(533) Dear Glaucon, I said, you will not be able to follow me here, though I would do my best, and you should behold not an image only but the absolute truth, according to my notion. Whether what I told you would or would not have been a reality I cannot venture to say; but you would have seen something like reality; of that I am confident. Doubtless, he replied. But I must also remind you, that the power of dialectic alone can reveal this, and only to one who is a disciple of the previous sciences. Of that assertion you may be as confident as of the last. And assuredly no one will argue that there is any other method of comprehending by any regular process all true existence or of ascertaining what each thing is in its own nature; for the arts in general are concerned with the desires or opinions of men, or are cultivated with a view to production and construction, or for the preservation of such productions and constructions; and as to the mathematical sciences which, as we were saying, have some apprehension of true being—geometry and the like—they only dream about being, but never can they behold the waking reality so long as they leave the hypotheses which they use unexamined, and are unable to give an account of them. For when a man knows not his own first principle, and when the conclusion and intermediate steps are also constructed out of he knows not what, how can he imagine that such a fabric of convention can ever become science? Impossible, he said. Then dialectic, and dialectic alone, goes directly to the first principle and is the only science which does away with hypotheses in order to make her ground secure; the eye of

(42) In case the opposite of a thing is not known to us, because it has not as yet been discovered by or made known to us, nevertheless in such case we are fully justified in ascribing to the unknown opposite the qualities and characteristics diametrically opposed to the known opposite. The rule is this: "Whatever is affirmed of one of a pair of opposites must be denied to the other"; and " whatever is denied to the one, must be affirmed of the other." So true and infallible is this rule that it may be applied and employed as the basis of logical reasoning from the known to the unknown, for the purpose of discovering the latter.

(525) how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply 4 , taking care that one shall continue one and not become lost in fractions. That is very true. Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,—what would they answer? They would answer, as I should conceive, that they were speaking of those numbers which can only be realized in thought. Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth? Yes; that is a marked characteristic of it. And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been. Very true, he said.

(15) Endeavour not to conceal your errors by words, but to remedy them by reproofs. Pythagoras. Stob. p. 146.

It is not so difficult to err, as not to reprove him who errs. Pythagoras. Stob. p. 147.

(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.

(2) In the first place, therefore, we shall divide the genera of the proposed problems, in order that we may know the quantity and quality of them. And, in the next place, we shall show from what theologies the doubts are assumed, and according to what sciences they are investigated. For some things that are badly confused, require a certain distinction; others are conversant with the cause through which they subsist, and are apprehended; others, which we propose according to a certain contrariety, draw our decision on both sides; and some things require from us the whole development of mystic doctrines. Such, therefore, being the nature of the subjects of discussion, they are assumed from many places, and from different sciences. For some things introduce animadversions from what the wise men of the Chaldeans have delivered; others produce objections from what the prophets of the Egyptians teach; and there are some that, adhering to the theory of philosophers, make inquiries conformably to them. There are now likewise some, that from other opinions, which do not deserve to be mentioned, elicite a certain dubitation; and others originate from the common conceptions of mankind. These things, therefore, are of themselves variously disposed, and are multiformly connected with each other. Hence, through all these causes, a certain discussion is requisite for the management of them in a becoming manner.

(1) It is necessary that you should become scientific, either by learning from another person, or by discovering yourself the things of which you have a scientific knowledge. If, therefore, you learn from another person, that which you learn is foreign; but what you discover yourself is through yourself, and is your own. Moreover, if you investigate, discovery will be easy, and soon obtained; but if you do not know how to investigate, discovery will be to you impossible. And [right] reasoning indeed, when discovered, causes sedition to cease, and increases concord. For through this the inexhaustible desire of possessing is suppressed, and equality prevails; since by this we obtain what is just in contracts. Hence, on account of this, the poor receive from those who are able to give; and the rich give to those that are in want, both of them believing that through this they shall obtain the equal.

This however will be a rule and an impediment to those that act unjustly, viz. that men who possess scientific knowledge will appease their anger, prior to the commission of an injury, being persuaded that the perpetrators of it will not be concealed when it is committed; but that those who do not possess scientific knowledge, becoming manifest in the commission of an injury, will be restrained from acting unjustly.

(13) 'By the words "He is," is he to be apprehended, and by (admitting) the reality of both (the invisible Brahman and the visible world, as coming from Brahman). When he has been apprehended by the words "He is," then his reality reveals itself'

(95) This I set down here only for a manuduction; you will find it demonstrated more at large concerning the creation of this world.

(7) But this much is enough, since it is impossible for anyone to dispute the nature of the words I have just spoken about the blessed, imperishable, true God. Now, if anyone wants to believe the words set down (here), let him go from what is hidden to the end of what is visible, and this Thought will instruct him how faith in those things that are not visible was found in what is visible. This is a principle of knowledge.

(6) The theorems of philosophy are to be enjoyed as much as possible, as if they were ambrosia and nectar . For the pleasure arising from them is genuine, incorruptible, and divine. They are also capable of producing magnanimity; and though they cannot make us eternal beings, yet they enable us to obtain a scientific knowledge of eternal natures.

(103) Of which much were to be written, but [in the present book] you will find it more apprehensibly at the place concerning the creation: For there are to be found living testimonies enough, so that none need doubt whether the things be so or no.

(1) In proof of these things, and witnessing faithfully are the Universe, the Year of time, and Man himself, the Microcosm. He fixed these as testimonies of the Triad, the Heptad, and the Dodecad; the twelve constellations rulers of the world, the Dragon (THELE) Tali which environs the universe, and the microcosm, man. The triad, fire, water, and air; the fire above, the water below, and the air in the midst. The proof of which is that air is a participator with both.

(51) To the wise, whose hearts are enlightened, The mere sound of that voice proves its truth." "When you say to a thirsty man, 'Come quickly; This is water in the cup, take and drink it,' Does the thirsty man say, 'This is a vain pretension; Go, remove yourself from me, O vain pretender, Or proceed to give proofs and evidence That this is generic water, and concrete water thereof'? Or when a mother cries to her sucking babe,

(511) they start from hypotheses and do not ascend to a principle, those who contemplate them appear to you not to exercise the higher reason upon them, although when a first principle is added to them they are cognizable by the higher reason. And the habit which is concerned with geometry and the cognate sciences I suppose that you would term understanding and not reason, as being intermediate between opinion and reason. You have quite conceived my meaning, I said; and now, corresponding to these four divisions, let there be four faculties in the soul—reason answering to the highest, understanding to the second, faith (or conviction) to the third, and perception of shadows to the last—and let there be a scale of them, and let us suppose that the several faculties have clearness in the same degree that their objects have truth. I understand, he replied, and give my assent, and accept your arrangement.

(31) And thence ascertains the man's faith. Such an one needs not the evidence of fair acts and words "Beware of such, they spy out the heart." Require this evidence of act and word only from one But the light of the traveler arrived at the goal, That witness of his is exempt from bearing witness, Since the brilliance of that jewel beams forth, Wherefore require not from him act and word evidence, What is this evidence but manifestation of hidden things, Whether it be evidence in word, or deed, or otherwise?

(9) 'That doctrine is not to be obtained by argument, but when it is declared by another, then, O dearest, it is easy to understand. Thou hast obtained it now; thou art truly a man of true resolve. May we have always an inquirer like thee!'

(1) If, however, it be necessary, dismissing these particulars, to speak what appears to me to be the truth, you do not rightly infer “ that a knowledge of this mathematical science cannot be obtained, because there is much dissonance concerning it, or because Chæremon, or some other, has written against it .” For if this reason were admitted, all things will be incomprehensible. For all sciences have ten thousand controvertists, and the doubts with which they are attended are innumerable. As, therefore, we are accustomed to say in opposition to the contentious, that contraries in things that are true are naturally discordant, and that it is not falsities alone that are hostile to each other; thus, also, we say respecting this mathematical science, that it is indeed true; but that those who wander from the scope of it, being ignorant of the truth, contradict it. This, however happens not in this science alone, but likewise in all the sciences, which are imparted by the Gods to men.

(2) Thus we may not look for the Intellectual objects outside of the Intellectual-Principle, treating them as impressions of reality upon it: we cannot strip it of truth and so make its objects unknowable and non-existent and in the end annul the Intellectual-Principle itself. We must provide for knowledge and for truth; we must secure reality; being must become knowable essentially and not merely in that knowledge of quality which could give us a mere image or vestige of the reality in lieu of possession, intimate association, absorption.

The only way to this is to leave nothing out side of the veritable Intellectual-Principle which thus has knowledge in the true knowing , cannot forget, need not go wandering in search. At once truth is there, this is the seat of the authentic Existents, it becomes living and intellective: these are the essentials of that most lofty Principle; and, failing them, where is its worth, its grandeur?

Only thus is it dispensed from demonstration and from acts of faith in the truth of its knowledge: it is its entire self, self-perspicuous: it knows a prior by recognising its own source; it knows a sequent to that prior by its self-identity; of the reality of this sequent, of the fact that it is present and has authentic existence, no outer entity can bring it surer conviction.

Thus veritable truth is not accordance with an external; it is self-accordance; it affirms and is nothing other than itself and is nothing other; it is at once existence and self-affirmation. What external, then, can call it to the question, and from what source of truth could the refutation be brought? Any counter affirmation must fall into identity with the truth which first uttered itself; brought forward as new, it has to appear before the Principle which made the earlier statement and to show itself identical with that: for there is no finding anything truer than the true.

(4) There was, however, a certain person named Hippomedon, an Ægean, a Pythagorean and one of the Acusmatici, who asserted that Pythagoras gave the reasons and demonstrations of all these precepts, but that in consequence of their being delivered to many, and these such as were of a more sluggish genius, the demonstrations were taken away, but the problems themselves were left. Those however of the Pythagoreans that are called Mathematici , acknowledge that these reasons and demonstrations were added by Pythagoras, and they say still more than this, and contend that their assertions are true, but affirm that the following circumstance was the cause of the dissimilitude. Pythagoras, say they, came from Ionia and Samos, during the tyranny of Polycrates, Italy being then in a florishing condition; and the first men in the city became his associates.

But, to the more elderly of these, and who were not at leisure [for philosophy], in consequence of being occupied by political affairs, the discourse of Pythagoras was not accompanied with a reasoning process, because it would have been difficult for them to apprehend his meaning through disciplines and demonstrations; and he conceived they would nevertheless be benefited by knowing what ought to be done, though they were destitute of the knowledge of the why : just as those who are under the care of physicians, obtain their health, though they do not hear the reason of every thing which is to be done to them. But with the younger part of his associates, and who were able both to act and learn,—with these he conversed through demonstration and disciplines.

These therefore are the assertions of the Mathematici, but the former, of the Acusmatici. With respect to Hippasus however especially, they assert that he was one of the Pythagoreans, but that in consequence of having divulged and described the method of forming a sphere from twelve pentagons, he perished in the sea, as an impious person, but obtained the renown of having made the discovery. In reality, however, this as well as every thing else pertaining to geometry, was the invention of that man ; for thus without mentioning his name, they denominate Pythagoras. But the Pythagoreans say, that geometry was divulged from the following circumstance: A certain Pythagorean happened to lose the wealth which he possessed; and in consequence of this misfortune, he was permitted to enrich himself from geometry.

But geometry was called by Pythagoras Historia . And thus much concerning the difference of each mode of philosophising, and the classes of the auditors of Pythagoras. For those who heard him either within or without the veil, and those who heard him accompanied with seeing, or without seeing him, and who are divided into interior and exterior auditors, were no other than these. And it is requisite to arrange under these, the political, economic and legislative Pythagoreans.