Bekker (Berlin, 1831) · Joachim (1908)
Joachim (1908)
Chapter 1 (968a1–972b33)
968a
1 Ἆρά γ' εἰσὶν ἄτομοι γραμμαί, καὶ ὅλως ἐν ἅπασι τοῖς
ποσοῖς ἐστί τι ἀμερές, ὥσπερ ἔνιοί φασιν; εἰ γὰρ ὁμοίως
ὑπάρχει τό τε πολὺ καὶ τὸ μέγα καὶ τὰ ἀντικείμενα τούτοις,
τό τε ὀλίγον καὶ τὸ μικρόν, τὸ δ' ἀπείρους σχεδὸν διαιρέσεις
5 ἔχον οὐκ ἔστιν ὀλίγον ἀλλὰ πολύ, φανερὸν ὅτι πεπερασμένας
ἕξει τὰς διαιρέσεις τὸ ὀλίγον καὶ τὸ μικρόν·
εἰ δὲ πεπερασμέναι αἱ διαιρέσεις, ἀνάγκη τι εἶναι ἀμερὲς
μέγεθος, ὥστε ἐν ἅπασιν ἐνυπάρξει τι ἀμερές, ἐπείπερ καὶ
τὸ ὀλίγον καὶ τὸ μικρόν. ἔτι εἰ ἔστιν ἰδέα γραμμῆς, ἡ δ'
10 ἰδέα πρώτη τῶν συνωνύμων, τὰ δὲ μέρη πρότερα τοῦ ὅλου
τὴν φύσιν, διαιρετὴ ἂν εἴη αὐτὴ ἡ γραμμή, τὸν αὐτὸν
δὲ τρόπον καὶ τὸ τετράγωνον καὶ τὸ τρίγωνον καὶ τὰ ἄλλα
σχήματα, καὶ ὅλως ἐπίπεδον αὐτὸ καὶ σῶμα· συμβήσεται
γὰρ πρότερ' ἄττα εἶναι τούτων. ἔτι εἰ σώματός ἐστι
15 στοιχεῖα, τῶν δὲ στοιχείων μηδὲν πρότερον, τὰ δὲ μέρη τοῦ
ὅλου πρότερα, ἀδιαίρετον ἂν εἴη τὸ πῦρ καὶ ὅλως τῶν τοῦ
σώματος στοιχείων ἕκαστον, ὥστ' οὐ μόνον ἐν τοῖς νοητοῖς
ἀλλὰ καὶ ἐν τοῖς αἰσθητοῖς ἐστί τι ἀμερές. ἔτι δὲ κατὰ
τὸν Ζήνωνος λόγον ἀνάγκη τι μέγεθος ἀμερὲς εἶναι, εἴπερ
20 ἀδύνατον μὲν ἐν πεπερασμένῳ χρόνῳ ἀπείρων ἅψασθαι,
καθ' ἕκαστον ἁπτόμενον, ἀνάγκη δ' ἐπὶ τὸ ἥμισυ πρότερον
ἀφικνεῖσθαι τὸ κινούμενον, τοῦ δὲ μὴ ἀμεροῦς πάντως ἔστιν
ἥμισυ. εἰ δὲ καὶ ἅπτεται τῶν ἀπείρων ἐν πεπερασμένῳ
χρόνῳ τὸ ἐπὶ τῆς γραμμῆς φερόμενον, τὸ δὲ θᾶττον ἐν
25 τῷ ἴσῳ χρόνῳ πλεῖον διανύει, ταχίστη δ' ἡ τῆς διανοίας
κίνησις, κἂν ἡ διάνοια τῶν ἀπείρων ἐφάπτοιτο καθ' ἕκαστον
1ARE there indivisible lines? And, generally, is there a simple unit in every class of quanta Ὁ} § 1. Some people maintain this thesis on the following grounds :— (i) If we recognize the validity of the predicates ‘ big’ and ‘great’, we must equally recognize the validity of their opposites, ‘little’ and ‘small’. Now that which admits practically an infinite number of divisions, is ‘big’ not ‘little’ (or ‘great’ not ‘small’).? Hence, the ‘little’ quantum and the ‘small’ quantum will clearly admit only a finite number of divisions.2 But if the divisions are finite in number, there must be a simple magnitude. Hence in all classes of quanta there will be found a simple unit, since in all of them the predicates ‘little’ and ‘small’ apply.
9 (ii) Again, if there is an Idea of line, and if the Idea is first of the things 5called by its name’:—then, since the parts are by nature prior to their whole, the Ideal Line must be indivisible? And, on the same principle, the Ideal Square, the . Ideal Triangle, and all the other Ideal Figures—and, generalizing, the Ideal Plane and the Ideal Solid—must be without parts: for otherwise it will result that there are elements prior to each. of them.
14 (iii) Again, if Body consists of elements,* and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indivisible: for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought.* (iv) Again, Zeno’s argument proves that there must be simple magnitudes.'_ For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any ‘stretch’ which is not simple, motion—unless there be simple magnitudes—involves that the moving body 10touches successively one-by-one an infinite number of points in a finite time: which is impossible.?
But even if the body, which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the ‘stretch’ which it traverses in an equal time: and since the movement of thought is quickest of all movements :—it follows that thought too will come successively into contact with an infinity of objects in a finite time. ° And since ‘thought’s coming into contact with objects oneby-one’ is counting, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible, there must be such a thing as an ‘indivisible line’.* popular view, which regarded Earth, Air, Fire, and Water as the ‘ Letters’ of the Alphabet of Reality, and the physical universe as a complex of ‘ Syllables’ and ‘Words’ in which these four Letters are variously combined. But the grinciple of the argument 15would apply to the more refined forms which the theory assumes in the Zzsaeus of Plato and in Aristotle’s physical writings. The primordial triangles of the 7imaeus, gud Elements of all bodies, are presumably without physical parts, i.e. physically indivisible. And the Earth, Air, Fire, and Water, which (according to Aristotle) are the chemical constituents of all ὁμοιομερῆ---ηα therefore the primary constituents of all composite bodies—, are ‘ ra ἁπλᾶ σώματα᾽, although the character of each of them is dual, i.e. is exhibited in two of the four fundamental qualities. (For Aristotle’s theory of the Elements, cf. my article on ‘ Aristotle’s Conception of Chemical Combination’, Journal of Philology, No. 57.) ; (v) Again, the being of ‘indivisible lines’ (it is maintained) follows from the Mathematicians’ own statements. For if we accept their definition of ‘commensurate’ lines as those which are measured by the same unit of measurement,! and if we suppose that all commensurate lines actually are being measured,’ 20there will be some actual length, by which all of them will be measured.* And this length must be indivisible. For if it is divisible, its parts—since they are commensurate with the whole—will involve some unit of measurement measuring both them and their whole.. And thus the original 5: cf. also % 19) cannot be given by the English ‘there must be an indivisible line’ or ‘a line which is indivisible’. We must translate either as above, or by the plural ‘there must be indivisible lines’.
The argument (#23-4) is directed against a particular view of thought and of counting. ‘ Assume’—the writer says in effect—‘ that the moving body does in fact touch an infinity of points one-by-one in a finite time. According to your view that thought is the quickest of all movements, it will follow @ fortzor¢ that thought touches an infinity of objects one-by-one in a finite time: i.e. (according to your definition of counting) that we can count an infinite number in a finite time. But this is impossible. And the only way to avoid 25this absurdity, whz/st recognizing the fact of motion, is to postulate “ indivisible lines’’,’ The theory that thinking is a movement of the Soul was not held by Aristotle: for he argues in the de Anima (A.
9 (ii) Again, if there is an Idea of line, and if the Idea is first of the things 5called by its name’:—then, since the parts are by nature prior to their whole, the Ideal Line must be indivisible? And, on the same principle, the Ideal Square, the . Ideal Triangle, and all the other Ideal Figures—and, generalizing, the Ideal Plane and the Ideal Solid—must be without parts: for otherwise it will result that there are elements prior to each. of them.
14 (iii) Again, if Body consists of elements,* and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indivisible: for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought.* (iv) Again, Zeno’s argument proves that there must be simple magnitudes.'_ For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any ‘stretch’ which is not simple, motion—unless there be simple magnitudes—involves that the moving body 10touches successively one-by-one an infinite number of points in a finite time: which is impossible.?
But even if the body, which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the ‘stretch’ which it traverses in an equal time: and since the movement of thought is quickest of all movements :—it follows that thought too will come successively into contact with an infinity of objects in a finite time. ° And since ‘thought’s coming into contact with objects oneby-one’ is counting, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible, there must be such a thing as an ‘indivisible line’.* popular view, which regarded Earth, Air, Fire, and Water as the ‘ Letters’ of the Alphabet of Reality, and the physical universe as a complex of ‘ Syllables’ and ‘Words’ in which these four Letters are variously combined. But the grinciple of the argument 15would apply to the more refined forms which the theory assumes in the Zzsaeus of Plato and in Aristotle’s physical writings. The primordial triangles of the 7imaeus, gud Elements of all bodies, are presumably without physical parts, i.e. physically indivisible. And the Earth, Air, Fire, and Water, which (according to Aristotle) are the chemical constituents of all ὁμοιομερῆ---ηα therefore the primary constituents of all composite bodies—, are ‘ ra ἁπλᾶ σώματα᾽, although the character of each of them is dual, i.e. is exhibited in two of the four fundamental qualities. (For Aristotle’s theory of the Elements, cf. my article on ‘ Aristotle’s Conception of Chemical Combination’, Journal of Philology, No. 57.) ; (v) Again, the being of ‘indivisible lines’ (it is maintained) follows from the Mathematicians’ own statements. For if we accept their definition of ‘commensurate’ lines as those which are measured by the same unit of measurement,! and if we suppose that all commensurate lines actually are being measured,’ 20there will be some actual length, by which all of them will be measured.* And this length must be indivisible. For if it is divisible, its parts—since they are commensurate with the whole—will involve some unit of measurement measuring both them and their whole.. And thus the original 5: cf. also % 19) cannot be given by the English ‘there must be an indivisible line’ or ‘a line which is indivisible’. We must translate either as above, or by the plural ‘there must be indivisible lines’.
The argument (#23-4) is directed against a particular view of thought and of counting. ‘ Assume’—the writer says in effect—‘ that the moving body does in fact touch an infinity of points one-by-one in a finite time. According to your view that thought is the quickest of all movements, it will follow @ fortzor¢ that thought touches an infinity of objects one-by-one in a finite time: i.e. (according to your definition of counting) that we can count an infinite number in a finite time. But this is impossible. And the only way to avoid 25this absurdity, whz/st recognizing the fact of motion, is to postulate “ indivisible lines’’,’ The theory that thinking is a movement of the Soul was not held by Aristotle: for he argues in the de Anima (A.
968b
1 ἐν πεπερασμένῳ χρόνῳ, ὥστε εἰ τὸ καθ' ἕκαστον ἅπτεσθαι
τὴν διάνοιαν ἀριθμεῖν ἐστίν, ἐνδέχεται ἀριθμεῖν τὰ
ἄπειρα ἐν πεπερασμένῳ χρόνῳ. εἰ δὲ τοῦτο ἀδύνατον, εἴη
ἄν τις ἄτομος γραμμή. ἔτι καὶ ἐξ ὧν αὐτοὶ οἱ ἐν τοῖς
5 μαθήμασι λέγουσιν, εἴη ἄν τις ἄτομος γραμμή, ὡς φασίν,
εἰ σύμμετροί εἰσιν αἱ τῷ αὐτῷ μέτρῳ μετρούμεναι·
ὅσαι δ' εἰσὶ σύμμετροι, πᾶσαί εἰσι μετρούμεναι. εἴη γὰρ
ἄν τι μῆκος ᾧ πᾶσαι μετρηθήσονται. τοῦτο δ' ἀνάγκη
ἀδιαίρετον εἶναι. εἰ γὰρ διαιρετόν, καὶ τὰ μέρη μέτρου
10 τινὸς ἔσται· σύμμετρα γὰρ τῷ ὅλῳ. ὥστε μέρους τινὸς εἴη
διπλασία τὴν ἡμίσειαν, ἐπειδὴ τοῦτ' ἀδύνατον ἂν εἴη μέτρον.
ὡσαύτως δὲ καὶ αἱ μετρούμεναι ἅπαξ ὑπ' αὐτοῦ,
ὥσπερ πᾶσαι αἱ ἐκ τοῦ μέτρου σύνθετοι γραμμαί, ἐξ ἀμερῶν
σύγκεινται. τὸ δ' αὐτὸ συμβήσεται κἀν τοῖς ἐπιπέδοις·
15 πάντα γὰρ τὰ ἀπὸ τῶν ῥητῶν γραμμῶν σύμμετρα
ἀλλήλοις, ὥστε ἔσται τὸ μέτρον αὐτῶν ἀμερές. ἀλλὰ μὴν
εἴ τι τμηθήσεται μέτρον τινὰ τεταγμένην καὶ ὡρισμένην
γραμμήν, οὐκ ἔσται οὔτε ῥητὴ οὔτ' ἄλογος, οὔτε τῶν ἄλλων
οὐδεμία ὧν νῦν δὴ εἴρηται, οἷον ἀποτομὴν ἐκ δυοῖν ὀνομάτοιν·
20 ἀλλὰ καθ' αὑτὰς μὲν οὐδέ τινας ἕξουσι φύσεις, πρὸς
ἀλλήλας δὲ ἔσονται ῥηταὶ καὶ ἄλογοι. ἢ πρῶτον μὲν οὐκ
ἀνάγκη τὸ ἀπείρους ἔχον διαιρέσεις μὴ εἶναι μικρὸν καὶ
ὀλίγον· καὶ γὰρ τόπον καὶ μέγεθος καὶ ὅλως τὸ συνεχὲς
μικρὸν μὲν λέγομεν, καὶ ἐφ' ὧν μὲν ἁρμόττει τὸ ὀλίγον,
25 οὐ μὴν ἀλλ' ἀπείρους διαιρέσεις φαμὲν ἔχειν. ἔτι δ' εἰ
ἐν τῷ συνθέτῳ γραμμαί, κατὰ τούτων τῶν ἀτόμων λέγεται
1ch. 3) against all attempts to define the Soul as ‘that which moves itself,, and maintains that ‘itis impossible that movement should be a property of the Soul’ (l.c. 406% 2 ff.). Certain speculations of Plato in the 7zmaeus (which Aristotle criticizes, 1], c. 406” 26 ff.) regard thought as a movement: and Theophrastus and his pupil, Strato, are known to have maintained that thought was a movement of the Soul (cf. Apelt, Bectrage &c., p. 270). But we must not infer—as Apelt (l.c.) does—that Aristotle is not the author of the present treatise : still less that it was written by Theophrastus or Strato. For we are here dealing with an argumentum ad hominem, and the writer is not himself committed to the view that thought is a movement of the Soul. ᾿ unit of measurement would turn out 5to be twice one of its parts, viz. twice its half But since this is impossible, there must be an indivisible unit of measurement.? And just as all the lines, which are compounded of the unit, are composed of ‘simples’, so also the lines, which the unit measures once, consist of ‘ simples’.® And the same can be shown to follow in the plane figures too. For all the squares, which are drawn on the. rational lines, are-commensurate with one another ; and therefore (by the preceding argument) their unit of measurement will be simple.‘ But if (per cmpossibile) any such unit-square be cut along any prescribed and determinate line, that line will be neither ‘rational’ nor ‘irrational’, nor any of the recognized kinds of (irrational) lines which produce rational squares, such as the ‘apotome’ or the ‘line ex duobus n®minibus’. Such lines, at which the unit-square might be divided, will have no nature of their own at all; though, relatively to one another, they will be rational or 10irrational.’
consist ultimately of a finite number of minimal squares, not themselves divisible into any smaller plane figures.
In order to understand the argument, and the fallacy on which it rests, it will be necessary to explain certain technical terms of Greek geometry. (1) The expression τὰ ἀπὸ τῶν ῥητῶν γραμμῶν (1. 15) must—in accordance with Euclid’s invariable usage—mean ‘ the squares on the ῥηταὶ ypappai ’, The noun implied is τετράγωνα : but τὸ ἀπό followed by the genitive is constantly used without τετράγωνον, and always means the square on suchand-such a line. (Hence Apelt is wrong in translating ‘ Alle F/achen mit rationalen Seitenlinien’.) (2) The proper meaning of ῥηταὶ γραμμαί will be seen from the following definitions of Euclid (Z/em. X) :—def. 3 ‘... given any straight line, there are an infinity of straight lines commensurate with it and an infinity incommensurate with it—incommensurate either in length only, ov both in length and in respect to the areas which 15they and it produce if squared (ai μὲν μήκει μόνον, ai δὲ καὶ δυνάμει: see below). Let the given straight line, and all the straight lines which are commensurate with it (whether commensurate both μήκει and δυνάμει, or δυνάμει only), be called “‘ Rational” (ῥηταί) : and let the straight lines, which are incommensurate with it, be called “ Irrational ” (ἄλογοι) ᾽ : def. 4 ‘ And let the square on the given straight line, and all the squares commensurate therewith, be called “ Rational”: and let the squares incommensurate with it be called “Irrational” ...’ (3) Any straight lines, which are multiples of the same unit of length, are said to be σύμμετροι μήκει. Ife.g. the unit of measurement be # ποδιαία (the line one foot long), all lines containing a whole number of feet are σύμμέτροι μήκει. But lines which donot contain a whole number of the same unit of length are said to be σύμμετροι δυνάμει, if they form squares containing a whole number of the same unit of area. All lines, 20which are σύμμετροι μήκει, are necessarily also σύμμετροι duvdyec—but the converse does not hold (Eucl. Z/em. X, prop. 9, Coroll.).
We are now in a position to understand the argument of »14-16. The writer extends the relative term ‘rational’ illegitimately (making it absolute), just as before he illegitimately extended the relative term ‘commensurate’. All ‘rational’ lines are by definition δυνάμει σύμμετροι : and therefore all squares on rational lines are commensurate. And if we suppose them actually measured, there will be an actual minimal square, the unit of measurement of them all (cf. above, ” 6-8): and this minimal square can be shown to be indivisible—i.e. not to contain smaller plane figures—as before the unit-line was shown to be ἀδιαίρετον (" 8-12). But—unless we assume that all lines consist of indivisible and equal unttlines—we cannot assume that a@// lines are ‘rational’ in Euclid’s sense, nor that a// squares are commensurate with one another.
§ 2. To these 25arguments we must make the following answers :— att debeihs (i) (a) In the first place, it does not follow that the quantum, which admits an infinite number of divisions, is not ‘small’ or ‘little’.
consist ultimately of a finite number of minimal squares, not themselves divisible into any smaller plane figures.
In order to understand the argument, and the fallacy on which it rests, it will be necessary to explain certain technical terms of Greek geometry. (1) The expression τὰ ἀπὸ τῶν ῥητῶν γραμμῶν (1. 15) must—in accordance with Euclid’s invariable usage—mean ‘ the squares on the ῥηταὶ ypappai ’, The noun implied is τετράγωνα : but τὸ ἀπό followed by the genitive is constantly used without τετράγωνον, and always means the square on suchand-such a line. (Hence Apelt is wrong in translating ‘ Alle F/achen mit rationalen Seitenlinien’.) (2) The proper meaning of ῥηταὶ γραμμαί will be seen from the following definitions of Euclid (Z/em. X) :—def. 3 ‘... given any straight line, there are an infinity of straight lines commensurate with it and an infinity incommensurate with it—incommensurate either in length only, ov both in length and in respect to the areas which 15they and it produce if squared (ai μὲν μήκει μόνον, ai δὲ καὶ δυνάμει: see below). Let the given straight line, and all the straight lines which are commensurate with it (whether commensurate both μήκει and δυνάμει, or δυνάμει only), be called “‘ Rational” (ῥηταί) : and let the straight lines, which are incommensurate with it, be called “ Irrational ” (ἄλογοι) ᾽ : def. 4 ‘ And let the square on the given straight line, and all the squares commensurate therewith, be called “ Rational”: and let the squares incommensurate with it be called “Irrational” ...’ (3) Any straight lines, which are multiples of the same unit of length, are said to be σύμμετροι μήκει. Ife.g. the unit of measurement be # ποδιαία (the line one foot long), all lines containing a whole number of feet are σύμμέτροι μήκει. But lines which donot contain a whole number of the same unit of length are said to be σύμμετροι δυνάμει, if they form squares containing a whole number of the same unit of area. All lines, 20which are σύμμετροι μήκει, are necessarily also σύμμετροι duvdyec—but the converse does not hold (Eucl. Z/em. X, prop. 9, Coroll.).
We are now in a position to understand the argument of »14-16. The writer extends the relative term ‘rational’ illegitimately (making it absolute), just as before he illegitimately extended the relative term ‘commensurate’. All ‘rational’ lines are by definition δυνάμει σύμμετροι : and therefore all squares on rational lines are commensurate. And if we suppose them actually measured, there will be an actual minimal square, the unit of measurement of them all (cf. above, ” 6-8): and this minimal square can be shown to be indivisible—i.e. not to contain smaller plane figures—as before the unit-line was shown to be ἀδιαίρετον (" 8-12). But—unless we assume that all lines consist of indivisible and equal unttlines—we cannot assume that a@// lines are ‘rational’ in Euclid’s sense, nor that a// squares are commensurate with one another.
§ 2. To these 25arguments we must make the following answers :— att debeihs (i) (a) In the first place, it does not follow that the quantum, which admits an infinite number of divisions, is not ‘small’ or ‘little’.
969a
1 τὸ μικρόν, καὶ ἄπειροι στιγμαὶ ἐνυπάρχουσιν. ᾗ δὲ γραμμή,
διαίρεσις κατὰ στιγμήν, καὶ ὁμοίως καθ' ὁποιανοῦν ἀπείρους
ἂν ἔχοι διαιρέσεις ἅπασα ἡ μὴ ἄτομος. ἔνιαι δὲ τούτων
εἰς μακρὰ καὶ ἄπειροι οἱ λόγοι. πᾶσαν δὲ τμηθῆναι τὸν
5 ἐπιταχθέντα δυνατὸν τὴν μὴ ἄτομον. ἔτι εἰ τὸ μέγα ἐκ
μικρῶν τινῶν σύγκειται, ἢ οὐθὲν ἔσται τὸ μέγα, ἢ τὸ πεπερασμένας
ἔχον διαιρέσεις οὐ μέγα ἔσται. τὸ γὰρ ὅλον
τὰς τῶν μερῶν ἔχει διαιρέσεις ὁμοίως. εὔλογον δ' ἐστὶ τό
τε σμικρὸν πεπερασμένας ἔχειν διαιρέσεις καὶ τὸ μέγα
10 ἀπείρους, οὕτως ἀξιοῦσιν. ὥστε φανερὸν ὅτι οὐκ ἐν τούτῳ λέγοιτο
τὸ μέγα καὶ τὸ μικρόν, τῷ πεπερασμένας ἔχειν καὶ
ἀπείρους διαιρέσεις. εἰ δ' ὅτι καὶ ἐν ἀριθμοῖς τὸ ὀλίγον
πεπερασμένας ἔχει διαιρέσεις, καὶ ἐν γραμμαῖς τις ἀξιοίη
τὸ μικρόν, εὔηθες. ἐκεῖ μὲν γὰρ ἐξ ἀμερῶν τε ἡ γένεσις,
15 καὶ ἔστι τι ὃ τῶν ἀριθμῶν ἀρχή ἐστι, καὶ πᾶς ὁ μὴ ἄπειρος
πεπερασμένας ἔχει διαιρέσεις· ἐπὶ δὲ τῶν μεγεθῶν οὐχ
ὁμοίως. οἱ δ' ἐν τοῖς εἴδεσι τὰς ἀτόμους κατασκευάζοντες
τοὔλαττον ἴσως ἀξίωμα λαμβάνουσι τοῦ προκειμένου, τὸ τιθέναι
τούτων ἰδέας· καὶ τρόπον τινὰ ταῦτ' ἀναιροῦσι δι' ὧν
20 δεικνύουσιν. καὶ γὰρ διὰ τούτων τῶν λόγων ἀναιρεῖται τὰ
εἴδη. πάλιν δὲ τῶν σωματικῶν στοιχείων εὔηθες τὸ ἀμερῆ
ἀξιοῦν. εἰ γὰρ αὖ καὶ ἀποφαίνονταί τινες οὕτως, ἀλλὰ πρός
γε τὴν ὑποκειμένην σκέψιν αὐτὸ τὸ ἐξ ἀρχῆς λαμβάνουσιν.
μᾶλλον δὲ ὅσῳ μᾶλλον τὸ ἐξ ἀρχῆς δόξειαν ἀναλαμβάνεσθαι,
25 τόσῳ μᾶλλον δοκεῖ διαιρετὸν εἶναι σῶμα καὶ
μῆκος καὶ τοῖς ὄγκοις καὶ τοῖς διαστήμασιν. ὁ δὲ τοῦ Ζήνωνος
λόγος οὐ συμβιβάζει οὐ συμπεπερασμένῳ χρόνῳ τῶν
ἀπείρων ἅπτεσθαι τὸ φερόμενον ὡδὶ τὸν αὐτὸν τρόπον. ὁ
γὰρ χρόνος καὶ τὸ μῆκος ἄπειρον καὶ πεπερασμένον λέγεται,
30 καὶ τόσας ἔχει διαιρέσεις. οὐδὲ δὴ τὸ καθ' ἕκαστον
ἅπτεσθαι τῶν ἀπείρων τὴν διάνοιαν οὐκ ἔστιν ἀριθμεῖν, εἰ
ἄρα τις καὶ νοήσειεν οὕτως ἐφάπτεσθαι τῶν ἀπείρων τὴν
διάνοιαν. ὅπερ ἴσως ἀδύνατον· οὐ γὰρ ἐν συνεχέσι καὶ
1For we apply the predicate ‘small’ to place and magnitude, and generally to the continuous (and in some quanta the predicate ‘little’ is suitably applied)! ; and nevertheless regards the text, I adopt Apelt’s reading in 1. 19, ὧν δυνάμεις ῥηταί, οἷον ἀποτομὴ ἢ ἡ ἐκ δυοῖν dvoudrow for the MSS. ὧν δὴ viv [viv δή N] εἴρηται, οἷον ἀποτομὴν ἐκ δυοῖν ὀνομάτοιν. .
The lines called ἐκ δυοῖν ὀνομάτοιν and ἀποτομή are two types of Irrationals (i.e. μήκει ἀσύμμετροι, but δυνάμει σύμμετροι) which play a large part in Euclid, Z7em. Bk. X.
The line ἐκ δυοῖν ὀνομάτοιν is defined in Prop. 36 thus : —‘ If two rational straight lines, which are commensurate δυνάμει only, be added together, the whole line is irrational: let it be called “the line ἐκ δύο ὀνομάτων "᾿ :—i.e. the A B Ὁ line AC is 5that type of ‘ Irrational’ (irrational relatively to 42 and BC) which is called ‘ex duobus nominibus’, if it is such, that 4B? is commensurate with BC’, but AB τ ἘΠ ΤΕΥ ΘΟΒΉΒΡΝΣ with BC. AB and BC are called the “ ὀνόματα ᾿ ο : :
The ἀποτομή is defined in Prop. 73 thus :—‘ If from a rational line there be taken a rational line commensurate with the whole line δυνάμει only, the remainder is irrational: let it be called an “ dworouy”’ :--i.e. if the line AB be divided at C, so that 44? is commensurate with CS” but AB is x Το τ Β incommensurate (μήκει) with CZ, then AC is called an ἀποτομή. The complementary part of the whole lin (viz. CB) is called relatively to AC its προσαρμόζουσα (cf. Propp. 79-84). We might illustrate these two types of ‘Irrationals’ thus 5 Let the two ὀνόματα be 1 and 4/5, Then the whole line, 42+ BC,=(14+ 4/5).
κοντα να ‘we affirm that these quanta admit an infinite number of divisions.
25. (i) (b) Moreover, if in the composite magnitude there are 10contained (indivisible) lines,! the predicate ‘ small’ is applied to these indivisible lines, and each of them contains an infinite number of points. But each of them, gud line, admits of division at a point, and equally at any and every point: hence each of these indivisible lines would admit an infinite number of divisions just like the non-indivisible lines.2 Moreover, some amongst the non-indivisible lines are ‘small’. But every non-indivisible line admits of division in accordance with any prescribed ratio: and the ratios, (in accordance with which any such line may be divided), are infinite in number.* however, omitted by Z*.) Apelt defends the MSS. reading, but interprets καὶ ἐφ᾽ ὧν---ὀλίγον as part of the subject of the sentence, taking μικρόν as predicate of the whole. This seems difficult, because (a) the μὲν [ἐφ᾽ ὧν μὲν] is purely gratuitous, and (4) there is no reason why the writer - { should over-ride the distinction between μικρόν and ὀλίγον.
If the τό be 15retained, the clause must, I think, be treated as parenthetical and interpreted as above.
(i) (c) Again, since the ‘great’ is compounded of certain ‘smalls’, the ‘great’ will either be nothing, or it will be identical with that which admits a finite number of divisions.! For the whole admits the divisions admitted by its parts: i.e. 2ts divisions are finite or infinite, according as their divisions are finite or infinite.* It is unreasonable that, whilst the small admits a finite number of divisions only, the great should admit an infinite number; and yet this is what the advocates of the theory postulate.® It is clear, therefore, that it is not φημ admitting a finite and an infinite number of divisions that quanta are called ‘ small’ and ‘great’ respectively. And to argue that, because zz numbers the ‘little’ number admits a finite number of divisions, therefore zz Jines the ‘small’ line must admit only a finite number of divisions, is childish. For in numbers the more 20complex are developed out of ‘simples’, and there is a determinate something from which the whole series of the numbers starts, and every number which is not infinite admits The argument of the whole passage (>25-%5) I take to be as follows:—‘Every composite length contains lines. According to the theory, some amongst these lines are ‘‘indivisible”. But every one of these lines, gud line, contains an infinity of points, and admits therefore an infinity of divisions: for a point is that at which a line can be divided. Yet by comparison with the whole (composite) length, all the “indivisible” lines, and at least some of the divisible lines, are “small”. Hence infinitely-divisible quanta may be “small™,’ The λόγοι (% 4) are, I presume, the numerical ratios in which any line may be divided.
or a finite number of divisions ; but in magnitudes the case is not parallel.
17 (ii) As to those who try to establish the being of the indivisible lines by arguments drawn from the Ideal 25Lines, we may perhaps say that, in positing Ideas of these quanta, they are assuming a premiss too narrow to carry their conclusion ; and, by arguing thus, they in a sense destroy the premisses which they use to prove their conclusion. For their arguments destroy the Ideas.?
(iii) Again, as to the corporeal elements,* it is childish to postulate them as ‘simple’. For even though some physicists do as a matter of fact make this statement about them, yet to assume this for the present inquiry * is a petitio principit. Or rather, the more obviously the argument would appear to involve a petitio principiz, the more the opinion is confirmed that Solids and Lengths® are divisible in bulk and distance.° ἡ that, because you are unable to solve Zeno’s argument, you should make yourselves slaves of your inability, and should commit yourselves to still greater errors, in the endeavour to support your incompetence.' 6 (v) As to what they say about ‘commensurate lines ’—that all 30lines, because commensurate *, are measured by one and the same actual unit of measurement—this is sheer sophistry ; nor is it in the least in accordance with the mathematical assumption as to commensurability. For the mathematicians do not make the assumption in this form, nor is it of pase use to them.
Moreover, it is actually 3 inconsistent to postulate both that every line becomes commensurate, and that there is a common measure of all commensurate lines.‘ te ade ' This and the preceding argument are directed against the fourth argument ( 18—54) of the advocates of indivisible lines.
The lines called ἐκ δυοῖν ὀνομάτοιν and ἀποτομή are two types of Irrationals (i.e. μήκει ἀσύμμετροι, but δυνάμει σύμμετροι) which play a large part in Euclid, Z7em. Bk. X.
The line ἐκ δυοῖν ὀνομάτοιν is defined in Prop. 36 thus : —‘ If two rational straight lines, which are commensurate δυνάμει only, be added together, the whole line is irrational: let it be called “the line ἐκ δύο ὀνομάτων "᾿ :—i.e. the A B Ὁ line AC is 5that type of ‘ Irrational’ (irrational relatively to 42 and BC) which is called ‘ex duobus nominibus’, if it is such, that 4B? is commensurate with BC’, but AB τ ἘΠ ΤΕΥ ΘΟΒΉΒΡΝΣ with BC. AB and BC are called the “ ὀνόματα ᾿ ο : :
The ἀποτομή is defined in Prop. 73 thus :—‘ If from a rational line there be taken a rational line commensurate with the whole line δυνάμει only, the remainder is irrational: let it be called an “ dworouy”’ :--i.e. if the line AB be divided at C, so that 44? is commensurate with CS” but AB is x Το τ Β incommensurate (μήκει) with CZ, then AC is called an ἀποτομή. The complementary part of the whole lin (viz. CB) is called relatively to AC its προσαρμόζουσα (cf. Propp. 79-84). We might illustrate these two types of ‘Irrationals’ thus 5 Let the two ὀνόματα be 1 and 4/5, Then the whole line, 42+ BC,=(14+ 4/5).
κοντα να ‘we affirm that these quanta admit an infinite number of divisions.
25. (i) (b) Moreover, if in the composite magnitude there are 10contained (indivisible) lines,! the predicate ‘ small’ is applied to these indivisible lines, and each of them contains an infinite number of points. But each of them, gud line, admits of division at a point, and equally at any and every point: hence each of these indivisible lines would admit an infinite number of divisions just like the non-indivisible lines.2 Moreover, some amongst the non-indivisible lines are ‘small’. But every non-indivisible line admits of division in accordance with any prescribed ratio: and the ratios, (in accordance with which any such line may be divided), are infinite in number.* however, omitted by Z*.) Apelt defends the MSS. reading, but interprets καὶ ἐφ᾽ ὧν---ὀλίγον as part of the subject of the sentence, taking μικρόν as predicate of the whole. This seems difficult, because (a) the μὲν [ἐφ᾽ ὧν μὲν] is purely gratuitous, and (4) there is no reason why the writer - { should over-ride the distinction between μικρόν and ὀλίγον.
If the τό be 15retained, the clause must, I think, be treated as parenthetical and interpreted as above.
(i) (c) Again, since the ‘great’ is compounded of certain ‘smalls’, the ‘great’ will either be nothing, or it will be identical with that which admits a finite number of divisions.! For the whole admits the divisions admitted by its parts: i.e. 2ts divisions are finite or infinite, according as their divisions are finite or infinite.* It is unreasonable that, whilst the small admits a finite number of divisions only, the great should admit an infinite number; and yet this is what the advocates of the theory postulate.® It is clear, therefore, that it is not φημ admitting a finite and an infinite number of divisions that quanta are called ‘ small’ and ‘great’ respectively. And to argue that, because zz numbers the ‘little’ number admits a finite number of divisions, therefore zz Jines the ‘small’ line must admit only a finite number of divisions, is childish. For in numbers the more 20complex are developed out of ‘simples’, and there is a determinate something from which the whole series of the numbers starts, and every number which is not infinite admits The argument of the whole passage (>25-%5) I take to be as follows:—‘Every composite length contains lines. According to the theory, some amongst these lines are ‘‘indivisible”. But every one of these lines, gud line, contains an infinity of points, and admits therefore an infinity of divisions: for a point is that at which a line can be divided. Yet by comparison with the whole (composite) length, all the “indivisible” lines, and at least some of the divisible lines, are “small”. Hence infinitely-divisible quanta may be “small™,’ The λόγοι (% 4) are, I presume, the numerical ratios in which any line may be divided.
or a finite number of divisions ; but in magnitudes the case is not parallel.
17 (ii) As to those who try to establish the being of the indivisible lines by arguments drawn from the Ideal 25Lines, we may perhaps say that, in positing Ideas of these quanta, they are assuming a premiss too narrow to carry their conclusion ; and, by arguing thus, they in a sense destroy the premisses which they use to prove their conclusion. For their arguments destroy the Ideas.?
(iii) Again, as to the corporeal elements,* it is childish to postulate them as ‘simple’. For even though some physicists do as a matter of fact make this statement about them, yet to assume this for the present inquiry * is a petitio principit. Or rather, the more obviously the argument would appear to involve a petitio principiz, the more the opinion is confirmed that Solids and Lengths® are divisible in bulk and distance.° ἡ that, because you are unable to solve Zeno’s argument, you should make yourselves slaves of your inability, and should commit yourselves to still greater errors, in the endeavour to support your incompetence.' 6 (v) As to what they say about ‘commensurate lines ’—that all 30lines, because commensurate *, are measured by one and the same actual unit of measurement—this is sheer sophistry ; nor is it in the least in accordance with the mathematical assumption as to commensurability. For the mathematicians do not make the assumption in this form, nor is it of pase use to them.
Moreover, it is actually 3 inconsistent to postulate both that every line becomes commensurate, and that there is a common measure of all commensurate lines.‘ te ade ' This and the preceding argument are directed against the fourth argument ( 18—54) of the advocates of indivisible lines.
969b
1 ὑποκειμένοις ἡ τῆς διανοίας κίνησις, ὥσπερ ἡ τῶν φερομένων.
εἰ δ' οὖν καὶ ἐγχωρεῖ κινεῖσθαι οὕτως, οὐκ ἔστι τοῦτο
ἀριθμεῖν· τὸ γὰρ ἀριθμεῖν ἐστὶ τὸ μετὰ ἐπιστάσεως. ἀλλ'
ἄτοπον ἴσως τὸ μὴ δυναμένους λύειν τὸν λόγον δουλεύειν
5 τῇ ἀσθενείᾳ, καὶ προσεξαπατᾶν ἑαυτοὺς μείζους ἀπάτας,
βοηθοῦντας τῇ ἀδυναμίᾳ. τὸ δ' ἐπὶ τῶν συμμέτρων γραμμῶν,
ὡς ὅτι αἱ πᾶσαι τῷ αὐτῷ τινὶ καὶ ἑνὶ μετροῦνται,
κομιδῇ σοφιστικὸν καὶ ἥκιστα κατὰ τὴν ὑπόθεσιν τὴν ἐν
τοῖς μαθήμασιν· οὔτε γὰρ ὑποτίθενται οὕτως, οὔτε χρήσιμον
10 αὐτοῖς ἐστίν. ἅμα δὲ καὶ ἐναντίον πᾶσαν μὲν γραμμὴν
σύμμετρον γίνεσθαι, πασῶν δὲ τῶν συμμέτρων κοινὸν μέτρον
εἶναι ἀξιοῦν. ὥστε γελοῖον τὸ κατὰ τὰς ἐκείνων δόξας
καὶ ἐξ ὧν αὐτοὶ λέγουσι φάσκοντες δείξειν, εἰς ἐριστικὸν
ἅμα καὶ σοφιστικὸν ἐκκλίνειν λόγον, καὶ ταῦθ' οὕτως
15 ἀσθενῆ. πολλαχῇ γὰρ ἀσθενής ἐστι καὶ πάντα τρόπον διαφυγεῖν
καὶ τὰ παράδοξα καὶ τοὺς ἐλέγχους. ἔτι δ' ἄτοπον
ἂν εἴη διὰ μὲν τὸν Ζήνωνος λόγον παραπεπεῖσθαί
τινας ἀτόμους ποιεῖν γραμμάς, τῷ μὴ ἔχειν ἀντειπεῖν,
διὰ δὲ τῆς εὐθείας εἰς τὴν ἡμιόλιον κίνησιν, ἣν ἀναγκαῖον
20 εὐθὺς τέμνειν ἀπείρων μεταξὺ πιπτουσῶν περιφερειῶν καὶ
διαστημάτων ὄντων, καὶ πάλιν διὰ τὴν τῶν ἴσων κύκλων
εὔπειστον, ὅτι ἀνάγκη ἂν ὅτι κινηθῇ, μεῖζον ἡμικύκλιον
κινεῖσθαι, καὶ ὅσα ἄλλα τοιαῦτα τεθεώρηται περὶ τὰς
γραμμὰς μὴ οἷόν τε ἐνδέχεσθαι τοιαύτην δή τινα γενέσθαι
25 κίνησιν ὥστ' ἐφ' ἑκάστην τῶν μεταξὺ μὴ πίπτειν πρότερον·
πολὺ γὰρ ταῦτα μᾶλλον ὁμολογούμενα ἐκείνων. ὅτι μὲν
οὖν ἔκ γε τῶν εἰρημένων λόγων οὔτ' ἀναγκαῖον ἀτόμους
εἶναι γραμμὰς οὔτε πιθανόν, φανερόν. ἔτι δὲ καὶ ἐκ τῶνδε
γένοιτ' ἂν φανερώτερον. πρῶτον μὲν ἐκ τῶν ἐν τοῖς μαθήμασι
30 δεικνυμένων καὶ τιθεμένων, ἃ οὐ δίκαιον ἢ πιστοτέροις
λόγοις κινεῖν. οὔτε γὰρ ὁ τῆς γραμμῆς οὔτε ὁ τῆς
εὐθείας ὅρος ἐφαρμόσει τῇ ἀτόμῳ διὰ τὸ μήτε μεταξὺ
τινῶν εἶναι μήτ' ἔχειν μέσον. ἔπειτα πᾶσαι αἱ γραμμαὶ
1BI The writer urges (i) that Zeno’s argument involves a fallacy, which the advocates of indivisible lines have failed to detect (* 26-30). (ii) That the movement of thought (‘psychical process’) is not analogous to the movement of a body. The latter is essentially conditioned by the continuity of the path traversed and the continuity of the body moving : for physical movement takes place in a material swdstratum—i.e. a solid material body—and along a path in space. (iii) That if the movement of thought were analogous to the movement of a body, more than this would be required to constitute ‘counting’. For to ‘count’ is not merely to traverse a continuous path, coming into instantaneous contact with the infinite succession of points, into which that path may be mathematically resolved: to ‘count’ essentially 5involves Jausing at the successive steps of the process. (iv) That the argument drawn from ‘counting’ is an extravagant supposition by which the advocates of ‘ indivisible lines’ are endeavouring to support themselves in an erroneous position—a position really due to their incompetence in failing to detect Zeno’s fallacy.
Bee The MSS. read ὡς ὅτι ai πᾶσαι. This presumably means ‘e.g. that’ or ‘viz. that’, But it is very doubtful whether ὡς ὅτι could be used in this way as equivalent to the ordinary οἷον ὅτι. I propose to read ὡς, ὅτι (σύμμετροι), ai πᾶσαι.
καὶ ἐναντίον.
* b6-12. This is directed against the fifth argument of the advocates of indivisible lines (cf. above, 4-14).
It is difficult to be sure of the meaning of » 10-iz, owing to the obscurity of the argument which is being attacked. I think the point of the criticism is as follows. The mathematical definition of commensurate lines can always be satisfied, in the sense that, given any line 42, you ἡ can always find a line ‘commensurate’ 10with it: i.e. any line can become ‘commensurate’ with some line. But though αὐ lines are ‘commensurate’ in this sense, they are not all commensurate with one another, and have not got one and the same common measure. Yet the advocates of ‘indivisible’ lines maintain doth (i) that any line can become ‘ commenHence their procedure is ridiculous, since, whilst professing that they are going to demonstrate their thesis in accordance with the opinions of the mathematicians,-and by premisses drawn from the mathematicians’ own ‘statements, they lapse into an argument which is a mere piece of contentious and sophistical dialectic—and such a feeble piece of sophistry too ! For it zs feeble in many respects, and totally (unable) to escape paradox on the one side, and destructive scientific criticism on the other.
Moreover, it would be absurd for people to be led astray by Zeno’s argument, and to be persuaded—because they cannot refute it—to invent indivisible lines: and yet to pay no attention to all those theorems 15concerning lines, in which it is proved that it is impossible for a movement to be generated such that in it the moving thing does zot fall successively on each of © the intervening points before reaching the end-point. . For the theorems in question are far better established, and more generally admitted, than the arguments of Zeno.”
surate’, avd (ii) that all commensurate lines have a common measure: and these two propositions are inconsistent. For (i) is true only if ‘commensurate’ be used in a relative sense; and then (ii) is false. Whilst (ii) is true only if ‘commensurate’ be used in an absolute sense ; and then (i) is false.
Ὁ by2-16. Bekker reads ὥστε γελοῖον τὸ [τὸ om. W®*] κατὰ [καὶ N] ras ἐκείνων δόξας καὶ ἐξ ὧν αὐτοὶ λέγουσι φάσκοντες δείξειν, εἰς ἐριστικὸν ἅμα καὶ σοφιστικὸν ἐκκλίνειν [ἐγκλίνειν LPW® ἐγκλίναι N] λόγον, καὶ ταῦθ᾽ οὕτως ἀσθενῆ. πολλαχῇ [πολλαχῶς ΤΟΡ 8] γὰρ ἀσθενής ἐστι καὶ πάντα τρόπον διαφυγεῖν καὶ τὰ παράδοξα καὶ τοὺς ἐλέγχους.
By reading φάσκοντας in ]. 13 very tolerable 20sense may be made of the first sentence. Apelt follows N and reads τὸ καὶ ras rd... ἐγκλίναι .... * ridiculum est et illorum (sc. mathematicorum) placita et ea, quibus ipsi argumenta sua superstruunt, in sophisticas captiones detorquere.’ But αὐτοί (cf. ” 4, to which this refers) is most naturally taken as ‘the mathematicians’: and in any case Apelt’s interpretation is not convincing.
- The last sentence seems to be corrupt. The general sense of the passage would be satisfied by πάντα τρόπον ἀδύνατος (or ἀδυνατεῖ) διαφυγεῖν ...1 but I hesitate to propose any reading. The point seems to be that the advocates of indivisible lines are exposed to a double fire. They are using as an argument what to common sense is ridiculous paradox, and what to professional mathematicians is demonstrably unscientific.
_ 2 by6-26. In the above paraphrase I think I have reproduced the general drift of this passage. Zeno showed that if a body is to move from A to B, it must touch all the intermediate points before reaching 2: 25i.e. it must traverse an infinity in a finite time. And he argued that motion is impossible. The advocates of indivisible lines replied: ‘ Motion is a ae 26 § 3. It is clear, then, that the being of indivisible lines is neither demonstrated nor rendered plausible—at any rate by the arguments which we have quoted. And this conclusion will grow clearer in the light of the following considerations :— 29 (A) Inthe first place,’ our result will be confirmed by reflection on the conclusions proved in mathematics, and on the assumptions? there laid down—conclusions and assumptions fact, and therefore—since Zeno’s argument is sound—the line 4B must consist of a finite number of indivisible unit-lines.” The writer here rejoins: ‘Geometry proves that there can be no motion without the phenomenon to which Zeno called attention. A motion, such as your theory requires—a motion in which the moving body does not traverse successively all the intermediate points—does not, and cannot, occur. And the theorems, in which 30geometry establishes this, are far more convincing than the arguments of Zeno.’
In other words :—Geometry, assuming motion to be a fact, shows that the moving thing does traverse an infinity of intervening points, and shows that there can be no motion in which this does not take place. The advocates of indivisible lines have made no attempt to refute these geometrical proofs. Their postulate of ‘indivisible lines’, even if it evaded Zeno, collides with these far more solid facts of geometry : for the kind of motion which would occur, if there were indivisible lines, is shown by geometry to be impossible.
Bee The MSS. read ὡς ὅτι ai πᾶσαι. This presumably means ‘e.g. that’ or ‘viz. that’, But it is very doubtful whether ὡς ὅτι could be used in this way as equivalent to the ordinary οἷον ὅτι. I propose to read ὡς, ὅτι (σύμμετροι), ai πᾶσαι.
καὶ ἐναντίον.
* b6-12. This is directed against the fifth argument of the advocates of indivisible lines (cf. above, 4-14).
It is difficult to be sure of the meaning of » 10-iz, owing to the obscurity of the argument which is being attacked. I think the point of the criticism is as follows. The mathematical definition of commensurate lines can always be satisfied, in the sense that, given any line 42, you ἡ can always find a line ‘commensurate’ 10with it: i.e. any line can become ‘commensurate’ with some line. But though αὐ lines are ‘commensurate’ in this sense, they are not all commensurate with one another, and have not got one and the same common measure. Yet the advocates of ‘indivisible’ lines maintain doth (i) that any line can become ‘ commenHence their procedure is ridiculous, since, whilst professing that they are going to demonstrate their thesis in accordance with the opinions of the mathematicians,-and by premisses drawn from the mathematicians’ own ‘statements, they lapse into an argument which is a mere piece of contentious and sophistical dialectic—and such a feeble piece of sophistry too ! For it zs feeble in many respects, and totally (unable) to escape paradox on the one side, and destructive scientific criticism on the other.
Moreover, it would be absurd for people to be led astray by Zeno’s argument, and to be persuaded—because they cannot refute it—to invent indivisible lines: and yet to pay no attention to all those theorems 15concerning lines, in which it is proved that it is impossible for a movement to be generated such that in it the moving thing does zot fall successively on each of © the intervening points before reaching the end-point. . For the theorems in question are far better established, and more generally admitted, than the arguments of Zeno.”
surate’, avd (ii) that all commensurate lines have a common measure: and these two propositions are inconsistent. For (i) is true only if ‘commensurate’ be used in a relative sense; and then (ii) is false. Whilst (ii) is true only if ‘commensurate’ be used in an absolute sense ; and then (i) is false.
Ὁ by2-16. Bekker reads ὥστε γελοῖον τὸ [τὸ om. W®*] κατὰ [καὶ N] ras ἐκείνων δόξας καὶ ἐξ ὧν αὐτοὶ λέγουσι φάσκοντες δείξειν, εἰς ἐριστικὸν ἅμα καὶ σοφιστικὸν ἐκκλίνειν [ἐγκλίνειν LPW® ἐγκλίναι N] λόγον, καὶ ταῦθ᾽ οὕτως ἀσθενῆ. πολλαχῇ [πολλαχῶς ΤΟΡ 8] γὰρ ἀσθενής ἐστι καὶ πάντα τρόπον διαφυγεῖν καὶ τὰ παράδοξα καὶ τοὺς ἐλέγχους.
By reading φάσκοντας in ]. 13 very tolerable 20sense may be made of the first sentence. Apelt follows N and reads τὸ καὶ ras rd... ἐγκλίναι .... * ridiculum est et illorum (sc. mathematicorum) placita et ea, quibus ipsi argumenta sua superstruunt, in sophisticas captiones detorquere.’ But αὐτοί (cf. ” 4, to which this refers) is most naturally taken as ‘the mathematicians’: and in any case Apelt’s interpretation is not convincing.
- The last sentence seems to be corrupt. The general sense of the passage would be satisfied by πάντα τρόπον ἀδύνατος (or ἀδυνατεῖ) διαφυγεῖν ...1 but I hesitate to propose any reading. The point seems to be that the advocates of indivisible lines are exposed to a double fire. They are using as an argument what to common sense is ridiculous paradox, and what to professional mathematicians is demonstrably unscientific.
_ 2 by6-26. In the above paraphrase I think I have reproduced the general drift of this passage. Zeno showed that if a body is to move from A to B, it must touch all the intermediate points before reaching 2: 25i.e. it must traverse an infinity in a finite time. And he argued that motion is impossible. The advocates of indivisible lines replied: ‘ Motion is a ae 26 § 3. It is clear, then, that the being of indivisible lines is neither demonstrated nor rendered plausible—at any rate by the arguments which we have quoted. And this conclusion will grow clearer in the light of the following considerations :— 29 (A) Inthe first place,’ our result will be confirmed by reflection on the conclusions proved in mathematics, and on the assumptions? there laid down—conclusions and assumptions fact, and therefore—since Zeno’s argument is sound—the line 4B must consist of a finite number of indivisible unit-lines.” The writer here rejoins: ‘Geometry proves that there can be no motion without the phenomenon to which Zeno called attention. A motion, such as your theory requires—a motion in which the moving body does not traverse successively all the intermediate points—does not, and cannot, occur. And the theorems, in which 30geometry establishes this, are far more convincing than the arguments of Zeno.’
In other words :—Geometry, assuming motion to be a fact, shows that the moving thing does traverse an infinity of intervening points, and shows that there can be no motion in which this does not take place. The advocates of indivisible lines have made no attempt to refute these geometrical proofs. Their postulate of ‘indivisible lines’, even if it evaded Zeno, collides with these far more solid facts of geometry : for the kind of motion which would occur, if there were indivisible lines, is shown by geometry to be impossible.
970a
1 σύμμετροι ἔσονται. πᾶσαι γὰρ ὑπὸ τῶν ἀτόμων μετρηθήσονται,
αἵ τε μήκει σύμμετροι καὶ αἱ δυνάμει. αἱ δὲ
ἄτομοι σύμμετροι πᾶσαι μήκει· ἴσαι γάρ· ὥστε καὶ δυνάμει.
εἰ δὲ τοῦτο, διαιρετὸν ἔσται τὸ τετράγωνον. ἔτι εἰ ἡ
5 περὶ τὴν μείζω τὸ πλάτος ποιεῖ παραβαλλομένη, τὸ ἴσον
τῶν ἀπὸ τῆς ἀτόμου καὶ τῆς ποδιαίας παραβαλλομένων
περὶ τὴν δίπουν ἔλαττον ποιήσει τὸ πλάτος τῆς ἀμεροῦς·
ἔσται ἔλαττον τὸ περὶ τῆς ἀτόμου. ἔτι εἰ ἐκ τριῶν δοθεισῶν
εὐθειῶν συνίσταται τρίγωνον, καὶ ἐκ τῶν ἀτόμων συσταθήσεται.
10 ἐν ἅπαντι δὲ ἰσοπλεύρῳ ἡ κάθετος ἐπὶ μέσην
πίπτει, ὥστε καὶ ἐπὶ τὴν ἄτομον. ἔτι εἰ τὸ τετράγωνον τῶν
ἀμερῶν διὰ μέσου ἐμπεσούσης καὶ καθέτου ἀχθείσης, ἡ τοῦ
τετραγώνου πλευρὰ τὴν κάθετον δύναται καὶ τὴν ἡμίσειαν
τῆς διαμέτρου, ὥστε οὐκ ἐλαχίστη. οὐδὲ διπλάσιον τὸ ἀπὸ
15 τῆς διαμέτρου χωρίον ἔσται τοῦ ἀπὸ τῆς ἀτόμου. ἀφαιρεθέντος
γὰρ τοῦ ἴσου ἡ λοιπὴ ἔσται ἐλάσσων τῆς ἀμεροῦς.
εἰ γὰρ ἴσως τετραπλάσιον ἂν ἔγραψεν ἡ διάμετρος, ἄλλα
δ' ἄν τις καὶ ἕτερα τοιαῦτα συνάγοι· πᾶσι γὰρ ὡς εἰπεῖν
ἐναντιοῦται τοῖς ἐν τοῖς μαθήμασιν. πάλιν τοῦ μὲν ἀμεροῦς
20 μία ἡ σύναψις, γραμμῆς δὲ δύο· καὶ γὰρ ὅλη ὅλης
ἅπτεται, καὶ κατὰ τὸ πέρας ἐξ ἐναντίας. ἔτι γραμμὴ
προστεθεῖσα οὐ ποιεῖ μείζω τὴν ὅλην· τὰ γὰρ ἀμερῆ συντιθέμενα
οὐ ποιήσει μεῖζον. ἔτι ἐκ δυοῖν ἀμεροῖν μηδὲν
γίνεσθαι συνεχὲς διὰ τὸ πλείους διαιρέσεις ἔχειν ἅπαν τὸ
25 συνεχές· ἅπασα δὲ γραμμὴ παρὰ τὴν ἄτομον συνεχὴς
οὐκ ἂν εἴη γραμμὴ ἄτομος. ἔτι εἰ ἅπασα γραμμὴ παρὰ
τῆς ἀτόμου καὶ ἴσα καὶ ἄνισα διαιρεῖται, καὶ μὴ ἐκ τριῶν
ἀτόμων καὶ ὅλως περιττῶν, ὥστ' ἀδιαίρετος ἡ ἄτομος.
ὁμοίως δὲ κἂν εἰ δίχα τέμνεται· πᾶσα γὰρ ἡ ἐκ τῶν
30 περιττῶν. εἰ δὲ δίχα μὲν μὴ πᾶσα τέμνεται ἀλλ' ἡ ἐκ
τῶν ἀρτίων, τὴν δὲ δίχα διαιρουμένην καὶ ὅσα δυνατὸν τέμνειν,
διαιρεθήσεται καὶ οὕτως ἡ ἄτομος, ὅταν ἡ ἐκ τῶν
ἀρτίων εἰς ἄνισα διαιρῆται. πάλιν εἰ τὸ κεκινημένον ἐν ᾧ
1The text of this passage is so corrupt that it seems hopeless to make out the details of the argument.
In Il. 19-21 the writer is clearly referring to the movement of a straight line about one of its terminal points, whereby a semicircle (and, ultimately, a circle) is generated. διάστημα is the regular term in Euclid for the distance at which, from a given point as centre, the circumference of a circle isdrawn. Cf. e.g. Eucl. Elem. 1.22... κέντρῳ μὲν τῷ Z, διαστήματι δὲ τῷ ΖΔ κύκλος γεγράφθω 6 ΔΚΛ. .., and so constantly. (διάστημα in fact=‘ radius’ a In 1. 19 we should read with Apelt διὰ δὲ(τὴν) τῆς εὐθείας εἰς τὸ ἡμικύκλιον [so NZ*: the other MSS. read ἡμιόλιον] κίνησιν, . . « But Apelt (in the Prolegg. to his text) proposes other emendations for the rest of the 5passage, which are not convincing. It is best to recognize that the passage is hopeless, until somebody can discover the exact geometrical theorems to which the writer is referring.
which we have no right to reject except on more convincing arguments than those adduced by the advocates of indivisible lines.
For (i) neither the definition of ‘line’, nor that of ‘ straight line’, will apply to the indivisible line, since the latter is not between any terminal points, and does not possess a middle.”
(ii) Secondly, all lines will be commensurate. For all lines -—both those which are commensurate in length, and those which produce commensurate squares—will be measured by the indivisible lines.
And the indivisible lines are all of them commensurate in length (for they are all equal to one another), and therefore also they all produce commensurate squares. But if so, then the square on any line will always be rational.* 4 (iii) Again, since, in a rectangle, the line 10applied at right angles tothe longerside determines the breadth of the figure: the rectangle, which is equal in area to the square on the indivisible j line (v.g. on the line one foot long), will, if applied to a line i double the indivisible line (v.g. to a line two feet long), have a breadth determined by a line shorter than the indivisible line: for its breadth will be less than the breadth of the square on the indivisible line.’
are all, gud infinitesimal, equal’: hence all commensurate μήκει, and therefore also commensurate δυνάμει ( 2-4). The point of the criticism is that the doctrine annihilates the mathei matical conceptions of Commensurate and Incommensurate, Rational and i Irrational. The passage should be compared with Euclid, E/em. X, deff. 2, 3 and 4 (see above, note on 96814) : and with Plato, 7) heaet. 147 D-148B. In the + Theaetetus, Theaetetus and Socrates the Younger are represented as having generalized certain results of the mathematician 15Theodorus (their master), and having divided all numbers into two series, thus :— Series 1: Those numbers which, if regarded as the areas of rectangular figures, are squares with whole numbers as their sides, e.g. 4, 9, 16, 25, ἄς.
The roots of these square numbers are what we should call ‘ rational’ or the sides of the squares are lines σύμμετροι μήκει, viz. containing tA numbers of the unit of length (the line one foot long).
Theaetetus and Socrates called the sides containing the squares in this series ‘ μήκη.
Series 2: Those numbers which, if regarded as the areas of rectangular figures with whole numbers as their sides, as E are oblongs; or, if regarded as squares, have not whole numbers as their sides. To this series belong e.g. 3, 5, 6, 7, 8, &c.: and the sides containing these squares—e.g. 4/3, /5, »/6, &c.—were called by Theaetetus and his friend ‘ duvdpes’, ie. δυνάμει σύμμετροι. (Cf. Theaet. 147D ἥ τε τρίπους καὶ ἡ πεντέπους δυνάμεις are not μήκει 20σύμμετροι τῇ ποδιαίᾳ. 710.,. 148B ὡς μήκει μὲν οὐ συμμέτρους ς D ἐκείναις, τοῖς δ᾽ ἐπιπέδοις ἃ δύνανται.) We should call the ‘sides’ of this series of squares ‘irrational square roots’ or ‘ surds’.
* 94-8. In this passage I adopt Apelt’s reading and interpretation throughout: v. Apelt, Avistotelis quae feruntur, &c., prolegg. pp. xiv, xv.
If we suppose the ‘ indivisible line’ to be one foot long (cf. Arist. Met. 1052. 33—ev ταῖς γραμμαῖς χρῶνται ws ἀτόμῳ τῇ ποδιαίᾳ), then a rectangle, applied to a line two feet A G B long, must—if its area is to be equal to the square on the indivisible line—have as its other side a line shorter than the indivisible line: which is absurd.
Let AB be the indivisible line, one foot long. Let BE be the line, two (iv) Again, since any three given straight lines can be combined to form a triangle, a triangle can also be formed by combining three given indivisible lines. Such a triangle will be equilateral: but in every equilateral 25triangle the perpendicular dropped from the apex bisects the base. Hence, in ᾿ the equilateral triangle whose sides are the indivisible lines, the ‘ indivisible’ base will be bisected by the perpendicular dropped from its apex.!
(v) Again, if the square can be constructed of Simples (i.e. with indivisible lines as its sides), then let its diagonal be drawn, and a perpendicular dropped from one angle on to the diagonal. The square on the side (i.e. the original square constructed feet long. Let CABD be the square on ABZ. If to the line BE there be applied a rectangular figure GF EB equal in area to CABD, FE or GB will be less than 42.
Though I accept Apelt’s interpretation, there are one or two difficulties to which attention should be called. (1) παραβάλλειν is the technical term constantly used in Euclid (cf. e.g. Elem. 1. 44, &c.) for ‘applying’ a rectangle or a parallelogram to a given line: i.e. for constructing such a figure with a given line as one of 30its sides. But (so far as I know) it is always the figure which ‘ παραβάλλεται᾽, not the side. Hence παραβαλλομένη here (9705) is suspicious.
(2) Euclid constantly uses the technical expression ‘Adros ποιεῖ τὴν AB’ to mean ‘[a rectangle applied to such-and-such a given line] makes as its other side the line 42’. But, whatever may have been the original significance of the phrase, there is no implication in Euclid’s usage that the side thus produced is shorter than the given line. So far as I have been able to discover, πλάτος ποιεῖν in Euclid (a) always has the accusative (e.g.
In Il. 19-21 the writer is clearly referring to the movement of a straight line about one of its terminal points, whereby a semicircle (and, ultimately, a circle) is generated. διάστημα is the regular term in Euclid for the distance at which, from a given point as centre, the circumference of a circle isdrawn. Cf. e.g. Eucl. Elem. 1.22... κέντρῳ μὲν τῷ Z, διαστήματι δὲ τῷ ΖΔ κύκλος γεγράφθω 6 ΔΚΛ. .., and so constantly. (διάστημα in fact=‘ radius’ a In 1. 19 we should read with Apelt διὰ δὲ(τὴν) τῆς εὐθείας εἰς τὸ ἡμικύκλιον [so NZ*: the other MSS. read ἡμιόλιον] κίνησιν, . . « But Apelt (in the Prolegg. to his text) proposes other emendations for the rest of the 5passage, which are not convincing. It is best to recognize that the passage is hopeless, until somebody can discover the exact geometrical theorems to which the writer is referring.
which we have no right to reject except on more convincing arguments than those adduced by the advocates of indivisible lines.
For (i) neither the definition of ‘line’, nor that of ‘ straight line’, will apply to the indivisible line, since the latter is not between any terminal points, and does not possess a middle.”
(ii) Secondly, all lines will be commensurate. For all lines -—both those which are commensurate in length, and those which produce commensurate squares—will be measured by the indivisible lines.
And the indivisible lines are all of them commensurate in length (for they are all equal to one another), and therefore also they all produce commensurate squares. But if so, then the square on any line will always be rational.* 4 (iii) Again, since, in a rectangle, the line 10applied at right angles tothe longerside determines the breadth of the figure: the rectangle, which is equal in area to the square on the indivisible j line (v.g. on the line one foot long), will, if applied to a line i double the indivisible line (v.g. to a line two feet long), have a breadth determined by a line shorter than the indivisible line: for its breadth will be less than the breadth of the square on the indivisible line.’
are all, gud infinitesimal, equal’: hence all commensurate μήκει, and therefore also commensurate δυνάμει ( 2-4). The point of the criticism is that the doctrine annihilates the mathei matical conceptions of Commensurate and Incommensurate, Rational and i Irrational. The passage should be compared with Euclid, E/em. X, deff. 2, 3 and 4 (see above, note on 96814) : and with Plato, 7) heaet. 147 D-148B. In the + Theaetetus, Theaetetus and Socrates the Younger are represented as having generalized certain results of the mathematician 15Theodorus (their master), and having divided all numbers into two series, thus :— Series 1: Those numbers which, if regarded as the areas of rectangular figures, are squares with whole numbers as their sides, e.g. 4, 9, 16, 25, ἄς.
The roots of these square numbers are what we should call ‘ rational’ or the sides of the squares are lines σύμμετροι μήκει, viz. containing tA numbers of the unit of length (the line one foot long).
Theaetetus and Socrates called the sides containing the squares in this series ‘ μήκη.
Series 2: Those numbers which, if regarded as the areas of rectangular figures with whole numbers as their sides, as E are oblongs; or, if regarded as squares, have not whole numbers as their sides. To this series belong e.g. 3, 5, 6, 7, 8, &c.: and the sides containing these squares—e.g. 4/3, /5, »/6, &c.—were called by Theaetetus and his friend ‘ duvdpes’, ie. δυνάμει σύμμετροι. (Cf. Theaet. 147D ἥ τε τρίπους καὶ ἡ πεντέπους δυνάμεις are not μήκει 20σύμμετροι τῇ ποδιαίᾳ. 710.,. 148B ὡς μήκει μὲν οὐ συμμέτρους ς D ἐκείναις, τοῖς δ᾽ ἐπιπέδοις ἃ δύνανται.) We should call the ‘sides’ of this series of squares ‘irrational square roots’ or ‘ surds’.
* 94-8. In this passage I adopt Apelt’s reading and interpretation throughout: v. Apelt, Avistotelis quae feruntur, &c., prolegg. pp. xiv, xv.
If we suppose the ‘ indivisible line’ to be one foot long (cf. Arist. Met. 1052. 33—ev ταῖς γραμμαῖς χρῶνται ws ἀτόμῳ τῇ ποδιαίᾳ), then a rectangle, applied to a line two feet A G B long, must—if its area is to be equal to the square on the indivisible line—have as its other side a line shorter than the indivisible line: which is absurd.
Let AB be the indivisible line, one foot long. Let BE be the line, two (iv) Again, since any three given straight lines can be combined to form a triangle, a triangle can also be formed by combining three given indivisible lines. Such a triangle will be equilateral: but in every equilateral 25triangle the perpendicular dropped from the apex bisects the base. Hence, in ᾿ the equilateral triangle whose sides are the indivisible lines, the ‘ indivisible’ base will be bisected by the perpendicular dropped from its apex.!
(v) Again, if the square can be constructed of Simples (i.e. with indivisible lines as its sides), then let its diagonal be drawn, and a perpendicular dropped from one angle on to the diagonal. The square on the side (i.e. the original square constructed feet long. Let CABD be the square on ABZ. If to the line BE there be applied a rectangular figure GF EB equal in area to CABD, FE or GB will be less than 42.
Though I accept Apelt’s interpretation, there are one or two difficulties to which attention should be called. (1) παραβάλλειν is the technical term constantly used in Euclid (cf. e.g. Elem. 1. 44, &c.) for ‘applying’ a rectangle or a parallelogram to a given line: i.e. for constructing such a figure with a given line as one of 30its sides. But (so far as I know) it is always the figure which ‘ παραβάλλεται᾽, not the side. Hence παραβαλλομένη here (9705) is suspicious.
(2) Euclid constantly uses the technical expression ‘Adros ποιεῖ τὴν AB’ to mean ‘[a rectangle applied to such-and-such a given line] makes as its other side the line 42’. But, whatever may have been the original significance of the phrase, there is no implication in Euclid’s usage that the side thus produced is shorter than the given line. So far as I have been able to discover, πλάτος ποιεῖν in Euclid (a) always has the accusative (e.g.
970b
1 χρόνῳ κινεῖται τὴν ὅλην ἐν τῷ ἡμίσει τὴν ἡμίσειαν κινηθήσεται,
καὶ ἐν τῷ ἐλάττονι ἔλαττον ἢ τὴν ἡμίσειαν, ὥστ'
εἰ μὲν περιττῶν σύγκειται τῶν ἀτόμων τὸ μῆκος, ἀναιρεθήσεται
ἡ μέση τομὴ τῶν ἀτόμων, εἴπερ ἐν τῷ ἡμίσει
5 χρόνῳ τὸ ἥμισυ δίεισιν· ὁμοίως γὰρ ὅ τε χρόνος καὶ ἡ
γραμμὴ τμηθήσεται. ὥστε οὐδεμία τῶν συγκειμένων τμηθήσεται
εἰς ἴσα καὶ ἄνισα, οὐδ' ὁμοίως τοῖς χρόνοις τμηθήσονται.
οὐκ ἔσονται ἄτομοι γραμμαί. τὰ δὲ τοῦ αὐτοῦ
λόγου ἐστί, καθάπερ ἐλέχθη, τὸ πάντα ταῦτα ποιεῖν ἐξ
10 ἀμερῶν. ἔτι ἅπασα ἡ μὴ ἄπειρος δύο ἔχει πέρατα·
γραμμὴ γὰρ ὥρισται τούτοις. ἡ δὲ ἄτομος οὐκ ἄπειρος, ὥστε
ἕξει πέρας. διαιρετὴ ἄρα· τὸ γὰρ πέρας ἄλλο καὶ οὗ
πέρας. ἢ ἔσται τις οὔτ' ἄπειρος οὔτε πεπερασμένη γραμμὴ
παρὰ ταύτας. ἔτι οὐκ ἐν ἁπάσῃ γραμμῇ στιγμὴ ἔσται. ἐν
15 μὲν γὰρ τῇ ἀτόμῳ οὐκ ἔστιν· εἰ μὲν γὰρ μία μόνη, ὑπάρξει
γραμμή, εἶτα στιγμή· εἰ δὲ πλείους, διαιρετὴ ἡ γραμμή.
εἰ μὲν οὖν ἐν τῇ ἀτόμῳ μὴ ἐνυπάρχει στιγμή, οὐδ' ὅλως
ἐν γραμμῇ ἔσται· αἱ γὰρ ἄλλαι ἐκ τῶν ἀτόμων. ἔτι ἢ
μηθὲν τῶν στιγμῶν ἔσται μεταξὺ ἢ γραμμή· εἰ δὲ μεταξὺ
20 γραμμή, ἐν ἁπάσαις δὲ πλείους στιγμαί, οὐκ ἔσται ἄτομος.
ἔτι οὐχ ἁπάσης ἔσται γραμμῆς τετράγωνον· ἕξει γὰρ μῆκος
καὶ πλάτος, ὥστε διαιρετόν, ἐπεὶ τὸ μέν, τὸ δέ τι. εἰ
δὲ τὸ τετράγωνον, καὶ ἡ γραμμή. ἔτι τὸ πέρας τῆς γραμμῆς
στιγμὴ ἔσται, ἀλλ' οὐ γραμμή. πέρας μὲν γάρ, τὸ
25 ἔσχατον δὲ ἡ ἄτομος. εἰ γὰρ στιγμή, τὸ πέρας τῇ ἀτόμῳ
ἔσται στιγμή, καὶ ἔσται γραμμὴ γραμμῆς στιγμῇ μείζων.
εἰ δ' ἐνυπάρχει τῇ ἀτόμῳ ἡ στιγμή, διὰ τὸ ταὐτὸ πέρας
τῶν συνεχουσῶν γραμμῶν, ἔσται τι πέρας τῆς ἀμεροῦς.
ὅλως τε τί διοίσει στιγμὴ γραμμῆς; οὐδὲν γὰρ ἴδιον ἕξει ἡ
30 ἄτομος γραμμὴ παρὰ τὴν στιγμὴν πλὴν τοὔνομα. ἔτι
ὁμοίως μένει ἐπίπεδον καὶ σῶμά ἐστιν ἄτομον. ἑνὸς γὰρ
ὄντος ἀδιαιρέτου καὶ τἆλλα συνακολουθήσει διὰ τὸ θάτερον
διῃρῆσθαι κατὰ θάτερον. σῶμα οὐκ ἔσται ἀδιαίρετον διὰ τὸ
1“τὴν AB’) expressing the line resulting, and (4) does not mean ‘determines the breadth’, but simply ‘ makes as its containing side (other than the given line)’. Cf. e.g. Euclid, vem. X. 60, where the line thus produced is the longer of the two containing sides: and so often. But here (9707 5, 37) the writer speaks of a line ‘making the breadth’ (τὸ πλάτος ποιεῖ), and the expression must be distinguished from the technical phrase in Euclid.
(3) In 6 Apelt reads τῷ ἀπὸ τῆς ἀτόμου καὶ τῆς ποδιαίας. τὸ ἀπὸ τῆς ἀτόμου means ‘the square on the indivisible line’ (εξ, above, note on " 14): and we are to take the καί as illustrative or explanatory. There is no serious difficulty here, though this introduction of the one-foot line is a little sudden. But the words in 1.8 are very difficult. Apelt 5there reads ἔσται (yap) ἔλαττον τοῦ ἀπὸ τῆς ἀτόμου, and the words ought to mean ‘For it’—presumably, ‘the breadth’ —‘ will be less than the square on the indivisible line’, As this is nonsense, and as the alternative rendering (‘for it’, viz. the rectangle, ‘is less than the square’) gives a meaning irrelevant to the argument, we have to translate ‘ For the breadth of the rectangle will be less than that of the square’. But I cannot say that the Greek justifies this translation.
with Simples as its sides) will be equal to the square on the perpendicular together with the square on half the diagonal. Hence the side of the square—i.e. the ‘ indivisible’ line—will not be the smallest line.* Nor will the area, which is the square on the diagonal, be double the square on the indivisible line. For (suppose it to be so: then,) if from the diagonal a length equal to the side of the original square be subtracted, the remaining portion of the diagonal will be less than the ‘simple’ line. For if the 10remaining portion of the diagonal were (not less than, but) equal to the ‘simple’ line, the square on the diagonal would have been four times the original square.”
“a νων And one might collect other similar absurdities to which the doctrine leads ; for indeed it conflicts with practically everything in mathematics.’
(B) Then again (the following arguments support our criticism of the doctrine) :—?
(i) The Simple admits of only one mode of conjunction, but a line admits of two: for one line may be conjoined to another either by contact along the whole length of both lines, or by contact at either of its opposite terminal points.* (ii) Further, the addition of a line will not (on the theory) make the whole line any longer than the original line to which the addition was made: for Simples will not, by being added together, produce an increased total magnitude.* (iii) Further, every continuous quantum admits more divisions than one, and therefore no continuous quantum can be formed out of 15two Simples. And since every line (other than the indivisible line) is admittedly continuous, there can be no indivisible line: (for if there were, a continuous quantum— viz. the line formed by the conjunction of two indivisible lines —would be formed out of two Simples.)° In 1. 16 ἀφαιρεθέντος γάρ τοῦ ἴσου, we should presumably understand Be 917, The MSS. read ἄλλα δ᾽ ἄν τις καὶ ἕτερα κτλ. Apelt conjectures ἄλογα δ᾽ ἄν κτλ. There should, of course, be a full stop between διάμετρος and ἄλλα (or ἄλογα).
2 810. This begins a second series of arguments (in support of the writer’s rejection of indivisible lines). πάλιν here corresponds to πρῶτον pev...(909” 29), which introduced the series of arguments just concluded.
3. 810-21, What is ‘ simple’ or ‘ without parts’ can be conjoined with anything else only in one fashion. But a line can be (a) laid alongside of another line, or (ὁ) conjoined with it, end to end. (Cf. de Caelo, 299 25).
The words in 821 κατὰ τὸ πέρας ἐξ ἐναντίως (ἐναντίου LP) 20are obscure. I take them to mean ‘at either of its contrary terminal points’, The mode A B of σύναψις is the same whether the line x y x y xy be conjoined with the line 4B at A or at B, and at ror at y. ᾿ : .
4 821-23. Apelt conjectures (from Pachymeres) ἔτι γραμμὴ ζγραμμῇ) προστεθεῖσα. .. ᾿ The addition of γραμμῇ makes the Greek easier, but does not seem absolutely necessary. ΗΝ Ὑγδ: Ri a23-26. I adopt Apelt’s reading ἔτι (el) ἐκ δυοῖν ἀμεροῖν μηδὲν γίνεται “(γίνεσθαι MSS.), and also his punctuation, but not his interpretation.
I have paraphrased freely, so as to bring out the argument as I under26 (iv) Further, if every line (other than the indivisible line) can be divided both into equal and into unequal parts—every line, even if it consist of three or any odd number of indivisible lines—it will follow that the ‘ indivisible’ line is divisible.’
stand it. The writer assumes (ἅπασα δὲ γραμμὴ παρὰ τὴν ἄτομον συνεχής) that even the advocates of indivisible lines admit that all o¢#er lines 25are continuous: and argues that a line compounded of two indivisible lines would, on their admission, have to be continuous, but could not be so on the principle that every continuum admits more than one division.
And the same will result if every line admits of bisection : for then every line consisting of an odd number of indivisible lines will admit of bisection, and this will involve the division of the ‘indivisible’ line.?
it would not be possible to divide 4A into # and 4, nor into3and}. But by triply bisecting 4Z, and eliminating 4th, the remainder AJ could be divided into AG = 3 and GJ = 4: whilst, by eliminating 2th, the remainder AF could be divided into 44 = 3 and HF =}, _ There is rfo evidence in this passage that the writer knew of the following method for dividing any given line into any number of parts :—Let it be required to divide 42 into (e.g.) three equal parts. From 8 draw BC εξ = AB, produce BC to D, making CD = 4.8: and produce BD to £, making DE = AB, Join EA; 30and from 7) and C draw DF and CG, each parallel to HA, to the points fF and Gon AB. AF, FG,and GB will then be, each of them, 4rd of 42.
If we assume that the writer was unaware of this latter method, it is obvious (a) that no line consisting of an odd number of unit-lines could be ‘ divided into unequal parts’, for the first bisection would divide the middle unit-line: and (6) that there would be a limit to the ‘division into unequal parts’ of lines consisting of an even number of unit-lines, since no such line could be progressively bisected ad /iéitum without dividing the unit-line (cf. % 33).
(3) In 6 Apelt reads τῷ ἀπὸ τῆς ἀτόμου καὶ τῆς ποδιαίας. τὸ ἀπὸ τῆς ἀτόμου means ‘the square on the indivisible line’ (εξ, above, note on " 14): and we are to take the καί as illustrative or explanatory. There is no serious difficulty here, though this introduction of the one-foot line is a little sudden. But the words in 1.8 are very difficult. Apelt 5there reads ἔσται (yap) ἔλαττον τοῦ ἀπὸ τῆς ἀτόμου, and the words ought to mean ‘For it’—presumably, ‘the breadth’ —‘ will be less than the square on the indivisible line’, As this is nonsense, and as the alternative rendering (‘for it’, viz. the rectangle, ‘is less than the square’) gives a meaning irrelevant to the argument, we have to translate ‘ For the breadth of the rectangle will be less than that of the square’. But I cannot say that the Greek justifies this translation.
with Simples as its sides) will be equal to the square on the perpendicular together with the square on half the diagonal. Hence the side of the square—i.e. the ‘ indivisible’ line—will not be the smallest line.* Nor will the area, which is the square on the diagonal, be double the square on the indivisible line. For (suppose it to be so: then,) if from the diagonal a length equal to the side of the original square be subtracted, the remaining portion of the diagonal will be less than the ‘simple’ line. For if the 10remaining portion of the diagonal were (not less than, but) equal to the ‘simple’ line, the square on the diagonal would have been four times the original square.”
“a νων And one might collect other similar absurdities to which the doctrine leads ; for indeed it conflicts with practically everything in mathematics.’
(B) Then again (the following arguments support our criticism of the doctrine) :—?
(i) The Simple admits of only one mode of conjunction, but a line admits of two: for one line may be conjoined to another either by contact along the whole length of both lines, or by contact at either of its opposite terminal points.* (ii) Further, the addition of a line will not (on the theory) make the whole line any longer than the original line to which the addition was made: for Simples will not, by being added together, produce an increased total magnitude.* (iii) Further, every continuous quantum admits more divisions than one, and therefore no continuous quantum can be formed out of 15two Simples. And since every line (other than the indivisible line) is admittedly continuous, there can be no indivisible line: (for if there were, a continuous quantum— viz. the line formed by the conjunction of two indivisible lines —would be formed out of two Simples.)° In 1. 16 ἀφαιρεθέντος γάρ τοῦ ἴσου, we should presumably understand Be 917, The MSS. read ἄλλα δ᾽ ἄν τις καὶ ἕτερα κτλ. Apelt conjectures ἄλογα δ᾽ ἄν κτλ. There should, of course, be a full stop between διάμετρος and ἄλλα (or ἄλογα).
2 810. This begins a second series of arguments (in support of the writer’s rejection of indivisible lines). πάλιν here corresponds to πρῶτον pev...(909” 29), which introduced the series of arguments just concluded.
3. 810-21, What is ‘ simple’ or ‘ without parts’ can be conjoined with anything else only in one fashion. But a line can be (a) laid alongside of another line, or (ὁ) conjoined with it, end to end. (Cf. de Caelo, 299 25).
The words in 821 κατὰ τὸ πέρας ἐξ ἐναντίως (ἐναντίου LP) 20are obscure. I take them to mean ‘at either of its contrary terminal points’, The mode A B of σύναψις is the same whether the line x y x y xy be conjoined with the line 4B at A or at B, and at ror at y. ᾿ : .
4 821-23. Apelt conjectures (from Pachymeres) ἔτι γραμμὴ ζγραμμῇ) προστεθεῖσα. .. ᾿ The addition of γραμμῇ makes the Greek easier, but does not seem absolutely necessary. ΗΝ Ὑγδ: Ri a23-26. I adopt Apelt’s reading ἔτι (el) ἐκ δυοῖν ἀμεροῖν μηδὲν γίνεται “(γίνεσθαι MSS.), and also his punctuation, but not his interpretation.
I have paraphrased freely, so as to bring out the argument as I under26 (iv) Further, if every line (other than the indivisible line) can be divided both into equal and into unequal parts—every line, even if it consist of three or any odd number of indivisible lines—it will follow that the ‘ indivisible’ line is divisible.’
stand it. The writer assumes (ἅπασα δὲ γραμμὴ παρὰ τὴν ἄτομον συνεχής) that even the advocates of indivisible lines admit that all o¢#er lines 25are continuous: and argues that a line compounded of two indivisible lines would, on their admission, have to be continuous, but could not be so on the principle that every continuum admits more than one division.
And the same will result if every line admits of bisection : for then every line consisting of an odd number of indivisible lines will admit of bisection, and this will involve the division of the ‘indivisible’ line.?
it would not be possible to divide 4A into # and 4, nor into3and}. But by triply bisecting 4Z, and eliminating 4th, the remainder AJ could be divided into AG = 3 and GJ = 4: whilst, by eliminating 2th, the remainder AF could be divided into 44 = 3 and HF =}, _ There is rfo evidence in this passage that the writer knew of the following method for dividing any given line into any number of parts :—Let it be required to divide 42 into (e.g.) three equal parts. From 8 draw BC εξ = AB, produce BC to D, making CD = 4.8: and produce BD to £, making DE = AB, Join EA; 30and from 7) and C draw DF and CG, each parallel to HA, to the points fF and Gon AB. AF, FG,and GB will then be, each of them, 4rd of 42.
If we assume that the writer was unaware of this latter method, it is obvious (a) that no line consisting of an odd number of unit-lines could be ‘ divided into unequal parts’, for the first bisection would divide the middle unit-line: and (6) that there would be a limit to the ‘division into unequal parts’ of lines consisting of an even number of unit-lines, since no such line could be progressively bisected ad /iéitum without dividing the unit-line (cf. % 33).
971a
1 εἶναι ἐν αὐτῷ βάθος καὶ πλάτος, οὐδ' ἂν γραμμὴ εἴη
ἀδιαίρετος· σῶμα μὲν γὰρ κατ' ἐπίπεδον, ἐπίπεδον δὲ
κατὰ γραμμήν. ἐπεὶ δὲ οἵ τε λόγοι δι' ὧν ἐπιχειροῦσι
πείθειν ἀσθενεῖς εἰσί, καὶ ψευδεῖς ἐναντίαι δόξαι πᾶσαι τοῖς
5 ἰσχύουσι πρὸς πίστιν, φανερὸν ὅτι οὐκ ἂν εἴη γραμμὴ ἄτομος.
δῆλον δ' ἐκ τούτων ὅτι οὐδ' ἂν ἐκ στιγμῶν εἴη γραμμή. σχεδὸν
γὰρ οἱ πλεῖστοι τῶν λόγων οἱ αὐτοὶ ἁρμόσουσιν. ἀνάγκη
γὰρ διαιρεῖσθαι τὴν στιγμήν, ὅταν ἢ ἐκ περιττῶν τέμνηται
ἴσα ἢ ἐξ ἀρτίων τὰ ἄνισα. καὶ τὸ τῆς γραμμῆς μέρος μὴ
10 εἶναι γραμμήν, μηδὲ τὸ τοῦ ἐπιπέδου ἐπίπεδον. καὶ γραμμὴ
δὲ γραμμῆς στιγμῇ εἶναι μείζων· ἐξ ὧν γὰρ σύγκειται,
τούτοις καὶ ὑπερέξει. τοῦτο δ' ὅτι ἀδύνατον, ἔκ τε τῶν ἐν
τοῖς μαθήμασι δῆλον, καὶ ἔτι συμβήσεται τὴν στιγμὴν ἐν
χρόνῳ δὴ εἶναι τὸ φερόμενον, εἴπερ τὴν μείζω μὲν ἐν
15 πλείονι χρόνῳ, τὴν δ' ἴσην ἐν ἴσῳ, ἡ δὲ τοῦ χρόνου ὑπεροχὴ
χρόνος. ἀλλ' ἴσως καὶ ὁ χρόνος ἐστὶν ἐκ τῶν νῦν,
καὶ τοῦ αὐτοῦ λόγου λέγειν ἄμφω. εἰ δὴ τὸ νῦν ἀρχὴ καὶ
πέρας τοῦ χρόνου καὶ ἡ γραμμὴ στιγμῆς, μή ἐστι δὲ συνεχὴς
ἡ ἀρχὴ καὶ τὸ πέρας ἀλλ' ἔχουσί τι μεταξύ, οὐκ
20 ἂν εἴη οὔτε τὰ νῦν οὔτε στιγμαὶ ἀλλήλοις συνεχεῖς. ἔτι ἡ
μὲν γραμμὴ μέγεθός τι, ἡ δὲ τῶν στιγμῶν σύνθεσις οὐδὲν
ποιεῖ μεγέθος διὰ τὸ μηδ' ἐπὶ πλείω τόπον ἔχειν. ὅταν
γὰρ ἐπὶ γραμμὴν γραμμὴ τεθῇ καὶ ἐφαρμόσῃ, οὐδὲν γίνεται
μεῖζον τὸ πλάτος. ἐν δὲ τῇ γραμμῇ καὶ στιγμαὶ
25 ἐνυπάρχουσιν, οὐδ' ἂν αἱ στιγμαὶ πλείω κατέχοιεν τόπον,
ὥστε οὐκ ἂν ποιοῖεν μέγεθος. ἔτι εἰ ἅπαντα ἅπτεται παντὸς
ἢ ὅλον ὅλου ἢ τινὶ τινὸς ἢ ὅλον τινός, ἡ δὲ στιγμὴ
ἀμερὴς ὅλως ἅπτοιτο. τὸ δ' ὅλον ὅλου ἁπτόμενον ἀνάγκη
ἓν εἶναι. εἰ γάρ τι ἐστὶν ἢ θάτερον μή ἐστιν, οὐκ ἂν ὅλον
30 ὅλου ἅπτοιτο. εἰ δ' ἅμα ἐστὶ τὰ ἀμερῆ, τὸν αὐτὸν κατέχει
τόπον πλείων ὃν καὶ πρότερον τὸ ἕν· τῶν γὰρ ἅμα
1wo And if not every line, but only lines consisting of an even number of units admit of bisection: still, even so, the ‘ indivisible’ line will be divided, when the line consisting of an even number of units is divided into unequal parts (by progressive bisection).
(C) Again,? {the following arguments must be considered against the doctrine) :— (i) If a body has been set in motion and takes a certain time to traverse a certain stretch, and half that time to traverse half that stretch, it will traverse less than half the stretch in less than half the time.* Hence if* the stretch be a length consisting of an odd number of indivisible unit-lines, we shall here again find ® the bisection of the ‘indivisible’ lines, since the body will traverse half the stretch in the half time: for the time and the 5line will be correspondingly divided.® So that none of the composite lines will admit of division both into equal and into unequal parts, nor will they admit of © division corresponding to the division of the times, if there are to be ‘indivisible’ lines... And yet (as we said) the truth is, that the same argument, which leads to the view that lines consist of Simples, leads by logical necessity to the view that all these things (composite times, e.g., as well as composite lines) consist of Simples.?
(ii) Further, every line which is not infinite has two terminal points: for line is defined by these. Now, the ‘indivisible’ line is not infinite, and will therefore have a terminal point. Hence it is divisible: for the terminal point and that which it terminates are different from one another. Otherwise there will be a third kind of line, which is neither finite nor infinite?
(iii) Further, there will not be a point contained in every line. For there will be no point contained in the indivisible 10line ; since, if it contains one point only, a line will be a point, whilst if it contains more than one point it will be divisible. And if* there is no point in the indivisible line, neither will there be a point in any line at all: for all the other lines are made up out of the indivisible lines.® Moreover, if there are points in the indivisible line, there will either be nothing between the points, or a line. But if there is a line between them, and ifall lines contain more points than one, the unit-line will not be indivisible.
(iv) Again, it will not be possible to construct a square on every line. For a square will always possess length and breadth, and will therefore be divisible, since each of its dimensions— its length and its breadth—is a determinate something. But if the square is divisible, then so will be the line on which it is constructed.?
23 -(v) Again, the limit of the line will be a line and not a point.® For it is the ultimate thing which is a limit, and it is the ‘ 15indi- ‘visible line’ which is ultimate.* For if the ultimate thing be ‘point ’, then the limit to the indivisible line will be a point, and one line will be longer than another by a point. But if it be urged that the limiting point is contained w#thin the μι posites of these. For the geometrical principle cf. Arist. Post. Anal. καὶ εἰ ἐν πάσῃ γραμμῇ στιγμή...
indivisible line, on the ground that two lines united so as to form a continuous line have one and the same limit at their juncture, then the simple line (i.e. the line without parts) will after all have a limit belonging to it.! And, indeed, how will a point differ at all from a line on their theory? For the indivisible line will possess nothing characteristic to distinguish it from the point, except the name.” (vi) Again, if there be indivisible lines, there must, by parity of reasoning, be indivisible planes and solids too. For the being of an indivisible unit in one dimension will carry with it the being of indivisibles in the remaining 20dimensions too,* since it is at a plane that a solid is divided, and at a line that a plane is divided. But there is no indivisible solid: for a solid contains depth and breadth. Hence neither can there be an ° indivisible line.2 For a solid is divisible at a plane, and a plane is divisible at a line.® 3. But since the arguments by which they endeavour to convince us are weak and false, and since the opinions (which they are trying to establish) conflict with all the most convincing arguments, it is clear that there can be no indivisible line.!
§ 4. And it is further clear from the above considerations that a line can no more be composed of points than of indivisible lines. For the same arguments, or most of them, will apply equally against both views.
For (i) it will necessarily follow that the point is divided, when the line composed of an odd number of points is divided into equal parts, or when the line composed of an even number — of points is divided into unequal parts.?
the planes 25bounded by those lines—and if there are simple planes there must be simple solids, viz. the solids contained by those planes. For to divide a solid is to divide it at a plane, and thus to divide all the planes at right angles to the plane of division. And to divide a plane (cf. above, ? 21-23) is to divide it at a line, and thus to divide all the lines at right angles to the line of division. Hence if every solid, however minute, is A p divisible, every plane must be divisible too: and if every plane, however small, is divisible, every line must be,divisible too.
This appears to be the argument: but the reason given (% 1) for the divisibility of every solid is obscure, in the same way as the reason given for the divisibility of every square (° 23) was not convincing. And could not the advocates of ‘indivisible lines’ have insisted that a plane figure, though divisible, might yet have as ove of its containing sides an ‘indivisible line’? The oblong B . ABCD, e.g.,might be divisible along its 30length 4S, ’ and yet indivisible in respect to its breadth 4D: 1. 6. AD might be an ‘ indivisible line’.
And (ii) it will follow that the part of a line is not a line, nor the part of a plane a plane.!
(C) Again,? {the following arguments must be considered against the doctrine) :— (i) If a body has been set in motion and takes a certain time to traverse a certain stretch, and half that time to traverse half that stretch, it will traverse less than half the stretch in less than half the time.* Hence if* the stretch be a length consisting of an odd number of indivisible unit-lines, we shall here again find ® the bisection of the ‘indivisible’ lines, since the body will traverse half the stretch in the half time: for the time and the 5line will be correspondingly divided.® So that none of the composite lines will admit of division both into equal and into unequal parts, nor will they admit of © division corresponding to the division of the times, if there are to be ‘indivisible’ lines... And yet (as we said) the truth is, that the same argument, which leads to the view that lines consist of Simples, leads by logical necessity to the view that all these things (composite times, e.g., as well as composite lines) consist of Simples.?
(ii) Further, every line which is not infinite has two terminal points: for line is defined by these. Now, the ‘indivisible’ line is not infinite, and will therefore have a terminal point. Hence it is divisible: for the terminal point and that which it terminates are different from one another. Otherwise there will be a third kind of line, which is neither finite nor infinite?
(iii) Further, there will not be a point contained in every line. For there will be no point contained in the indivisible 10line ; since, if it contains one point only, a line will be a point, whilst if it contains more than one point it will be divisible. And if* there is no point in the indivisible line, neither will there be a point in any line at all: for all the other lines are made up out of the indivisible lines.® Moreover, if there are points in the indivisible line, there will either be nothing between the points, or a line. But if there is a line between them, and ifall lines contain more points than one, the unit-line will not be indivisible.
(iv) Again, it will not be possible to construct a square on every line. For a square will always possess length and breadth, and will therefore be divisible, since each of its dimensions— its length and its breadth—is a determinate something. But if the square is divisible, then so will be the line on which it is constructed.?
23 -(v) Again, the limit of the line will be a line and not a point.® For it is the ultimate thing which is a limit, and it is the ‘ 15indi- ‘visible line’ which is ultimate.* For if the ultimate thing be ‘point ’, then the limit to the indivisible line will be a point, and one line will be longer than another by a point. But if it be urged that the limiting point is contained w#thin the μι posites of these. For the geometrical principle cf. Arist. Post. Anal. καὶ εἰ ἐν πάσῃ γραμμῇ στιγμή...
indivisible line, on the ground that two lines united so as to form a continuous line have one and the same limit at their juncture, then the simple line (i.e. the line without parts) will after all have a limit belonging to it.! And, indeed, how will a point differ at all from a line on their theory? For the indivisible line will possess nothing characteristic to distinguish it from the point, except the name.” (vi) Again, if there be indivisible lines, there must, by parity of reasoning, be indivisible planes and solids too. For the being of an indivisible unit in one dimension will carry with it the being of indivisibles in the remaining 20dimensions too,* since it is at a plane that a solid is divided, and at a line that a plane is divided. But there is no indivisible solid: for a solid contains depth and breadth. Hence neither can there be an ° indivisible line.2 For a solid is divisible at a plane, and a plane is divisible at a line.® 3. But since the arguments by which they endeavour to convince us are weak and false, and since the opinions (which they are trying to establish) conflict with all the most convincing arguments, it is clear that there can be no indivisible line.!
§ 4. And it is further clear from the above considerations that a line can no more be composed of points than of indivisible lines. For the same arguments, or most of them, will apply equally against both views.
For (i) it will necessarily follow that the point is divided, when the line composed of an odd number of points is divided into equal parts, or when the line composed of an even number — of points is divided into unequal parts.?
the planes 25bounded by those lines—and if there are simple planes there must be simple solids, viz. the solids contained by those planes. For to divide a solid is to divide it at a plane, and thus to divide all the planes at right angles to the plane of division. And to divide a plane (cf. above, ? 21-23) is to divide it at a line, and thus to divide all the lines at right angles to the line of division. Hence if every solid, however minute, is A p divisible, every plane must be divisible too: and if every plane, however small, is divisible, every line must be,divisible too.
This appears to be the argument: but the reason given (% 1) for the divisibility of every solid is obscure, in the same way as the reason given for the divisibility of every square (° 23) was not convincing. And could not the advocates of ‘indivisible lines’ have insisted that a plane figure, though divisible, might yet have as ove of its containing sides an ‘indivisible line’? The oblong B . ABCD, e.g.,might be divisible along its 30length 4S, ’ and yet indivisible in respect to its breadth 4D: 1. 6. AD might be an ‘ indivisible line’.
And (ii) it will follow that the part of a line is not a line, nor the part of a plane a plane.!
971b
1 ὄντων καὶ μὴ ἐχόντων ἐπέκτασιν κατὰ ταὐτὰ ὁ αὐτὸς
ἀμφοῖν τόπος. τὸ δ' ἀμερὲς οὐκ ἔχει διάστασιν, ὥστ' οὐκ
ἂν εἴη μέγεθος συνεχὲς ἐξ ἀμερῶν. οὐκ ἄρα οὔθ' ἡ γραμμὴ
ἐκ στιγμῶν οὔθ' ὁ χρόνος ἐκ τῶν νῦν. ἔτι εἰ ἔστιν ἐκ στιγμῶν,
5 ἅψεται στιγμὴ στιγμῆς· ἐὰν οὖν ἐκ τοῦ Κ ἐκβληθῇ
ἡ ΑΒ καὶ ΓΔ, ἅψεται τοῦ Κ καὶ ἡ ἐν τῇ ΚΔ στιγμή.
ὥστε καὶ ἄλλῳ τινί· τὸ γὰρ ἀμερὲς τοῦ ἀμεροῦς ὅλον ὅλου
ἐφάπτεται. ὥστε τὸν αὐτὸν ἐφέξει τόπον τοῦ Κ, καὶ ἁπτόμεναι
στιγμαὶ ἐν τῷ αὐτῷ τόπῳ ἀλλήλαις. εἰ δ' ἐν τῷ
10 αὐτῷ, καὶ ἅπτονται· τὰ γὰρ ἐν τῷ αὐτῷ τόπῳ ὄντα
πρῶτα ἅπτεσθαι ἀναγκαῖον, εἶθ' οὕτως εὐθεῖα εὐθείας ἅψεται
κατὰ δύο στιγμάς. ἡ γὰρ ἐν τῇ ΑΚ στιγμὴ καὶ τῇ
ΚΓ καὶ ἑτέρας ἅπτεται στιγμῆς. ὥστε ἡ ἐκ τῆς ΓΔ
κατὰ πλείους ἅπτεται στιγμάς. ὁ αὐτὸς δὲ λόγος καὶ εἰ
15 μὴ δι' ἀλλήλων ἀλλ' ὁπωσοῦν ἥψατο γραμμῆς. ἔτι καὶ
ἡ τοῦ κύκλου τῆς εὐθείας ἅψεται κατὰ πλείω. τῆς γὰρ
συναφῆς καὶ ἡ ἐν τῷ κύκλῳ καὶ ἡ ἐν τῇ εὐθείᾳ ἅπτεται
καὶ ἀλλήλων. εἰ δὲ τοῦτο μὴ δυνατόν, οὐδὲ τὸ ἅπτεσθαι
στιγμὴν στιγμῆς· εἰ δὲ μὴ ἅπτεσθαι, οὐδ' εἶναι τὴν γραμμὴν
20 στιγμήν· οὐδὲ γὰρ ἅπτεσθαι ἀναγκαῖον. ἔτι πῶς ποτὲ
ἔσται εὐθεῖα γραμμὴ καὶ περιφερής; οὐδὲν γὰρ διοίσει ἡ
σύναψις τῶν στιγμῶν ἐν τῇ εὐθείᾳ καὶ τῇ περιφερεῖ. τὸ
γὰρ ἀμερὲς τοῦ ἀμεροῦς ὅλον ὅλου ἅπτεται, καὶ οὐκ ἔστιν
ὅλως ἅπτεσθαι. εἰ οὖν αἱ μὲν γραμμαὶ διάφοροι, ἡ δὲ
25 σύναψις ἀδιάφορος, οὐκ ἔσται δὴ γραμμὴ ἐκ τῆς συνάψεως,
ὥστ' οὐδ' ἐκ στιγμῶν. ἔτι ἀναγκαῖον ἢ ἅπτεσθαι ἢ
μὴ ἅπτεσθαι τὰς στιγμὰς ἀλλήλων. εἰ μὲν οὖν τὸ ἐφεξῆς
ἅπτεσθαι ἀνάγκη, ὁ αὐτὸς ἔσται λόγος· εἰ δὲ ἐνδέχεται
ἐφεξῆς τι εἶναι μὴ ἁπτόμενον, τὸ δὲ συνεχὲς οὐδὲν ἄλλο
30 λέγομεν ἢ τὸ ἐξ ὧν ἐστὶν ἁπτομένων· ὥστε καὶ οὕτως ἀνάγκη
τὰς στιγμὰς ἅπτεσθαι ἀλλήλων, ἢ εἶναι γραμμὴν συνεχῆ.
1Further (iii) it will follow that one line is longer than another by a point?: for it is by its constituent elements that one line will exceed another. But that it is impossible for one line to be longer than another by a point, is clear both from what is proved in mathematics and from the following argument. For, if it were possible, the absurd consequence would result that the moving body would take a time to traverse the point.® For, as it traverses the equal line in an equal time, it will traverse the longer line in a greater time: and that by which the greater time exceeds the equal time is itself a time.
Perhaps, however, we are to suppose that just as a line consists of points, so also time consists of ‘nows’, and both theses belong to the same way of thinking. (Let us then 5examine the doctrine that a line, or generally continua, like times and lengths, consist of discrete elements. )‘ In 1. 9 τὰ ἄνισα is strange: Z* omits τά.
The reference is to the obscure argument at * 26-33.
(a) Since, then, the Now is a beginning and end of a! time, and the Point a beginning and end of a line; and since the beginning of anything is not ‘continuous’ with its end, but they have an interval between them ; it follows that neither Nows nor Points can be continuous with one another.?
(0) Again, a line* is a magnitude: but the ‘ composition’ of points constitutes no magnitude, because several points put together occupy no more space than one. For when one line is superimposed on another and coincides * with it, the breadth is in no wise increased. And since points too are contained in the line thus superimposed, it follows that neither would points, by being superimposed on points, occupy more space. Hence points would not constitute a magnitude by composition.° Of these 10four alternatives σύνθεσις is used by Aristotle as the general term to express any kind of combination of a manifold: cf. e.g. Zop. Z 13, 150 22, Z 14, 151* 20-32. Here, however, as we shall see, the writer appears to use it to express one special kind of combination. The remaining alternatives are treated by Aristotle as exhausting the ways in which points might be supposed to cohere to form a line: cf. Arist. Phys. 18 ff. Aristotle’s definitions (PAys. l.c.), which the writer here assumes, are ‘ συνεχῆ μὲν ὧν τὰ ἔσχατα ἕν, ἁπτόμενα δ᾽ ὧν ἅμα, ἐφεξῆς δ᾽ ὧν μηδὲν μεταξὺ συγγενές᾽. :
218, τοῦ χρόνου, i.e. any given period of time.
2 817-20, Two things are called ‘continuous’ when the end of one is identical with the beginning of the other. But the Nows and the Points are themselves Ends and Beginnings, or Extremes (ἔσχατα), and cannot therefore be ‘ continuous’ with one another.
3 821, ἡ μὲν γραμμή ‘the line’, i.e. any and every line: cf. ° 18, τοῦ χρόνου.
* 823. For this use of 15ἐφαρμόζειν cf. e.g. Euclid, E/em. 1. 4, “ ἐφαρμύσει kat τὸ Β σημεῖον ἐπὶ TOE...”
5 420-26. In this argument the writer seems to be excluding a view that point is applied to point so as to ‘compound’ a line. Line is length without breadth: and if line be applied to line, the two coincide, fall on one another, and do not produce a surface, i.e. do not ‘ increase the breadth’ of the first line. So point is position without magnitude, and no application (composition or addition) of point to point can produce magnitude—i.e. length. If the line 4Z be applied to the line CD, the points in AB will coincide with the points in CD: and as the line CD is A Y B no ‘broader’ than it was before, neither will any point x in CD beὉ x D come a length by ‘ composition’ with i the corresponding point yin AB. There is some difficulty in the text. In % 22 the MSS. read διὰ τὸ μηδ᾽ ἐπὶ πλείω τόπον ἔχειν. Should, we perhaps read διὰ τὸ μηδ᾽ ἔτι πλείω τόπον κατέχειν’ In 1. 24 I retain the MSS. reading ἐν δὲ τῇ 20γραμμῇ . « - (Apelt’s emendation εἰ δὲ τῇ γραμμῇ... does not suit the movement of the argument.) But I read (c) Again, whenever one thing is ‘ contiguous’ with another, the contact is either whole-with-whole, or part-with-part, or whole-with-part. But the point is without parts. Hence the contact of point with point must be a contact wholewith-whole.!
But if one thing is in contact with another whole-with-whole, the two things must be one. For if either of them is anything in any respect in which the other is not, they would not be in contact whole-with-whole.?
But if the Simples {when in contact) are (not ‘ one’, but) ‘coincident ’, then a plurality occupies the same place which was formerly occupied by one: for if two things are coincident and neither admits of being extended beyond the coincidence, just so far the place occupied by both isthe same. And since ° the Simple has no dimension, it follows that a continuous magnitude cannot be composed of Simples. Hence neither can a 25line consist of Points nor a time of Nows.® in 1.25 οὐδ᾽ ἂν (ἄρ᾽) ai στιγμαὶ. . ., and alter the punctuation, so that the whole passage runs as follows :— + + » μεῖζον τὸ πλάτος" ἐν δὲ τῇ γραμμῇ καὶ στιγμαὶ ἐνυπάρχουσιν" οὐδ᾽ ἂν (ἄρ᾽) αἱ στιγμαὶ πλείω κατέχοιεν τόπον, ὥστε οὐκ ἂν ποιοῖεν μέγεθος.
‘ In 327, 28 I read with Apelt (after Hayduck) ἡ δὲ στιγμὴ ἀμερής, ὅλως (av) ἅπτοιτο.
The principle that all contact must be whole-with-whole, or partwith-part, or whole-with-part, is enunciated by Aristotle (Piys. 231 2), and applied similarly to ἀδιαίρετα and specially to points, 2 829. The MSS. read εἰ γάρ τι [τις NZ*] ἐστὶν ἢ θάτερον μή ἐστιν... .: I read ἣ θάτερον (cf. the Latin transl. ‘si quid remanet quod alteri non coniungatur ἢ).
Apelt conjectures εἰ yap dis (or δύ᾽) ἐστὶν... ‘si totum bis est vel non simul alterum complectitur .. .’
a26->4. The outline of the argument is as follows :—The contact of Points, gzé Simples, must be whole-with-whole. Now two things are ‘contiguous’ 30when their extremities are ἅμα, ‘ coincident’ or ‘ together’. But since Simples have no parts—no extremities in distinction from the rest of themselves—the contact of Simples must mean absolute unity.
Perhaps, however, we are to suppose that just as a line consists of points, so also time consists of ‘nows’, and both theses belong to the same way of thinking. (Let us then 5examine the doctrine that a line, or generally continua, like times and lengths, consist of discrete elements. )‘ In 1. 9 τὰ ἄνισα is strange: Z* omits τά.
The reference is to the obscure argument at * 26-33.
(a) Since, then, the Now is a beginning and end of a! time, and the Point a beginning and end of a line; and since the beginning of anything is not ‘continuous’ with its end, but they have an interval between them ; it follows that neither Nows nor Points can be continuous with one another.?
(0) Again, a line* is a magnitude: but the ‘ composition’ of points constitutes no magnitude, because several points put together occupy no more space than one. For when one line is superimposed on another and coincides * with it, the breadth is in no wise increased. And since points too are contained in the line thus superimposed, it follows that neither would points, by being superimposed on points, occupy more space. Hence points would not constitute a magnitude by composition.° Of these 10four alternatives σύνθεσις is used by Aristotle as the general term to express any kind of combination of a manifold: cf. e.g. Zop. Z 13, 150 22, Z 14, 151* 20-32. Here, however, as we shall see, the writer appears to use it to express one special kind of combination. The remaining alternatives are treated by Aristotle as exhausting the ways in which points might be supposed to cohere to form a line: cf. Arist. Phys. 18 ff. Aristotle’s definitions (PAys. l.c.), which the writer here assumes, are ‘ συνεχῆ μὲν ὧν τὰ ἔσχατα ἕν, ἁπτόμενα δ᾽ ὧν ἅμα, ἐφεξῆς δ᾽ ὧν μηδὲν μεταξὺ συγγενές᾽. :
218, τοῦ χρόνου, i.e. any given period of time.
2 817-20, Two things are called ‘continuous’ when the end of one is identical with the beginning of the other. But the Nows and the Points are themselves Ends and Beginnings, or Extremes (ἔσχατα), and cannot therefore be ‘ continuous’ with one another.
3 821, ἡ μὲν γραμμή ‘the line’, i.e. any and every line: cf. ° 18, τοῦ χρόνου.
* 823. For this use of 15ἐφαρμόζειν cf. e.g. Euclid, E/em. 1. 4, “ ἐφαρμύσει kat τὸ Β σημεῖον ἐπὶ TOE...”
5 420-26. In this argument the writer seems to be excluding a view that point is applied to point so as to ‘compound’ a line. Line is length without breadth: and if line be applied to line, the two coincide, fall on one another, and do not produce a surface, i.e. do not ‘ increase the breadth’ of the first line. So point is position without magnitude, and no application (composition or addition) of point to point can produce magnitude—i.e. length. If the line 4Z be applied to the line CD, the points in AB will coincide with the points in CD: and as the line CD is A Y B no ‘broader’ than it was before, neither will any point x in CD beὉ x D come a length by ‘ composition’ with i the corresponding point yin AB. There is some difficulty in the text. In % 22 the MSS. read διὰ τὸ μηδ᾽ ἐπὶ πλείω τόπον ἔχειν. Should, we perhaps read διὰ τὸ μηδ᾽ ἔτι πλείω τόπον κατέχειν’ In 1. 24 I retain the MSS. reading ἐν δὲ τῇ 20γραμμῇ . « - (Apelt’s emendation εἰ δὲ τῇ γραμμῇ... does not suit the movement of the argument.) But I read (c) Again, whenever one thing is ‘ contiguous’ with another, the contact is either whole-with-whole, or part-with-part, or whole-with-part. But the point is without parts. Hence the contact of point with point must be a contact wholewith-whole.!
But if one thing is in contact with another whole-with-whole, the two things must be one. For if either of them is anything in any respect in which the other is not, they would not be in contact whole-with-whole.?
But if the Simples {when in contact) are (not ‘ one’, but) ‘coincident ’, then a plurality occupies the same place which was formerly occupied by one: for if two things are coincident and neither admits of being extended beyond the coincidence, just so far the place occupied by both isthe same. And since ° the Simple has no dimension, it follows that a continuous magnitude cannot be composed of Simples. Hence neither can a 25line consist of Points nor a time of Nows.® in 1.25 οὐδ᾽ ἂν (ἄρ᾽) ai στιγμαὶ. . ., and alter the punctuation, so that the whole passage runs as follows :— + + » μεῖζον τὸ πλάτος" ἐν δὲ τῇ γραμμῇ καὶ στιγμαὶ ἐνυπάρχουσιν" οὐδ᾽ ἂν (ἄρ᾽) αἱ στιγμαὶ πλείω κατέχοιεν τόπον, ὥστε οὐκ ἂν ποιοῖεν μέγεθος.
‘ In 327, 28 I read with Apelt (after Hayduck) ἡ δὲ στιγμὴ ἀμερής, ὅλως (av) ἅπτοιτο.
The principle that all contact must be whole-with-whole, or partwith-part, or whole-with-part, is enunciated by Aristotle (Piys. 231 2), and applied similarly to ἀδιαίρετα and specially to points, 2 829. The MSS. read εἰ γάρ τι [τις NZ*] ἐστὶν ἢ θάτερον μή ἐστιν... .: I read ἣ θάτερον (cf. the Latin transl. ‘si quid remanet quod alteri non coniungatur ἢ).
Apelt conjectures εἰ yap dis (or δύ᾽) ἐστὶν... ‘si totum bis est vel non simul alterum complectitur .. .’
a26->4. The outline of the argument is as follows :—The contact of Points, gzé Simples, must be whole-with-whole. Now two things are ‘contiguous’ 30when their extremities are ἅμα, ‘ coincident’ or ‘ together’. But since Simples have no parts—no extremities in distinction from the rest of themselves—the contact of Simples must mean absolute unity.
972a
1 ἔτι εἰ ἄτοπον στιγμὴ ἐπὶ στιγμῆς, ἵν' ᾖ γραμμὴ καὶ ἐπὶ
στιγμῇ, ἐπεὶ ἡ γραμμὴ ἐπίπεδον, ἀδύνατον τὰ εἰρημένα
εἶναι. εἴτε γὰρ ἐφεξῆς αἱ στιγμαί εἰσι, τμηθήσεται ἡ
γραμμὴ κατ' οὐδετέραν τῶν στιγμῶν, ἀλλ' ἀνὰ μέσον·
5 εἴθ' ἅπτονται, γραμμὴ ἔσται τῆς μιᾶς στιγμῆς χώρα.
τοῦτο δ' ἀδύνατον. ἔτι διαιροῖτ' ἂν ἅπαντα καὶ ἀναλύοιτο
εἰς στιγμάς, καὶ ἡ στιγμὴ μέρος σώματος, εἴπερ τὸ μὲν
σῶμα ἐξ ἐπιπέδων, τὸ δ' ἐπίπεδον ἐκ γραμμῶν, αἱ δὲ
γραμμαὶ ἐκ στιγμῶν. εἰ δ' ἐξ ὧν πρώτων ἐνυπαρχόντων
10 ἕκαστά ἐστι, στοιχεῖά ἐστι ταῦτα, αἱ στιγμαὶ ἂν εἴησαν στοιχεῖα
σωμάτων. ὥστε συνώνυμα στοιχεῖα οὐδέτερα τῷ εἴδει.
φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι οὐκ ἔστι γραμμὴ ἐκ στιγμῶν.
ἀλλ' οὐδ' ἀφαιρεθῆναι οἷόν τε στιγμὴν ἀπὸ γραμμῆς.
εἰ γὰρ ἐνδέχεται ἀφαιρεθῆναι, καὶ προστεθῆναι δυνατόν·
15 προστεθέντος δέ τινος τὸ προστεθὲν μεῖζον ἔσται τοῦ
ἐξ ἀρχῆς, ἐὰν τοιοῦτον ᾖ τὸ προστιθέμενον ὥστε ἓν ὅλον
ποιεῖν. ἔσται γραμμὴ γραμμῆς στιγμῇ μείζων. τοῦτο δ'
ἀδύνατον. ἀλλὰ καθ' ἑαυτὴν μὲν οὐχ οἷόν τε, κατὰ συμβεβηκὸς
δ' ἐνδέχεται στιγμὴν ἀπὸ γραμμῆς ἀφελεῖν, τῷ
20 ἐνυπάρχειν ἐν τῇ ἀφαιρουμένῃ γραμμῇ. εἰ τοῦ ὅλου ἀφαιρουμένου
καὶ ἡ ἀρχὴ καὶ τὸ πέρας ἀφαιρεῖται, γραμμῆς
δ' ἦν ἡ ἀρχὴ καὶ τὸ πέρας στιγμή, καὶ γραμμῆς ἐγχωρεῖ
ἀφαιρεῖν καὶ στιγμὴν ἐνδέχοιτο. αὕτη δ' ἡ ἀφαίρεσις
κατὰ συμβεβηκός. εἰ δὲ τὸ πέρας ἅπτεται, οὔτε πέρας ἢ
25 αὐτοῦ ἢ τῶν ἐκείνου τινός. ἡ δὲ στιγμή, ᾗ πέρας γραμμῆς,
ἅπτεται. ᾗ μὲν οὖν γραμμῆς ἔσται στιγμὴ μείζων, ἡ δὲ
στιγμὴ ἐκ στιγμῶν· τῶν γὰρ ἁπτομένων οὐδὲν ἀνὰ μέσον.
ὁ αὐτὸς λόγος καὶ ἐπὶ τῆς τομῆς, εἰ ἡ τομὴ στιγμῆς καὶ
ἡ τομὴ ἅπτεταί τινος καὶ ἐπὶ στερεοῦ καὶ ἐπιπέδου· ὡσαύτως
30 δὲ καὶ τὸ στερεὸν ἐξ ἐπιπέδων καὶ γραμμῶν. οὐκ
ἀληθὲς δὲ κατὰ στιγμὴν εἰπεῖν, οὐδ' ὅτι ἐλάχιστον τῶν ἐκ
γραμμῆς εἰς τὸ ἐλάχιστον τῶν ἐνυπαρχόντων εἴρηται.
τὸ δὲ ἐλάχιστον, ὧν ἐστὶν ἐλάχιστον, καὶ ἔλαττόν ἐστιν.
1If this be denied, and it be maintained that the ‘contiguous’ Simples are ‘ coincident’, but remain ‘two’: it will follow that two or more Simples can be ‘ coincident’ without taking up more place than one Simple, and therefore (since oze Simple has no dimension, i.e. no inner extension) no continuous magnitude can be composed of Simples. And a corollary of this is, that a line cannot consist of points, nor a time of ‘nows’.
In ° 1 I read, with LPW2Z%, ἐπέκτασιν, κατὰ ταῦτα ὁ αὐτὸς κτλ, Apelt’s conjecture (ἐπέκτασιν καθ᾽ ἕαυτά, 6 αὐτὸς ...) is tempting, but unnecessary.
In ° 2 d:acracis=dimension, cf. Bonitz, Index, 189% 30 ff.
Io (d) Further, if the line consists of points, point will be in contact with point. If, then, from K there be drawn the lines AB and CD, the point B in the line A(B)X and the point 5C in the line K(C)D will both be in contact with X.1 So that the points B and C will also be in contact with one another : for the Simple, when in contact with the Simple, is in contact whole-with-whole. So that the points will occupy the same place as K, and, gud in contact with X, will be in the same place with one another. But if they are in the same place with one another, they must also be in contact with one another : for things which are in the same ‘continent’ place must be in contact.” But, if this is so, one straight line will touch another straight line in two points. For the point (8) in the line 4K touches both the point AC and another (viz. the point contiguous to Cin the line K(C)D). Hence the line AK touches the line CD in more points than one."
And the same argument would apply not only in the case supposed, where two lines were in contact with one another at the point KX, but also if there had been any number of lines touching one another at K.?
Lal in contact with AK whole-with-whole—must 10have one and the same ‘continent place’ as A, and therefore as one another: and therefore must be in contact with one another. The nerve of the argument is contained in the words ‘and ¢he fotnts, because in contact with Α΄": but Apelt’s reading could only be translated ‘ Therefore the points which are in contact with K will also be in the same place as one another ’ ᾿ (Apelt’s note on 1. 9 εἰ δ᾽ ἐν τῷ αὐτῷ. . . “ scribendum potius videtur yap’, shows that he has failed to follow the writer’s argument.)
15 (e) Further, if a line consist of points in contact with one 2. A ας τανε another, the circumference of a circle will touch the tangent at more pointsthan one. For both the point on the circumference and the point in the tangent touch the point of junction and also touch one another. But since this is not possible, neither is it possible for point to touch point. And if point cannot touch point, neither can the line consist of points: for if it did, they would necessarily be in contact.”
(f) Moreover, 15how—on the supposition that the line consists of points—will there any longer be straight azd curved lines ἢ ‘For the conjunction of the points in the straight line will not differ in any way from their conjunction in the curved line. For the contact of Simple with Simple is contact whole-withwhole, and Simples admit no other mode of contact. Since, then, the straight and curved lines are different, but the conjunction of points is invariably the same, clearly a line will not be curved or straight because of the conjunction : hence neither will a line consist of points.* (g) Further, the points (of which the line consists) must either touch or not touch one another. Now if ‘the next’ in a series must touch the preceding term, the same arguments, which were advanced above, will apply: but if there can be ‘a next’ without its being in contact (with its predecessor or successor), yet by ‘the continuous’ we mean nothing but a composite whose constituents are in contact. So that the points forming the line 20must be in contact, in so far as the line must be continuous, even though we suppose the points to be a ‘series’, (h) 1 ἔτι εἰ ἄτοπον στιγμὴ ἐπὶ στιγμῆς [ἐπιστήμη Z*], ἵν᾽ ἡ ° [ἡ PZ*] γραμμὴ καὶ ἐπὶ στιγμῇ, [γραμμὴ καὶ ἐπιστήμης ΝΜ", ἐπιστήμη καὶ γραμμή Z*], ἐπεὶ ἡ γραμμὴ ἐπίπεδον, ἀδύνατον τὰ εἰρημένα εἶναι. + For if the points form a series without lines from a difference in the mode of contact of their points. And so the theory that lines consist of points in contact breaks down: for it cannot account for the difference between straight and curved.’
In >25 one may suspect some corruption in the text. The MSS. read οὐκ ἔσται δὴ γραμμὴ ἐκ τῆς συνάψεως. The sense required is given in Rota’s translation—‘ non fiet ex punctorum contactu linea circularis et recta.’
contact, the line will be divided not at either of the points, but between them: whilst if they are in contact, a line will be the place of the single point. And this is impossible.} 6 (Ὁ) Further, all things would be divided, i.e. be dissolved, 25into points; and the point would be a part of a solid, since the solid—on the theory—consists of planes, the plane of lines, and the lines of points. And since those constituents, of which (as their primary immanent factors) the various groups of things are composed, are ‘elements’, points would be ‘elements’ of bodies. Hence ‘ elements’ would be identical in nature as well as in name, and not even specifically different.”
12 ᾧ 5. It is clear, then, from the above arguments that a line does not consist of points.® (a) But neither is it possible to subtract a point from a line. For, if a point can be subtracted, it can also be added. But if anything is added, that to which it was added will be bigger than it was at first, if that which is added be such as to coalesce and form one whole with it. Hence a line will be bigger than another line by a point. And this is impossible. But though it is not possible to subtract a point as such from a line, one may subtract it zzcidentally, viz. in so far as a point 30he proposes to read ἂν ἢ γραμμὴ καὶ ἐπὶ στιγμῇ, which he translates ‘wenn auch eine Linie auf einem Punkte sein kann’: but one may envy, without wishing to imitate, this free-and-easy attitude to Greek Grammar. It seemed best to own myself defeated, and simply to print the original Greek.
is contained in the line which one is subtracting from another line. For since, if the whole be subtracted, its beginning and its end are subtracted too ; and since the beginning and the end of a line are points: then, if it be possible to subtract a line from a line, it will be possible also thereby to subtract a point.
In ° 1 I read, with LPW2Z%, ἐπέκτασιν, κατὰ ταῦτα ὁ αὐτὸς κτλ, Apelt’s conjecture (ἐπέκτασιν καθ᾽ ἕαυτά, 6 αὐτὸς ...) is tempting, but unnecessary.
In ° 2 d:acracis=dimension, cf. Bonitz, Index, 189% 30 ff.
Io (d) Further, if the line consists of points, point will be in contact with point. If, then, from K there be drawn the lines AB and CD, the point B in the line A(B)X and the point 5C in the line K(C)D will both be in contact with X.1 So that the points B and C will also be in contact with one another : for the Simple, when in contact with the Simple, is in contact whole-with-whole. So that the points will occupy the same place as K, and, gud in contact with X, will be in the same place with one another. But if they are in the same place with one another, they must also be in contact with one another : for things which are in the same ‘continent’ place must be in contact.” But, if this is so, one straight line will touch another straight line in two points. For the point (8) in the line 4K touches both the point AC and another (viz. the point contiguous to Cin the line K(C)D). Hence the line AK touches the line CD in more points than one."
And the same argument would apply not only in the case supposed, where two lines were in contact with one another at the point KX, but also if there had been any number of lines touching one another at K.?
Lal in contact with AK whole-with-whole—must 10have one and the same ‘continent place’ as A, and therefore as one another: and therefore must be in contact with one another. The nerve of the argument is contained in the words ‘and ¢he fotnts, because in contact with Α΄": but Apelt’s reading could only be translated ‘ Therefore the points which are in contact with K will also be in the same place as one another ’ ᾿ (Apelt’s note on 1. 9 εἰ δ᾽ ἐν τῷ αὐτῷ. . . “ scribendum potius videtur yap’, shows that he has failed to follow the writer’s argument.)
15 (e) Further, if a line consist of points in contact with one 2. A ας τανε another, the circumference of a circle will touch the tangent at more pointsthan one. For both the point on the circumference and the point in the tangent touch the point of junction and also touch one another. But since this is not possible, neither is it possible for point to touch point. And if point cannot touch point, neither can the line consist of points: for if it did, they would necessarily be in contact.”
(f) Moreover, 15how—on the supposition that the line consists of points—will there any longer be straight azd curved lines ἢ ‘For the conjunction of the points in the straight line will not differ in any way from their conjunction in the curved line. For the contact of Simple with Simple is contact whole-withwhole, and Simples admit no other mode of contact. Since, then, the straight and curved lines are different, but the conjunction of points is invariably the same, clearly a line will not be curved or straight because of the conjunction : hence neither will a line consist of points.* (g) Further, the points (of which the line consists) must either touch or not touch one another. Now if ‘the next’ in a series must touch the preceding term, the same arguments, which were advanced above, will apply: but if there can be ‘a next’ without its being in contact (with its predecessor or successor), yet by ‘the continuous’ we mean nothing but a composite whose constituents are in contact. So that the points forming the line 20must be in contact, in so far as the line must be continuous, even though we suppose the points to be a ‘series’, (h) 1 ἔτι εἰ ἄτοπον στιγμὴ ἐπὶ στιγμῆς [ἐπιστήμη Z*], ἵν᾽ ἡ ° [ἡ PZ*] γραμμὴ καὶ ἐπὶ στιγμῇ, [γραμμὴ καὶ ἐπιστήμης ΝΜ", ἐπιστήμη καὶ γραμμή Z*], ἐπεὶ ἡ γραμμὴ ἐπίπεδον, ἀδύνατον τὰ εἰρημένα εἶναι. + For if the points form a series without lines from a difference in the mode of contact of their points. And so the theory that lines consist of points in contact breaks down: for it cannot account for the difference between straight and curved.’
In >25 one may suspect some corruption in the text. The MSS. read οὐκ ἔσται δὴ γραμμὴ ἐκ τῆς συνάψεως. The sense required is given in Rota’s translation—‘ non fiet ex punctorum contactu linea circularis et recta.’
contact, the line will be divided not at either of the points, but between them: whilst if they are in contact, a line will be the place of the single point. And this is impossible.} 6 (Ὁ) Further, all things would be divided, i.e. be dissolved, 25into points; and the point would be a part of a solid, since the solid—on the theory—consists of planes, the plane of lines, and the lines of points. And since those constituents, of which (as their primary immanent factors) the various groups of things are composed, are ‘elements’, points would be ‘elements’ of bodies. Hence ‘ elements’ would be identical in nature as well as in name, and not even specifically different.”
12 ᾧ 5. It is clear, then, from the above arguments that a line does not consist of points.® (a) But neither is it possible to subtract a point from a line. For, if a point can be subtracted, it can also be added. But if anything is added, that to which it was added will be bigger than it was at first, if that which is added be such as to coalesce and form one whole with it. Hence a line will be bigger than another line by a point. And this is impossible. But though it is not possible to subtract a point as such from a line, one may subtract it zzcidentally, viz. in so far as a point 30he proposes to read ἂν ἢ γραμμὴ καὶ ἐπὶ στιγμῇ, which he translates ‘wenn auch eine Linie auf einem Punkte sein kann’: but one may envy, without wishing to imitate, this free-and-easy attitude to Greek Grammar. It seemed best to own myself defeated, and simply to print the original Greek.
is contained in the line which one is subtracting from another line. For since, if the whole be subtracted, its beginning and its end are subtracted too ; and since the beginning and the end of a line are points: then, if it be possible to subtract a line from a line, it will be possible also thereby to subtract a point.
972b
1 ἐν δὲ τῇ γραμμῇ οὐδὲν ἄλλο ἢ στιγμαὶ καὶ γραμμαὶ ἐνυπάρχουσιν.
ἡ δὲ γραμμὴ τῆς στιγμῆς οὐκ ἔστι μείζων· οὐδὲ
γὰρ αὖ τὸ ἐπίπεδον τῆς γραμμῆς. ὥστ' οὐκ ἔσται στιγμὴ
τὸ ἐν γραμμῇ ἐλάχιστον. εἰ δὲ συμβλητὸν τῇ γραμμῇ
5 ἡ στιγμή, τὸ δὲ ἐλάχιστον ἐν τρισὶ προσώποις, οὐκ ἔσται ἡ
στιγμὴ τῶν ἐν τῇ γραμμῇ ἐλάχιστον. καὶ ἄλλ' ἄττα ἐνυπάρχει
παρὰ τὰς στιγμὰς καὶ τὰς γραμμὰς ἐν τῷ μήκει·
οὐ γὰρ ἐκ στιγμῶν. εἰ δὲ τὸ ἐν τόπῳ ὂν ἡ στιγμὴ μῆκος
ἢ ἐπίπεδον ἢ στερεὸν ἐκ τούτων τι, ἐξ ὧν δ' ἐστὶν ἡ γραμμή,
10 ἐκεῖνα ἐν τόπῳ (καὶ γὰρ ἡ γραμμή), καὶ μήτε σῶμα
μήτ' ἐπίπεδον μήτε ἐκ τούτων τι ἐνυπάρχει τῇ γραμμῇ,
οὐκ ἔσται οὐθὲν ὅλως παρὰ τὰς στιγμὰς καὶ τὰς γραμμὰς
ἐν τῷ μήκει. ἔτι εἰ τοῦ ἐν τόπῳ ὄντος τὸ μεῖζον λεγόμενον
μῆκος ἡ ἐπιφάνεια στερεόν, ἡ δὲ στιγμὴ ἐν τόπῳ, τὸ
15 δ' ἐν τῷ μήκει ὑπάρχον παρὰ τὰς στιγμὰς καὶ τὰς γραμμὰς
οὐθὲν τῶν προειρημένων, ὥστ' οὐκ ἔσται ἡ στιγμὴ τῶν
ἐνυπαρχόντων ἐλάχιστον. ἔτι εἰς ὃ ἐλάχιστόν τι τῶν ἐν τῇ
οἰκίᾳ, μήτε τῆς οἰκίας συμβαλλομένης πρὸς αὐτὸ λέγεται·
ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων· οὐδὲ τὸ ἐν γραμμῇ ἐλάχιστον
20 πρὸς γραμμὴν συγκρινόμενον ἔσται. ὥστε οὐχ ἁρμόσει
τὸ ἐλάχιστον, ἐπεὶ τὸ μὴ ὂν ἐν τῇ οἰκίᾳ μή ἐστι τῶν
ἐν τῇ οἰκίᾳ ἐλάχιστον. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων.
ἐνδέχεται γὰρ στιγμὴν αὐτὴν καθ' αὑτὴν εἶναι. οὐκ ἔσται
κατὰ ταύτης ἀληθὲς εἰπεῖν ὅτι τὸ ἐν γραμμῇ ἐλάχιστον,
25 ὅτι οὐκ ἔστιν ἡ στιγμὴ ἄρθρον ἀδιαίρετον. τὸ μὲν γὰρ ἄρθρον
ἀεὶ δυοῖν ὅρος, ἡ δὲ στιγμὴ καὶ μιᾶς γραμμῆς ὅρος
ἐστίν. ἔτι ἡ μὲν πέρας, τὸ δὲ διαίρεσίς ἐστι μᾶλλον. ἔτι
ἡ γραμμὴ καὶ τὸ ἐπίπεδον ἄρθρα ἔσονται· ἀνάλογον γὰρ
ἔχουσιν, ὅτι τὸ ἄρθρον διάφορόν πως ἐστίν, διὸ καὶ Ἐμπεδοκλῆς
30 ἐποίησε διὸ δεῖ ὀρθῶς. ἡ δὲ στιγμὴ καὶ τὸ ἐν τοῖς
ἀκινήτοις. ἔτι οὐδεὶς ἔχει ἄπειρα ἄρθρα ἐν τῷ σώματι ἢ
τῇ χειρί, στιγμὰς δ' ἀπείρους. ἔτι λίθου ἄρθρον οὐκ ἔστιν,
οὐδ' ἔχει, στιγμὰς δὲ ἔχει.
1But such a subtraction of a point is ncidental or per accidens (Ὁ) But if the limit couches that of which it is the limit (touches either 7¢ or some one of its parts), and if the point, qué limit of the line, touches the line, then the line will be greater than another line bya point, and the point will consist of points. For there is nothing between two things in contact.
The same argument applies in the case of division, since the ‘division ’ is a point and, gué dividing-point, is in contact with something. It applies also in the case of a solid and a plane. And the solid must consist of planes, the plane of lines, just as (on the theory) the line consists of points.
" 820-24. I follow Hayduck and Apelt in reading εἰ (yap) τοῦ ὅλου ἀφαιρουμένου καὶ ἡ ἀρχὴ καὶ τὸ πέρας 5ἀφαιρεῖται, γραμμῆς δ᾽ ἦν ἡ ἀρχὴ καὶ τὸ πέρας στιγμή, καὶ εἰ γραμμῆς {ζγραμμὴν) ἐγχωρεῖ ἀφαιρεῖν, καὶ στιγμὴν (ἂν) ἐνδέχοιτο.
3. 824-27. The writer shows that it is wrong to conceive the limit as ‘in contact’ with that which it limits, and the point as ‘in contact’ with the line or any part of it.
In ]. 24 I read (with Apelt) οὗ τὸ πέρας for the MSS. οὔτε πέρας.
In 1. 25 I punctuate . .. ἐκείνου τινός, ἡ δὲ στιγμή, ἣ πέρας, γραμμῆς ἅπτεται, and in 1, 26 I adopt Apelt’s conjecture ἡ μὲν οὖν (γραμμὴ) γραμμῆς ἔσται στιγμῇ μείζων for the MSS. ἡ μὲν οὖν γραμμῆς ἔσται στιγμὴ μείζων [N ἡ μὲν οὖν γραμμὴ ἔσται στιγμῆς μείζων].
If the point C decomes the limit of the line «1.8, and is therefore ‘in contact’ with 4B, then (i) BR4+Cis > BA by the point C, and (ii) the terminal point © ls B C of the line CAB is the composite point C+A: for C and A are in contact whole-with-whole, and there is nothing between them.
% 428-30. This passage is obscure owing to its brevity. In]. 28 I read 10(with NW2) ὁ (8°) αὐτὸς λόγος. . ., but perhaps we ought to retain the asyndeton, in spite of its harshness. The writer’s style, especially at the end of the treatise, is abrupt and compressed in the extreme. In]. 28 I read εἰ ἡ τομὴ στιγμὴ [so Z*: the other MSS. read στιγμῆς] καὶ, 7 [MSS. ἡ] τομή, amrerai twos, and in 1. 30 I accept Apelt’s conjecture καὶ {τὸ ἐπίπεδον) ἐκ γραμμῶν. :
If a line consists of points in contact, division of a line—the actual ‘cut ’—is itself a point, and (gwd@ dividing-point) is in contact with the adjacent points, or halves of a point, which it separates. But if so, we shall be led to the same absurdities as before (cf. " 24-27). Hence (c) Neither? is it true to say of a point that it is ‘the smallest constituent of a line’.
(i) For if it be called ‘the smallest of the things contained in the line’, what is ‘smallest’ is also smad/er than those things. of which it is the smallest. But in the line there is contained nothing but points and 15lines: and the line is not bigger than the point, for neither is the plane bigger than the line.2» Hence the point will not be the smallest of the constituents in the line.® (ii) And if the point is comparable in magnitude with the line, yet, since ‘the smallest’ involves three degrees of comparison,* the point will not be the ssad/est of the constituents of the line: or 5 there will be other things in the length besides we must not regard division as ‘ dividing a point’, or as itself a ‘ point of dividing’. But if not, how can a line—which ex hyfothesz is nothing but ‘points in contact ’—be ‘ divided’?
The writer then briefly reminds us that, if a line consists of points in contact, on the same principle a plane is a sum of lines, a solid a sum of planes, in contact with one another: and if we thus conceive solids and planes, ‘the same argument’ will apply to them. One plane, e.g., will be greater than another by a line, one solid greater than another by a plane, if 20we are able to ‘subtract’ a line from a plane, and a plane from a solid ; and we shall get into difficulties with ‘division ’.
the points and lines, so that it will not consist of points. But, since that which is in place is either a point or a length ora plane or a solid, or some compound of these: and since the constituents of a line are in place (for the line is in place): and since neither a solid nor a plane, nor anything compounded of these, is contained in the line :—there can be absolutely nothing in the length except points and lines.?
(iii) Further, since that which is called ‘ greater’ than that which is in place is a length or a surface or a solid: then, since the point is in place, and since that which is contained in the length besides points and lines is none of the aforementioned : —the point cannot be the smallest of the constituents of a length.® (iv) Further, since ‘the smallest of the things contained in a house’ is so called, without in the least 25comparing the house with it, and so in all other cases :—neither will the smallest of the constituents in the line be determined by comparison with to be required by the logic of the passage. The writer propounds a dilemma :— (1) If there are only two kinds of constituent in the line, one of those kinds (viz. the point) cannot be the ‘ smallest’ ; (2) If, on the other hand, there are more than two kinds of constituent in the line, there must be something other than points and lines contained in it. This he shows to be impossible in the following argument.
the line. Hence the term ‘smallest’ applied to the point will not be suitable.!
(v) Further, that which is not in the house is not the smallest of the constituents of the house, and so in all other cases. Hence, since the point can exist per se, it will not be true to say of it that it is ‘the smallest thing in the line’.
(4) Lastly, the point is not an ‘ indivisible joint ’.* For (i) the joint is always a limit of 30two things, but the point is a limit of ove line as well as of two. Moreover (ii) the point is an end, but the joint is more of the nature of a division.
Again (iii) the line and the plane will be ‘joints’ (too) : for they are analogous to the point. Again (iv) the joint zs in a sense on account of movement (which explains the verse of Empedocles 5) : but the point is found also in the immovable things.° (v) Again, nobody has an infinity of joints in his body or his hand, but he has an infinity of points.® (vi) Moreover, there is no joint of a stone, nor has it any: but it has points.
The same argument applies in the case of division, since the ‘division ’ is a point and, gué dividing-point, is in contact with something. It applies also in the case of a solid and a plane. And the solid must consist of planes, the plane of lines, just as (on the theory) the line consists of points.
" 820-24. I follow Hayduck and Apelt in reading εἰ (yap) τοῦ ὅλου ἀφαιρουμένου καὶ ἡ ἀρχὴ καὶ τὸ πέρας 5ἀφαιρεῖται, γραμμῆς δ᾽ ἦν ἡ ἀρχὴ καὶ τὸ πέρας στιγμή, καὶ εἰ γραμμῆς {ζγραμμὴν) ἐγχωρεῖ ἀφαιρεῖν, καὶ στιγμὴν (ἂν) ἐνδέχοιτο.
3. 824-27. The writer shows that it is wrong to conceive the limit as ‘in contact’ with that which it limits, and the point as ‘in contact’ with the line or any part of it.
In ]. 24 I read (with Apelt) οὗ τὸ πέρας for the MSS. οὔτε πέρας.
In 1. 25 I punctuate . .. ἐκείνου τινός, ἡ δὲ στιγμή, ἣ πέρας, γραμμῆς ἅπτεται, and in 1, 26 I adopt Apelt’s conjecture ἡ μὲν οὖν (γραμμὴ) γραμμῆς ἔσται στιγμῇ μείζων for the MSS. ἡ μὲν οὖν γραμμῆς ἔσται στιγμὴ μείζων [N ἡ μὲν οὖν γραμμὴ ἔσται στιγμῆς μείζων].
If the point C decomes the limit of the line «1.8, and is therefore ‘in contact’ with 4B, then (i) BR4+Cis > BA by the point C, and (ii) the terminal point © ls B C of the line CAB is the composite point C+A: for C and A are in contact whole-with-whole, and there is nothing between them.
% 428-30. This passage is obscure owing to its brevity. In]. 28 I read 10(with NW2) ὁ (8°) αὐτὸς λόγος. . ., but perhaps we ought to retain the asyndeton, in spite of its harshness. The writer’s style, especially at the end of the treatise, is abrupt and compressed in the extreme. In]. 28 I read εἰ ἡ τομὴ στιγμὴ [so Z*: the other MSS. read στιγμῆς] καὶ, 7 [MSS. ἡ] τομή, amrerai twos, and in 1. 30 I accept Apelt’s conjecture καὶ {τὸ ἐπίπεδον) ἐκ γραμμῶν. :
If a line consists of points in contact, division of a line—the actual ‘cut ’—is itself a point, and (gwd@ dividing-point) is in contact with the adjacent points, or halves of a point, which it separates. But if so, we shall be led to the same absurdities as before (cf. " 24-27). Hence (c) Neither? is it true to say of a point that it is ‘the smallest constituent of a line’.
(i) For if it be called ‘the smallest of the things contained in the line’, what is ‘smallest’ is also smad/er than those things. of which it is the smallest. But in the line there is contained nothing but points and 15lines: and the line is not bigger than the point, for neither is the plane bigger than the line.2» Hence the point will not be the smallest of the constituents in the line.® (ii) And if the point is comparable in magnitude with the line, yet, since ‘the smallest’ involves three degrees of comparison,* the point will not be the ssad/est of the constituents of the line: or 5 there will be other things in the length besides we must not regard division as ‘ dividing a point’, or as itself a ‘ point of dividing’. But if not, how can a line—which ex hyfothesz is nothing but ‘points in contact ’—be ‘ divided’?
The writer then briefly reminds us that, if a line consists of points in contact, on the same principle a plane is a sum of lines, a solid a sum of planes, in contact with one another: and if we thus conceive solids and planes, ‘the same argument’ will apply to them. One plane, e.g., will be greater than another by a line, one solid greater than another by a plane, if 20we are able to ‘subtract’ a line from a plane, and a plane from a solid ; and we shall get into difficulties with ‘division ’.
the points and lines, so that it will not consist of points. But, since that which is in place is either a point or a length ora plane or a solid, or some compound of these: and since the constituents of a line are in place (for the line is in place): and since neither a solid nor a plane, nor anything compounded of these, is contained in the line :—there can be absolutely nothing in the length except points and lines.?
(iii) Further, since that which is called ‘ greater’ than that which is in place is a length or a surface or a solid: then, since the point is in place, and since that which is contained in the length besides points and lines is none of the aforementioned : —the point cannot be the smallest of the constituents of a length.® (iv) Further, since ‘the smallest of the things contained in a house’ is so called, without in the least 25comparing the house with it, and so in all other cases :—neither will the smallest of the constituents in the line be determined by comparison with to be required by the logic of the passage. The writer propounds a dilemma :— (1) If there are only two kinds of constituent in the line, one of those kinds (viz. the point) cannot be the ‘ smallest’ ; (2) If, on the other hand, there are more than two kinds of constituent in the line, there must be something other than points and lines contained in it. This he shows to be impossible in the following argument.
the line. Hence the term ‘smallest’ applied to the point will not be suitable.!
(v) Further, that which is not in the house is not the smallest of the constituents of the house, and so in all other cases. Hence, since the point can exist per se, it will not be true to say of it that it is ‘the smallest thing in the line’.
(4) Lastly, the point is not an ‘ indivisible joint ’.* For (i) the joint is always a limit of 30two things, but the point is a limit of ove line as well as of two. Moreover (ii) the point is an end, but the joint is more of the nature of a division.
Again (iii) the line and the plane will be ‘joints’ (too) : for they are analogous to the point. Again (iv) the joint zs in a sense on account of movement (which explains the verse of Empedocles 5) : but the point is found also in the immovable things.° (v) Again, nobody has an infinity of joints in his body or his hand, but he has an infinity of points.® (vi) Moreover, there is no joint of a stone, nor has it any: but it has points.