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sider this, O most excellent Cratylus, of which I am often dreaming,* can we in truth affirm that there are such realities as the Beautiful, the Good,” &c. ? και εγώ πολλάκις ονειρώττω, πότερον φώμέν τι είναι αυτο ΤΟ ΚΑΛΟΝ και 'ΑΓΑΘΟΝ και εν έκαστον των όντων όντως

In this philosophy, too, he shows that there could be no true moral or political science, no law, no real State, no social or civil rights, with their corresponding obligations. See the Theætetus, 172, B., and the remarks thereupon, page 138. There could be no science, he affirms, of any kind, for it must necessarily be grounded on the eternal and immutable. Αίσθησις would take the place of επιστήμη, and nothing could be really known : 'Αλλά μήν ουδ' αν γνώσθείη γε υπ' ουδενός ουδέν άμα γαρ αν επιόντος του γνωσομένου άλλο και άλλοϊον γίγνοιτο· και εκ τούτου του λόγου ούτε το γνωσόμενον ούτε το γνωσθησόμενον αν είη. Hence he draws the sublime conclusion, that, since the very laws of our being compel us to affirm the real, and not merely relative existence of these ideas, therefore there is something which is eternal and immutable, or, in the language of the ancient schools, all things do not flow, but some things stand. Ει δε έστι μεν αεί το γιγνώσκον, έστι δε το αει γιγνωσκόμενον, έστι δε το KAΛON, έστι δε το ΑΓΑΘΟΝ, έστι δε ΤΟ ΔΙΚΑΙΟΝ, ού μου φαίνεται ταύτα όμοια όντα ροή ουδέν ουδε φορά. But if there is something which eternally knows, and something which is eternally knownif there is THE BEAUTIFUL, and THE GOOD, and THE JUST, then things do not all seem to me to be similar to motion or a flowing stream.Cratylus, 440, B.

* ονειρώττω. No word could better express that peculiar state of mind in which Socrates (or Plato) often contemplated his favourite doctrine of ideas. Sometimes he seems to be perfectly assured of the real existence of the każòv, &c., the Fair, the Just, and the Good. Again, he appears perplexed with doubt, and, at other times, seems to have but a glimpse, as in a dream, of some such bright reminiscences of a better state.

XXI.

Mathematical Use of the Word λόγος. , PAGE 24, LINE 6. åvà Nóyov. The common reading is åváhoyov; the other, however, is unquestionably to be preferred. It would signify here proportionally, certa quadam ratione. This is called in Latin ratio, and in Greek hóyos (especially in all mathematical writings), because a simple quantity or magnitude, irrespective of the relation it bears to another as a multiple or a divisor, cannot be an object of science, or be contemplated by the mind. It remains only an object of sense, aloontóv, being, to the intellect, ããoyov, and therefore äyvwotov. See the Theætetus, 202, B. It is this relation or ratio which becomes the true vontòv, or real object of the mind, while the sensible figure serves only as the diagram by which it is exhibited. Hence it is styled the aóyos, ratio, or reason. It is that which is predicated of its subject, and hence is its dóyos, or word, as well as reason ; because, when viewed as sim. ple quantity or magnitude, nothing can be said about it, no truth affirmed respecting it.

This dóyos, or reason, ever implies a third thing or middle term, namely, the common measure or divisor to which both quantities must be referred, and by which we are enabled to predicate the one as a part, or multiple, or any certain ratio of the other. The dóyol or ratios are absolute and immutable verities of science, as all vontà must be, while the aloontà by which they are suggested are muta. ble, flowing, and without anything which can be styled absolute. They likewise are capable of being compared among themselves, and thus give rise to others-ratios of ratios, ad infinitum. In modern works the simple radical meaning of the term is lost sight of, because we use the Latin ratio without any reference to its primary sense, as the same with the Greek ióyos, and hence the great vagueness which prevails in most minds respecting this plain mathematical idea. In some of our older mathematical works, such as the English editions of Euclid's Elements by Dee and Barrow respectively, our own word reason is everywhere properly employed instead of ratio: By this means the metaphysical notion of ratio is kept before the mind as the intelligible, by which what would otherwise be merely, as magnitude, an object of sense, becomes known to the intellect as an object of science. See Proclus, Commentary on Euclid's Elements, lib. i.

All mathematical truths, and especially the geometrical, are ultimately to be resolved into a comparison of ratios. For even parallelism, and other properties which would seem to have no connexion with it, do, after all, depend upon certain equalities or correspondences, from which they derive their dóyos, notion, or definition. So that all mathematical science is finally brought down to those innate ideas of the Tò loov, &c., which are discussed in the Phædon, and of which visible magnitude is only suggestive. Even a straight line involves this idea of the loov, or simplest ratio. It is that which lies evenly, equally, or, as it is expressed by Euclid, éšícov, between its extreme points ; that is, having nothing capable of being predicated of the one side and not of the other. Playfair and others seem to have entirely misunderstood the expression, and to have greatly bungled in their efforts to amend, by substituting a far more coinplex idea for this old and perfect definition of Euclid. Any one who is capable of consulting his own consciousness, must acknowledge that the language of Euclid best expresses that innate idea of straightness, which we ever apply, as the perfect ideal exemplar, to the determination of visible figure.

From this use of the word lóyos it is, that those magnitudes and numbers whose ratio cannot be expressed by other numbers—that is, which have no common divisor by which one may be predicated as any arithmetical part or multiple of another -- are called aroya, and in modern works, irrational. Two magnitudes, however, may be arithmetically incommensurable or irrational, like the side and diagonal of the square, the circumference and diameter of the circle, or what are styled surds among numbers; and yet, in all these cases, there may be, and often is, a geometrical representation which renders them rational, and may be styled the expression of the ratio, nóyos, or reason, just as well as though they were embraced by some common numerical divisor.

Much on this subject of quantities, styled aroya, or irrational, may be found in Euclid's Laws of the Musical Can. on, as contained in Meibomius. All concords, let it be remarked, are founded on rational numbers, while the irra. tional ever produce discords under all circumstances. The first have a dóyos or reason, and the soul, when the sounds suggest it, perceives this reason in its supersensual being, although unconscious of the intellectual process on which it depends; and hence a delight which mere sense could never furnish. Where this process is made objective, and thus presented to the mind, it is called science. It would not be difficult to refer to the same ideas of equality and ratio all the fundamental elements of the beauty of figure and motion.

XXII.

Paradox of Circular Motion. PAGE 24, LINE 8. Διό δη των θαυμαστών απάντων πηγή yéyovev. This is stated as a sort of strange paradox, that one motion should be at the same time greater and less, or should give rise to different velocities, according as the revolution was nearer to, or more remote from, the centre, while there was but one impulse distributing itself proportionally, ávà lóyov, to every part. The paradox, however, arises from confounding circular, or angular, with rectilineal motion. The idea of the latter arises from a compound comparison of two elements, namely, the space passed over, and the time employed in the passage. Hence, there being no absolute measure of space, there can be nothing absolute about rectilineal motion. The other must be always refer. red to the centre of motion, and the time occupied in one revolution; or, in other words, one must be referred to space and time, the other to time only. The latter

may also be said to have something absolute about it, since there is an absolute standard of angular space. Hence the motions of the inner concentric circles of the same great circle, moving on one centre, identical with the centre of the circle, are all the same when thus measured, although varying infinitely when referred to other points. The velocity of the hour hand of a watch, that revolves once in twenty-four hours, is the same with that of the earth on its axis. If the same hour hand could be conceived of as extending to the moon, the tangential velocity of its extremity would be greater than the orbit motion of that body-exceeding many thousand miles a minute—and yet its absolute velo. city, taken as a whole, would be that same slow and almost imperceptible motion which appears in our timepieces..

XXIII. The Words φθίσης, γένεσις, πάθος, αnd φθορά. PAGE 25, LINE 5. φθίνει.. αμφότερα απόλλυται. This word φθίνει (φθίσεις) is applied to a diminution of the number of parts or particles of which a body is composed, with. out a change of the essential idea, law, or nature. It is

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