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call very ridiculously by the name of geometry.1 But the similitude of numbers, that are naturally not similar to each other, becomes conspicuous, when applied to the properties of plain surfaces; which wonderful thing, not of human but divine origin, will appear very clear to him, who is able to think. And after this, those numbers, that are increased by a triple2 (ratio), and are similar to the nature of a solid, and those, that are on the other hand dissimilar, and are by another art similar to this, which those, who are conversant with it, call stereometry,3 (to be considered):4 which is indeed a thing divine and wonderful to those, who look into it; that, 5 while the power is ever revolving about the double, and that which is from the opposite to this, according to each analogy does every nature fashion out for itself a species and genus.5 Now the first power of the double, according to number,6 1 proceeds, according to pro

1 Alluding to its name, literally, earth-measuring; which is a mechanical operation; while the geometry here intended is a speculative science. T.

* Bekker was the first to edit i-pie for rpcig, which Taylor had expressed in his " triple increase," and Sydenham suggests in Not. MSS. Ficinns has "in tres usque dimensiones—"

3 I have adopted ortptontrpiav, found in the best MS. L, in lieu of yttii^iirpiav.

* Ficinus alone has, what the sense requires, and is adopted by Taylor, "considerandi sunt," as if his MS. had Okittt£ov after ytyovores

55 Such is the literal version of the unintelligible Greek. Ast conceives that the author meant to say something to this effect, " As numbers are doubled, by passing from simple to compound, so nature, by preserving a kind of ratio in all things, fashions both genus and species."

* Of numbers, some represent lines, others superficies, and others solid and cubic quantities. To the first belongs the number 2; to the second, 4, which is the square of 2; and to the third, 8, which is the cube of 2. Double proportion was considered likewise by the ancients as perfect. First, because it is the first proportion, produced between 1 and 2; and secondly, because it contains all proportions within itself; for the sesquialter sesquitertian (1J), and the other proportions are, as it were, parts below double proportion. The numbers, which the author here adduces, are 1, 2, 4, 6, 8, 12. The ratio of 4 to 2 is double ; and that of 8 to 4 is also double. Now these two excesses are equal in ratio; for each is double; but they are not equal in number; for 8 exceeds 4 by 4, but 4 exceeds 2 only by 2. Again, if we compare 6 to 4, and afterwards to 8, in the first case we have a sesquialter, and in the second a sesquitertian ratio; but these excesses are unequal in ratio, although equal in number. For the ratio of 6 to 4 = U, and the ratio of 8 to 6 = 1J; but 6 exceeds 4 by 2, and is exceeded by 8 by 2. Again, compare 12 to 6, which is a double ratio, and between these compare 8 to each. Then, 12 to 8 will be a sesquialter ratio, and 8 to 6 will be a sesquitertian ratio; but a double ratio

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portion through one to two,1 2 possessing a double by power.2 But that, which, as regards the solid and tangible, is again a double, proceeds from one to eight. 3But that of the double quantity to the middle, and perhaps, what is more than the less, and less than the greater;3 4 while the other by the same part surpasses, and is surpassed by the extremes.4 But in the middle of six to twelve, there is found the sesquialter and sesquitertian proportions. And in the middle of these, a power, 5 turned to both,5 has distributed to men a use,6 where voice and measure are combined, for the sake of sports, rhythm, and harmony, after having been granted to the happy dancing of the Muses.

[13.] Let all these then be held to take place in this way, and let them exist. But as regards the finish to this, let us proceed to the divine generation and the most beautiful and divine nature of things visible, as far as a deity has granted to man to look upon them; 7 which nature, no one, after having beheld, will boast of having received with facility without the particulars mentioned above.7 Besides this, in our several in

arises from 12 to 6; while the excesses between 12 and 8, and 8 and 6, are unequal both in ratio and number. T.

'—1 Such is the literal version of the Latin of Ficinus, who seems to have found in his MS., tart St iv, instead of h, —Ast too says that one would have expected af' ivoc, but that iv is the subject of the words, 'f H'tv n-piirij Tov StirXaaiov. But how this could be, I confess I cannot understand.

22 Such is the version of the Latin of Ficinus, "duplum potentia possidens." Taylor has "being double according to power," which would be in Greek, Stirkaaiov Kara Svva/itv ovaa. But ItirXdatov 17 Kara divaptv oiaa would mean, "that according to power being double," words I confess I cannot understand. I could have understood, had the author written to this effect, "But the double of one, as regards a superficies, proceeds to four."

33 Such is the literal version of the Greek, where I am quite at a loss. Ast explains it by a paraphrase, "With regard to the power of the double, as regards the middle number (4), it exceeds by as much the lesser number (1), as it is exceeded by the larger number (8)."

*—* Ast considers all the words between the numerals as an interpolation, for they merely repeat the idea already expressed in the preceding sentence.

58 Here again I am at a loss to understand the expression " turned to both;" for it was sufficient to say simply that there is 9, a middle number between 6 and 12, applicable to the 9 Muses; for such, I presume, is what the author meant to say.

6 Ficinus has translated xpt/av by "usum commodumque," for he was uncertain here, as elsewhere, how he ought to render it by one word.

'—7 Such is the literal version of the Greek, which I cannot under

tercourse we must refer every individual thing to its species, '(and all things to one,)1 by asking questions and disproving what has been not correctly asserted. For this is truly a touchstone the most beautiful and thoroughly the first amongst men; but in the case of such as are not (touchstones), and only pretend to be, there is a labour the most vain of all.

Further still, the accuracy of time must be considered by us, and how exactly it completes all that takes place in heaven; so that he, who believes the assertion to be true, that soul is a thing older and more divine than body, would also conceive it has been very beautifully and sufficiently said, that all things are full of gods; and that we have never been neglected through the forgetfulness or carelessness of superior beings. But as regards.all such things as these, we should bear this in mind, that, if any one apprehends correctly each of these matters, there will be a great benefit to him, who has apprehended them; but if not, that it will be better for him to be ever calling upon a god, 2 according to method.2 And let this be the method—for it is necessary to say so much at least as this—Every diagram, system of number, and composition of harmony, together with the one agreement of all the stars in their revolutions, ought to be apparent to him, who learns in a proper manner. And that, of which we are speaking, will become apparent, if a person rightly learns, looking to one thing. For to those, who think upon the matter, there will appear to be naturally one bond to all of these. But if a person will take the matter in hand in any other way, he must, as we have said, call upon fortune. For, without these, no nature will become lucky in states. But this is the method, (and) this the nurture, and through these subjects of instruction we must proceed, whether they are difficult or easy. Nor is it lawful to neglect the gods; since the happy report, relating to all of them, has, according to a manner, become apparent. And I call him, who thus apprehends all these points, the man the most truly wise; who, I stoutly affirm, both in jest and

stand; nor could, I think, Ficinus, whose abridged version is "quam nunquam sine dictis artibu's assequemur."

'—1 The words between the brackets are found only in the version of Ficinus, " omnia denique in unum—"

1' I have transposed Kara rpoirov from the end of the preceding sentence to its present place, as required by the words immediately following. earnest, will, when he shall have filled up by death his allotted portion in things of this kind, 1 if he be still almost dying,1 neither share any longer in many of his senses then, as at present; and he will, after being a partaker of one destiny alone, and becoming one out of many, be fortunate, and, at the same time, most wise and blessed; whether any one lives blessed on the continent, or in islands; and that he will participate in a fortune, which ever happens to be of this kind; and that, whether any one studies these questions, living a public or a private life, he will meet with the same fate and in a similar manner from the gods. But what we said at the beginning, the same assertion appears even now to be really true; that it is not possible for men to be perfectly blessed and happy, except a few. And this is rightly asserted by us. For such as are divine and at the same time prudent men, and naturally participate in the rest of virtue, and in addition have acquired all, that is closely connected with a blessed instruction, and such things as we have mentioned, to these alone have the gifts of fortune fallen by lot, 2 and are in a sufficient state.2 To those then, who have laboured in this way upon such points, we say privately and lay down publicly as a law, that the greatest offices ought to be given to those, who have arrived at the period of an old man; and that all the others ought to follow them, and with good words hymn all the gods and goddesses; and lastly, that all of us, after having known and sufficiently examined the nocturnal assembly, most correctly exhort it to this wisdom.

11 The words between the numerals Ficinus, followed by Taylor, has omitted, either because they were not in his MS., or, what is more probable, because he could not understand them; nor, in fact, do I see how they can stand here, unless Bavarip just before be omitted.

22 I have translated as if the Greek were fi\ix* Te Kal ncavme i\ct— not IsavAg *'^>7X* Tl Kai 'XEl- Ficinus has rather loosely—" iis solum modo satis ad felicitatem omnia se habere videntur."

INTKODUCTION TO THE AXIOCHUS.

Although the Axiochus and five following dialogues, all equally spurious, have been generally appended to the complete editions of Plato, yet, strange to say, they have never been translated into English. This fact, as regards the Axiochus especially, is the more remarkable, as that dialogue has been so great a favourite with scholars of different countries, that twelve translations have been made of it into Latin, four into German, and two into French. For though Cousin asserts that his own is the only French version, yet he might have known from Fabricius and Fischer, that Dolet had preceded him in 1544; whose tiny volume, that contains a translation likewise of the Hipparchus, is so scarce, that no copy of it is to be found in the National Library at Paris, as is stated distinctly in a modern reprint of it; nor is it mentioned, I may add, in the different Catalogues of the British Museum.

Of these twelve translations nine have appeared in print; but the remaining three are to be found only in MSS. from the pens respectively of Cincius Bomanus, Bainutius, and Leonardus Aretinus; unless it be said that to one of these three is to be referred the copy, once in the possession of Swart of Altorf, in the Catalogue of whose library, says Fischer, it is described in P. 2, p. 277, n. 871, under the title of " Axiochus Platonis de contemnenda morte. Venundatur ah Alex. Haliatte, Mediolanensi, s. a. 4." Of the version by Cincius MSS. are to be found in Archbishop Parker's library at C. C Camb., as stated in the Catalogue, p. 65, and in the British Museum, Burney MSS., 226, and in the National Library at Paris, Cod. 6729; while a copy of the version by Rainutius is in MS. Harl. 4923, and in Arundel MSS. 277. To these however I have paid to attention; as they are done too loosely to enable one to ascertain

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