sition extends not to time, though time be a continuous subject. How, indeed, should the parts of time have position, which are so far from being permanent, that they fly as fast as they arrive? Here, therefore, we are rather to look for a sequel in just order;' for a continuity not by position, as in the limbs of an animal, but for a continuity by succession : Velut unda supervenit undam. Horat. Epist. ii. 2. 176. And thus are the two species of quantity, the continuous and the discrete, distinguished from each other. Besides this, among the continuous themselves there is a further distinction. Body and its attributes, the superficies and the line, are continuous quantities, capable all of them of being divided; and by being divided, of becoming a multitude; and by becoming a multitude, of passing into quantity discrete. But those continuous quantities, time and place, admit not, like the others, even the possibility of being divided. For grant place to be divided, as Germany is divided from Spain; what interval can we suppose, except it be other place? Again: suppose time to be divided, as the age of Sophocles from that of Shakspeare; what interval are we to substitute, except it be other time! Place, therefore, and time, though continuous like the rest, are incapable of being divided, because they admit not, like the rest, to have their continuity broken." But to proceed. Let us imagine, as we are walking, that at a distance we view a mountain, and at our feet a molehill: the mountain we call great, the molehill little; and thus we have 1 Ο δὲ μή ἐστιν ὑπομένον, πῶς ἂν τοῦτο θέσιν τινὰ ἔχοι; ἀλλὰ μᾶλλον τάξιν τινὰ εἴποις ἂν ἔχειν, τῷ τὸ μὲν πρότερον εἶναι TOû Xpóvov, Tò dè űσTepov. Arist. Præd. p. 32. edit. Sylb. They cannot be divided actually, from the reasons here given; but they may be be divided in power, else they could not be continuous; nor could there exist such terms as a month, a year, a cubit, a furlong, &c. In this sense of potential division they may be divided infinitely, as appears from the following theorem : A. moves quicker B moves slower have moved through the same space in a less time. Let it have moved through it in the time ζ θ. It is thus the sphere A divides the time. Again: inasmuch as the quicker A has in the time ( passed through the whole space yd, the slower B in the same time will have passed through a smaller space. Let this be y K. It is thus the sphere B divides the space. Again: inasmuch as the slower sphere B in the time (0 has passed through the space y K, the quicker sphere A will have passed through it in a less time; so that the time ( will be again divided by the quicker body. But this being so divided, the space y will be divided also by the slower body, according to the same ratio. And thus it will always be, as often as we repeat successively what has been already demonstrated: for the quicker body will after this manner divide the time, and the slower body will divide the space; and that, in either case, to infinite, because their continuity is infinitely divisible in power. See the original of this theorem in Aristotle's Physics, lib. vi. cap. . 2. p. 111. edit. Sylb. "EoTW Tò μèv èp' & α K. T.A. two opposite attributes in quantity continuous. Again in a meadow we view a herd of oxen grazing, in a field we see a yoke of them ploughing the land: the herd we call many, the yoke we call few; and thus have we two similar opposites in quantity discrete. Of these four attributes, great and many fall under the common name of excess; little and few under the common name of defect. Again: excess and defect, though they include these four, are themselves included under the common name of inequality. Further still, even inequality itself is but a species of diversity; as its opposite, equality, is but a species of identity. They are subordinate species confined always to quantity, while identity and diversity (their genera) may be found to pass through all things.* Now it is here, namely, in these two, equality and inequality, that we are to look for that property by which this genus is distinguished. It is from quantity only that things are denominated equal or unequal. Further still: whatever is equal, is equal to something else; and thus is equality a relative term. Again if we resolve inequality into its several excesses and defects, it will be apparent that each of these is a relative term also. It is with reference to little that great is called great; with reference to few that many are called many; and it is by the same habitudes inverted exist little and few. And thus is it that, through the property here mentioned, the attribute of quantity passes insensibly into that of relation; a fact not unusual in other attributes as well as these, from the universal sympathy and congeniality of nature. Nay, so merely relative are many of these excesses and defects, that the same subject, from its different relations, may be found susceptible of both at once. The mountain, which by its relation to the molehill was great, by its relation to the earth is The following characters of the three first great arrangements, or universal genera, are thus described by Aristotle: Taurà μèv γὰρ, ὧν μία ἡ οὐσία· ὅμοια δ', ὧν ἡ ποιότης μία· ἴσα δὲ, ὧν τὸ ποσὸν ἕν: “Things are the same, of which the substance is one; similar, of which the quality is one; equal, of which the quantity one." Metaph. A. KEP. Le'. p. 88. edit. Sylb. * Ιδιον δὲ μάλιστα τοῦ ποσοῦ, τὸ ἴσον καὶ ἄνισον λέγεσθαι. Arist. Præd. p. 34. 2 Aristotle says expressly of the things here mentioned, that no one of them is quantity, but exists rather among the tribe of relatives, inasmuch as nothing is great or little of itself, but merely with reference to something else. Τούτων δὲ οὐδέν ἐστι που σὸν, ἀλλὰ μᾶλλον τῶν πρός τι, οὐδὲν γὰρ AUTÒ KAť AUTÒ, K.T.λ. Arist. Præd. p. 33. edit. Sylb. This may be true with regard to mountains and molehills, and the other more indefinite parts of nature; but with regard to the more definite parts, such as vegetables and animals, here the quantities are not left thus vague, but are, if not ascertained precisely, at least ascertained in some degree. Thus Aristotle: Ἔστι γάρ τι πᾶσι τοῖς ζώοις πέρας τοῦ μεγέθους· διὸ καὶ τῆς τῶν ὀστῶν αὐξήσεως. Εἰ γὰρ ταῦτ ̓ εἶχεν αὔ ξησιν ἀεὶ, καὶ τῶν ζώων ὅσα ἔχει ὀστοῦν ἢ τὸ ἀνάλογον, ἠυξάνετ ̓ ἂν ἕως ἔζη: “ ΑΠ animals have a certain bound or limit to their bulk; for which reason the bones have a certain bound or limit to their growth. Were the bones, indeed, to grow for ever, then, of course, as many animals as have bone, or something analogous to it, would continue to grow as long as they lived." X little; and the herd, which were many by their relation to the single yoke, are few by their relation to the sands of the seashore. And hence it appears that the excesses and defects which belong to quantity are not of a relative nature only, but of an indefinite one likewise. The truth of this will become still more evident, when it is remembered that every magnitude is infinitely divisible, and that every multitude is infinitely augmentable. What, then, is to be done? How is it possible that such attributes should become the objects of science? It is then only we are said to know, when our perception is definite; since whatever falls short of this, is not knowledge, but opinion. Can, then, the knowledge be definite, when its object is indefinite? Is not this the same, as if we were to behold an object as straight, which was in itself crooked; or an object as quiescent, which was in itself moving? We may repeat, therefore, the question, and demand, what is to be done? It may be answered as follows: quantity continuous is circumscribed by figure, which, being the natural boundary both of the superficies and the solid, gives them the distinguishing names of triangle, square, or circle; of pyramid, cube, or sphere, &c. By these figures, not only the infinity of magnitude is limited, but the means also are furnished for its most exact mensuration. Again; the infinity of quantity discrete is ascertained by number, the very definition of which is ĥlos wρioμévov, that is, "multitude circumscribed or defined." Thus, if, in describing a battle, we are told that many of the enemy were slain, and but few saved; our knowledge (if it deserve the name) is perfectly vague and indefinite. But if these indefinite multitudes are defined by number, and we are Arist. de Anim. Gener. ii. 6. p. 227. edit. What follows from Simplicius is to the same purpose; only where he mentions form, we must understand that efficient animating principle described in the sixth chapter of this work. Εκαστον εἶδος συνυπάγει, μετὰ τῆς oiκείας ἰδιότητος, καὶ ποσοῦ τι μέτρον σύμ μετρον τῇ ἰδιότητι· οὐ γὰρ σχῆμα μόνον ἐπιφέρει μεθ ̓ ἑαυτοῦ τὸ εἶδος, ἀλλὰ καὶ μέγεθος, ὁ μετὰ διαστάσεως εἰς τὴν ὕλην παραγίγνεται. Πλάτος δὲ ἔχει καὶ τοῦτο ἐνθάδε διὰ τὸ ἀόριστον πῶς τῆς ἐνύλου φύσεως. Ἐὰν δὲ πολὺ τὸν ὅρον παραλλάξῃ, ἢ πρὸς τὸ μεῖζον, ἢ πρὸς τὸ ἔλαττον, τέρας voμíCeraι: "Every form introduces, along with its own original peculiarity, a certain measure of quantity, bearing proportion to that peculiarity; for it brings with itself, not a figure only, but a magnitude also, which passes into the matter by giving it extent. Now even here this magnitude has a sort of latitude, from the indefinite nature of the material principle [with which it is united.] But yet, notwithstanding if it change the bound or limit, either as to greater or to less, in a remarkable degree, the being [by such deviation] is esteemed a monster." Simplic. in Præd. p. 37. A. edit. Basil. Simplicius gives examples of this deviation in the case of giants and of dwarfs. b Aristotle's instance goes further, and shews how a smaller number may be called many, a larger number be called few. Ἐν μὲν τῇ κώμῃ πολλοὺς ἀνθρώπους φαμὲν εἶναι, ἐν ̓Αθήναις δὲ ὀλίγους, πολλαπλασίους αὐτῶν ὄντας καὶ ἐν μὲν τῇ οἰκίᾳ πολλοὺς, ἐν δὲ τῷ θεάτρῳ ὀλίγους, πολλῷ πλείους αὐτῶν ὄντας: “We say, there are many men in a village, and but few in Athens, though the number in this last be many times larger; so, too, we say, there are many persons in a house, and but few in the theatre, though the number in this last may be many times more. Ibid. See before, page 254, and Hermes, p. 223. told that the slain were a thousand, the saved a hundred; in such case our knowledge becomes adequate and complete. d It is in the contemplation of these two quantities thus defined, the continuous by figure, the discrete by number, that we behold them rendered subjects for the two noblest of sciences, the first of them for geometry, the second for arithmetic; from which two, (and not from mere experiments, as some have hastily asserted,) both the knowledge of nature, and the utilities of common life, are in the greatest part derived. It is here we see the rise of those mathematical sciences, arithmetic, geometry, music, &c. which the ancients esteemed so essential to a liberal education. Nor can we believe there is any one now but must acknowledge, that a mind properly tinged with such noble speculations, (supposing there be no want of genius, or of courage,) is qualified to excel in every superior scene of life. Far more honourable they surely are, than the arts of riding a horse, or of wielding a sword, those accomplishments usually assigned to our youth of distinction, and for the sake of which alone they are often sent into distant countries, as if there were nothing to be taught them at home, nor any thing in a gentleman worth cultivating but his body. We would not undervalue these bodily accomplishments, (for perfection of every sort is certainly worth aiming at ;) but we would wish them to be rated as much below the mental, as the body itself is inferior to the mind. There is an elegant account of the sciences above mentioned in the Republic of Plato. Glaucus (one of the persons of the dialogue) takes paius to recommend them from their usefulness in human life: arithmetic for accounts and distributions; geometry for encampments and mensurations; music for solemn festivals in honour of the gods; and astronomy for agriculture, for navigation, and the like. Socrates, on his part, denies not the truth of all this, but still insinuates, that they were capable of answering an end more sublime. "You are pleasant," says he, "in your seeming to fear the multitude, lest you should be thought to enjoin certain sciences that are useless. It is, indeed, no contemptible matter, though a difficult one, to believe, that through these particular sciences the soul has an organ purified and enlightened, which is destroyed and blinded by studies of other kind; an organ better worth saving than a thousand eyes; inasmuch as truth becomes visible through this alone."e These, that we have here mentioned, appear to be the only d See Hermes, p. 218, and note, p. 222. • The above is an attempt to translate the following elegant passage of Plato: Ηδὺς εἶ, ὅτι ἔοικας δεδιότι τοὺς πολλοῦς, μὴ δοκῇς ἄχρηστα μαθήματα προστάττειν Tò d'otiv où пávu paûλov, åλλà xaλendν πιστεῦσαι, ὅτι ἐν τούτοις τοῖς μαθήμασιν ἑκαστοῖς ὅργανόν τι ψυχῆς ἐκκαθαίρεται, καὶ ἀναζωπυρεῖται, ἀπολλύμενον καὶ τυφ λούμενον ὑπὸ τῶν ἄλλων ἐπιτηδευμάτων, κρεῖττον ὂν σωθῆναι μυρίων ὀμματων· μόνῳ γὰρ αὐτῷ ἀλήθεια ὁρᾶται. Plat. de Repub. lib. vii. p. 527. edit Serran. Hermes, page 202. species of quantity; inasmuch as other things are called quantities, not from themselves, but with reference to these. Thus we say, that there is much white, because the superficies, which it covers, is much; and that an action was long, because the time was long during which it was transacted. And hence it is, that, if any one is to explain the quantity of an action, as, for example, the length of the Trojan war, he explains it by the time, saying, it was a war of ten years. So when we give the quantity of any thing white, we define it by the superficies, because, as that is in quantity, so also is the white.' We further observe, that quantity continuous and discrete may be said to blend themselves with all things. Thus in substances, let Mount Athos represent the former; the army of Xerxes, the latter. In colours, let us view the former in the uniform blueness of a clear sky; the latter, in the many and diversified tints of a rainbow. In sounds we find quantity discrete belonging to speech or language, it being the essence of articulation, that every syllable should be distinct. The continuous, on the contrary, naturally suggests itself to our ears, when we hear yellings, howlings, and heavy psalmody. In motions, when a grasshopper moves by leaps, we behold quantity discrete; when a ship sails smoothly, we behold quantity continuous. The motion of all animals, that have feet, (whether they leap or not,) by being alternate, is of the discrete kind: but it is fabled of the gods, that, when they moved as gods, it was under one continued progression of their whole frame together; to which Virgil, they say, alludes, in speaking of Venus, Et vera incessu patuit dea. En. i. 411. The mind, though devoid of corporeal extension, admits what is analogous to these two species of quantity, and recognises their force even within the sacred recesses of itself. For what can be more truly united in perfect continuity, than the terms which compose a self-evident truth? And how is this continuity still further extended, when by the union of two such truths there is produced a third, under the indissoluble connection of a demonstrative syllogism? If there was not this syllogistic continuity, there might indeed be other continuities, but it would never be in our power to prove any thing concerning them. Again, when we consider either many propositions, without reference to a syllogism; or many independent terms, without reference to a proposition; what have we then but quantity discrete? Philosophical arrangements? Treasures, as capable of being numbered, estimated, and recorded, as those which the miser commits to his coffers. 1 Κυρίως δὲ ποσὰ ταῦτα λέγεται μόνα τὰ εἰρημένα, τὰ δὲ ἄλλα πάντα κατὰ συμβεβηκός· εἰς ταῦτα γὰρ ἀποβλέποντες καὶ τὰ ἄλλα ποσὰ λέγομεν· οἷον πολὺ τὸ λευκον λέγεται, τῷγε τὴν ἐπιφάνειαν πολλὴν εἶναι· καὶ ἡ πρᾶξις μακρὰ, τῷγε τὸν χρόνον, κ. τ. λ. Aristot. Praed. p. 32. edit. Sylb. |