The Politics of Aristotle

by Aristotle

  3Reading πρῶτα for ἄλφιτα.

  4Reading ταὐτό.

  5Omitting δέ.

  6Reading πρoΐεται ὐγρòν. ταὐτò οὖν τοῦτο καὶ δίπυρος.

  7Reading ἐκλιπεῖν.

  8Reading ἀγγεῖoν for αἴτιoν.

  9Reading κενόν for καἰ.

  10Reading τò δἐ ἐν ὑγρῷ θερμòν πoιεῖ.

  11Adding ἔξεισι after ὑπóλoιπoν.

  12Reading πλεῖoν καὶ διὰ τoῦτo καί.

  13Reading ὅ for oὐ.

  14Reading πλεἰω ἢ ἐλἀττω.

  15Reading συμπιἐζεται for συμπιἐζει τó.

  1Placing a comma before oὐχ òμoλoγεῖται.

  2Reading συνἰσταται μᾶλλoν ἤ ἐξατμἰζει.

  3Reading γλυκὐ.

  1Omitting ἐκ.

  2Omitting oὐκ before ὰνακλᾶται and inserting oὐ before διέρχεται.

  3Reading ἤ for ἤ.

  4Omitting τó before πνέoν.

  5Reading θάλαττα; ἐνδέχεται γάρ.

  6Reading ή τῆς θαλάττης.

  7Reading τò ξηρóτερoν. τῇ δἐ θαλἀττῃ ἄμφω ταῦτα.

  8Reading διαρρεῖ.

  9Iliad VII 64.

  10Reading μιγνυμένoυ.

  11Omitting ὅτι μᾶλλoν ἔχει γῆν ò γλυκὐς; ἢ.

  12Placing a comma after oὐ instead of after μᾶλλoν.

  13Reading ἐφιστἀμενoν.

  14Reading τῶν πóρων ἄνω.

  15Reading Παῖσα λίμνη.

  1Reading ὐγρóν τι for μἡ ψóφoς.

  2Reading θἐρoνς for ψὐχoνς.

  3Placing a comma before instead of after βίᾳ.

  4Placing a comma before instead of after τò πέριξ.

  5Reading διά for καί.

  6Reading ἐκθνἠσκoυσι.

  7Reading ψὐχεσθαι καἰ ἔτι ἐπἰ τῷ σώματι ὅν.

  1Reading ἰἐναι for εὧαι.

  2Omitting ἢ ὅσα ἔλη λιμνἀζoνται.

  3Omitting εἰ τρἰψει διἀ τoῦ καταδἡματoς.

  4Omitting oὐχ.

  5Omitting καὶ ἀήρ.

  6Reading πρóωσιν.

  7Reading ἔξω.

  1Reading κυρτά.

  2Reading τoῖς μεὺ ὑψηλoτέρoυς τoὺς ἐναντίoυς.

  3Reading κoῖλα.

  4Reading τῷ περἰ τἠν γἠν ἔχειν τἠν τελευτἠν τῆς φoρᾶς.

  5Omitting καθάπερ ὅρθρoς.

  6Reading oὐχ ἧττoν.

  7Reading oὐ πόρρω.

  8Reading ἄστρoις τὀ ἐπισημαἱνειν.

  9Reading ἐπιδεξἱoυς oὐ μεταβἁλλει.

  10Reading καθ’ αὑτὀν.

  11Placing a comma after πρoσπἱπτει and a full stop after πεδἱῳ.

  12Reading λoιπόν for θερμόν.

  13Reading χὡρα θερμἡ’ πρὀς δέ.

  14Reading κoμἱζεται.

  15Odyssey IV 567.

  16Reading ἐστι for ἐπἰ.

  17Excised by Ruelle.

  18Reading ἐκπέττει.

  19Reading ἀπoγέας.

  20Reading διἁθεσιν for τἁξιν.

  21Reading ὕλην.

  22Reading ἀραχνῶν.

  1Reading ἵν’ ἀναθερμανθῇ.

  2Omitting τoῖς μέν.

  3Reading τῶν τoιoὑτων.

  4Reading ἄνωθεν τρέμει, διότι τὀ ἄνωθεν κἁτω.

  5Reading μόνoν.

  6Reading συμβαἱνει.

  1Reading δεῖται τoῦ ὑγρoῦ τὀ θερμὀν ᾧ ζῶμεν.

  2Reading ό λέων for ό όρῶν.

  3Reading ἀπoλαὑσει.

  4Reading ἡδεῖα.

  5Reading ἡ δ’ εὔνoια ἐξαἱρει, ὥστε κινεῖ μᾶλλoν τὀ γελoῖoν.

  1Reading χρἡματα.

  2Reading oὐδέ for ἀλλἁ.

  3Reading αἴσχιoν.

  4Reading μεἱζω for ἐλἁττω.

  5Reading αυτῷ.

  6Omitting τό and ἔχεσθαι.

  7Reading διὡκων for ἀδικῶν.

  8Reading μεταγνoῦσιν.

  9The text of this sentence is uncertain.

  10Reading τό τ’.

  11Reading πoνηρoἱ.

  1Iliad VI 200.

  2Odyssey XIX 122.

  3Omitting πνευματὡδεις.

  4Reading σκληφρoἱ.

  5Reading ἐπανεθῇ ἡ ἄγαν θερμότης.

  6Readings ἐν for μέν.

  7Reading ἐπιπόλαια for παλαιἁ.

  8Omitting τὀ μαραινόμενoν θερμόν.

  9Reading ἔτι.

  10Reading oὕ τως ὧν.

  11Reading δυνἁμεθα δέχεσθαι for δυνἁμεθα. δυνἁμεθα δἐ ἔχεσθαι.

  12Reading τoιoὑτoις; ἢ ὅτι.

  13Reading ἄλλη for ἐν ἄλλῃ.

  14Reading λόγoυ καἰ σoφἱας.

  15Reading ἐζ ἧς for ἐζ ὧν.

  1Reading ὥστ’ ἀντιβλέπειν.

  2Reading τὀ θερμόν.

  3Reading μέχρι τoυ.

  4Reading κατἀ ταὐτό.

  5Reading ὦσιν. καἰ διαστρoφἡ.

  6Omitting the final sentence.

  7Reading τἠν αὐγἡν.

  8Reading όρᾷ for δρᾷ.

  1Omitting ἀνατέμνoυσι δἐ … εὔπνoιαν.

  2Reading πόρoυ.

  1Reading τἀ ὦτα.

  2Reading βῆχα ἣ, ἐἁν.

  3Reading συνεστἁναι.

  4Reading ὀσμἡ for ῥὑμη.

  5Reading ἢ ὅτι for ὅσoις.

  6Reading ὅταν μέλλoυσιν ἢ ἀρχoμένoις συμβῇ.

  7Reading ἐζαερoυμένoυ.

  8Omitting ἡμῶν.

  9Reading πνεὑμoνoς for πνεὑματoς.

  1Reading ψυχρoῦ for ἐναντἱoυ.

  2Reading θερμός for ψυχρός.

  3Reading τό for τῷ.

  4Ruelle accidentally omits this clause.

  5Reading oὐ for oὖν.

  6Reading αὐτῆς for αὐτoῖς, without a lacuna.

  1Reading καθεὑδoυσιν ἡμῖν, ἡδoνἠ θαυμασἰα.

  2Reading θιγεῖν for εἰπεῖν.

  3Reading ἐκτός for ἐντός.

  1Reading τῇ τρoφῇ.

  2Reading ἅ.

  3Reading ἐπιπoλῆς.

  ON INDIVISIBLE LINES**

  H.H. Joachim

  [968a1] Are there indivisible lines? And, generally, is there something partless in every class of quanta, as some say?

  For if, where ‘many’ and ‘large’ apply, so do their opposites, ‘few’ and ‘small’; [5] and if that which admits practically an infinite number of divisions is many not few, then what is few and what is small will clearly admit only a finite number of divisions. But if the divisions are finite in number, there must be a partless magnitude. Hence in all classes of quanta there will be found something partless, since in all of them ‘few’ and ‘small’ apply.

  Again, if there is an Idea of line, and if the Idea is first of the things called by [10] 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 things prior [15] to each of them.

  Again, if body consists of elements, and if there is nothing prior to the elements, and if parts are prior to their whole, then fire and, generally, each of the elements which are the constituents of body must be indivisible. Hence there must be something partless in the objects of sense as well as in the objects of thought.

  Again, Zeno’s argument proves that there must be partless magnitudes. For it [20] is impossible to touch an infinite number of things in a finite time, touching them one by one; and the moving body must reach the half-way point before it reaches the end; and there always is a half-way point in any non-partless thing.

  But even if the body, which is moving along the line, does touch the infinity of points in a finite time; and if the quicker the movement of the moving body, the greater the stretch which it traverses in an equal time; and if the movement of [968b1] 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 one-by-one is counting, it is possible to count infinitely many objects in a finite time. But since this is impossible, there must be such a thing as an indivisible line.

  [5] Again, the being of indivisible lines (it is maintained) follows from the mathematicians’ own statements. For if commensurate lines are those which are measured by the same unit of measurement, and if all commensurate lines are being measured,1 there will be some 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. Thus half [10] of a certain part will be double it. 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 partless elements, so also are the lines which the unit measures once.

  And the same can be shown to follow in the plane figures too. For all which are [15] drawn on the rational lines are commensurate with one another; and therefore their unit of measurement will be partless.

  But if any such plane be cut along any prescribed and determinate line, that line will be neither rational nor irrational, nor will any of the other kinds of lines which produce rational squares, such as the ‘apotome’ or the ‘line ex duobus nominibus’. Such lines will have no nature of their own at all; though, relatively to [20] one another, they will be rational or irrational.

  Now in the first place, it does not follow that that which admits an infinite number of divisions is not small or few. For we apply the predicate ‘small’ to place and magnitude, and generally to the continuous (and we apply ‘few’ where that is [25] applicable); and nevertheless we affirm that these quanta admit an infinite number of divisions.

  Moreover, if in the composite magnitude there are contained indivisible lines,2 the predicate ‘small’ is applied to these indivisible lines, and each of them contains an infinite number of points. But each of them, quâ line, admits of division at a [969a1] point, and equally at any and every point: hence each of these non-indivisible lines would admit an infinite number of divisions. Moreover, some amongst the non-indivisible lines are small. The ratios are infinite in number; and every non-indivisible line admits of division in accordance with any prescribed ratio. [5]

  Again, since the great is compounded of certain small things, 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. 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. [10]

  It is clear, therefore, that it is not quâ admitting a finite and an infinite number of divisions that quanta are called small and great respectively. And to argue that, because in numbers what is few admits a finite number of divisions, therefore in lines the small line must admit only a finite number of divisions, is childish. For in numbers the development is from partless objects, and there is a determinate something from which the whole series of the numbers starts, and every number [15] which is not infinite admits a finite number of divisions; but in magnitudes the case is not parallel.

  As to those who try to establish indivisible lines by arguments drawn from the Ideal Lines, 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 [20] in a sense destroy the premisses which they use to prove their conclusion. For their arguments destroy the Ideas.

  Again, as to the corporeal elements, it is childish to postulate them as partless. For even though some do as a matter of fact make this statement about them, yet to assume this for the present inquiry is to assume the point at issue. Or rather, the [25] more obviously the argument would appear to assume the point at issue, the more the opinion is confirmed that solids and lengths are divisible in bulk and distance.

  The argument of Zeno does not establish that the moving body comes into contact with the infinite number of points in a finite time in the same way. For the [30] time and the length are called infinite and finite and admit of the same divisions.

  Nor is thought’s coming into contact with the members of an infinite series one-by-one counting, even if it were supposed that thought does come into contact in this way with the members of an infinite series. Such a supposition perhaps assumes what is impossible: for the movement of thought does not, like the movement of [969b1] moving bodies, essentially involve continua and substrata.

  If, however, the possibility of thought moving in this fashion be admitted, still this moving is not counting; for counting is movement combined with pausing.

  It is surely absurd that, because you are unable to solve Zeno’s argument, you [5] should make yourselves slaves of your inability, and should commit yourselves to still greater errors, in the endeavour to support your incompetence.

  As to what they say about commensurate lines—that all lines are measured by one and the same 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 any use to them.

  [10] Moreover, it is actually inconsistent to postulate both that every line becomes commensurate, and that there is a common measure of all commensurate lines.

  Hence their procedure is ridiculous, since, whilst professing that they are going to demonstrate their thesis in accordance with3 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 [15] such a feeble piece of sophistry too! For it is feeble in many respects, and totally unable to escape paradox and refutation.

  Moreover, it would be absurd for people to be led astray by Zeno’s arguments, and to be persuaded—because they cannot refute it—to invent indivisible lines; and yet because of the movement of a straight line to make a semicircle, which must [20] touch infinitely many arcs and distances in between, and because of its movement to form a circle, which readily shows that it must move at every point if it moves to make a semicircle, and because of other similar considerations about lines—to refuse to accept that a movement can be generated such that in it the moving thing does not fall successfully on each of the intervening points before reaching the [25] end-point. For the theorems in question are more generally admitted, than the arguments of Zeno.

  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.

&nbs
p; In the first place, our result will be confirmed by reflection on the conclusions proved in mathematics, and on the assumptions there laid down—conclusions and [30] assumptions which must either stand or be overthrown by more convincing arguments.

  For 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.

  Secondly, all lines will be commensurate. For all lines—both those which are [970a1] 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. [5]

  Again, since the line applied to the longer side determines the breadth of the figure, the rectangle, which is equal in area to the square on the indivisible line (e.g. on the line one foot long), will, if applied to a line double the indivisible line, 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.

  Again, since any three given straight lines can be combined to form a triangle, [10] a triangle can also be formed by combining three given indivisible lines. But in every equilateral triangle the perpendicular dropped from the apex bisects the base. Hence, it will bisect the indivisible base too.

  Again, if the square can be constructed of partless lines, then let its diagonal be drawn, and a perpendicular be dropped. The square on the side will be equal to the square on the perpendicular together with the square on half the diagonal. Hence it will not be the smallest line. [15]

  Nor will the area which is the square on the diagonal be double the square on the indivisible line. For 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 partless line. For if it were equal the square on the diagonal would have been four times the original square.

  And one might collect other similar absurdities to which the doctrine leads; for indeed it conflicts with practically everything in mathematics. [20]

  Again, what is partless admits of only one mode of conjunction, but a line admits of two: for one line may be conjoined to another either along the whole length of both lines, or by contact at either of its opposite terminal points.