Chapter XLI: Appendix: B

ON PLATO'S SURVEY OF THE SCIENCES.

(_Cam. Phil. Soc._ APRIL 23, 1855.)

A survey by Plato of the state of the Sciences, as existing in his time, may be regarded as hardly less interesting than Francis Bacon's Review of the condition of the Sciences of _his_ time, contained in the _Advancement of Learning_. Such a survey we have, in the seventh book of Plato's _Republic_; and it will be instructive to examine what the Sciences then were, and what Plato aspired to have them become; aiding ourselves by the light afforded by the subsequent history of Science.

In the first place, it is interesting to note, in the two writers, Plato and Bacon, the same deep conviction that the large and profound philosophy which they recommended, had not, in their judgment, been pursued in an adequate and worthy manner, by those who had pursued it at all. The reader of Bacon will recollect the passage in the _Novum Organon_ (Lib. I. Aphorism 80) where he speaks with indignation of the way in which philosophy had been degraded and perverted, by being applied as a mere instrument of utility or of early education: "So that the great mother of the Sciences is thrust down with indignity to the offices of a handmaid;--is made to minister to the labours of medicine or mathematics; or again, to give the first preparatory tinge to the immature minds of youth[327]."

In the like spirit, Plato says (_Rep._ VI. § 11, Bekker's ed.):

"Observe how boldly and fearlessly I set about my explanation of my assertion that philosophers ought to rule the world. For I begin by saying, that the State must begin to treat the study of philosophy in a way opposite to that now practised. Now, those who meddle at all with this study are put upon it when they are children, between the lessons which they receive in the farm-yard and in the shop[328]; and as soon as they have been introduced to the hardest part of the subject, are taken off from it, even those who get the most of philosophy. By the hardest part, I mean, the discussion of principles--Dialectic[329]. And in their succeeding years, if they are willing to listen to a few lectures of those who make philosophy their business, they think they have done great things, as if it were something foreign to the business of life. And as they advance towards old age, with a very few exceptions, philosophy in them is extinguished: extinguished far more completely than the Heraclitean sun, for theirs is not lighted up again, as that is every morning:" alluding to the opinion which was propounded, by way of carrying the doctrine of the _unfixity_ of sensible objects to an extreme; that the Sun is extinguished every night and lighted again in the morning. In opposition to this practice, Plato holds that philosophy should be the especial employment of men's minds when their bodily strength fails.

What Plato means by _Dialectic_, which he, in the next Book, calls the highest part of philosophy, and which is, I think, what he here means by the hardest part of philosophy, I may hereafter consider: but at present I wish to pass in review the Sciences which he speaks of, as leading the way to that highest study. These Sciences are Arithmetic, Plane Geometry, Solid Geometry, Astronomy and Harmonics.

The view in which Plato here regards the Sciences is, as the instruments of that culture of the philosophical spirit which is to make the philosopher the fit and natural ruler of the perfect State--the Platonic Polity. It is held that to answer this purpose, the mind must be instructed in something more stable than the knowledge supplied by the senses;--a knowledge of objects which are constantly changing, and which therefore can be no real permanent Knowledge, but only Opinion. The real and permanent Knowledge which we thus require is to be found in certain sciences, which deal with _truths necessary and universal_, as we should now describe them: and which therefore are, in Plato's language, a knowledge of that which really _is_[330].

This is the object of the Sciences of which Plato speaks. And hence, when he introduces Arithmetic, as the first of the Sciences which are to be employed in this mental discipline, he adds (VII. § 8) that it must be not mere common Arithmetic, but a science which leads to speculative truths[331], seen by Intuition[332]; not an Arithmetic which is studied for the sake of buying and selling, as among tradesmen and shopkeepers, but for the sake of pure and real Science[333].

I shall not dwell upon the details with which he illustrates this view, but proceed to the other Sciences which he mentions.

Geometry is then spoken of, as obviously the next Science in order; and it is asserted that it really does answer the required condition of drawing the mind from visible, mutable phenomena to a permanent reality. Geometers indeed speak of their visible diagrams, as if their problems were certain practical processes; to erect a perpendicular; to construct a square: and the like. But this language, though necessary, is really absurd. The figures are mere aids to their reasonings. Their knowledge is really a knowledge not of visible objects, but of permanent realities: and thus, Geometry is one of the helps by which the mind may be drawn to Truth; by which the philosophical spirit may be formed, which looks upwards instead of downwards.

Astronomy is suggested as the Science next in order, but Socrates, the leader of the dialogue, remarks that there is an intermediate Science first to be considered. Geometry treats of plane figures; Astronomy treats of solids in motion, that is, of spheres in motion; for the astronomy of Plato's time was mainly the doctrine of the sphere. But before treating of solids in motion, we must have a science which treats of solids simply. After taking space of two dimensions, we must take space of three dimensions, length, breadth and depth, as in cubes and the like[334]. But such a Science, it is remarked, has not yet been discovered. Plato "notes as deficient" this branch of knowledge; to use the expression employed by Bacon on the like occasions in his Review. Plato goes on to say, that the cultivators of such a science have not received due encouragement; and that though scorned and starved by the public, and not recommended by any obvious utility, it has still made great progress, in virtue of its own attractiveness.

In fact, researches in Solid Geometry had been pursued with great zeal by Plato and his friends, and with remarkable success. The five Regular Solids, the Tetrahedron or Pyramid, Cube, Octahedron, Dodecahedron and Icosahedron, had been discovered; and the curious theorem, that of Regular Solids there can be just so many, these and no others, was known. The doctrine of these Solids was already applied in a way, fanciful and arbitrary, no doubt, but ingenious and lively, to the theory of the Universe. In the _Timæus_, the elements have these forms assigned to them respectively. Earth has the Cube: Fire has the Pyramid: Water has the Octahedron: Air has the Icosahedron: and the Dodecahedron is the plan of the Universe itself. This application of the doctrine of the Regular Solids shows that the knowledge of those figures was already established; and that Plato had a right to speak of Solid Geometry as a real and interesting Science. And that this subject was so recondite and profound,--that these five Regular Solids had so little application in the geometry which has a bearing on man's ordinary thoughts and actions,--made it all the more natural for Plato to suppose that these solids had a bearing on the constitution of the Universe; and we shall find that such a belief in later times found a ready acceptance in the minds of mathematicians who followed in the Platonic line of speculation.

Plato next proceeds to consider Astronomy; and here we have an amusing touch of philosophical drama. Glaucon, the hearer and pupil in the Dialogue, is desirous of showing that he has profited by what his instructor had said about the real uses of Science. He says Astronomy is a very good branch of education. It is such a very useful science for seamen and husbandmen and the like. Socrates says, with a smile, as we may suppose: "You are very amusing with your zeal for utility. I suppose you are afraid of being condemned by the good people of Athens for diffusing Useless Knowledge." A little afterwards Glaucon tries to do better, but still with no great success. He says, "You blamed me for praising Astronomy awkwardly: but now I will follow your lead. Astronomy is one of the sciences which you require, because it makes men's minds look upwards, and study things above. Any one can see that." "Well," says Socrates, "perhaps any one can see it except me--I cannot see it." Glaucon is surprised, but Socrates goes on: "Your notice of 'the study of things above' is certainly a very magnificent one. You seem to think that if a man bends his head back and looks at the ceiling he 'looks upwards' with his mind as well as his eyes. You may be right and I may be wrong: but I have no notion of any science which makes the _mind_ look upwards, except a science which is about the permanent and the invisible. It makes no difference, as to that matter, whether a man gapes and looks up or shuts his mouth and looks down. If a man merely look up and stare at sensible objects, his mind does not look upwards, even if he were to pursue his studies swimming on his back in the sea."

The Astronomy, then, which merely looks at phenomena does not satisfy Plato. He wants something more. What is it? as Glaucon very naturally asks.

Plato then describes Astronomy as a real science (§ 11). "The variegated adornments which appear in the sky, the visible luminaries, we must judge to be the most beautiful and the most perfect things of their kind: but since they are mere visible figures, we must suppose them to be far inferior to the true objects; namely, those spheres which, with their real proportions of quickness and slowness, their real number, their real figures, revolve and carry luminaries in their revolutions. These objects are to be apprehended by reason and mental conception, not by vision." And he then goes on to say that the varied figures which the skies present to the eye are to be used as _diagrams_ to assist the study of that higher truth; just as if any one were to study geometry by means of beautiful diagrams constructed by Dædalus or any other consummate artist.

Here then, Plato points to a kind of astronomical science which goes beyond the mere arrangement of phenomena: an astronomy which, it would seem, did not exist at the time when he wrote. It is natural to inquire, whether we can determine more precisely what kind of astronomical science he meant, and whether such science has been brought into existence since his time.

He gives us some further features of the philosophical astronomy which he requires. "As you do not expect to find in the most exquisite geometrical diagrams the true evidence of quantities being equal, or double, or in any other relation: so the true astronomer will not think that the proportion of the day to the month, or the month to the year, and the like, are real and immutable things. He will seek a deeper truth than these. We must treat Astronomy, like Geometry, as a series of problems suggested by visible things. We must apply the intelligent portion of our mind to the subject."

Here we really come in view of a class of problems which astronomical speculators at certain periods have proposed to themselves. What is the real ground of the proportion of the day to the month, and of the month to the year, I do not know that any writer of great name has tried to determine: but to ask the reason of these proportions, namely, that of the revolution of the earth on its axis, of the moon in its orbit, and of the earth in its orbit, are questions just of the same kind as to ask the reason of the proportion of the revolutions of the planets in their orbits, and of the proportion of the orbits themselves. Now who has attempted to assign such reasons?

Of course we shall answer, Kepler: not so much in the Laws of the Planetary motions which bear his name, as in the Law which at an earlier period he thought he had discovered, determining the proportion of the distances of the several Planets from the Sun. And, curiously enough, this solution of a problem which we may conceive Plato to have had in his mind, Kepler gave by means of the Five Regular Solids which Plato had brought into notice, and had employed in his theory of the Universe given in the _Timæus_.

Kepler's speculations on the subject just mentioned were given to the world in the _Mysterium Cosmographicum_ published in 1596. In his Preface, he says "In the beginning of the year 1595 I brooded with the whole energy of my mind on the subject of the Copernican system. There were three things in particular of which I pertinaciously sought the causes; why they are not other than they are: the number, the size, and the motion of the orbits." We see how strongly he had his mind impressed with the same thought which Plato had so confidently uttered: that there must be some reason for those proportions in the scheme of the Universe which appear casual and vague. He was confident at this period that he had solved two of the three questions which haunted him;--that he could account for the number and the size of the planetary orbits. His account was given in this way.--"The orbit of the Earth is a circle; round the sphere to which this circle belongs describe a dodecahedron; the sphere including this will give the orbit of Mars. Round Mars inscribe a tetrahedron; the circle including this will be the orbit of Jupiter. Describe a cube round Jupiter's orbit; the circle including this will be the orbit of Saturn. Now inscribe in the Earth's orbit an icosahedron: the circle inscribed in it will be the orbit of Venus. Inscribe an octahedron in the orbit of Venus; the circle inscribed in it will be Mercury's orbit. This is the reason of the number of the planets;" and also of the magnitudes of their orbits.

These proportions were only approximations; and the Rule thus asserted has been shown to be unfounded, by the discovery of new Planets. This Law of Kepler has been repudiated by succeeding Astronomers. So far, then, the Astronomy which Plato requires as a part of true philosophy has not been brought into being. But are we thence to conclude that the demand for such a kind of Astronomy was a mere Platonic imagination?--was a mistake which more recent and sounder views have corrected? We can hardly venture to say that. For the questions which Kepler thus asked, and which he answered by the assertion of this erroneous Law, are questions of exactly the same kind as those which he asked and answered by means of the true Laws which still fasten his name upon one of the epochs of astronomical history. If he was wrong in assigning reasons for the number and size of the planetary orbits, he was right in assigning a reason for the proportion of the motions. This he did in the _Harmonice Mundi_, published in 1619: where he established that the squares of the periodic times of the different Planets are as the cubes of their mean distances from the central Sun. Of this discovery he speaks with a natural exultation, which succeeding astronomers have thought well founded. He says: "What I prophesied two and twenty years ago as soon as I had discovered the five solids among the heavenly bodies; what I firmly believed before I had seen the _Harmonics_ of Ptolemy; what I promised my friends in the title of this book (_On the perfect Harmony of the celestial motions_), which I named before I was sure of my discovery; what sixteen years ago I regarded as a thing to be sought; that for which I joined Tycho Brahe, for which I settled in Prague, for which I devoted the best part of my life to astronomical contemplations; at length I have brought to light, and have recognized its truth beyond my most sanguine expectations." (_Harm. Mundi_, Lib. V.)

Thus the Platonic notion, of an Astronomy which deals with doctrines of a more exact and determinate kind than the obvious relations of phænomena, may be found to tend either to error or to truth. Such aspirations point equally to the five regular solids which Kepler imagined as determining the planetary orbits, and to the Laws of Kepler in which Newton detected the effect of universal gravitation. The realities which Plato looked for, as something incomparably more real than the visible luminaries, are found, when we find geometrical figures, epicycles and eccentrics, laws of motion and laws of force, which explain the appearances. His Realities are Theories which account for the Phenomena, Ideas which connect the Facts.

But, is Plato right in holding that such Realities as these are _more real_ than the Phenomena, and constitute an Astronomy of a higher kind than that of mere Appearances? To this we shall, of course, reply that Theories and Facts have each their reality, but that these are realities of different kinds. Kepler's Laws are as real as day and night; the force of gravity tending to the Sun is as real as the Sun; but not more so. True Theories and Facts are equally real, for true Theories _are_ Facts, and Facts are familiar Theories. Astronomy is, as Plato says, a series of Problems suggested by visible Things; and the Thoughts in our own minds which bring the solutions of these Problems, have a reality in the Things which suggest them.

But if we try, as Plato does, to separate and oppose to each other the Astronomy of Appearances and the Astronomy of Theories, we attempt that which is impossible. There are no Phenomena which do not exhibit some Law; no Law can be conceived without Phenomena. The heavens offer a series of Problems; but however many of these Problems we solve, there remain still innumerable of them unsolved; and these unsolved Problems have solutions, and are not different in kind from those of which the extant solution is most complete.

Nor can we justly distinguish, with Plato, Astronomy into transient appearances and permanent truths. The theories of Astronomy are permanent, and are manifested in a series of changes: but the change is perpetual just _because_ the theory is permanent. The perpetual change _is_ the permanent theory. The perpetual changes in the positions and movements of the planets, for instance, manifest the permanent machinery: the machinery of cycles and epicycles, as Plato would have said, and as Copernicus would have agreed; while Kepler, with a profound admiration for both, would have asserted that the motions might be represented by ellipses, more exactly, if not more truly. The cycles and epicycles, or the ellipses, are as real as space and time, _in_ which the motions take place. But we cannot justly say that space and time and motion are more real than the bodies which move in space and time, or than the appearances which these bodies present.

Thus Plato, with his tendency to exalt Ideas above Facts,--to find a Reality which is more real than Phenomena,--to take hold of a permanent Truth which is more true than truths of observation,--attempts what is impossible. He tries to separate the poles of the Fundamental Antithesis, which, however antithetical, are inseparable.

At the same time, we must recollect that this tendency to find a Reality which is something beyond appearance, a permanence which is involved in the changes, is the genuine spring of scientific discovery. Such a tendency has been the cause of all the astronomical science which we possess. It appeared in Plato himself, in Hipparchus, in Ptolemy, in Copernicus, and most eminently in Kepler; and in him perhaps in a manner more accordant with Plato's aspirations when he found the five Regular Solids in the Universe, than when he found there the Conic Sections which determine the form of the planetary orbits. The pursuit of this tendency has been the source of the mighty and successful labours of succeeding astronomers: and the anticipations of Plato on this head were more true than he himself could have conceived.

When the above view of the nature of true astronomy has been proposed, Glaucon says:

"That would be a task much more laborious than the astronomy now cultivated." Socrates replies: "I believe so: and such tasks must be undertaken, if our researches are to be good for anything."

After Astronomy, there comes under review another Science, which is treated in the same manner. It is presented as one of the Sciences which deal with real abstract truth; and which are therefore suited to that development of the philosophic insight into the highest truth, which is here Plato's main object. This Science is _Harmonics_, the doctrine of the mathematical relations of musical sounds. Perhaps it may be more difficult to explain to a general audience, Plato's views on this than on the previous subjects: for though Harmonics is still acknowledged as a Science including the mathematical truths to which Plato here refers, these truths are less generally known than those of geometry or astronomy. Pythagoras is reported to have been the discoverer of the cardinal proposition in this Mathematics of Music:--namely, that the musical notes which the ear recognizes as having that definite and harmonious relation which we call an _octave_, a _fifth_, a _fourth_, a _third_, have also, in some way or other, the numerical relation of 2 to 1, 3 to 2, 4 to 3, 5 to 4. I say "some way or other," because the statements of ancient writers on this subject are physically inexact, but are right in the essential point, that those simple numerical ratios are characteristic of the most marked harmonic relations. The numerical ratios really represent the rate of vibration of the air when those harmonics are produced. This perhaps Plato did not know: but he knew or assumed that those numerical ratios were cardinal truths in harmony: and he conceived that the exactness of the ratios rested on grounds deeper and more intellectual than any testimony which the ear could give. This is the main point in his mode of applying the subject, which will be best understood by translating (with some abridgement) what he says. Socrates proceeds:

(§ 11 near the end.) "Motion appears in many aspects. It would take a very wise man to enumerate them all: but there are two obvious kinds. One which appears in astronomy, (the revolutions of the heavenly bodies,) and another which is the echo of that[335]. As the eyes are made for Astronomy, so are the ears made for the motion which produces Harmony[336]: and thus we have two sister sciences, as the Pythagoreans teach, and we assent.

(§ 12.) "To avoid unnecessary labour, let us first learn what _they_ can tell us, and see whether anything is to be added to it; retaining our own view on such subjects: namely this:--that those whose education we are to superintend--real philosophers--are never to learn any imperfect truths:--anything which does not tend to that point (exact and permanent truth) to which all our knowledge ought to tend, as we said concerning astronomy. Now those who cultivate music take a very different course from this. You may see them taking immense pains in measuring musical notes and intervals by the ear, as the astronomers measure the heavenly motions by the eye.

"Yes, says Glaucon, they apply their ears close to the instrument, as if they could catch the note by getting near to it, and talk of some kind of recurrences[337]. Some say they can distinguish an interval, and that this is the smallest possible interval, by which others are to be measured; while others say that the two notes are identical: both parties alike judging by the ear, not by the intellect.

"You mean, says Socrates, those fine musicians who torture their notes, and screw their pegs, and pinch their strings, and speak of the resulting sounds in grand terms of art. We will leave them, and address our inquiries to our other teachers, the Pythagoreans."

The expressions about the small interval in Glaucon's speech appear to me to refer to a curious question, which we know was discussed among the Greek mathematicians. If we take a keyed instrument, and ascend from a key note by two _octaves_ and a _third_, (say from _A__{1} to _C__{3}) we arrive at the _same nominal note_, as if we ascend four times by a _fifth_ (_A__{1} to _E__{1}, _E__{1} to _B__{2}, _B__{2} to _F__{2}, _F__{2} to _C__{3}). Hence one party might call this the _same_ note. But if the Octaves, Fifths, and Third be perfectly true intervals, the notes arrived at in the two ways will not be really the same. (In the one case, the note is ½ × ½ × ⅘; in the other ⅔ × ⅔ × ⅔ × ⅔; which are ⅕ and 16/81, or in the ratio of 81 to 80). This small interval by which the two notes really differ, the Greeks called a _Comma_, and it was the smallest musical interval which they recognized. Plato disdains to see anything important in this controversy; though the controversy itself is really a curious proof of his doctrine, that there is a mathematical truth in Harmony, higher than instrumental exactness can reach. He goes on to say:

"The musical teachers are defective in the same way as the astronomical. They do indeed seek numbers in the harmonic notes, which the ear perceives: but they do not ascend from them to the Problem, What are harmonic numbers and what are not, and what is the reason of each[338]?" "That", says Glaucon, "would be a sublime inquiry."

Have we in Harmonics, as in Astronomy, anything in the succeeding History of the Science which illustrates the tendency of Plato's thoughts, and the value of such a tendency?

It is plain that the tendency was of the same nature as that which induced Kepler to call his work on Astronomy _Harmonice Mundi_; and which led to many of the speculations of that work, in which harmonical are mixed with geometrical doctrines. And if we are disposed to judge severely of such speculations, as too fanciful for sound philosophy, we may recollect that Newton himself seems to have been willing to find an analogy between harmonic numbers and the different coloured spaces in the spectrum.

But I will say frankly, that I do not believe there really exists any harmonical relation in either of these cases. Nor can the problem proposed by Plato be considered as having been solved since his time, any further than the recurrence of vibrations, when their ratios are so simple, may be easily conceived as affecting the ear in a peculiar manner. The imperfection of musical scales, which the _comma_ indicates, has not been removed; but we may say that, in the case of this problem, as in the other ultimate Platonic problems, the duplication of the cube and the quadrature of the circle, the impossibility of a solution has been already established. The problem of a perfect musical scale is impossible, because no power of 2 can be equal to a power of 3; and if we further take the multiplier 5, of course it also cannot bring about an exact equality. This impossibility of a perfect scale being recognized, the practical problem is what is the system of _temperament_ which will make the scale best suited for musical purposes; and this problem has been very fully discussed by modern writers.

FOOTNOTES:

[Footnote 327: Accedit et illud quod naturalis philosophia in iis ipsis viris, qui ei incubuerunt, vacantem et integrum hominem, præsertim his recentioribus temporibus, vix nacta sit; nisi forte quis monachi alicujus in cellula, aut nobilis in villula lucubrantis, exemplum adduxerit; sed facta est demum naturalis philosophia instar transitus cujusdam et pontisternii ad alia. Atque magna ista scientiarum mater ad officia ancillæ detrusa est; quæ medicinæ aut mathematicis operibus ministrat, et rursus quæ adolescentium immatura ingenia lavat et imbuat velut tinctura quadam prima, ut aliam postea felicius et commodius excipiant.]

[Footnote 328: μεταξὺ οἰκονομίας καὶ χρεματισμοῦ, between house-keeping and money-getting.]

[Footnote 329: τὸ περὶ τοὺς λόγους.]

[Footnote 330: The Sciences are to draw the mind from that which grows and perishes to that which really is: μάθημα ψυχῆς ὁλκὸν ἀπὸ τοῦ γιγνομένου ἐπι τὸ ὅν.]

[Footnote 331: ἐπὶ θέαν τῆς τῶν ἀριθμῶν φύσεως.]

[Footnote 332: τῇ νοηήσει αὐτῇ.]

[Footnote 333: He adds "and for the sake of war;" this point I have passed by. Plato does not really ascribe much weight to this use of Science, as we see in what he says of Geometry and Astronomy.]

[Footnote 334: ἀρθῶς ἕχει ἑξῆς μετὰ δευτέραν αὕξην τρίτην λαμβάνειν, ἕστι δέ που τοῦτο περὶ τὴν τῶν κύβων αύξην καὶ τὸ βάθους μέτεχον.]

[Footnote 335: ἀντίστροφον αὐτοῦ.]

[Footnote 336: πρὸς ἐναρμόνιον φορὰν ὦτα παγῆναι.]

[Footnote 337: πυκνώματα ἄ ττα.]

[Footnote 338: τίνες ξύμφωνοι ἀριθμοὶ, &c.]