Of the word ΕΙ engraven over the gate of Apollo's temple at Delphi (6-10)

Nicander having delivered these words, our friend Theon, whom you know, asked Ammonius if he might have liberty to plead for logic, which was so highly injured. And Ammonius bidding him speak and defend it, he said: Now that this God is a most expert logician many of his oracles show; for it is, to wit, the part of the same artist to dissolve and frame ambiguities. Moreover, as Plato said, when an oracle was given to the Greeks that they should double the altar in Delos, which is a work of the utmost perfection in geometry, that the God did not order the doing of that very thing, but commanded the Greeks to apply themselves to geometry; so the same God, by giving ambiguous oracles, honors and recommends logic,

as necessary to those who desire to understand him aright. Now this conjunction El, or if, has a very great efficacy in logic, as forming the most rational proposition; for how can it be otherwise, since the very brutes have indeed the knowledge of the substance of things, but to man only has Nature given the consideration and judgment of consequence? For that there is both day and light, wolves and dogs and birds are sensible. But that if it is day there must be light, no other animal understands but man, who only has the conception of antecedent and consequent, of the significance and connection of these things with one another, and of their habitude and difference, from which demonstrations take their principal beginnings. Now since philosophy is conversant about truth, since the light of truth is demonstration, and the beginning of demonstration this connection of propositions; the faculty which contains and effects this was by wise men, with good reason, consecrated to the God who most of all loves truth. Now the God indeed is a prophet, and the art of prophesying is a divination concerning the future from things that are present and past. For neither is the original of any thing without a cause, nor the foreknowledge of any thing without reason. But since all things that are done follow and are connected to those that have been done, and those that shall be done to those that are done, according to the progress proceeding from the beginning to the end; he who knows how to look into the causes of this together, and naturally connect them one with another, knows also and divines

What things now are, shall be, or e’er have been.[*]

And Homer indeed excellently well places first things that are present, and afterwards what is future and past. For by the very nature of the connection the argument is based on that which now is. Thus, if this is, that preceded; and again, if this is, that shall be. For the knowledge of the consequence is, as has been said, an artificial and rational thing; but sense gives the assumption to reason. Whence (though it may seem indecent to say it) I will not be afraid to aver this, that the tripod of truth is reason, which recognizes the dependence of the consequent on the antecedent, and then, assuming the reality of the antecedent, infers the conclusion of the demonstration. If then the Pythian Apollo delights in music, and is pleased with the singing of swans and the harmony of the lute, what wonder is it that, for the sake of logic, he embraces and loves this argumentative particle, which he sees the philosophers so much and so frequently to use? Hercules indeed, not having yet unbound Prometheus, nor conversed with the sophisters that were with Chiron and Atlas, but being still a young man and a plain Boeotian, at first abolished logic and derided this word EI; but afterwards he seemed by force to have seized on the tripod, and contended with our God himself for the pre-eminence in this art; for being grown up in age, he appeared to be the most expert both in divination and logic.

Theon having ended his speech, I think it was Eustrophus the Athenian who said to us: Do you not see how valiantly Theon vindicates logic, having, in a manner, got on the lion’s skin? So it is not right even for us—who comprehensively place all the affairs, nature, and principles of things both divine and human in number, and make it most especially the author and lord of honest and estimable things—to be at quiet, but we must willingly offer the first-fruits of our dear mathematics to the God; since we think that this letter E does not of itself differ from the other letters either in power, figure, or expression, but that it has been preferred as being the sign of that great number which has an influence over all things, called the Quinary (or Pemptas), from which the Sages

have expressed the art of numbering by the verb πεμπτάζειν (signifying to account by fives). Now Eustrophus spake these things to us, not in jest, but because I did at that time studiously apply myself to the mathematics, and perhaps also in every thing to honor that saying, Nothing too much, as having been conversant in the Academy.

I answered therefore that Eustrophus has excellently solved the difficulty by number. For (said I) since all number is distributed into even and odd, unity is in efficacy common to them both,—for that being added to an even number, it makes it odd, and to an odd, it makes it even, two constituting the beginning of the even, and three of the odd. Now the number of five, composed of these two, is deservedly honored, as being the first compound made of the first simple numbers, and is called the marriage, for the resemblance of the odd with the male, and the even with the female. For in the divisions of the numbers into equal parts, the even, being wholly separated, leaves a certain capacious beginning and space in itself; but in the odd, suffering the same thing, there always remains a middle, of generative distribution, by which it is more fruitful than the other, and being mixed is always master, never mastered. For by the mixture of both, even and odd together, there is never produced an even number but always an odd. But which is more, either of them added to and compounded with itself shows the difference; for no even joined with another even ever produced an odd, or went forth of its proper nature, being through weakness unable to generate another and imperfect. But odd numbers mixed with odd do, through their being every way fruitful, produce many even ones. Time does not now permit us to set down the other powers and differences of numbers. Therefore have the Pythagoreans, through a certain resemblance, said that five is the marriage of the first male and the first female number. This

also is it for which it is called Nature, by the multiplication of itself determining again into itself. For as Nature, taking a grain of wheat for seed and diffusing it, produces many forms and species between, by which she brings her work to an end, but at last she shows again a grain of wheat, restoring the beginning in the end of all; so, while the rest of the numbers, when they are multiplied into others, terminate by the increase only those of five and six, multiplied by themselves, bring back and preserve themselves. For six times six makes thirty-six, and five times five makes twenty-five. And again, six does this once, and only after one manner, to wit, when it is squared. But this indeed befalls five both by multiplication and by composition with itself, to which being added, it alternatively makes itself and ten; and this as far as all number can extend, this number imitating the beginning or first Cause which governs the universe. For as that first Cause, preserving the world by itself, does reciprocally perfect itself by the world, as Heraclitus says of fire,

Fire turns to all things, and all things to fire;

and as money is changed for gold, and gold for money; so the congress of five with itself is framed by Nature to produce nothing imperfect or strange, but has limited changes; for it either generates itself or ten, that is, either what is proper to itself, or what is perfect.

But these things have perhaps had more than sufficient time spent on them. This, however, is evident, that they properly attribute to this God the number of five, saying that it sometimes of itself produces itself like fire, and other whiles the number of ten, like the world. But do we think that this number is not also concerned with music, which is of all things most acceptable to this God For the chiefest operation of harmony is, as one may say, about symphonies. Now that these are five and no more, reason convinces even him who will by his senses without reasoning make trial either on strings or pipe-holes. For all these accords take their original in proportions of number; and the proportion of the symphony diatessaron is sesquitertial, of diapente sesquialter, of diapason duple, of diapason with diapente triple, and of disdiapason quadruple. But as for that which, transcending all measures, the musicians add to these, naming it diapason with diatessaron, it is not fit we should receive it, gratifying the unreasonable pleasure of the ear against proportion, which is as the law. I may therefore let pass the five positions of the tetrachords, and also the five first,—whether they are to be called tones, tropes, or harmonies,—according to the changes of which by rising or falling, either to more or less, the rest are bases or trebles. And, whereas there are many or rather infinite intervals, are not five of them only used in music, to wit, diesis, hemitonion, tons, triemitonion, and ditonon? Nor is there in the voice any other space, either greater or less, that, being distinguished by base or treble, comes into melody.