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 aλoya, 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, λóyoç, or reason, just as well as though they were embraced by some common numerical divisor. Much on this subject of quantities, styled äλoya, or irrational, may be found in Euclid's Laws of the Musical Canon, 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 2óyoç 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. Διὸ δὴ τῶν θαυμαστῶν ἁπάντων πηγὴ yéyovεv. 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 rev olution was nearer to, or more remote from, the centre, while there was but one impulse distributing itself proportionally, ȧvà λóyov, 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 referred 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 velocity, taken as a whole, would be that same slow and almost imperceptible motion which appears in our timepieces. XXIII. The Words φθίσις, γένεσις, πάθος, and φθορά. PAGE 25, LINE 5. φθίνει . . ἀμφότερα ἀπόλλυται. This word φθίνει (φθίσις) is applied to a diminution of the number of parts or particles of which a body is composed, without a change of the essential idea, law, or nature. It is 168 THE WORDS φθίσις, γένεσις, πάθος, AND φθορά. the opposite of αὐξάνεται, αὔξησις. 'Απόλλυται is used where the very law, nature, or idea of a thing (that which makes it what it is) is taken away. It is the opposite of yiyveral. The one would express the difference between a fat man and a lean one, the other between a living man and a dead body. 0ívw is generally intransitive, but is sometimes used in a transitive sense, as in the Iliad, vi., 407: Δαιμόνιε φθίσει σε τὸ σὸν μένος. Φθίνω and φθίμενος are applied by the poets to the dead, but more in a metaphorical than a strictly philosophical sense. When thus poetically used, they still retain something of their primary meaning, and suggest the conception of the wasted, the emaciated, the weak (åμɛvηvà kápηva), as though the ghostly state were but a diminution of their former life. In the same manner the poets use kaμovтes, the wearied, the deceased. The Hebrews seem to have had something of the same metaphorical conception in their רְפָאִים word Mere increment or diminution is said to take place as long as the kαƐσтηêvĩa ëğıç, the constituting state, remains (diaμévη, continues through, or survives the change). But it may be destroyed both ways (aupóтepa), that is, by píois, or avšŋois, when carried so far that the law of the body becomes affected. No increment or decrement which does not take away that eğıç, or state, which makes a thing what it is, can ever amount to that great change denoted by p0opà, and the verb ảπóλλvμ. Until this takes place, the real or essential identity remains, although that phenomenal identity may have been affected, which depends on the numerical sum or aggregate. So that we may say, that, if every particle of matter has been removed and replaced by others, still, if during the process the kalɛoтηkvĩa ëğıç is preserved, it is the same body, although not the same matter; and if, on the other hand, no single material particle be lost, yet 66 if, in any way, this ëğıç has been destroyed, it is no longer the same, but there has taken place the yéveσis of something else, having a different name, a different law, and a different identity, or, as is said a few lines below, μɛтabaλὸν εἰς ἄλλην ἕξιν διέφθαρται παντελῶς, “ passing into another state, it is utterly destroyed;" the thing which before was, no longer is, whatever may have taken its place. See Note XXV., on the difference between yévɛois and ἀλλοίωσις. The next question is, What is yévɛois, or generation? It is rather abruptly put by the supposed interlocutor, yet still is naturally enough suggested by what precedes: Tíyvetaι δὴ πάντων γένεσις ἡνίκ ̓ ἂν τί πάθος ή ; to which the succeeding answer is given: Δῆλον ὡς ὁπόταν ἀρχὴ λαβοῦσα aʊğŋv, k. t. λ., “It is evident (that generation takes place, or that the peculiar ráðoç under which it takes place is) whenever a principle (åpxò, here put for the originating idea, the principium, or law of life to anything) receiving growth, (that is, being developed in the outward or material) passes into the second change, and from this into the next, and so on, until, coming as far as to three, it arrives at such a state as to become an object of sensation." This is certainly rather obscure, but perhaps as well expressed as was possible, in the attempt to set forth the transition from the law of life to its material organic development. Tptov would seem to refer to the three mathematical dimensions, length, breadth, and thickness, which every object of sense must possess in some degree; or it may be intended as an indefinite number, representing the stages, be they more or less, through which the thing generated must pass, until it become an object of sensation, visible, tangible, &c. P XXIV. Philosophy of the Verb TO BE. Platonic Use of eiuí and γίγνομαι. PAGE 25, LINE 11. Μεταβάλλον μὲν οὖν οὕτω καὶ μετακινούμενον γίγνεται πᾶν. ἔστι δὲ ὄντως ἂν ὁπόταν μένῃ· μεταβαλὸν δὲ εἰς ἄλλην ἕξιν διέφθαρται παντελῶς. The tenses here are emphatic, and must have their precise meaning. "While thus changing and moving, it is in the act of being generated. It really is, when it becomes fixed and stands; but after it has passed into another state, it (that is, the former thing) is utterly destroyed." Mévy is to be taken here in its philosophical sense, as opposed to μεταβάλλει, and for ἕστηκε, in the language of the schools, as opposed to kiveitai, or to that which is in a constant flux or motion. This, however, can only strictly be applied to the law or idea, and in this sense it includes what Plato so often expresses by the phrase dɛì KATÀ TAVTά, &c., as that which remains unaffected amid the material mutations to which it is constantly subject. It may be, however, that övτws öv is not to be taken here in the highest philosophical sense, as opposed to yɩyvóμɛvov, but more according to the vulgar usage of the substantive verb, as signifying the real being, not simply of the law or idea, but of the generated material object itself, during that period in which it suffers no máloç, or change of state. It is because they are always suffering change or flux, like a river ever passing away, and never for two successive moments preserving the same numerical or aggregate identity, that even some of the ancient philosophers who were theists denied that generated material things were at all entitled to the epithet ovτws övтa. Plato, however, clearly regards their identity as not depending upon number and the aggregate mass; but as long as they suffer |