The works of Plato · Plato
SOCRATES, TIMÆUS, CRITIAS, HERMOCRATES.
Footnote 314: ἐξ οὗ τὸ ἰσόπλευρον τρίγωνον ἐκ τρίτου συνέστηκε. The sense of the whole passage is explained geometrically in Stallbaum’s note. We here give only what seems the correct rendering of the Greek. Footnote 315: Namely, the _tetrahedron_ or pyramid, _octahedron_, and _icosahedron_, which are generated from six equal-sided triangles, and fourthly, the _cube_, generated from an equilateral triangle. See Stallbaum, _ad locum_. Footnote 316: Gr. γενόμενος εἷς ἀριθμὸς ἑνὸς ὄγκου μέγα ἀποτελέσειεν ἂν ἄλλο εἶδος ἕν, lit. _one number being formed will complete one separate and large kind of single mass_. ----- XXIX.—We must speak next in order concerning the quality of each class individually, and show from what composition of numbers each was formed. The first, then, indeed, is that composed from the smallest triangles, its element being that which has its subtending side twice the length of the shorter. Now, two of these triangles being mutually brought together to form a diagonal diameter, and this being thrice repeated, so that the diagonals and shorter sides shall meet in the same point, as in a centre, the result will be one equilateral triangle composed of six triangles. But four equilateral triangles form by the union of three plane angles one solid angle, the size of which exceeds that of the most obtuse plane angle; and thus by forming a figure that comprises four of these angles, we constitute that first species of solid, [the tetrahedron,] which divides into equal and similar parts the entire sphere in which it is inscribed. The second species of solid, [the octahedron,] is formed from the same triangles, which unite to form eight equilateral triangles, and form one solid angle from four plane angles, six solid angles being requisite to constitute this second class of solids. The third, [the icosahedron,] is formed from the union of one hundred and twenty elements, so united as to form twelve solid angles, each formed by five plane equilateral triangles, and it has for its bases (or faces) twenty equilateral triangles. These are the only solids that can be formed from this element, [viz. the scalene triangle]. It was the isosceles triangle, however, that produced the fourth elementary figure,—four of them being so united with their right angles at the centre, as to form one equilateral four-sided plane; and six of these again united forming eight solid angles, each of which is formed by the combination of three plane right angles;—the figure of the body thus composed being cubical, having six plane quadrangular equilateral bases. There was yet a certain fifth combination, [the regular dodecahedron;] and this the Deity employed in tracing the plan of the universe.[317]
XXX.—Should any one then, after careful consideration of all these points, be at a loss to determine whether the number of worlds is infinite or finite,[318] let him consider that to admit an infinite number thereof, would be the notion only of one who is ignorant of all that he should best know. Still it may with much propriety be questioned whether there is in reality only one world, or whether there are five. According to our opinion, indeed, which is founded on probable reasoning, there is but one world; though others perhaps, regarding the question in another point of view, may be of a different opinion. Let us now leave alone further speculations of this kind, and returning to the elementary forms that have now been created in our discussion, let us assign them respectively to fire, earth, water, and air. To earth, indeed, let us assign a cubical form; for of all the four kinds earth is the most immoveable, and is of all bodies the most easy to model,—such being necessarily the case with that which has the most secure bases.
Among the triangles, then, that we originally mentioned, that which has equal sides possesses firmer bases than one having unequal sides; and of the two equilateral plane figures thus formed, the square has necessarily a firmer base than the triangle, whether considered with reference to its parts or the whole. On this account, in assigning this figure to the earth, we still preserve probability;—and we shall do this also by assigning to water that figure which is the least moveable of the whole, to fire that which is most so, and to air that which is intermediate between the two,—assigning also the smallest body to fire, the greatest to water, and to air one of a size intermediate between fire and water;—and again, the most subtile body to fire, the next in this respect to air, and the third to water. Among all these then, that body which possesses the fewest bases must necessarily be the most easily moveable, as well as most penetrating, and in every way most acute, being also exceedingly light, from being composed of the smallest possible number of elements:—and so also the second has these properties in a secondary degree, and the third in the third degree. Let it be agreed then, that, according both to strict and probable reasoning, the solid form of the pyramid is the element and germ of fire; that the second form described is air; and the third water. All these elements then, we must conceive to be so minutely small, that the individual parts of each kind may from their smallness escape notice, and yet when many of them are collected together, they become from their bulk the objects of our perception.[319] Moreover, as respects their relative proportions, numbers, motions, and other properties, the Deity, so far as the nature of necessity has willingly and obediently given way, has firmly established and united them together in just proportion.[320]
Again, let us consider this matter as follows:—When any one of the other forms is enclosed by fire, and is cut by the acuteness of its angles and sides, it escapes further division by passing into the nature of fire:—for among bodies that are uniform and similar to each other, no one by itself can cause any change in one of the same class, or experience any itself, with respect to that which it resembles; whereas, when one class of bodies is contained within another, and the weak one contends with the stronger, it will not fail to be destroyed. And again, when the smaller, being comprehended in the greater, and the few by the many, are broken in pieces and extinguished, if they be disposed to adopt the form of the prevailing nature, they cease to be extinguished; and air becomes generated from fire,—water from air:—but if they attack and fall into contention with any of the rest that they may meet, the agitated parts continue to be dissolved, till being every way repulsed and dissolved, they fly to their kindred nature; or else, being overcome, and joined into one out of many similar to the prevailing power, they abide there in familiar union.
And especially, as the result of these affections, all things whatever are mutually changing their positions;—for the numerous individuals of each class are distinguished according to their respective places through the motion of their receptacle, while those that are mutually dissimilar, but yet resemble others, are hurried on by the concussion [of other bodies] to the place occupied by the bodies they resemble. All unmixed and primary bodies then are generated from such causes as these:—but that other classes of bodies are naturally inherent in these forms, is owing to the composition of each rudimental part (στοιχεῖον); which not only at first produces one triangle, possessed of a certain magnitude, but others also, both greater and smaller, equal in number to the various classes existing in the forms themselves;—and hence, these being mingled with themselves and each other, are infinite in their variety,—-a fact which every one should consider, who means to argue with probability respecting the nature of things.
XXXII.—Concerning motion, then, and position, unless a person can clearly understand in what manner and in conjunction with what these two take place, he will meet with many hindrances in the subsequent part of this discourse. This matter indeed we have already partly discussed; but besides this, we must still inquire whether it be true, that motion never willingly resides in smoothness, inasmuch as it is difficult, or rather impossible, to conceive the existence of anything moving without a mover, or of a mover without something in motion;—motion being impossible while these are away, and it being equally impossible that these should be equable and smooth. Thus, then, we must assign a state of rest to smoothness, and motion to that which is unequal and rough; inequality indeed being the cause again of a rough uneven nature. Now, as regards the generation of inequality, that we have already discussed; but how the several bodies, when divided according to their classes, do not cease from their mutual courses and motions, this we did not explain:—and so we will once more make it the subject of inquiry.
The revolution of the universe, since it comprehends the various classes of things, and is of circular shape and naturally desirous of uniting with itself, compresses all things together, and suffers no place to remain void. On this account is it, that fire most of all penetrates through everything, and air, in the next degree, being second to the former in subtility and tenuity;—and the rest in the same way [according to their degree:]—for such as are composed of very large parts have a very large vacuity in their composition,—those, on the contrary, that are very small, a very small vacuity. The union, then, resulting from compression drives together the small parts into the intervals of the larger; and thus, the small parts being placed side by side with the large, the smaller dividing the greater, while the latter compress the smaller, they are all borne upwards and downwards to the places respectively suitable to each;—for each, on changing its magnitude, changes its position likewise: and so, from these causes the production of diversity is constantly maintained, giving that perpetual motion to these bodies, which both now subsists and ever will continue. XXXIII.—In the next place, we must understand that there are many kinds of fire;—as for instance, flame,—that which emanates from flame,[321] which without burning furnishes light to the eyes,—and that which abides in ignited bodies, even after the flame has been extinguished. In like manner, with respect to air, one kind is of the greatest purity, that denominated _æther_,—a second most turbulent, cloudy, and dark; and besides these, there are other nameless kinds formed by the inequality of the triangles. As respects water, again, it admits of a twofold division, one kind being liquid, the other fusible.
The liquid kind, therefore, being composed of very small and unequal parts of water, becomes moveable, both of itself and by other bodies, through the inequality of its composition and peculiar shape of its figure; whereas the fusible kind, which is composed of large and smooth parts, is more stable than the former, as well as heavier and more compact, owing to its smoothness;[322]—and when fire enters into and dissolves its substance, it becomes more moveable, from losing its uniformity; and when it is thus rendered easily moveable, and is repelled by the surrounding air, and extended on the earth, it is said to be _liquefied_, by way of expressing the division of its masses, and is said to _flow_ also, in order to show its extension over the earth:—and these two words express both the changes which it undergoes. Again, when fire escapes from this body, it does not retire into a vacuum, but the surrounding air repelled [by the fire] drives the easily-moveable mass of moisture into the place before occupied by the fire, with which at the same time it becomes mingled;—but when the mass by close compression once more becomes equable and smooth, it then resumes its smoothness and consistency, because fire, the artificer of inequality, has taken its departure;—and this departure of fire we denominate _cooling_, while the union that takes place without the presence of fire is termed a _condensation_ [or hardening]. But among all those which we term fusible fluids, there is one that becomes most dense, though formed from the most subtile and equable parts, and is of an uniform character, and partaking of a lustrous yellow colour:—it is that most highly prized possession,—gold, which is produced by filtration through a rock. And a node (ὄζος) of gold, rendered by its density very hard and turned black, is called _adamant_.
But that which consists of parts greatly resembling gold, and has more than one species, which surpasses gold in density, and that it may become the harder, contains but a small and insignificant portion of earth, though at the same time it is lighter, owing to the great intervals between its internal atoms;—this is a separate kind of lustrous and dense fluids, termed _brass_. But an earthy nature is therewith mingled, which, when through age the two parts become separated soon becomes visible of itself, and is denominated _rust_. All other such matters also, it would be no hard task further to discuss by pursuing the plan of probable arguments;—and any one, who by way of recreation interrupts for a while his reasonings on eternal being, and enters into probable speculations about the generation of material things, will by these means acquire a pleasure of which he need not repent, and establish for himself a moderate and wise recreation during life. ----- Footnote 321: Gr. τό τε ἀπὸ τῆς φλογὸς ἀπιόν. The old editions read ἁπτόν. Footnote 322: The distinction between τὸ ὑγρόν and τὸ χυτόν is not easily conveyed in translation. The former means an elastic fluid like water,—the latter, a mass of greater density and less elasticity, such as jelly, syrup, or oil. We have rendered it _fused_, because in the next page the term is used respecting metals.
Sometimes also, even without losing its moisture, the earth is fused by the fire, and on cooling becomes a stone of a black colour.[326] In the same manner, when this earth is deprived of the water it holds in mixture, but yet has small particles and is of a saline nature, it forms a half-solid body, still capable of solution in water,—either nitre, which purifies both oil and earth, or else salt, a substance so well adapted to give flavours pleasing to the palate, and deemed by the law as agreeable to the gods. The compounds of these two substances are not soluble by water, but yet can be melted by fire, for the following reasons. Neither fire nor air liquefies masses of earth; because, being naturally composed of elements smaller than the interstices (or pores) of earth, they easily penetrate through these capacious pores without subjecting it either to dissolution or liquefaction. The parts of water, on the other hand, as they are larger, strive to force a passage, and so dissolve and liquefy the earthy mass:—and hence, when the earth has no strong consistency, water alone will dissolve it, whereas if in a compact state, nothing will affect it but fire, which is the only body that can find an entrance.
As for water, again, its strong cohesiveness of parts, [when frozen], can be overcome only by fire, whereas, when the cohesion is less strong, it can be equally decomposed by fire and air, the latter entering its interstices and the former separating even its constituent triangles. Nothing, however, can dissolve air strongly condensed, without attacking its composing elements; though when it coheres less strongly, it may be dissolved, though only by fire. Again, in bodies composed of water and earth, while the water occupies all the interstices in its state of compression, the particles of water from without, not finding egress, flow round the entire mass without suffering decomposition; whereas the particles of fire that enter into the interstices of water, as water into those of earth, and have the same effect on water that fire has on air, alone possess the power of dissolving and liquefying the compound body. And among these, some contain less water than earth; such as all kinds of glass, and such stones as are termed fusible; while others, on the contrary, contain more water, such as wax and aromatic substances. ----- Footnote 326: Lindau conjectures this to be _basalt_. It is probably lava, or some similar volcanic product. The same commentator conceives the λίτρον mentioned in the succeeding sentence to be _potash_ or some alkali suitable for mingling with oil to form soap.
First, then, let us inquire why fire is called _hot_, the reason of which we shall perceive by considering its penetrating and cutting power about our bodies. Now, that this affection is a certain sharpness[327] is nearly evident to all; and as regards the tenuity of its sides, the sharpness of its angles, the smallness of its particles, and the velocity of its motion, through all which it becomes violent, penetrating and capable of instantly dividing whatever it meets; this we should carefully consider, recalling to mind the generation of its figure;—inasmuch as it is this, and no other nature, which separates and parcels out our bodies into small portions, and produces in us that affection which is very properly denominated _heat_. Now the contrary to this is sufficiently manifest; but still we must not pass it without explanation:—for in truth, among the humid particles surrounding the body, those that have the largest elementary parts enter and drive out the smaller; but being unable to occupy their places, they compress our bodily humours, and from being uneven and in a state of agitation, fix them and render them motionless through their evenness and density;—whereas things brought into contact against nature are naturally opposed and mutually repel each other. From this contest and agitation then there resulted a certain trembling and numbness; and the whole of this affection, as well as the cause that produces it, has the name of cold. Again we call those bodies _hard_ to which our flesh yields; and those _soft_, which yield to the pressure of our flesh;—thus using the terms comparatively, with reference to each other. Bodies also yield to pressure when placed on a small base, while those resting on quadrangular bases are the least impressible, owing to their very firm position, and because from their own extreme density they strongly resist all opposing pressure.
But when all the parts are naturally so similar, how can any one with propriety assign to them contrary names?—For supposing there were any regular solid body in the centre of the universe, it would never be carried to any of the extremities, owing to their entire uniformity;—but on the other hand, if any one were to move in a circle round this solid, he would often stand with his feet opposite to where they before were, and so call the same part of himself both the upside and down. Since the universe, therefore, as we have just observed, is of a spherical figure, no prudent man should assert that it has any part which is [absolutely] either up or down;—yet, as regards the origin and usual application of these terms which we thus transfer to the universe, this we agree to be a proper subject of investigation. If, in that spot of the universe which specially belongs to fire, and where the principal mass is collected with which it has a tendency to unite, any one were to ascend thither and being possessed of the requisite power, were to take up the particles, place them in a balance, and raising the scale, were forcibly to draw the fire towards the air, with which it has no affinity, it is clear that in this case the smaller mass would be more easily impelled than the larger. Indeed, when two things are simultaneously raised by one and the same power, the smaller quantity must of course yield more readily than the greater to the impulsive force by which they are constrained; and hence, the one is called _heavy_ and gravitating downwards, the other _light_ and tending upwards. The same thing also is observable in what we ourselves do, who inhabit this place [the earth].
For when, in walking on the earth, we separate particles of an earthy nature, and sometimes even portions of the earth itself, forcibly and unnaturally drawing them towards the air which is dissimilar,—then that which is smaller yielding more readily to our impulse is sooner attracted towards the foreign element:—this then we call _light_ and the place towards which it is drawn, _up_ (or _above_), giving to their contraries the terms _heavy_ and _down_ (or _below_) respectively. Hence these must mutually differ from each other, owing to the contrary positions that the several elements occupy:—for that which is light in one place is contrary to one corresponding with it in an opposite position, and so also to a heavy body another that is heavy, and to bodies placed above or below others, opposed to them in their respective positions;—and they will all be found, whether in a state of becoming or actually existing, to be contrary, transverse, and every way differing from each other. This one thing, moreover, is to be understood concerning all these matters, that the tendency of each towards a body of similar nature gives to the body so attracted the name _heavy_, and the place to which it tends, _down_ (or _below_); and thus to contrary things are assigned contrary appellations. Such are the causes that we assign to these phenomena. And again, as to the cause of the impression of smoothness and roughness, every one who has investigated it will be able to disclose it to others; for roughness comes from hardness joined with unevenness, while smoothness is the united effect of uniformity and density.
XXXVIII.—It remains for us now to consider what is most important in those affections common to the whole body, which are the chief causes of pleasure and pain, and to inquire how it is that certain impressions excite through the parts of the body certain sensations invariably attended by pleasure and pain. Thus then let us examine all our impressions, whether sensible or not, calling to mind the distinctions that we before made between bodies moved with ease and those with difficulty;—for this is the way to arrive at the point that we wish to determine. When a body by nature easily moveable has received an impression ever so slight, the several parts communicate it to the parts placed around them, producing on these parts the same effect, until at length they reach the intellect itself, to which they announce the power of the agents producing such impression;—whereas a body, which on the contrary is firm and stable and has no circular motion, is simply affected by itself without moving any of the surrounding bodies; and hence, as their components do not mutually communicate the first received impression, the entire animal remains unmoved, and experiences no sensation.
This is the case indeed as respects the bones and hair, and such other parts of the body as are chiefly of an earthy nature; whereas the phenomena above described principally refer to the organs of sight and hearing, which contain an abundance of fire and air. This is what we should hold then concerning pleasure and pain:—an impression produced in us contrary to nature, and with violence, causes pain,—one that is conformable to our nature, however strong, pleasure;—whereas an impression that is gentle and gradual is unperceived, while the contrary to these produces contrary effects. An impression, again, the whole of which is easily produced, is pre-eminently an object of sensation, but is not affected by pleasure and pain:—and of this kind are the affections belonging to the sight; which indeed, as we have above asserted, is a body of a nature daily becoming allied to ourselves.[328] For in this way the impressions caused by cuttings and burnings, and other similar accidents, do not cause pain to the sight; nor again, does it experience pleasure, from returning to its previous condition:—whereas the sensations that are strongest and most clear do this, so far as any one is affected by any object; and this is the reason why there is no violence whatever either in its expansion or contraction.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.