CHAPTER IX.

CONCERNING QUANTITY-ITS TWO SPECIES-THEIR CHARACTERS. TIME AND PLACE THEIR CHARACTERS. PROPERTY OF QUANTITY, WHAT. QUANTITIES RELATIVE. FIGURE AND NUMBER, THEIR EFFECT UPON QUANTITY-IMPORTANCE OF THIS EFFECT. SCIENCES MATHEMATICAL APPERTAIN TO IT-THEIR USE, ACCORDING TO PLATO. HOW OTHER BEINGS PARTAKE OF QUANTITY. ANALOGY, FOUND IN MIND. COMMON SENSE AND GENIUS, HOW DISTINGUISHed. AMAZING EFFICACY OF THIS GENUS IN AND THROUGH THE world. ILLUSTRATIONS.

THE attribute of substance, standing in arrangement next to quality, is quantity; the former having precedence, as being supposed more universal; while the latter, at least in appearance, seems not to extend beyond body.

Out of natural bodies is the visible world composed, and we may contemplate them in different manners; either one body, taken by itself and alone; or many bodies, taken collectively and at once. When Virgil says of the oak,

Quantum vertice ad auras

Ætherias, tantum radice ad Tartara tendit;

or when Milton informs us, that

Behemoth, biggest born of earth, unheaved
His vastness;

Geor. ii. 291.

Par. Lost, vii. 471.

in these instances we have only one body, taken by itself and alone, and this naturally suggests the idea of magnitude. But when in Virgil we read,

Quam multa in sylvis autumni frigore primo
Lapsa cadunt folia;

or when in Milton,

Thick as autumnal leaves, that strew the brooks
In Vallombrosa;

Æn. vi. 309.

Par. Lost, i. 302.

in these instances we have many bodies taken collectively and at once, and this naturally suggests the idea of multitude.

Horace gives the two species together in his fine address to Augustus:

Cum tot sustineas et tanta negotia.

Horat. Epist. 1. ii. 1.

Now in magnitude and multitude we behold these two primary, these two grand and comprehensive species, into which the genus of quantity is divided; magnitude, from its union, being called quantity continuous; multitude, from its separation, quantity discrete.P

- Τοῦ δὲ ποσοῦ τὸ μέν ἐστι διωρισμένον, τὸ δὲ συνεχές. Aristot. Præd. p. 30. edit. Sylb.

Of the continuous kind is every solid; also the bound of every solid, that is, a superficies; and the bound of every superficies, that is, a line; to which may be added those two concomitants of every body, namely, time and place. Of the discrete kind are fleets, armies, herds, flocks, the syllables of sounds articulate, &c.

We have mentioned formerly, when we treated of time, that every now or present instant was a boundary or term at which the past ended and the future began; and that it was in the perpetuity of this connection that time became continuous. In like manner within every line may be assumed infinite such connectives, under the character of points; and within every superficies, under the character of lines; and within every solid, under the character of superficies; to which connectives these quantities owe their continuity. And hence a specific distinction, attending all quantities continuous, that their several parts everywhere coincide in a common boundary or connective."

It is not so with quantities discrete; for here such coincidents is plainly impossible. Let us suppose, for example, a multitude of squares, x, y, z. &c.

Here if the line AB, where the square x ends, were the same with the line CD, where the square y begins, and EF in like manner the same with GH, they would no longer be a multitude of squares, but one continuous parallelogram; such as

Another specific character belonging to the solid body, the superficies, and the line, (all of which are quantities continuous,) is, that their parts have a definite position within some definite whole; while in quantities discrete, that is in multitudes, such position is no way requisite. In the most perfect continuous quantities, such as beams of timber, blocks of marble, &c. it is with difficulty the parts can change position, without destruction to the quantity, taken as continuous. But a herd of cattle, or an army of soldiers, may change position as often as they please, and no damage arise to the multitude, considered as a multitude. It must be remembered, however, that this character of po

See Hermes, lib. i. c. 7. p. 146.

See Arist. Prædic. p. 31. edit. Sylb. Ἡ δὲ γραμμὴ συνεχής ἐστιν, κ.τ.λ. This character is described to be πρός τινα κοινὸν ὅρον συνάπτειν. Ibid.

• Ετι, τὰ μὲν ἐκ θέσιν ἐχόντων πρὸς λλnλa tŵv èv avtoîs μopíwv ovvéστηKE' οἷον τὰ μὲν τῆς γραμμῆς μόρια θέσιν ἔχει πρὸς ἄλληλα, κ. τ. λ. Arist. Præd. p. 31. edit. Sylb.

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

ι “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 de voTepov. 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:

γ

A

moves quicker

B

Let A and B be two spheres that are moving, and let A be the quicker moving sphere, B the slower; and let the slower have moved through the space y d in the time n; it is evident that the quicker will

have moved through the same space in a less time. Let it have moved through it in the time (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 y d, 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 o has passed through the space yκ, the quicker sphere A will have passed through it in a less time; so that the time (0 will be again divided by the quicker body. But this being so divided, the space y k 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' is as

K. T.λ.

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 uèv 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 is one." Metaph. A. Kep. e. p. 88. edit. Sylb.

† Ἴδιον δὲ μάλιστα τοῦ ποσοῦ, τὸ ἴσον καὶ ἄνισον λέγεσθαι. Arist. Præed. p. 34.

• 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 τὸν, 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 deinite 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 ἔξη: “ 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."

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 nos proμévov, 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.
Sylb.

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 οἰκείας ἰδιότητος, καὶ ποσοῦ τι μέτρον σύμμετρον τῇ ἰδιότητι· οὐ γὰρ σχῆμα μόνον ἐπιφέρει μεθ ̓ ἑαυτοῦ τὸ εἶδος, ἀλλὰ καὶ μέγεθος, δ 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 voulçera: "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.

223.

See before, page 254, and Hermes, p.