Chapter 49

Quantity is an accumulation of units—for the unit is not called quantity. When one unit and one unit are combined, they become two. Thus quantity is not division, but an accumulation and addition of units. For, to divide two into separate units of one, this is division; but to say that one and one are two, this, rather, is addition.

One must know that quantity is the measure itself and the number—that which measures and that which numbers. Quanta, however, are subject to number and measure; in other words, they are the thing that are measured and numbered. Of the quanta, some are discrete and some are continuous. The quantum is continuous when one thing is measured, as when we have one piece of wood two or three cubits long, or a stone, or something of the sort. Being one, it is measured, and for this reason it is called continuous. Quanta, however, are discrete which are separated from each other, as in the case of ten stones or ten palm trees, for these are separated from each other. These, then, are said to be numbered, unless because of their small size and great number they are measured by the measure of something of the sort, as is grain and the like.

Those things are defined as continuous whose parts touch upon a certain common limit. Thus, since a two-cubit piece of wood, that is to say, a piece two cubits long, is one piece, then the end of one cubit and the beginning of the other are one. For they are joined together and connected, and they are not divided from each other. Discrete things are those whose parts do not touch upon a common limit, as in the case of ten stones. For, should you count off five and five, they will have no common limit connecting them. And should you put something in between this five and that five, then there will be eleven and not ten. The terms themselves, continuous and discrete, make this plain.

Now, among the discrete quanta come number and speech. By number we here mean things which are counted. And things which are counted are absolutely discrete, as has been shown. Speech, too, is discrete, for speech is counted in its words, and its parts do not have a common connecting limit. Thus, if the sentences has ten words and you separate them into groups of five, then they have no common limit connecting them. And so, should you add something in the middle, then there will be eleven and not ten. In the same way, the word is counted in its syllables, since it has no common limit connecting them together. Take the word Socrates, for example. Between the syllable so and the syllable era there is no common limit to connect them.

There are five continuous quanta: solid, surface, line, space, and time. One should know that the point is quantityless. This is because, being dimensionless, it is neither measured nor counted. The line, however, has one dimension, for it is length without breadth. Consequently, it is reckoned a continuous quantum. Since it is one, it is measured and its parts do have a common limit connecting them, which is the point in between. Now, dmcpdcvsioc, or surface, which is the outer part of the solid, is derived from cjxxtvsoai, to appear. It has two dimensions: length and breadth. Since it is one, it is measured and its parts do have a common limit connecting them, which is the line in between. Moreover, one should know that the flat and even surface is called a plane, whereas that which is uneven and warped is just called a surface. The solid has three dimensions: length, breadth, and depth or thickness. Since it is one, it is measured and its parts do have a common limit connecting them, which is the plane. Space is the surface of the air, for the space in which you are is a surface, that is to say, the terminating surface of the air containing you. As a surface, it is reckoned a continuous quantum. Time also is measured in the past and the future, and its parts have a common connecting limit, which is the present instant of time. The instant is quantityless. Notice, then, that there are three things which are quantityless: the unit, the point, and the instant. The following seven are properly called quanta: (1) number; (2) speech; (3) time; (4) space; (5) line; (6) surface; and (7) solid.

Those things which are considered in quanta, such as action, movement, color, and the like, we call quanta per accidens. For example, if the action and motion take place over a great length of time, we speak of much action and much motion; if over a short space of time, then we speak of a little. Similarly, if there is whiteness in an extensive body, we say much white; if in a small body, then we say little.

Furthermore, the quantum may be finite or infinite. That, then, which can be measured or counted is finite. On the other hand, that is infinite which by some degree of excessiveness exceeds all measure and number. And the term great and very great are used in the sense of infinite, as when we speak of ‘the very great compassion of God or the ‘great mystery of the dispensation of God the Word.

One should know that in the category of relation Aristotle places great and small, much and little, greater and smaller, less and more, double and half, and the like. Now, we say that under different aspects it is possible to place the same thing in different categories. Thus, when number and measure signify what has been explained above, they are put under quantity. On the other hand, when they have a mutual relation and are spoken of in relation to each other, then they are put under relation. Thus, ‘great is great in relation to ‘small and ‘double is double in relation to ‘half, and so on with the rest. In so far as the solid is physical, it comes under substance; but, in so far as mathematical, that is to say, measurable, it comes under quantity. And again, size and numerical quantity belong to quantity. Thus, size is measured and numerical quantity counted. And the term ‘how great refers to size, whereas ‘how many refers to numerical quantity.

There are three properties of the quantum, and they are called consequences. The first is the property of its not having any contrary in itself. Thus, in itself the solid has no contrary. However, in so far as it may happen to be white, it will have some contrary, namely, the black. One must furthermore know that there is no other number which is contrary to the number two, for, if there is any, there will be many of them. This is because all the other numbers would be contrary, in which case nature would have been unjust in opposing several contraries of one thing. For it is impossible for there to be several contraries to one thing.

The second property is that of not admitting of more or less. Thus, two palm trees cannot be more than two palm trees, and neither can two men be more than two men.

That which has no contrary does not admit of more or less.

The third property is that to every quantum and to quantum alone there may be equal and inequal. Thus, a line may be equal to a line or not equal to it.

[35] {Περὶ ποσοῦ καὶ ποσότητος.} Ποσότης ἐστὶ σωρεία μονάδων: τὴν μὲν γὰρ μονάδα οὔ φασι ποσότητα ἀλλ' ἀρχὴν ποσότητος. Μονάδος οὖν καὶ μονάδος συνερχομένων γίνονται δύο, ὥστε οὐ διαίρεσίς ἐστιν ἡ ποσότης ἀλλὰ σωρεία καὶ συνάφεια μονάδων: τὸ μὲν γὰρ τὰ δύο διελεῖν εἰς ἓν καὶ ἓν ἀνὰ μέρος, τοῦτο διαίρεσίς ἐστι, τὸ δὲ τὸ ἓν καὶ ἓν δύο εἰπεῖν, τοῦτο μᾶλλον συνάφειά ἐστιν. Ἰστέον δέ, ὅτι ποσότης μέν ἐστιν αὐτὸ τὸ μέτρον καὶ ὁ ἀριθμὸς ὁ μετρῶν καὶ ἀριθμῶν, ποσὰ δὲ τὰ τῷ ἀριθμῷ καὶ τῷ μέτρῳ ὑποκείμενα ἤγουν τὰ μετρούμενα καὶ ἀριθμούμενα. Τῶν δὲ ποσῶν τὰ μέν εἰσι διωρισμένα, τὰ δὲ συνεχῆ. Συνεχὲς μὲν οὖν ἐστιν, ὅτε ἕν ἐστι τὸ μετρούμενον, ὥσπερ ἓν ξύλον εὑρίσκεται δίπηχυ, τρίπηχυ, ἢ λίθος ἤ τι τῶν τοιούτων, καὶ ἓν ὑπάρχον μετρεῖται καὶ λέγεται συνεχές. Διωρισμένα δέ εἰσι τὰ ἀπ' ἀλλήλων κεχωρισμένα ὡς ἐπὶ δέκα λίθων ἢ δέκα φοινίκων: ταῦτα γὰρ κεχωρισμένα εἰσὶν ἀπ' ἀλλήλων. Ταῦτα οὖν ἀριθμεῖσθαι λέγονται, εἰ μὴ διὰ σμικρότητα καὶ πλῆθος μετρηθῶσι μοδίῳ ἤ τινι τῶν τοιούτων ὥσπερ σῖτος ἤ τι τοιοῦτον. Ὁρίζονται δὲ τὰ μὲν συνεχῆ, ὧν τὰ μόρια πρός τινα κοινὸν ὅρον συνάπτουσιν: ἑνὸς γὰρ ὄντος τοῦ ξύλου τοῦ διπήχεος ἤγουν δύο πήχεις ἔχοντος τὸ τέλος τοῦ ἑνὸς πήχεος καὶ ἡ ἀρχὴ τοῦ ἄλλου πήχεος μία ἐστί, κεκολλημέναι γάρ εἰσι καὶ συνημμέναι καὶ οὔκ εἰσιν ἀπ' ἀλλήλων διῃρημέναι: τὰ δὲ διωρισμένα, ὧν τὰ μόρια πρός τινα κοινὸν ὅρον οὐ συνάπτουσιν ὡς ἐπὶ δέκα λίθων. Ἐὰν γὰρ ἀριθμήσῃς πέντε καὶ πέντε, οὐκ ἔχουσι κοινὸν ὅρον τὸν συνάπτοντα αὐτούς: εἰ γὰρ δώσεις τι μεταξὺ τῶν πέντε καὶ τῶν πέντε, γίνονται ἕνδεκα καὶ οὐ δέκα: καὶ αὐτὸ δὲ τὸ ὄνομα δηλοῖ τοῦ συνεχοῦς καὶ τοῦ διωρισμένου. Ὑπὸ μὲν οὖν τὸ διωρισμένον ποσὸν ἀνάγεται ὁ ἀριθμὸς καὶ ὁ λόγος. Ἀριθμὸν δὲ λέγομεν ἐνταῦθα τὰ ἀριθμούμενα: τὰ γὰρ ἀριθμούμενα πάντα διωρισμένα εἰσίν, ὡς δέδεικται. Καὶ ὁ λόγος δὲ διωρισμένος ἐστίν: ὁ γὰρ λόγος ἀριθμούμενος ταῖς λέξεσιν οὐκ ἔχει κοινὸν ὅρον τὸν συνάπτοντα τὰ μόρια αὐτοῦ. Ἐὰν γὰρ ἔχῃ δέκα λέξεις ὁ λόγος καὶ διέλῃς αὐτὰς εἰς πέντε καὶ πέντε, οὐκ ἔχουσι κοινὸν ὅρον τὸν συνάπτοντα αὐτάς: ἐὰν γὰρ παρεντεθῇ τι ἐν τῷ μέσῳ, γίνονται ἕνδεκα καὶ οὐ δέκα. Ὁμοίως καὶ ἡ λέξις ἀριθμουμένη συλλαβαῖς οὐκ ἔχει κοινὸν ὅρον τὸν συνάπτοντα τὰς συλλαβάς, οἷον Σωκράτης: μεταξὺ τῆς « Σω » συλλαβῆς καὶ τῆς « κρα » οὐκ ἔστι κοινὸς ὅρος συνάπτων αὐτάς. Συνεχῆ δὲ ποσὰ πέντε: σῶμα, ἐπιφάνεια, γραμμή, τόπος, χρόνος. Χρὴ δὲ γινώσκειν, ὅτι ἡ στιγμὴ ἄποσός ἐστιν: οὐ γὰρ μετρεῖται οὐδὲ ἀριθμεῖται, ἐπειδὴ οὐκ ἔχει οὐδεμίαν διάστασιν. Ἡ δὲ γραμμὴ ἔχει μίαν διάστασιν, ἔστι γὰρ μῆκος ἀπλατὲς οἷον: αὕτη οὖν ὑπὸ τὸ συνεχὲς ποσὸν ἀνάγεται. Μία γὰρ οὖσα μετρεῖται καὶ τὰ μόρια αὐτῆς ἔχουσι κοινὸν ὅρον συνάπτοντα αὐτὰ τὴν μεταξὺ στιγμήν. Ἐπιφάνεια δέ ἐστι τὸ ἔξω μέρος τοῦ σώματος, παρὰ τὸ φαίνεσθαι. Ἔχει δὲ δύο διαστάσεις, μῆκος καὶ πλάτος. Καὶ αὕτη δὲ μία οὖσα μετρεῖται, καὶ τὰ μόρια αὐτῆς ἔχουσι κοινὸν ὅρον συνάπτοντα αὐτὰ τὴν μεταξὺ γραμμήν. Χρὴ δὲ γινώσκειν, ὅτι ἡ μὲν ὁμαλὴ καὶ ἴση ἐπιφάνεια ἐπίπεδος λέγεται, ἡ δὲ ἀνώμαλος καὶ σκολιὰ ἁπλῶς ἐπιφάνεια. Τὸ δὲ σῶμα ἔχει τρεῖς διαστάσεις: μῆκος, πλάτος, βάθος ἤγουν πάχος: καὶ ἓν ὑπάρχον μετρεῖται, καὶ τὰ μόρια αὐτοῦ ἔχουσι κοινὸν ὅρον συνάπτοντα αὐτὰ τὴν ἐπιφάνειαν. Καὶ ὁ τόπος δὲ ἐπιφάνειά ἐστι τοῦ ἀέρος: ὁ γὰρ τόπος σου ἐπιφάνεια ἤγουν τὸ τέλος τοῦ περιέχοντός σε ἀέρος ἐστὶ καὶ ὡς ἐπιφάνεια ὑπὸ τὸ συνεχὲς ποσὸν ἀνάγεται. Καὶ ὁ χρόνος δὲ μετρεῖται εἰς τὸν παρεληλυθότα καὶ εἰς τὸν μέλλοντα, καὶ ἔχουσι τὰ μόρια αὐτοῦ κοινὸν ὅρον συνάπτοντα αὐτὰ τὸ νῦν: τὸ δὲ νῦν ἄποσόν ἐστιν. Ἰδοὺ οὖν τρία εἰσὶν ἄποσα: ἡ μονὰς καὶ ἡ στιγμὴ καὶ τὸ νῦν. Καὶ κυρίως μὲν ταῦτα τὰ ἑπτὰ λέγονται ποσά: ἀριθμός, λόγος, χρόνος, τόπος, γραμμή, ἐπιφάνεια, σῶμα. Κατὰ συμβεβηκὸς δὲ λέγομεν ποσὰ καὶ τὰ ἐν αὐτοῖς θεωρούμενα, πρᾶξιν, κίνησιν, χρῶμα καὶ τὰ τοιαῦτα: οἷον, εἰ ἐν πολλῷ χρόνῳ γένηται ἡ πρᾶξις καὶ ἡ κίνησις, φαμὲν πολλὴν πρᾶξιν καὶ πολλὴν κίνησιν, εἰ δὲ ἐν ὀλίγῳ, ὀλίγην. Ὁμοίως καί, εἰ ἐν πολλῷ σώματι λευκότης εἴη, φαμὲν πολὺ λευκὸν, εἰ δὲ ἐν ὀλίγῳ, ὀλίγον. Ἔτι τοῦ ποσοῦ τὸ μέν ἐστιν ὡρισμένον, τὸ δὲ ἀόριστον. Ὡρισμένον μὲν οὖν ἐστι τὸ δυνάμενον μετρεῖσθαι ἢ ἀριθμεῖσθαι, τὸ δὲ ἀόριστον τὸ ὑπεροχῇ τινι ὑπερβάλλον πᾶν μέτρον καὶ πάντα ἀριθμόν: καὶ λέγεται μέγα καὶ πολὺ ἀορίστως, ὡς λέγομεν: Πολλὴ ἡ εὐσπλαγχνία τοῦ θεοῦ, μέγα τὸ μυστήριον τῆς τοῦ θεοῦ λόγου οἰκονομίας. Χρὴ δὲ γινώσκειν, ὅτι ὁ Ἀριστοτέλης τὸ μέγα καὶ μικρὸν πολύ τε καὶ ὀλίγον καὶ τὸ μεῖζον καὶ τὸ μικρότερον καὶ ὀλιγώτερον καὶ πλείω καὶ διπλάσιον καὶ ἥμισυ καὶ τὰ τοιαῦτα ὑπὸ τὰ πρός τι τίθησι. Λέγομεν οὖν, ὅτι δυνατὸν τὸ αὐτὸ πρᾶγμα κατ' ἄλλον καὶ ἄλλον σκοπὸν ὑπὸ ἄλλην καὶ ἄλλην κατηγορίαν ἀνάγεσθαι. Ὡς μὲν γὰρ ἀριθμὸν καὶ μέτρον δηλοῦντα τὰ προειρημένα ὑπὸ τὸ ποσὸν ἀνάγονται, ὡς δὲ σχέσιν ἔχοντα πρὸς ἄλληλα καὶ ὡς πρὸς ἄλληλα λεγόμενα ὑπὸ τὰ πρός τι: τὸ γὰρ μέγα πρὸς τὸ μικρὸν λέγεται μέγα καὶ τὸ διπλάσιον πρὸς τὸ ἥμισυ, ὁμοίως καὶ τὰ λοιπά. Τὸ δὲ σῶμα, καθὸ μὲν φυσικόν ἐστιν, ὑπὸ τὴν οὐσίαν ἀνάγεται, καθὸ δὲ μαθηματικὸν ἤγουν μετρούμενον, ὑπὸ τὸ ποσόν. Ἔτι τοῦ ποσοῦ τὸ μὲν μέγεθος, τὸ δὲ πλῆθος. Τὸ μὲν οὖν μέγεθος μετρεῖται, τὸ δὲ πλῆθος ἀριθμεῖται. Ἀκολουθεῖ δὲ τὸ πηλίκον τῷ μεγέθει, τὸ δὲ ποσὸν τῷ πλήθει. Ἴδια δέ εἰσι τοῦ ποσοῦ τρία, ἅτινα λέγονται παρακολουθήματα: πρῶτον τὸ μὴ ἔχειν τι ἐναντίον καθ' αὑτό: τὸ γὰρ σῶμα αὐτὸ καθ' αὑτὸ οὐκ ἔχει τι ἐναντίον, καθὸ δέ ἐστι τυχὸν λευκόν, ἔχει τι ἐναντίον, τὸ μέλαν. Χρὴ δὲ γινώσκειν, ὅτι τῷ δύο ἀριθμῷ οὐκ ἔστιν ἐναντίος ἄλλος ἀριθμός: εἰ γὰρ ἔσται, πολλοὶ ἔσονται. Πάντες γὰρ οἱ ἄλλοι ἀριθμοὶ ἐναντίοι ἔσονται, καὶ εὑρίσκεται ἡ φύσις ἄδικος ἑνὶ πολλὰ ἀντιτάξασα: οὐ γὰρ ἐνδέχεται ἑνὶ πολλὰ εἶναι ἐναντία. Δεύτερον τὸ μὴ ἐπιδέχεσθαι τὸ μᾶλλον καὶ ἧττον. Οὐδὲ γὰρ οἱ δύο φοίνικές εἰσι μᾶλλον δύο ἤπερ οἱ δύο ἄνθρωποι. Τὸ δὲ μὴ ἔχον ἐναντίον οὐκ ἐπιδέχεται τὸ μᾶλλον καὶ ἧττον. Τρίτον δέ, ὃ μόνῳ καὶ παντὶ τῷ ποσῷ, τὸ ἴσον καὶ ἄνισον. Γραμμὴ γὰρ γραμμῆς ἴση ἐστὶ καὶ ἄνισος.