(524) Is not their mode of operation on this wise—the sense which is concerned with the quality of hardness is necessarily concerned also with the quality of softness, and only intimates to the soul that the same thing is felt to be both hard and soft? You are quite right, he said. And must not the soul be perplexed at this intimation which the sense gives of a hard which is also soft? What, again, is the meaning of light and heavy, if that which is light is also heavy, and that which is heavy, light? Yes, he said, these intimations which the soul receives are very curious and require to be explained. Yes, I said, and in these perplexities the soul naturally summons to her aid calculation and intelligence, that she may see whether the several objects announced to her are one or two. True. And if they turn out to be two, is not each of them one and different? Certainly. And if each is one, and both are two, she will conceive the two as in a state of division, for if there were undivided they could only be conceived of as one? True. The eye certainly did see both small and great, but only in a confused manner; they were not distinguished. Yes. Whereas the thinking mind, intending to light up the chaos, was compelled to reverse the process, and look at small and great as separate and not confused. Very true. Was not this the beginning of the enquiry ‘What is great?’ and ‘What is small?’ Exactly so. And thus arose the distinction of the visible and the intelligible.

(71) Magnitude is divided into two parts--magnitude which is stationary and magnitude which is movable, the stationary pare having priority. Multitude is also divided into two parts, for it is related both to itself and to other things, the first relationship having priority. Pythagoras assigned the science of arithmetic to multitude related to itself, and the art of music to multitude related to other things. Geometry likewise was assigned to stationary magnitude, and spherics (used partly in the sense of astronomy) to movable magnitude. Both multitude and magnitude were circumscribed by the circumference of mind. The atomic theory has proved size to be the result of number, for a mass is made up of minute units though mistaken by the uninformed for a single simple substance.

(4) "Dialecticians of the day," replied the Spirit of the River, "all say that the infinitesimally small has no form, and that the infinitesimally great is beyond all measurement. Is that so?" "If we regard greatness as compared with that which is small," said the Spirit of the Ocean, "there is no limit to it; and if we regard smallness as compared with that which is great, it eludes our sight. The infinitesimal is a subdivision of the small; the colossal is an extension of the great. In this sense the two fall into different categories. "Both small and great things must equally possess form. The mind cannot picture to itself a thing without form, nor conceive a form of unlimited dimensions. The greatness of anything may be a topic of discussion, or the smallness of anything may be mentally realized. But that which can be neither a topic of discussion nor be realized mentally, can be neither great nor small.

(2) [Now,] seeing that the hollow roundness of the Cosmos is borne round into the fashion of a sphere; by reason of its [very] quality or form, it never can be altogether visible unto itself. So that, however high a place in it thou shouldest choose for looking down below, thou could’st not see from it what is at bottom, because in many places it confronts [the senses], and so is thought to have the quality [of being visible throughout]. For it is solely owing to the forms of species, with images of which it seems insculpted, that it is thought [to be] as though ’twere visible [throughout]; but as a fact ’tis ever to itself invisible.

(67) Timaeus: and that large motion produces “loud” sound, and motion of the opposite kind “soft” sound. The subject of concords of sounds must necessarily be treated in a later part of our exposition. We have still remaining a fourth kind of sensation, which we must divide up seeing that it embraces numerous varieties, which, as a whole, we call “colors.” This consists of a flame which issues from the several bodies, and possesses particles so proportioned to the visual stream as to produce sensation; and as regards the visual stream, we have already stated merely the causes which produced it.

(32) "Light," writes Edwin D. Babbitt, "reveals the glories of the external world and yet is the most glorious of them all. It gives beauty, reveals beauty and is itself most beautiful. It is the analyzer, the truth-teller and the exposer of shams, for it shows things as they are. Its infinite streams measure off the universe and flow into our telescopes from stars which are quintillions of miles distant. On the other hand it descends to objects inconceivably small, and reveals through the microscope objects fifty millions of times less than can be seen by the naked eye. Like all other fine forces, its movement is wonderfully soft, yet penetrating and powerful. Without its vivifying influence, vegetable, animal, and human life must immediately perish from the earth, and general ruin take place. We shall do well, then, to consider this potential and beautiful principle of light and its component colors, for the more deeply we penetrate into its inner laws, the more will it present itself as a marvelous storehouse of power to vitalize, heal, refine, and delight mankind." (See The Principles of Light and Color.)

(523) do not pass from one sensation to the opposite; inviting objects are those which do; in this latter case the sense coming upon the object, whether at a distance or near, gives no more vivid idea of anything in particular than of its opposite. An illustration will make my meaning clearer:—here are three fingers—a little finger, a second finger, and a middle finger. Very good. You may suppose that they are seen quite close: And here comes the point. What is it? Each of them equally appears a finger, whether seen in the middle or at the extremity, whether white or black, or thick or thin—it makes no difference; a finger is a finger all the same. In these cases a man is not compelled to ask of thought the question what is a finger? for the sight never intimates to the mind that a finger is other than a finger. True. And therefore, I said, as we might expect, there is nothing here which invites or excites intelligence. There is not, he said. But is this equally true of the greatness and smallness of the fingers? Can sight adequately perceive them? and is no difference made by the circumstance that one of the fingers is in the middle and another at the extremity? And in like manner does the touch adequately perceive the qualities of thickness or thinness, of softness or hardness? And so of the other senses; do they give perfect intimations of such matters?

(510) sphere of the visible consists of images. And by images I mean, in the first place, shadows, and in the second place, reflections in water and in solid, smooth and polished bodies and the like: Do you understand? Yes, I understand. Imagine, now, the other section, of which this is only the resemblance, to include the animals which we see, and everything that grows or is made. Very good. Would you not admit that both the sections of this division have different degrees of truth, and that the copy is to the original as the sphere of opinion is to the sphere of knowledge? Most undoubtedly. Next proceed to consider the manner in which the sphere of the intellectual is to be divided. In what manner? Thus:—There are two subdivisions, in the lower of which the soul uses the figures given by the former division as images; the enquiry can only be hypothetical, and instead of going upwards to a principle descends to the other end; in the higher of the two, the soul passes out of hypotheses, and goes up to a principle which is above hypotheses, making no use of images 14 as in the former case, but proceeding only in and through the ideas themselves. I do not quite understand your meaning, he said. Then I will try again; you will understand me better when I have made some preliminary remarks. You are aware that students of geometry, arithmetic, and the kindred sciences assume the odd and the even and the figures and three kinds of angles and the like in their several branches of science; these are their hypotheses, which they and every body are supposed to know, and therefore they do not deign to give any account of them either to themselves or others;

(35) of the sense perceptions. Not only is there a great deal more to light than anyone has ever seen but there are also unknown forms of light which no optical equipment will ever register. There are unnumbered colors which cannot be seen, as well as sounds which cannot be heard, odors which cannot be smelt, flavors which cannot be tasted, and substances which cannot be felt. Man is thus surrounded by a supersensible universe of which he knows nothing because the centers of sense perception within himself have not been developed sufficiently to respond to the subtler rates of vibration of which that universe is composed.

(21) But how deep or how large the place of this world is, no man knoweth, though some physicists or astrologers have undertaken to measure the deep with their measures of circles; their measuring is but conjectural, or a measuring of somewhat that is comprehensible or palpable; as if a man would grasp the wind in his fist.

(507) I do indeed wish, I replied, that I could pay, and you receive, the account of the parent, and not, as now, of the offspring only; take, however, this latter by way of interest 11 , and at the same time have a care that I do not render a false account, although I have no intention of deceiving you. Yes, we will take all the care that we can: proceed. Yes, I said, but I must first come to an understanding with you, and remind you of what I have mentioned in the course of this discussion, and at many other times. What? The old story, that there is a many beautiful and a many good, and so of other things which we describe and define; to all of them the term ‘many’ is applied. True, he said. And there is an absolute beauty and an absolute good, and of other things to which the term ‘many’ is applied there is an absolute; for they may be brought under a single idea, which is called the essence of each. Very true. The many, as we say, are seen but not known, and the ideas are known but not seen. Exactly. And what is the organ with which we see the visible things? The sight, he said. And with the hearing, I said, we hear, and with the other senses perceive the other objects of sense? True. But have you remarked that sight is by far the most costly and complex piece of workmanship which the artificer of the senses ever contrived? No, I never have, he said. Then reflect; has the ear or voice need of any third or

(7) Scientists have offered the illustration of a rapidly moving wheel, top, or cylinder, to show the effects of increasing rates of vibration. The illustration supposes a wheel, top, or revolving cylinder, running at a low rate of speed--we will call this revolving thing "the object" in following out the illustration. Let us suppose the object moving slowly. It may be seen readily, but no sound of its movement reaches the ear. The speed is gradually increased. In a few moments its movement becomes so rapid that a deep growl or low note may be heard. Then as the rate is increased the note rises one in the musical scale. Then, the motion being still further increased, the next highest note is distinguished. Then, one after another, all the notes of the musical scale appear, rising higher and higher as the motion is increased. Finally when the motions have reached a certain rate the final note perceptible to human ears is reached and the shrill, piercing shriek dies away, and silence follows. No sound is heard from the revolving object, the rate of motion being so high that the human ear cannot register the vibrations. Then comes the perception of rising degrees of Heat. Then after quite a time the eye catches a glimpse of the object becoming a dull dark reddish color. As the rate increases, the red becomes brighter. Then as the speed is increased, the red melts into an orange. Then the orange melts into a yellow. Then follow, successively, the shades of green, blue, indigo, and finally violet, as the rate of sped increases. Then the violet shades away, and all color disappears, the human eye not being able to register them. But there are invisible rays emanating from the revolving object, the rays that are used in photographing, and other subtle rays of light. Then begin to manifest the peculiar rays known as the "X Rays," etc., as the constitution of the object changes. Electricity and Magnetism are emitted when the appropriate rate of vibration is attained.

(18) Things seen have as their property manifestation, action, inertia. They form the basis of the elements and the sense-powers. They make for experience and for liberation.

(43) When the object dwells in the mind, clear of memory-pictures, uncoloured by the mind, as a pure luminous idea, this is perception without exterior or consideration.

(15) Presiding over the ear and the eye, the organs of touch, taste, and smell, and also over the mind, he experiences sense-objects.

(4) Besides, if rarity were of this dimness The cause thou askest, either through and through This planet thus attenuate were of matter, Or else, as in a body is apportioned The fat and lean, so in like manner this Would in its volume interchange the leaves. Were it the former, in the sun's eclipse It would be manifest by the shining through Of light, as through aught tenuous interfused. This is not so; hence we must scan the other, And if it chance the other I demolish, Then falsified will thy opinion be. But if this rarity go not through and through, There needs must be a limit, beyond which Its contrary prevents the further passing, And thence the foreign radiance is reflected, Even as a colour cometh back from glass, The which behind itself concealeth lead. Now thou wilt say the sunbeam shows itself More dimly there than in the other parts, By being there reflected farther back. From this reply experiment will free thee If e'er thou try it, which is wont to be The fountain to the rivers of your arts.

(1) Since, however, we are narrating the wisdom employed by Pythagoras in instructing his disciples, it will not be unappropriate to relate that which is proximate in a following order to this, viz. how he invented the harmonic science, and harmonic ratios. But for this purpose we must begin a little higher. Intently considering once, and reasoning with himself, whether it would be possible to devise a certain instrumental assistance to the hearing, which should be firm and unerring, such as the sight obtains through the compass and the rule, or, by Jupiter, through a dioptric instrument; or such as the touch obtains through the balance, or the contrivance of measures;—thus considering, as he was walking near a brazier’s shop, he heard from a certain divine casualty the hammers beating out a piece of iron on an anvil, and producing sounds that accorded with each other, one combination only excepted.

But he recognized in those sounds, the diapason, the diapente, and the diatessaron, harmony. He saw, however, that the sound which was between the diatessaron and the diapente was itself by itself dissonant, yet, nevertheless, gave completion to that which was the greater sound among them. Being delighted, therefore, to find that the thing which he was anxious to discover had succeeded to his wishes by divine assistance, he went into the brazier’s shop, and found by various experiments, that the difference of sound arose from the magnitude of the hammers, but not from the force of the strokes, nor from the figure of the hammers, nor from the transposition of the iron which was beaten. When, therefore, he had accurately examined the weights and the equal counterpoise of the hammers, he returned home, and fixed one stake diagonally to the walls, lest if there were many, a certain difference should arise from this circumstance, or in short, lest the peculiar nature of each of the stakes should cause a suspicion of mutation.

Afterwards, from this stake he suspended four chords consisting of the same materials, and of the same magnitude and thickness, and likewise equally twisted. To the extremity of each chord also he tied a weight. And when he had so contrived, that the chords were perfectly equal to each other in length, he afterwards alternately struck two chords at once, and found the before-mentioned symphonies, viz. a different symphony in a different combination. For he discovered that the chord which was stretched by the greatest weight, produced, when compared with that which was stretched by the smallest, the symphony diapason. But the former of these weights was twelve pounds, and the latter six. And, therefore, being in a duple ratio, it exhibited the consonance diapason; which the weights themselves rendered apparent.

But again, he found that the chord from which the greatest weight was suspended compared with that from which the weight next to the smallest depended, and which weight was eight pounds, produced the symphony diapente. Hence he discovered that this symphony is in a sesquialter ratio, in which ratio also the weights were to each other. And he found that the chord which was stretched by the greatest weight, produced, when compared with that which was next to it in weight, and was nine pounds, the symphony diatessaron, analogously to the weights. This ratio, therefore, he discovered to be sesquitertian; but that of the chord from which a weight of nine pounds was suspended, to the chord which had the smallest weight [or six pounds,] to be sesquialter.

For 9 is to 6 in a sesquialter ratio. In like manner, the chord next to that from which the smallest weight depended, was to that which had the smallest weight, in a sesquitertian ratio, [for it was the ratio of 8 to 6,] but to the chord which had the greatest weight, in a sesquialter ratio [for such is the ratio of 12 to 8.] Hence, that which is between the diapente and the diatessaron, and by which the diapente exceeds the diatessaron, is proved to be in an epogdoan ratio, or that of 9 to 8. But either way it may be proved that the diapason is a system consisting of the diapente in conjunction with the diatessaron, just as the duple ratio consists of the sesquialter and sesquitertian, as for instance, 12, 8, and 6; or conversely, of the diatessaron and the diapente, as in the duple ratio of the sesquitertian and sesquialter ratios, as for instance 12, 9, and 6.

After this manner, therefore, and in this order, having conformed both his hand and his hearing to the suspended weights, and having established according to them the ratio of the habitudes, he transferred by an easy artifice the common suspension of the chords from the diagonal stake to the limen of the instrument, which he called chordotonon . But he produced by the aid of pegs a tension of the chords analogous to that effected by the weights.

(530) them are obvious enough even to wits no better than ours; and there are others, as I imagine, which may be left to wiser persons. But where are the two? There is a second, I said, which is the counterpart of the one already named. And what may that be? The second, I said, would seem relatively to the ears to be what the first is to the eyes; for I conceive that as the eyes are designed to look up at the stars, so are the ears to hear harmonious motions; and these are sister sciences—as the Pythagoreans say, and we, Glaucon, agree with them? Yes, he replied. But this, I said, is a laborious study, and therefore we had better go and learn of them; and they will tell us whether there are any other applications of these sciences. At the same time, we must not lose sight of our own higher object. What is that? There is a perfection which all knowledge ought to reach, and which our pupils ought also to attain, and not to fall short of, as I was saying that they did in astronomy. For in the science of harmony, as you probably know, the same thing happens. The teachers of harmony compare the sounds and consonances which are heard only, and their labour, like that of the astronomers, is in vain. Yes, by heaven! he said; and ’tis as good as a play to hear them talking about their condensed notes, as they call them; they put their ears close alongside of the strings like persons catching a sound from their neighbour’s wall 5 —one set of them declaring that they distinguish an intermediate note and have found the least interval which should be the unit of measurement; the others insisting that the two sounds have passed into the same—either party setting

(510) but they begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion? Yes, he said, I know. And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on—the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind? That is true. And of this kind I spoke as the intelligible, although in the search after it the soul is compelled to use hypotheses; not ascending to a first principle, because she is unable to rise above the region of hypothesis, but employing the objects of which the shadows below are resemblances in their turn as images, they having in relation to the shadows and reflections of them a greater distinctness, and therefore a higher value.

(17) An object is perceived, or not perceived, according as the mind is, or is not, tinged with the colour of the object.