IN the preceding lecture we were principally occupied partly with the ascetic, and partly with the mystical elements of Platonism. We saw that the meditatio mortis of the Phaedo and the intellectualis amor of the Symposium are inspired by one and the same idea. The ultimate object is to reach those eternal and unseen realities to which the soul, or, strictly speaking, the rational part of the soul, is itself akin. The scheme of education which Plato in the Republic devises for the guardians of his ideal city is directed towards the same end, and on this account it requires to be considered in any attempt to understand the position of the Platonic philosophy in the history of religious thought.

To nothing, perhaps, does Plato assign so much importance as to education. It was to the practical work of an educator that he devoted the larger share of his energies throughout an unusually long and laborious life. The Academy which he founded is the earliest example of what we should call a College or University, the type and model of all the philosophical schools that followed it. “Among his contemporaries,” says Grote, “he must have exercised greater influence through his school than through his writings.”1 From the well-known passage of the Phaedrus where written discourse is depreciated in comparison with the living word, we may reasonably infer that Plato himself considered his literary productions of less importance than the work he endeavoured to accomplish as a teacher. The development of the intellect and concomitantly also of the character by means of oral discussion and debate, the mind of the teacher acting directly upon the mind of the pupil, without any intermediate vehicle such as deaf and speechless books supply—this, according to the Phaedrus, is the primary business of the educator. The writing of books may be useful as an innocent pastime, or to preserve the records of oral discussion against the forgetfulness of age, or by way of guidance to those who may afterwards pursue the same track; but literature is a much less efficient means of education than the spoken word.2

So much it appeared necessary to say with the view of guarding against the idea that on educational questions Plato was merely a theorist and nothing more. We should remember that the principal work of his life was to educate the students of the Academy; and that he had probably already tested by experiment many of the theories on education with which we meet throughout his writings.

Let us now proceed to consider Plato's doctrine of education, as it is unfolded in the Republic.

The Republic contains two discourses or treatises on education, the one concerned with the preliminary training of the character in boyhood and youth, the other with the training, primarily of the intellect, but secondarily, and as a consequence, also of the character, in youth and early manhood. It is the second of these two treatises that chiefly demands our attention; but inasmuch as the earlier scheme is intended to lead up to the later, and itself comprises not a few ideas which are of importance for Plato's moral and religious teaching, we must give to this also the consideration which it deserves.

Plato discusses the subject of “musical” education under two main heads, the first dealing with the content, and the second with the form of what is to be taught. It will be enough if we consider the leading principles that come to light in the course of the discussion, so far as it bears upon the subject of our lectures.

Concerning the being of the Godhead, the fundamental rule prescribed by Plato for the guidance of teachers of the young is that God must always be represented as He really is.3 Now, in the first place, God is good; and since that which is good can never be the cause, of evil, whatever there is of evil in the world must be assigned to another cause and not to God. The view here taken is in harmony with the dualism of the Timaeus and the Laws, where evil is attributed, in the one case to Necessity, and in the other to a malevolent World-soul;4 but in the Republic. Plato is more concerned to affirm the negative position that evil does not come from God, than to determine the nature of the principle from which it actually comes. As regards the problem of suffering, however, an alternative explanation is admitted. Suffering may also be regarded as a form of chastisement designed by God to improve and benefit the sufferer.5

The doctrine of the divine goodness, which in the preliminary stages of education Plato would inculcate as an article of belief, prepares the way for that intellectual apprehension of the metaphysical Idea of good, to which, when correct opinion is replaced by knowledge, the guardians of Plato's city are expected finally to attain. A similar remark holds good of the second doctrine on which Plato here insists—that of the changelessness of God; for the Idea is itself essentially uniform and changeless. It is particularly to be observed that Plato makes the immutability of God a result of his perfection. If God should suffer change, we are told, the operating cause must be either something external to himself, or his own free will. The former alternative is inadmissible; for analogy shows that things are liable to change in proportion as they are evil, and in God there is no evil at all. Neither will he desire to change himself, inasmuch as the only possibility of change for that which is the best is towards the worse; and no one, whether God or man, deliberately chooses to make himself worse than he was before. Our conclusion, therefore, is that God “ever abides immutably in his own form”—μϵ̂νϵι ἀϵὶ ἁπθω̑ς ϵ̓ν τῃ̑ ϵ̔αυτου̑ μορϕῃ̑6 Nor can it be supposed that the Godhead, while himself remaining immutable, is nevertheless prone to beguile mankind by false and unreal appearances or visions, or by means of a lie expressed in words. There is absolutely no taint of falsehood in the divine nature.7

These two dogmas—the goodness and the unchangeableness of God—are formulated by Plato in the course of a severe and frequently unfair attack upon the religious teaching of the poets. The feud between Philosophy and Poetry, kindled by Xenophanes, breaks out afresh in the Republic, and rages with more violence than ever. In Plato's criticism of the poetical theology the dominating idea is that the character of the Godhead must necessarily be such as to furnish a moral standard to mankind: for his own conception of the ethical end, as we have already seen, is “assimilation to God”—ὁμοίωσις τῳ̑ θϵ ῳ̑.8 He points out that Homer continually represents the Gods and heroes as lacking the virtue of self-control, prone to insubordination, lustful, avaricious, revengeful, gluttonous, and mean. The inevitable effect—so Plato maintains—of these and similar representations is to encourage the same vices in the young. No one, he says, will be otherwise than indulgent to his own iniquities, if he believes that the Gods and their kinsmen set the example of immorality.9 For this reason, as well as because they are impious and untrue, all such legends ought to be proscribed.

So much, then, for the theological beliefs which Plato desires to instil into his guardians during the period of childhood and adolescence. It is obvious that they are more in harmony with Christian thought than with the traditional theology of Greece. Proceeding in the next place to consider the form of the instruction to be imparted to the young, Plato again develops his own views by an attack upon Greek poetry. Without pursuing the subject into detail, we must be content to apprehend the nature of Plato's own conception of what Poetry and Art should be. According to the Platonic theory, the two antagonistic principles of beauty and ugliness, good and evil, surround us on every side; they reveal themselves in the works alike of nature and of man, and we can discern them also in human character and conduct. The business of the artist, whether poet, painter, sculptor, or architect, or whether he devotes himself to any of the subordinate departments of imitation, is wholly to ignore whatever is unbeautiful and base, and, searching out the beautiful and good, to embody this and this alone in the material on which he works. Thus it comes to pass that, “dwelling as it were in a healthful region, our youthful citizens will imbibe good influences from every quarter, whencesoever from fair works of art there smites upon their eyes or ears as it were a health-bringing breath from goodly places; unconsciously leading them from earliest childhood into likeness and friendship and harmony with the beauty of reason.” 10 In this way the mind becomes a “mansion for all lovely forms,” the memory a “dwelling-place for all sweet sounds and harmonies.” But the ultimate end of musical education has not been reached until we are able to recognise the beautiful and good, not only in the works of imitative art, but also in the originals from which these works are copied. Poetry and Art should lead the youthful mind to apprehend, and in apprehending to assimilate, the principles of beauty and goodness wherever they manifest themselves, either in the physical world or in the lives of human beings. Musical education, according to Plato, is consequently the means whereby we “learn to read in the moral world”: its object is only then attained when we have conceived an abiding passion for the beautiful in the comprehensive meaning attached to the word καλόν in ancient Greece—moral and spiritual beauty, as well as the beauty of material objects.11

Mr. Nettleship has pointed out that the Platonic conception of the true office and function of Poetry in a well-ordered commonwealth has been affirmed by no one in statelier or more impressive language than by Milton, in the famous passage where he expresses on his own behalf the hope of leaving “something so written to after-times, as they should not willingly let it die.” The poet's abilities, Milton declares, “wheresoever they be found, are the inspired gift of God, rarely bestowed, but yet to some (though most abuse) in every nation; and are of power, beside the office of a pulpit, to imbreed and cherish in a great people the seeds of virtue and public civility, to allay the perturbations of the mind, and set the affections in right tune; to celebrate in glorious and lofty hymns the throne and equipage of God's almightiness, and what he works.… Lastly, whatsoever in religion is holy and sublime, in virtue amiable or grave, whatsoever hath passion or admiration in all the changes of that which is called fortune from without, or the wily subtleties and refluxes of man's thoughts from within; all these things with a solid and treatable smoothness to paint out and describe.”12 In all essential features, this is just Plato's view of what Poetry ought to be.

It was stated at the outset of this lecture that the ultimate aim of education, as conceived by Plato, was to raise the soul out of the temporal and visible into the sphere of that invisible and eternal Being to which she herself, by right of birth, belongs.

But the preliminary discipline, regarded in and by itself, does not and cannot bring the soul into immediate contact with reality. We shall best understand its general character and efficacy, as well as the relation in which it stands to the intellectual discipline that follows, if we consider for a little the Platonic simile of the line, to which it is desirable also on other grounds that I should now direct your attention.

At the end of the sixth book of the Republic, Plato offers a classification of what (for want of a better word) we may perhaps call “apprehensibles,” meaning thereby the entire contents both of the phenomenal and of the ideal worlds, arranged in an ascending scale from lowest to highest, according to the degree of luminosity or truth which they possess. We are to take the line AB and divide it into two unequal parts at C; after which each of the two parts, AC and CB, is to be subdivided in the ratio of the original sections, so that AD is to DC and CE is to EB as AC is to CB. What may have been the object of Plato in dwelling on these particular proportions, we need not now inquire; it is enough for our purpose if we understand what the different segments of the line are intended to symbolise.

Of the two larger sections, AC represents the class of things which we apprehend by sense-perception and opinion, while CB stands for the objects of knowledge, which it is the province of Reason or the ratiocinative faculty to grasp. Here, as elsewhere, between the Kingdom of Sense and the Kingdom of Knowledge, Plato draws, you will observe, a clear and sharp demarcation. To the first of the two segments in the sphere of sense or opinion, namely, AD, belong ϵἰκόνϵς or “images,” the category in which there is least of light and truth. The example which Plato gives, shadows, reflections in water and the like, are all of them taken from the visible world; but inasmuch as he repeatedly speaks of AC as containing the objects, not only of sense, but also of opinion, and elsewhere uses the word “opinions” to denote canons or standards of morality, taste, and so forth, instilled by habituation or authority without scientific knowledge of the rational grounds on which they rest, we are justified in finding in the lowest section of the line shadows, reflections or imitations of opinables as well as sensibles. Now among the shadows of sensibles there should be no doubt that Plato included the products of so-called imitative art; for, according to the Platonic theory, painting and sculpture copy directly from the life,—imitate the visible and tangible things which men ignorantly call real,—forgetting that these are themselves but imitations of the invisible and intangible Essence which is the sole and ultimate reality. Where then are we to look for the shadows or reflections of opinables? Presumably these are nothing but the canons or opinions expressed or embodied in the writings and speeches of poets, rhetoricians, etc., in so far as these canons or opinions reflect and imitate the actual beliefs of the multitude or any other beliefs and “appearances” whatsoever, be they right or be they wrong.13 Poetry, not less than Art, is regarded by Plato as an imitation of that which is itself in turn an imitation of the truth; so that we are well within our rights, if, in agreement with Nettleship and other Platonic students, we include the creations of Poetry among the “images” appropriated to the lowest division of the line. The mental condition or state which is correlated with this class of apprehensibles is called by Plato ϵἰκασία, a word which ordinarily in Greek means “conjecture,” but which in this particular passage receives a new and quasi-technical meaning, being used with a play on ϵἰκόνϵς, “images,” to denote the condition of mind which acquiesces in images and accepts them as the only truth, the lowest of all the intellectual states, as its objects are the lowest of those to which the human mind can be directed.

The second section of the line, DC, consists of the originals which are copied or reflected in the first. Here again the instances cited by Plato belong to the sphere of sensibles: he speaks chiefly of living creatures and the other works of Nature, together with manufactured objects of every kind. But, as before, so now, we are justified in viewing this category also as embracing not only sensibles, but also opinables; and, thus regarded, it will of course embrace the originals of those reflected opinions which we have already found reason to assign to the lowest of the four grades; in other words, it will contain the canons or standards exemplified in the words and deeds of those who live not by the light of Reason, but in obedience to authority or unconscious habit, canons which are reflected, as we have seen, in Poetry and other forms of imitation, through the medium of language. Plato's name for the mental condition that takes the visible and opinable for true, refusing to penetrate into the region of the invisible, is the word which was destined to play so momentous a part in later religious thought—the word πίστις“belief” or “faith.” With Plato, Knowledge and not Faith is “the assurance of things hoped for, the test of things not seen”—πραγμάτωνἔλϵγχος οὐ βλϵπομϵ́νων.14 The Platonic πίστις is still concerned with the visible and opinable, although it possess a somewhat higher degree of clearness than ϵἰκασία or “Conjecture,” just as its objects contain more light and truth than the “images” of Poetry and Art.

If the above exposition is correct, it is manifest that the preliminary scheme of education in the Republic deals from beginning to end with the objects included in the two lower sections of the line. The aim of that discipline is to produce correct “opinion” or “belief,” not yet knowledge, although in the case of the guardians correct opinion will prove, of course, a stepping-stone to knowledge. Our pupils, Plato says, must first be moulded into unconscious harmony with the beauty of Reason, in order that, when Reason comes, they may welcome her with joy, in virtue of their affinity to her.15 Under the guidance of a purified form of Art and Poetry, the student who has assimilated the earlier discipline becomes able at last, by a kind of instinctive and unreasoning sense, to discriminate between right and wrong, fair and foul, as they show themselves in the world of nature and of man; but he is still concerned with the visible and opinable region of “becoming,” not yet with the invisible realm of Being; and the virtue he has acquired is only (Plato says) an ὑπογραϕή or “adumbration” of the true or philosophic virtue, inasmuch as it rests on a foundation of “correct opinion” and not knowledge.16 It follows that if education is to achieve its true and final purpose—if we are to raise our eyes from shadows to realities, from darkness to light—a further discipline is necessary; and this further discipline is represented by the remaining sections of the line (CE, EB). The soul does not desist from her journey until she has scaled the topmost summit of the intelligible land and beheld the Idea of Good. As Mr. Adamson has said, “Plato's theory of education ends where it began, with a revelation of the divine being.”17

We remark, in the first place, that the secondary course of training described in the Republic is confined to a small minority of those who received the earlier education. The others are excluded, not in any narrow spirit of intellectualism, but from the conviction that many persons are naturally incapable of rising from “correct opinion” to knowledge. Only the highest section of the Guardians, that is to say, the future rulers of the city, have access to the curriculum we are about to discuss. Plato begins by enumerating the more distinctively natural qualifications of his philosopher-king. The keynote of the philosophic temperament he declares to be the love of Wisdom or Truth, not of this or that portion of Truth, but of all Truth, everywhere and always. The philosopher is one who aspires to contemplate “all time and all existence”; unable to acquiesce in the partial or particular, his unwearied mind is ever soaring towards the universal, ever strives to grasp the totality of things, both human and divine—του̑ ὅλου καὶ παντός, θϵίου τϵ καὶ ἀνθρωπίνου18 From this consuming passion for truth and knowledge spring all the moral virtues which Plato ascribes to the truly philosophic nature—courage and high-mindedness, temperance, justice, kindness, and the rest.19 Here, as elsewhere in his dialogues,20 he draws a portrait of what, in his opinion, human character should be; and indeed the philosopher-king of the Republic is expressly intended as a picture of the perfectly just and righteous man. The tendency to describe the intellectual and moral ideal under the form of an imaginary personality, afterwards known among the post-Aristotelian schools as the “wise man” or sage, would seem to have originated with Plato.

It is to characters thus endowed, continues Plato, that we shall entrust the government of our State, after we have made them perfect by means of education.21 How then is one to lead them upwards into the light, “even as some are said to have ascended out of Hades into Heaven”?22 The general character of Plato's intellectual discipline is determined partly by his conception of the goal toward which the mind must travel, and partly by his view of the nature of the mind itself The goal, of course, is the Idea, and ultimately the Idea of Good. It is the Idea of Good which, to borrow the phrase of Nettleship, is “at once the keystone of knowledge and the polestar of conduct.” Of the Platonic Ideas, the “colourless, formless, intangible Essence, visible only to Nous, the pilot of the soul,”23 I will speak in a subsequent lecture; meantime let me remind you of some cardinal points in connection with Plato's doctrine of Nous. We have seen that Nous, according to Plato, is the part of human nature which is related to God; nay more, it is this which makes us distinctively and truly human by making us essentially divine.24 And further, the faculty of Nous—so Plato affirms—is present in every human being from the first. It constitutes the eye of soul (ὄμμα ψυχη̑ς), and the eye of soul can never wholly lose its power of seeing. Through its affinity with God, man's Reason, even when present in the body, retains an upward impulse, feels still a natural and spontaneous yearning toward the fountain of its being. But until education has come to the rescue, our spiritual insight is clouded by the darkness of the prison-house in which we live. The eye of the soul is turned unnaturally downwards, seeing only what is of the earth, earthy; at best, it gazes on what is no better than a shadow of the truth. And thus the soul, though “a heavenly and not an earthly plant,” draws its sustenance from earth, and not, as it has a right to do, from heaven.

Let us now see how these considerations about the nature and condition of the soul determine Plato's view of the scope and method of education. It will follow (says Plato) that

“education is not at all what certain of its professors declare it to be. They tell us that they put Knowledge into an empty soul, as though one should put sight into blind eyes. Our theory is of quite another kind. This faculty of Reason, present in every human soul, this organ wherewith each man learns…must, along with the entire soul, be turned round from the sphere of Becoming until it can endure to gaze upon Being, and the brightest part of being, that is, the Good. Education is therefore the art of converting (της πϵριαγωγη̑ς) the Reason in the easiest and most effectual way. It is not the art of putting sight into the soul's eye: believing, on the other hand, that sight is already present in the soul, but turned in the wrong direction and looking at the wrong things, it endeavours to remedy this defect.”25

Thus, according to the Platonic view, education does not consist in filling the soul with a mass of uncorrelated fact and dogma; it has nothing to do with what is popularly known as “cram ”—the travesty of educational method which Plato ascribes to some of the professional sophists of his own day. On the contrary, as Mr. Adamson has well said, “That it is the business of education to mature and develop something given, the germ of a personality, rather than impress it from without, is the very keynote and spirit of Plato's teaching. ‘There is a faculty residing in the soul of each person,’ he tells us, ‘an organ whose preservation is of more importance than a thousand eyes.’… The teacher must be content to efface himself, to stand aside. His business is to superintend the presentation of material and to guide his pupils to an orderly assimilation of it. But it is emphatically not his business to impress his ‘modes of thought’ so that they become a second nature in his pupils. Every bit of knowledge worth the name bears the private mark of the individual who has acquired it.”26 These are the words of a practical teacher who has tested by experience the value of Plato's principles in the earlier stages of mental and moral discipline. I have elsewhere attempted to explain the essential meaning of the Platonic conception by an illustration from the history of sculpture. Michael Angelo, himself, both in poetry and in statuary an exponent of the great Platonic thought which he expresses in the lines—

“Heaven-born, the soul a heav'n-ward course must hold, Beyond the visible world she soars to seek (For what delights the sense is false and weak) Ideal form, the universal mould27—

Michael Angelo “used to say that every block of marble contained a statue, and that the sculptor brings it to light by cutting away the encumbrances by which the ‘human face divine’ is concealed. In like manner, according to Plato, it is the business of the teacher to prune the soul of his pupil of those unnatural excrescences and incrustations which hide its true nature, until the human. soul divine stands out in all its pristine grace and purity.”28

Some of the figurative expressions employed by Plato to emphasise the distinctive character of his educational theory are of considerable interest and importance in connection with later religious thought. At one time the process is pictured as an ϵ̓πάνοδος or ascent of the soul into the realm of Being, a lifting of the eyes on high, a θϵ́α τω̑ ἄνω, “contemplation of that which is above.”29 The didactic art appears at other times as a kind of purification or purgation: its effect is to cleanse the soul from the defilement of the body and its senses, to lighten the soul of those leaden weights that drag it downwards to the earth.30 Or, again, it is a mole of deliverance (λύσις), a release from chains; or a quickening and rekindling (άναξωπυρϵι̑σθαι) of the spiritual vision. To several of these expressions interesting parallels occur in the New Testament; but the most striking analogy is furnished by Plato's description of the educational process as a πϵριαγωγή or “conversion” of the soul. The eye of the soul, Plato implies, must be turned from darkness to light (πρὸς τὸ ϕανὸν ϵ̓κ του̑ σκοτώδους),31 must pass from a day which is but night into the true day” (ϵ̓κ νυκτϵρινη̑ς τινοςη ἡμϵ́ρας ϵἰς ἀληθινήν).32 Nor is it merely the intellect which participates in this revolution; the character is also involved: for Plato expressly says that the revolution extends to the whole soul (ξὺν ὅλἼ̑ ψυχἼ̑).33 As with St. Paul, so also with Plato: conversion is the birth “of a new intellectual consciousness which transforms the will and is the source of a new moral life.”34 The whole personality of the pupil is to be transformed, to be reborn;35 as the light of truth shines ever clearer in his soul, “the inward man”—ὁ ϵ̓ντὸς ἄνθρωπος, as Plato would say36—is renewed unto knowledge after the idea of Good or God, until, so far as human nature admits, the assimilation is complete (ὁμοίωσις θϵἢ̑ κατὰ τὸ δυνατόν).37

But it is time for us to consider the actual curriculum of studies by means of which Plato hoped to lead his guardians out of the darkness of the visible into the brightness of the intelligible world. Reverting to the simile of the line, we have to ask what are the contents of CE, the section that lies between the realm of sense and the realm of Ideas. Plato's answer to this question is introduced by a somewhat elaborate discussion explaining the principle of the curriculum described in this part of the Republic. The originating cause of reflection or thought is declared to be the self-contradictory character of certain sense-perceptions. Thus, for example, we perceive one and the same object as both heavy and light, that is, heavy in comparison with a lighter, light in comparison with a heavier object. The contradiction produces a feeling of perplexity (ἀπορία—the old Socratic term) which the senses are powerless to assuage; and the intellect is consequently summoned to their aid. Observe now the way in which the intellect sets to work, when thus invoked by the senses. After disintegrating the impression into its component parts, the “heavy” and the “light,” it abstracts each of these two qualities from the material substance in which they inhere, and studies them as what we should call general notions or conceptions, apart from every element of sense-perception and corporeality. The question has ceased to be, “Is this particular object heavy or light?” and we have entered on a purely intellectual inquiry as to the οὐσία, or essential nature of heaviness and lightness. Such an inquiry when once fairly started will lead us farther and farther from the visible world, and nearer to the invisible Ideas which in Plato's way of thinking furnish the solution of this and every other problem.38

What then are the particular studies prescribed by Plato?

First in order comes ἀριθμητμητική, or the theory of numbers. Here, if anywhere,—so Plato holds,—the intellect is stimulated by the shock of self-contradictory perceptions; for number is an aggregate of units, and unity is never seen apart from multiplicity: we perceive one wood, for example, but many trees; one tree, but many branches, and so on. But the stimulus of perceptual contradictions can hardly count for much after the intellect is thoroughly aroused; and the important point, in view of the goal towards which the soul is travelling, is that we should realise the nature of the numbers with which the Platonic science of arithmetic professes to deal. The true arithmetician, according to Plato, although he may employ by way of illustration concrete numbers, such as one horse, two tables, three chairs and so on, is not really studying these material numbers at all: it is only with the abstract mathematical numbers 1, 2, 3, 4, etc., that he is concerned. To these mathematical numbers, following the usual bias of his thought, Plato seems to have ascribed a substantial, and not merely a conceptual existence. When the arithmetician defines his units as “indivisible and equal each to each,” he is not speaking of an “Unding,” but of real existences, compared with which their visible counterparts are imperfect and illusory; no two material and concrete units being ever either indivisible or exactly equal to one another. Mathematical numbers, as well as mathematical forms, appear to be included among those “imitations of the eternal existences, moulded from them in a mysterious and wondrous way,” of which we read in the Timaeus:39 they form, apparently, a kind of intermediate link between the visible numbers in DC, and the ideas of numbers in EB. On the one hand, they resemble visibles inasmuch as they are many and not one—that is to say, there is, for example a multiplicity of mathematical units, but only one Ideal unit. With the Ideas, on the other hand, they share the attributes of changelessness and eternity.40

The study of Number, according to Plato, if prosecuted on these lines, will insensibly lead the soul on high, away from the region of sense into the region of knowledge. Next in sequence follows Geometry. The geometrician, it is said, deals with τὸ ἀϵὶ ὄν, that which always is, eternal and necessary truth. In word, no doubt, he speaks of the visible and perishable triangle which he draws upon the board; but all the time he is thinking of the true mathematical triangle, and it is this whose properties he endeavours to explain.41 If I understand Plato rightly, he believes that mathematical triangles, circles, squares, etc., have a real or substantial existence, and occupy an intermediate position between the Ideas in whose likeness they are framed and the visible forms of which they are themselves the models. It is for this reason, among others, that the study of Geometry “compels the soul to turn towards the region where dwells the most blessed part of Being, which above all things she must behold.”42 But Geometry will have no such result unless it be pursued on exclusively theoretical lines; and we are told that Plato strongly deprecated the use of geometrical instruments and models. He is said to have reproved Eudoxus and others for this fault, maintaining that they “forfeited all the good of Geometry by allowing it to fall back upon sensibles rather than soar aloft and lay hold upon those eternal and incorporeal images upon which God by reason of his Godhead is evermore intent.”43

Plato's distrust of sense-perception is intelligible enough, so long as we are dealing with Plane Geometry; for we must remember that he is concerned with mathematics as an educative discipline, a means of forcing the pupil to use his reasoning faculties and think, instead of depending on the eye. Neither is it difficult to understand how the problems of Solid Geometry might be treated on similar lines, though here, perhaps, the student cannot so easily dispense with visible aids. At the time when Plato wrote the Republic, the study of Solid Geometry appears to have suddenly become popular in the Academy. He speaks of it as a singularly fascinating subject, and assigns to it the third place in his curriculum. But when we leave the region of pure mathematics and come to the concrete sciences of Astronomy and Harmonics, which form the last of the five preliminary studies, we must allow that Plato's attitude is more difficult to justify. Each of these two subjects is treated by him almost wholly from the standpoint of Geometry. As regards the first, he insinuates, perhaps with justice, that the astronomy of his own day was far too empirical. Plato himself flies to the opposite extreme. Refusing to allow that any study can lift the soul on high unless it deals with the invisible and real, he insists that the objects of true astronomy are not the celestial movements which we see, but the “movement wherewith essential speed and essential slowness, in true and genuine number and in all true forms, are moved in relation to each other and therewithal make that which is essentially in them to move: the true adornments, which are apprehended by reason and the mathematical intelligence, but not by sight.”44 The language of this sentence calls up in our minds the picture of a transcendental firmament analogous to the transcendental triangle in Geometry. It follows that just as the visible triangle is utilised by the geometrician for purposes of illustration and nothing more, so also the visible heavens should be employed as a moving diagram or orrery to facilitate our apprehension of the supra-celestial movements which they imitate. The true astronomer will “dispense with the starry heavens” (τὰ ϵ̓ν τῳ̑ οὐρνῳ̑ϵ̓άσομϵν), and cultivate astronomy by means of problems: only in this way, Plato adds—and the remark is highly significant of the aim and object of his whole curriculum—can he “make the natural intelligence of his soul useful, and not useless, as it was before.”45 This is the only kind of “utility” which the Republic admits. Here, as elsewhere in Book VII., Plato's aversion to the senses and their objects is profoundly real; nor could it well be otherwise, for in Platonism Truth lies yonder, in the realm of the Ideas.

Astronomy was looked upon by the Pythagoreans as the sister-science to Music or Harmonics. In this view Plato concurs; but his conception of Harmonics differs toto caelo from that of the Pythagoreans, and is in every respect analogous to his conception of Astronomy. There appear to have been two musical schools in the time of Plato, the so-called μουσικοί, who, as Mr. Monro remarks, “measured all intervals as multiples or fractions of the Tone,”46 selecting as their unit of measurement the quarter-tone or δίϵσις; and the Pythagorean or mathematical school, who investigated the mathematical ratios determining consonance and dissonance. The first of these two classes Plato dismisses with contempt as mere empiricists who “persecute and torture the strings, racking them upon the pegs.”47 For the second or Pythagorean school of theorists he has more respect; but they too are guilty of a fundamental error, inasmuch as it is only audible consonances whose ratios they examine; whereas they ought to have recourse to problems, inquiring which members are really concordant or discordant, and what is the reason in each case. It is difficult to follow out Plato's conception in detail; but we can clearly see that he regards certain mathematical ratios as possessing in themselves the quality of consonance, audible consonances being only sensible and therefore inadequate embodiments of these transcendental ratios, and, like the visible movements of the stars, useful merely for illustrative purposes and nothing more. The true musician, according to Plato, is one who penetrates into “the world of harmony beyond.”

So much, then, for the five preliminary studies forming the ϵ̓πάνοδος του̑ ὔντος, or ascent into the realm of Being. I have enumerated them in the order of their inception—theory of Number, Geometry, Stereometry, Astronomy, and theory of Music; but we are not to suppose that each preceding study is relinquished as soon as a new one begins. It will be observed that the complexity increases as we advance, except at the last: for the science of Harmonics does not seem to be more complex than Astronomy; the truth is rather that they are two complementary sides of the same subject, namely, the intelligible counterpart of movement, in the one case of visible movement, and in the other of audible.48 In the theory of Number, we are presumably dealing with one dimension, Number in antiquity being often represented by a line: Geometry adds a second dimension, Stereometry a third; and in Astronomy there is the further element of motion. The demand upon the intellect becomes correspondingly greater at each stage, particularly as all these sciences are to be treated from a purely abstract, perhaps we should rather say a transcendental, point of view.

It is deserving of notice that Plato's quadrivium of studies—for Stereometry may be viewed as a department of Geometry—is the historical prototype of liberal education in Europe. There is some reason to believe that the Pythagoreans had already established a course of study embracing Arithmetic, Music, Geometry, and Astronomy, and we have already seen that, towards the end of the fifth century B.C., the sophist Hippias taught these four subjects under the name of “Arts,” which is also the name applied by Plato to his propaedeutic studies. An allusion in Isocrates to “the education established in our day”49 has reference to the same curriculum. If Plato, as is not unlikely, owed something in this matter to his predecessors, he was probably the first to arrange the subjects according to a clearly conceived plan, and he certainly gave his own interpretation to them all. Nor can there be any doubt that when he installed the “Arts” along with Dialectic in his own Academy or University—for such it really was—he became the virtual founder of University education throughout the Middle Ages. The quadrivium of the Middle Ages consisted of these four subjects, which together with the trivium, Grammar, Dialectic, and Rhetoric, made up the seven liberales artes, proficiency in which was rewarded by the degree of artium baccalaureus.

To the student of language, the very name of “mathematics” speaks of Plato. It was in consequence of the position which he assigned to them in his Academy that mathematical pursuits came to be known in a special and peculiar sense as “studies” or μαθήματα: Plato himself, indeed, repeatedly so calls them. Thus in a passage of the Laws50 he writes: τοι̑ς ϵ̓λϵυθϵ́ροιςἔστι τρία μαθήματα “there are three studies suitable for freemen”—in other words, as we should say, three “liberal studies”: one of them is Calculation and Arithmetic: the measurement (μϵτρητική) of length, superficies, and depth is the second”—he means, of course, Geometry (γϵωμϵτρία), Plane and Solid: “and the third treats of the revolutions of the stars in their relations to one another.” By the time of Aristotle the mathematical use of μαθήματα is fully established; and “mathematics” is only a literal translation of τὰ μαθηματικά, the subject-matter of “learning” in the narrower and more restricted meaning of the word. So close are the links that bind our education to the past.

In conclusion, I would ask you to remember that, in making his preparatory discipline consist of mathematics, Plato is true to the principles expounded in the Timaeus. According to that dialogue, the Creator, who, in Plato's opinion, is always geometrising, constructs the soul and body alike of the Universe and man by means of mathematical ratios and forms. In a certain sense mathematical science, from Plato's point of view, is thus a revelation of the Deity. But the pupil must not be allowed to acquiesce in it as final. His teacher must never lose sight of the something beyond which alone gives meaning and value to the period of preparation. When, after years of patient effort, we finally attain an elevation from which we can take a synoptic view of the road by which we have travelled, apprehending the different studies in their mutual relationship and discerning the continuity of our progress from first to last, we are in a position to enter on the study of the Ideas: but the ultimate goal, the Idea of the Good, is still far distant. All that we have hitherto learnt, says Plato, is only the prelude to the song of dialectic. It will be our duty in the remaining lectures to endeavour to interpret the music of that song.

• 1.

  Plato i. p. 216.

• 2.

  275 D–277 A.

• 3.

  Rep. ii. 379 A

• 4.

  See p. 362f., and Laws x. 896 D ff. 449 f.

• 5.

  Rep. ii. 379 A–380 C. Cf. p.

• 6.

  ii. 381 C.

• 7.

  ii. 380 D–383 C.

• 8.

  See p. 18; Theaet. 176 B.

• 9.

  iii. 391 E.

• 10.

  Rep. iii. 401 C.

• 11.

  Rep. iii. 401 A-403 C.

• 12.

  The Reason of Church Government urged against Prelaty, Bk. 2.

• 13.

  See my edition of the Republic of Plato, vol. ii. p. 158.

• 14.

  Heb. xi 1.

• 15.

  Rep. iii. 402 A.

• 16.

  Rep. vi. 504 D.

• 17.

  Education in Plato's Republic p. 246.

• 18.

  Rep. vi. 486 A.

• 19.

  485 A-486 E.

• 20.

  Cf. Theaet. 173 C ff.

• 21.

  Rep. vi. 437 A.

• 22.

  Rep. vii 521 C.

• 23.

  Phaedr. 247 C.

• 24.

  See p. 377 f.

• 25.

  Rep. vii. 518 B-D.

• 26.

  Education in Plato's Republic pp. 78-81.

• 27.

  Wordsworth's translation.

• 28.

  The Republic of Plato, vol. ii p. 98.

• 29.

  Cf. Col. iii. 1, 2 (see p, 359 f.).

• 30.

  Cf. Heb. xii. 1, ὄγκον ἀποθϵ́μϵνοι πάντα, laying aside every weight.

• 31.

  Rep. vii. 518 C. Cf. Acts xxvi. 18, ϵ̓πστρϵ́ψαι ἀπὸ σκότονς ϵἰς ϕω̑ς

• 32.

  521 C.

• 33.

  518 C.

• 34.

  The quotation is from Sanday and Headlam, Romans p. 165.

• 35.

  Cf. Theaet. 168 A, ἵν̕ ἄλλοι γϵνόμϵνοι ἀπαλλαγω̑σι ιω̑ν οι̂ πρότϵρονἡ̑σαν

• 36.

  Rep. ix. 589 A. Cf. Rom. vii. 22; Eph. iii. 16.

• 37.

  Theaet. 176 B.

• 38.

  vii. 523 A-524 C.

• 39.

  50 C

• 40.

  Rep. vii. 524 C-526 C. See, however, p. 369 n.

• 41.

  Rep. vii. 526 C-527 C.

• 42.

  Rep. vii. 526 E.

• 43.

  Plutarch, Quaest. Conv. viii. 2. 718 F. The last clause has reference to the Platonic saying, θϵὸςἀϵὶ γϵωμϵτρνι̑

• 44.

  Rep. vii. 529 C.

• 45.

  Rep. vii. 530 C.

• 46.

  Smith's Dict. of Ant. ii. p. 193.

• 47.

  Rep. vii. 531 B

• 48.

  Rep. vii. 530 D.

• 49.

  Panath. 26.

• 50.

  vii. 817 E.