(528) solids in revolution, instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed. That is true, Socrates; but so little seems to be known as yet about these subjects. Why, yes, I said, and for two reasons:—in the first place, no government patronises them; this leads to a want of energy in the pursuit of them, and they are difficult; in the second place, students cannot learn them unless they have a director. But then a director can hardly be found, and even if he could, as matters now stand, the students, who are very conceited, would not attend to him. That, however, would be otherwise if the whole State became the director of these studies and gave honour to them; then disciples would want to come, and there would be continuous and earnest search, and discoveries would be made; since even now, disregarded as they are by the world, and maimed of their fair proportions, and although none of their votaries can tell the use of them, still these studies force their way by their natural charm, and very likely, if they had the help of the State, they would some day emerge into light. Yes, he said, there is a remarkable charm in them. But I do not clearly understand the change in the order. First you began with a geometry of plane surfaces? Yes, I said. And you placed astronomy next, and then you made a step backward? Yes, and I have delayed you by my hurry; the ludicrous state of solid geometry, which, in natural order, should have followed, made me pass over this branch and go on to

(32) Timaeus: and out of these materials, such in kind and four in number, the body of the Cosmos was harmonized by proportion and brought into existence. These conditions secured for it Amity, so that being united in identity with itself it became indissoluble by any agent other than Him who had bound it together. Now of the four elements the construction of the Cosmos had taken up the whole of every one. For its Constructor had constructed it of all the fire and water and air and earth that existed, leaving over, outside it, no single particle or potency of any one of these elements. And these were his intentions:

(56) Timaeus: since it is in all ways the sharpest and most acute of all; and it must also be the lightest, since it is composed of the fewest identical parts; and the second comes second in point of these same qualities, and the third third. Thus, in accordance with the right account and the probable, that solid which has taken the form of a pyramid shall be the element and seed of fire; the second in order of generation we shall affirm to be air, and the third water. Now one must conceive all these to be so small that none of them,

(53) Timaeus: of each of these Kinds which I must endeavor to explain to you in an exposition of an unusual type; yet, inasmuch as you have some acquaintance with the technical method which I must necessarily employ in my exposition, you will follow me. In the first place, then, it is plain I presume to everyone that fire and earth and water and air are solid bodies; and the form of a body, in every case, possesses depth also. Further, it is absolutely necessary that depth should be bounded by a plane surface; and the rectilinear plane is composed of triangles.

(55) Timaeus: And the third solid is composed of twice sixty of the elemental triangles conjoined, and of twelve solid angles, each contained by five plane equilateral triangles, and it has, by its production, twenty equilateral triangular bases. Now the first of the elemental triangles ceased acting when it had generated these three solids, the substance of the fourth Kind being generated by the isosceles triangle. Four of these combined, with their right angles drawn together to the center, produced one equilateral quadrangle; and six such quadrangles,

(54) Timaeus: the equilateral triangle is constructed as a third. The reason why is a longer story; but should anyone refute us and discover that it is not so, we begrudge him not the prize. Accordingly, let these two triangles be selected as those wherefrom are contrived the bodies of fire and of the other elements,— one being the isosceles, and the other that which always has the square on its greater side three times the square on the lesser side. Moreover, a point about which our previous statement was obscure must now be defined more clearly. It appeared as if the four Kinds,

(13) Pythagoras saith:—We posit another government which is not from another root, but it differs in name. And know, all ye seekers after this Science and ‘Wisdom, that whatsoever the envious may have enjoined in their books concerning the’ composition of natures which agree together,* in savour there is only one, albeit to sight they are as diverse as possible. Know, also, that the thing which they have described in so many ways follows and attains its companion without fire, even as the magnet follows the iron, to which the said thing is not vainly compared, nor to a seed, nor to a matrix, for it is also like unto these. And this same thing, which followst its companion without fire, causes many colours to appear when embracing it, for this reason, that the said one thing enters into every regimen, and is found everywhere, being a stone, and also not a stone; common ”* and precious; hidden and concealed, yet known by everyone; of one name and of many names, which is the Spume of the Moon. This stone, therefore, is not a stone, because it is more precious; without it Nature never operates anything; its name is one, yet we have called it by many names on account of the excellence of its nature.*

The Turba answereth:—O! Master! wilt thou not mention some of those names for the guidance of seekers?

And he:—It is called White Ethelia, White Copper, and that which flies from the fire and alone whitens copper. Break up, therefore, the White Stone, and afterwards coagulate it with milk.* Then pound the calx in the mortar, taking care that the humidity does not escape from the vessel; but coagulate it in the vessel until it shall become a cinder. Cook also with Spume of Luna and regulate. For ye shall find the stone broken, and already imbued with its own water.t This, therefore, is the stone which we call by all names, which assimilates the work and drinks it, and is the stone out of which also all colours appear. Take, therefore, that same gum,} which is from the scoriz, and mix with cinder of calx, which you have ruled, and with the feeces which you know, moistening with permanent water. Then look and see whether it has become a powder, but if not, roast in a fire stronger than the first fire, until it be pounded. Then imbue with permanent water, and the more the colours vary all the more suffer them to be heated. Know, moreover, that if you take white quicksilver, or the Spume of Luna, and do as ye are bidden, breaking up with a gentle fire, the same is coagulated, and becomes a stone. Out of this stone, therefore, when it is broken up, many colours will appear to you. But herein, if any ambiguity occur to you in our discourse, do as ye are bidden, ruling the same until: a white and coruscating stone shall be produced, and so ye find your purpose.

(32) Timaeus: had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow-terms; but now it is otherwise: for it behoved it to be solid of shape, and what brings solids into unison is never one middle term alone but always two. Thus it was that in the midst between fire and earth God set water and air, and having bestowed upon them so far as possible a like ratio one towards another—air being to water as fire to air, and water being to earth as air to water, —he joined together and constructed a Heaven visible and tangible. For these reasons

(6) We may now consider the question whether fire is the sole element existing in that celestial realm and whether there is any outgoing thence with the consequent need of renewal.

Timaeus pronounced the material frame of the All to consist primarily of earth and fire for visibility, earth for solidity- and deduced that the stars must be mainly composed of fire, but not solely since there is no doubt they are solid.

And this is probably a true account. Plato accepts it as indicated by all the appearances. And, in fact, to all our perception- as we see them and derive from them the impression of illumination- the stars appear to be mostly, if not exclusively, fire: but on reasoning into the matter we judge that since solidity cannot exist apart from earth-matter, they must contain earth as well.

But what place could there be for the other elements? It is impossible to imagine water amid so vast a conflagration; and if air were present it would be continually changing into fire.

Admitting that two self-contained entities, standing as extremes to each other need for their coherence two intermediaries; we may still question whether this holds good with regard to physical bodies. Certainly water and earth can be mixed without any such intermediate. It might seem valid to object that the intermediates are already present in the earth and the water; but a possible answer would be, "Yes, but not as agents whose meeting is necessary to the coherence of those extremes."

None the less we will take it that the coherence of extremes is produced by virtue of each possessing all the intermediates. It is still not proven that fire is necessary to the visibility of earth and earth to the solidarity of fire.

On this principle, nothing possesses an essential-nature of its very own; every several thing is a blend, and its name is merely an indication of the dominant constituent.

Thus we are told that earth cannot have concrete existence without the help of some moist element- the moisture in water being the necessary adhesive- but admitting that we so find it, there is still a contradiction in pretending that any one element has a being of its own and in the same breath denying its self-coherence, making its subsistence depend upon others, and so, in reality, reducing the specific element to nothing. How can we talk of the existence of the definite Kind, earth- earth essential- if there exists no single particle of earth which actually is earth without any need of water to secure its self-cohesion? What has such an adhesive to act upon if there is absolutely no given magnitude of real earth to which it may bind particle after particle in its business of producing the continuous mass? If there is any such given magnitude, large or small, of pure earth, then earth can exist in its own nature, independently of water: if there is no such primary particle of pure earth, then there is nothing whatever for the water to bind. As for air- air unchanged, retaining its distinctive quality- how could it conduce to the subsistence of a dense material like earth?

Similarly with fire. No doubt Timaeus speaks of it as necessary not to the existence but to the visibility of earth and the other elements; and certainly light is essential to all visibility- we cannot say that we see darkness, which implies, precisely, that nothing is seen, as silence means nothing being heard.

But all this does not assure us that the earth to be visible must contain fire: light is sufficient: snow, for example, and other extremely cold substances gleam without the presence of fire- though of course it might be said that fire was once there and communicated colour before disappearing.

As to the composition of water, we must leave it an open question whether there can be such a thing as water without a certain proportion of earth.

But how can air, the yielding element, contain earth?

Fire, again: is earth perhaps necessary there since fire is by its own nature devoid of continuity and not a thing of three dimensions?

Supposing it does not possess the solidity of the three dimensions, it has that of its thrust; now, cannot this belong to it by the mere right and fact of its being one of the corporeal entities in nature? Hardness is another matter, a property confined to earth-stuff. Remember that gold- which is water- becomes dense by the accession not of earth but of denseness or consolidation: in the same way fire, with Soul present within it, may consolidate itself upon the power of the Soul; and there are living beings of fire among the Celestials.

But, in sum, do we abandon the teaching that all the elements enter into the composition of every living thing?

For this sphere, no; but to lift clay into the heavens is against nature, contrary to the laws of her ordaining: it is difficult, too, to think of that swiftest of circuits bearing along earthly bodies in its course nor could such material conduce to the splendour and white glint of the celestial fire.

(8) Wherefore the wisest of the Egyptian priests decided that the temple of Athene should be hypaethral, just as the Hebrews constructed the temple without an image. And some, in worshipping God, make a representation of heaven containing the stars; and so worship, although Scripture says, "Let of Eurysus the Pythagorean, which is as follows, who in his book On Fortune, having said that the "Creator, on making man, took Himself as an exemplar," added, "And the body is like the other things, as being made of the same material, and fashioned by the best workman, who wrought it, taking Himself as the archetype." And, in fine, Pythagoras and his followers, with Plato also, and most of the other philosophers, were best acquainted with the Lawgiver, as may be concluded from their doctrine. And by a happy utterance of divination, not without divine help, concurring in certain prophetic declarations, and, seizing the truth in portions and aspects, in terms not obscure, and not going beyond the explanation of the things, they honoured it on as pertaining the appearance of relation with the truth. Whence the Hellenic philosophy is like the torch of wick which men kindle, artificially stealing the light from the sun. But on the proclamation of the Word all that holy light shone forth. Then in houses by night the stolen light is useful; but by day the fire blazes, and all the night is illuminated by such a sun of intellectual light.

(55) Timaeus: when joined together, formed eight solid angles, each composed of three plane right angles; and the shape of the body thus constructed was cubic, having six plane equilateral quadrangular bases. And seeing that there still remained one other compound figure, the fifth, God used it up for the Universe in his decoration thereof. Now in reasoning about all these things, a man might question whether he ought to affirm the existence of an infinite diversity of Universes or a limited number; and if he questioned aright he would conclude that the doctrine of an infinite diversity is that of a man unversed

(33) Timaeus: And he bestowed on it the shape which was befitting and akin. Now for that Living Creature which is designed to embrace within itself all living creatures the fitting shape will be that which comprises within itself all the shapes there are; wherefore He wrought it into a round, in the shape of a sphere, equidistant in all directions from the center to the extremities, which of all shapes is the most perfect and the most self-similar, since He deemed that the similar is infinitely fairer than the dissimilar. And on the outside round about, it was all made smooth with great exactness, and that for many reasons.

(5) Such, then, is the style of the example in arithmetic. And let the testimony of geometry be the tabernacle that was constructed, and the ark that was fashioned, - constructed in most regular proportions, and through divine ideas, by the gift of understanding, which leads us from things of sense to intellectual objects, or rather from these to holy things, and to the holy of holies. For the squares of wood indicate that the square form, producing fight angles, pervades all, and points out security. And the length of the structure was three hundred cubits, and the breadth fifty, and the height thirty; and above, the ark ends in a cubit, narrowing to a cubit from the broad base like a pyramid, the symbol of those who are purified and tested by fire. And this geometrical proportion has a place, for the transport of those holy abodes, whose differences are indicated by the differences of the numbers set down below.

(54) Timaeus: in being generated, all passed through one another into one another, but this appearance was deceptive. For out of the triangles which we have selected four Kinds are generated, three of them out of that one triangle which has its sides unequal, and the fourth Kind alone composed of the isosceles triangle. Consequently, they are not all capable of being dissolved into one another so as to form a few large bodies composed of many small ones, or the converse; but three of them do admit of this process. For these three are all naturally compounded of one triangle, so that when the larger bodies are dissolved many small ones will form themselves from these same bodies, receiving the shapes that befit them;

(1) Of his wisdom, however, the commentaries written by the Pythagoreans afford, in short, the greatest indication; for they adhere to truth in every thing, and are more concise than all other compositions, so that they savour of the ancient elegance of style, and the conclusions are exquisitely deduced with divine science. They are also replete with the most condensed conceptions, and are in other respects various and diversified both in the form and the matter. At one and the same time likewise, they are transcendently excellent, and without any deficiency in the diction, and are in an eminent degree full of clear and indubitable arguments, accompanied with scientific demonstration, and as it is said, the most perfect syllogism; as he will find to be the case, who, proceeding in such paths as are fit, does not negligently peruse them.

This science, therefore, concerning intelligible natures and the Gods, Pythagoras delivers in his writings from a supernal origin. Afterwards, he teaches the whole of physics, and unfolds completely ethical philosophy and logic. He likewise delivers all-various disciplines, and the most excellent sciences. And in short there is nothing pertaining to human knowledge which is not accurately discussed in these writings. If therefore it is acknowledged, that of the [Pythagoric] writings which are now in circulation, some were written by Pythagoras himself, but others consist of what he was heard to say, and on this account are anonymous, but are referred to Pythagoras as their author;—if this be the case, it is evident that he was abundantly skilled in all wisdom.

But it is said that he very much applied himself to geometry among the Egyptians. For with the Egyptians there are many geometrical problems; since it is necessary that from remote periods, and from the time of the Gods themselves, on account of the increments and decrements of the Nile, those that were skilful should have measured all the Egyptian land which they cultivated. Hence also geometry derived its name. Neither did they negligently investigate the theory of the celestial orbs, in which likewise Pythagoras was skilled. Moreover, all the theorems about lines appear to have been derived from thence. For it is said that what pertains to computation and numbers, was discovered in Phœnicia. For some persons refer the theorems about the celestial bodies to the Egyptians and Chaldeans in common.

It is said therefore, that Pythagoras having received and increased all these [theories,] imparted the sciences, and at the same time demonstrated them to his auditors with perspicuity and elegance. And he was the first indeed that denominated philosophy, and said that it was the desire, and as it were love of wisdom. But he defined wisdom to be the science of the truth which is in beings. And he said that beings are immaterial and eternal natures, and alone possess an efficacious power, such as incorporeal essences. But that the rest of things are only homonymously beings, and are so denominated through the participation of real beings, and such are corporeal and material forms, which are generated and corrupted, and never truly are.

And that wisdom is the science of things which are properly beings, but not of such as are homonymously so. For corporeal natures are neither the objects of science nor admit of a stable knowledge, since they are infinite and incomprehensible by science, and are as it were, non-beings, when compared with universals, and are incapable of being properly circumscribed by definition. It is impossible however to conceive that there should be science of things which are not naturally the objects of science. Hence it is not probable that there will be a desire of science which has no subsistence, but rather that desire will be extended to things which are properly beings, which exist with invariable permanency, and are always consubsistent with a true appellation.

For it happens that the perception of things which are homonymously beings, and which are never truly what they seem to be, follows the apprehension of real beings; just as the knowledge of particulars follows the science of universals. For he who knows universals properly, says Archytas, will also have a clear perception of the nature of particulars. Hence things which have an existence are not alone, nor only-begotten, nor simple, but they are seen to be various and multiform. For some of them are intelligible and incorporeal natures, and which are denominated beings; but others are corporeal and fall under the perception of sense, and by participation communicate with that which has a real existence. Concerning all these therefore, he delivered the most appropriate sciences, and left nothing [pertaining to them] uninvestigated.

He likewise unfolded to men those sciences which are common [ to all disciplines ,] as for instance the demonstrative, the definitive, and that which consists in dividing, as may be known from the Pythagoric commentaries. He was also accustomed to pour forth sentences resembling Oracles to his familiars in a symbolical manner, and which in the greatest brevity of words contained the most abundant and multifarious meaning, like the Pythian Apollo through certain oracles, or like nature herself through seeds small in bulk, the former exhibiting conceptions, and the latter effects, innumerable in multitude, and difficult to be understood. Of this kind is the sentence, The beginning is the half of the whole , which is an apothegm of Pythagoras himself.

But not only in the present hemistich, but in others of a similar nature, the most divine Pythagoras has concealed the sparks of truth; depositing as in a treasury for those who are capable of being enkindled by them, and with a certain brevity of diction, an extension of theory most ample and difficult to be comprehended, as in the following hemistich:

(53) Timaeus: Now all triangles derive their origin from two triangles, each having one angle right and the others acute ; and the one of these triangles has on each side half a right angle marked off by equal sides, while the other has the right angle divided into unequal parts by unequal sides. These we lay down as the principles of fire and all the other bodies, proceeding according to a method in which the probable is combined with the necessary; but the principles which are still higher than these are known only to God and the man who is dear to God.

(56) Timaeus: Wherefore, we are preserving the probable account when we assign this figure to earth, and of the remaining figures the least mobile to water, and the most mobile to fire, and the intermediate figure to air; and, further, when we assign the smallest body to fire, and the greatest to water, and the intermediate to air; and again, the first in point of sharpness to fire, the second to air, and the third to water. As regards all these forms, that which has the fewest bases must necessarily be the most mobile,

(8) Pyruacoras saith:—I affirm that God existed before all things, and with Him was nothing, as He was at first. But know, all ye Philosophers, that I declare this in order that I may fortify your opinion concerning these four elements and arcana, as well as in the sciences thereof, at which no one can arrive save by the will of God. Understand, that when God was alone, He created four things—fre, air, water, and earth, out of which things He afterwards created all others, both the sublime and the inferior, because He predestinated from the beginning that all creatures extracted from water should multiply and increase, that they might dwell in the world and perform His judgments therein. Consequently, before all, He created the four elements, out of which He afterwards created what He willed, that is to say, diverse creatures, some of which were produced from a single element.*

The Turba saith:—Which are these, O Master?

And he:—They are the angels, whom He created out of fire. But the Turspa:—Which, then, are created out of two?

And he:—Out of the elements of fire and air are the sun, moon, and stars composed. Hence the angels are more lucid than the sun, moon, and stars, because they are created from one substance, which is less dense than two, while the sun and the stars are created from a composition of fire and air.

The Turba saith: And what concerning the creation of Heaven?

Then he:—God created the Heaven out of water and air, whence this is also composed of two, namely, the second of the rarer things, which is air, and the second of the denser things, which is water. And they:—Master, continue thy discourse concerning these three, and rejoice our hearts with thy sayings, which are life to the dead. But the

other answereth:—I notify to you that God hath further made creatures out of three and out of four; out of three are created flying things, beasts, and 4 vegetables; some of these are created out of water, air, and earth, some out of fire, air, and earth. But the Turba saith:—Distinguish these divers creatures one from another.

And he:— Beasts are created out of fire, air, and earth; flying things out of fire, air, and water, because flying things, and all among vegetables which have a spirit, are created out of water, while all brute animals are from earth, air, and fire. Yet in vegetables there is no fire, for they are created out of earth, water, and air. Whereat the Turba saith:—Let us assume that a fire, with your reverence’s pardon, does reside in vegetables.

And he:—Ye have spoken the truth, and I affirm that they contain fire. And they:—Whence is that fire?

He answereth:—Out of the heat of the air which is concealed therein; for I have signified that a thin fire is present in the air, but the elementary fire concerning which you were in doubt is not produced, except in things which have spirit and soul. But out of four elements our father Adam and his sons were created,* that is, of fire, air, water, and likewise earth. Understand, all ye that are wise, how everything which God hath created out of one essence dies not until the Day of Judgment. The definition of death is the disjunction of the composite, but there is no disjunction of that which is simple, for it is one. Death» consists in the separation of the soul from the body, because anything formed out of two, three, or four components must disintegrate, and this is death. Understand, further, that no complex substance which lacks fire eats, drinks, or sleeps, because in all things which have a spirit fire is that which eats.*

The Turba answereth:—How is it, Master, that the angels, being created of fire, do not eat, seeing thou assertest that fire is that which eats! And he: Hence ye doubt, each having his opinion, and ye are become opponents, but if ye truly knew the elements, ye would not deny these things. I agree with all whose judgment it is that simple fire eats not, but thick fire. The angels, therefore, are not created out of thick fire, but out of the thinnest of very thin fire; being created, then, of that which is most simple and exceedingly thin, they neither eat, drink, nor sleep.

And the Turba:— Master, our faculties are able to perceive, for by God’s assistance we have exhausted thy sayings, but our faculties of hearing and of sight are unable to carry such great things. May God reward thee for the sake of thy disciples, since it is with the object of instructing future generations that thou hast summoned us together from our countries, the recompense of which thou wilt not fail to receive from the Judge to come!

ArisLEeus saith:— Seeing that thou hast gathered us together for the advantage of posterity, 1 think that no explanations will be more useful than definitions of those four elements which thou hast taught us to attain.

And he:—None of you are, I suppose, ignorant that all the Wise have propounded definitions in God. The

Tureva answereth:—Should your disciples pass over anything, it becomes you, O Master, to avoid omissions for the sake of future generations.

And he:—If it please you, I will begin the disposition here, since envious men in their books have separated that, or otherwise I will put it at the end of the book.* Whereat the

Turba saith:—Place it where you think it will be clearest for future generations.

And he:—I will place it where it will not be recognised by the foolish,+ nor ignored by the Sons of the Doctrine, for it is the key, the perfection and the end.

(3) Anaxacoras saith:—I make known that the beginning of all those things which God hath created is weight and proportion,* for weight rules all things, and the weight and spissitude of the earth is manifest in proportion; but weight is not found except in body. And know, all ye Turba, that the spissitude of the four elements reposes in the earth; for the spissitude of fire falls into air, the spissitude of air, together with the spissitude received from the fire, falls into water; the spissitude also of water, increased by the spissitude of fire and air, reposes in earth. Have you not observed how the spissitude of the four elements is conjoined in earth? The same, therefore, is more inspissated than all.

Then saith the Turba: —Thou hast well spoken. Verily the earth is more inspissated than are the rest. Which, therefore, is the most rare of the four elements and is most worthy to possess the rarity of these four?

He answereth:—Fire is the most rare among all, and thereunto cometh what is rare of these four. But air is less rare than fire, because it is warm and moist, while fire is warm and dry; now that which is warm and dry is more rare than the warm and moist. They say unto him: —Which element is of less rarity than air?

He answereth:—Water, since cold and moisture inhere therein, and every cold humid is of less rarity than a warm humid.

Then do they say unto him:—Thou hast spoken truly. What, therefore, is of less rarity than water?

He answereth:-—Earth, because it is cold and dry, and that which is cold and dry is of less rarity than that which is cold and moist.

PyTHAGoRAS saith:—Well have ye provided, O Sons of the Doctrine, the description of these four natures,* out of which God hath created all things. Blessed, therefore, is he who comprehends what ye have declared, for from the apex of the world he shall not find an intention greater than his own! Let us, therefore, make perfect our discourse.

They reply:—Direct every one to take up our speech in turn. Speak thou, O Pandolfus!

To

(55) Timaeus: and the most plastic body, and of necessity the body which has the most stable bases must be pre-eminently of this character. Now of the triangles we originally assumed, the basis formed by equal sides is of its nature more stable than that formed by unequal sides; and of the plane surfaces which are compounded of these several triangles, the equilateral quadrangle, both in its parts and as a whole, has a more stable base than the equilateral triangle.