every other line varies in direction in some of its successive parts. This is a direct appeal to intuition, without which we can make no beginning in the science of space. Such also is the axiom about parallel lines. Such is also the proof by superposition, to show that two triangles, if some of their measurements be the same, must wholly coincide.

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

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

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

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

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

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