Bekker (Berlin, 1831) · Forster (1913)
Forster (1913)
Chapter 1 (847a11–848a37)
847a
Θαυμάζεται τῶν μὲν κατὰ φύσιν συμβαινόντων, ὅσων
ἀγνοεῖται τὸ αἴτιον, τῶν δὲ παρὰ φύσιν, ὅσα γίνεται διὰ
τέχνην πρὸς τὸ συμφέρον τοῖς ἀνθρώποις. ἐν πολλοῖς γὰρ
ἡ φύσις ὑπεναντίον πρὸς τὸ χρήσιμον ἡμῖν ποιεῖ· ἡ μὲν
15 γὰρ φύσις ἀεὶ τὸν αὐτὸν ἔχει τρόπον καὶ ἁπλῶς, τὸ δὲ
χρήσιμον μεταβάλλει πολλαχῶς. ὅταν οὖν δέῃ τι παρὰ
φύσιν πρᾶξαι, διὰ τὸ χαλεπὸν ἀπορίαν παρέχει καὶ δεῖται
τέχνης. διὸ καὶ καλοῦμεν τῆς τέχνης τὸ πρὸς τὰς τοιαύτας
ἀπορίας βοηθοῦν μέρος μηχανήν. καθάπερ γὰρ ἐποίησεν
20 Ἀντιφῶν ὁ ποιητής, οὕτω καὶ ἔχει· τέχνῃ γὰρ κρατοῦμεν,
ὧν φύσει νικώμεθα. τοιαῦτα δέ ἐστιν ἐν οἷς τά τε ἐλάττονα
κρατεῖ τῶν μειζόνων, καὶ τὰ ῥοπὴν ἔχοντα μικρὰν κινεῖ
βάρη μεγάλα, καὶ πάντα σχεδὸν ὅσα τῶν προβλημάτων
μηχανικὰ προσαγορεύομεν. ἔστι δὲ ταῦτα τοῖς φυσικοῖς
25 προβλήμασιν οὔτε ταὐτὰ πάμπαν οὔτε κεχωρισμένα λίαν,
ἀλλὰ κοινὰ τῶν τε μαθηματικῶν θεωρημάτων καὶ τῶν
φυσικῶν· τὸ μὲν γὰρ ὣς διὰ τῶν μαθηματικῶν δῆλον, τὸ
δὲ περὶ ὃ διὰ τῶν φυσικῶν. περιέχεται δὲ τῶν ἀπορουμένων
Our wonder is excited, firstly, by phenomena which occur in accordance with nature but of which we do not know the cause, and secondly by those which are produced by art despite nature for the benefit of mankind. Nature often operates contrary to human expediency; for she 15always follows the same course without deviation, whereas human 1; expediency is always changing. When, therefore, we have to do something contrary to nature, the difficulty of it causes us perplexity and art has to be called to our aid. The kind of art which helps us in such perplexities we call Mechanical Skill. The words of the poet Antiphon 20are quite true: ‘Mastered by Nature, we o’ercome by Art.’ Instances of this are those cases in which the less prevails over the greater, and where forces of small motive power move great weights—in fact, practically all those problems which we call Mechanical Problems. They are not quite identical nor yet entirely unconnected with Natural Problems. 25They have something in common both with Mathematical and with Natural Speculations; for while Mathematics demonstrates how phenomena come to pass ; Natural Science demonstrates in what medium they occur.
847b
ἐν τῷ γένει τούτῳ τὰ περὶ τὸν μοχλόν. ἄτοπον γὰρ
εἶναι δοκεῖ τὸ κινεῖσθαι μέγα βάρος ὑπὸ μικρᾶς ἰσχύος,
καὶ ταῦτα μετὰ βάρους πλείονος· ὃ γὰρ ἄνευ μοχλοῦ κινεῖν
οὐ δύναταί τις, τοῦτο ταὐτὸ βάρος, προσλαβὼν ἔτι τὸ
15 τοῦ μοχλοῦ βάρος, κινεῖ θᾶττον. πάντων δὲ τῶν τοιούτων
ἔχει τῆς αἰτίας τὴν ἀρχὴν ὁ κύκλος. καὶ τοῦτο εὐλόγως
συμβέβηκεν· ἐκ μὲν γὰρ θαυμασιωτέρου συμβαίνειν τι
θαυμαστὸν οὐδὲν ἄτοπον, θαυμασιώτατον δὲ τὸ τἀναντία
γίνεσθαι μετ' ἀλλήλων. ὁ δὲ κύκλος συνέστηκεν ἐκ τοιούτων·
20 εὐθὺς γὰρ ἐκ κινουμένου τε γεγένηται καὶ μένοντος, ὧν ἡ
φύσις ἐστὶν ὑπεναντία ἀλλήλοις. ὥστ' ἐνταῦθα ἔστιν ἐπιβλέψασιν
ἧττον θαυμάζειν τὰς συμβαινούσας ὑπεναντιώσεις
περὶ αὐτόν. πρῶτον μὲν γὰρ τῇ περιεχούσῃ γραμμῇ τὸν
κύκλον, πλάτος οὐθὲν ἐχούσῃ, τἀναντία πως προσεμφαίνεται,
25 τὸ κοῖλον καὶ τὸ κυρτόν. ταῦτα δὲ διέστηκεν ἀλλήλων
ὃν τρόπον τὸ μέγα καὶ τὸ μικρόν· ἐκείνων τε γὰρ
μέσον τὸ ἴσον καὶ τούτων τὸ εὐθύ. διὸ μεταβάλλοντα εἰς
ἄλληλα τὰ μὲν ἀναγκαῖον ἴσα γενέσθαι πρότερον ἢ τῶν
Among questions of a mechanical kind are included those which are connected with the lever. It seems strange that a great weight can be moved with but little force, and that when the addition of more weight is involved ; for the very same weight, which one cannot move at all 15without a lever, one can move quite easily with it, in spite of the additional weight of the lever. The original cause of all such phenomena is the circle. It is quite natural that this should be so; for there is nothing strange in a lesser marvel being caused by a greater marvel, and it is a very great marvel that contraries should be present 20together, and the circle is made up of contraries. For to begin with, it is formed by motion and rest,} things which are by nature opposed to one another. Hence in examining the circle we need not be much astonished at the contradictions which occur in connexion with it. Firstly, in the line which encloses the circle, being without breadth, two 25contraries somehow appear, namely, the concave and the convex. These are as much opposed to one another as the great is to the small; the mean being Ϊ in the latter case the equal, in the former the straight.
848a
1 ἄκρων ὁποτερονοῦν, τὴν δὲ γραμμὴν εὐθεῖαν, ὅταν ἐκ κυρτῆς
εἰς κοῖλον ἢ πάλιν ἐκ ταύτης γίνηται κυρτὴ καὶ περιφερής.
ἓν μὲν οὖν τοῦτο τῶν ἀτόπων ὑπάρχει περὶ τὸν κύκλον,
δεύτερον δὲ ὅτι ἅμα κινεῖται τὰς ἐναντίας κινήσεις·
5 ἅμα γὰρ εἰς τὸν ἔμπροσθεν κινεῖται τόπον καὶ τὸν ὄπισθεν.
ἥ τε γράφουσα γραμμὴ τὸν κύκλον ὡσαύτως ἔχει· ἐξ
οὗ γὰρ ἄρχεται τόπου τὸ πέρας αὐτῆς, εἰς τὸν αὐτὸν τοῦτον τόπον
ἔρχεται πάλιν· συνεχῶς γὰρ κινουμένης αὐτῆς τὸ ἔσχατον
πάλιν ἀπῆλθε πρῶτον, ὥστε καὶ φανερὸν ὅτι μετέβαλεν
10 ἐντεῦθεν. διό, καθάπερ εἴρηται πρότερον, οὐδὲν ἄτοπον τὸ
πάντων εἶναι τῶν θαυμάτων αὐτὸν ἀρχήν. τὰ μὲν οὖν περὶ
τὸν ζυγὸν γινόμενα εἰς τὸν κύκλον ἀνάγεται, τὰ δὲ περὶ
τὸν μοχλὸν εἰς τὸν ζυγόν, τὰ δ' ἄλλα πάντα σχεδὸν τὰ
περὶ τὰς κινήσεις τὰς μηχανικὰς εἰς τὸν μοχλόν. ἔτι δὲ
15 διὰ τὸ μιᾶς οὔσης τῆς ἐκ τοῦ κέντρου γραμμῆς μηθὲν ἕτερον
ἑτέρῳ φέρεσθαι τῶν σημείων τῶν ἐν αὐτῇ ἰσοταχῶς, ἀλλ' ἀεὶ
τὸ τοῦ μένοντος πέρατος πορρώτερον ὂν θᾶττον, πολλὰ τῶν θαυμαζομένων
συμβαίνει περὶ τὰς κινήσεις τῶν κύκλων· περὶ
ὧν ἐν τοῖς ἑπομένοις προβλήμασιν ἔσται δῆλον. διὰ δὲ τὸ
20 τὰς ἐναντίας κινήσεις ἅμα κινεῖσθαι τὸν κύκλον, καὶ τὸ
μὲν ἕτερον τῆς διαμέτρου τῶν ἄκρων, ἐφ' οὗ τὸ Α, εἰς τοὔμπροσθεν
κινεῖσθαι, θάτερον δέ, ἐφ' οὗ τὸ Β, εἰς τοὔπισθεν,
κατασκευάζουσί τινες ὥστ' ἀπὸ μιᾶς κινήσεως πολλοὺς ὑπεναντίους
ἅμα κινεῖσθαι κύκλους, ὥσπερ οὓς ἀνατιθέασιν ἐν
25 τοῖς ἱεροῖς ποιήσαντες τροχίσκους χαλκοῦς τε καὶ σιδηροῦς.
εἰ γὰρ εἴη τοῦ ΑΒ κύκλου ἁπτόμενος ἕτερος κύκλος ἐφ' οὗ
ΓΔ, τοῦ κύκλου τοῦ ἐφ' οὗ ΑΒ κινουμένης τῆς διαμέτρου
εἰς τοὔμπροσθεν, κινηθήσεται ἡ ΓΔ εἰς τοὔπισθεν τοῦ κύκλου
τοῦ ἐφ' οὗ Α, κινουμένης τῆς διαμέτρου περὶ τὸ αὐτό. εἰς
30 τοὐναντίον ἄρα κινηθήσεται ὁ ἐφ' οὗ ΓΔ κύκλος τῷ ἐφ'
οὗ τὸ ΑΒ· καὶ πάλιν αὐτὸς τὸν ἐφεξῆς, ἐφ' οὗ ΕΖ, εἰς
τοὐναντίον αὑτῷ κινήσει διὰ τὴν αὐτὴν αἰτίαν. τὸν αὐτὸν δὲ
τρόπον κἂν πλείους ὦσι, τοῦτο ποιήσουσιν ἑνὸς μόνου κινηθέντος.
ταύτην οὖν λαβόντες ὑπάρχουσαν ἐν τῷ κύκλῳ τὴν
35 φύσιν οἱ δημιουργοὶ κατασκευάζουσιν ὄργανον κρύπτοντες
τὴν ἀρχήν, ὅπως ᾖ τοῦ μηχανήματος φανερὸν μόνον τὸ
θαυμαστόν, τὸ δ' αἴτιον ἄδηλον.
1Ἶ Therefore just as, if they are to change into one another, i the greater and smaller must become equal before they can ᾿ 8485 pass into the other extreme; so a line must become straight in passing from convex into concave, or on the other hand from concave into convex and curved. 5This, i is one peculiarity of the circle. Another peculiarity of the circle is that it moves in two contrary directions at the same time; for it moves simultaneously to a forward and a backward position.2 Such, too, is the nature of the radius which describes a circle. For its extremity comes back again to the same position from which it starts; for, when 10it moves continuously, its last position is a return to its original position, in such 10a way that it has clearly undergone a change from that position. Therefore, as has already been remarked, there is nothing strange in the circle being the origin of any and every marvel. The phenomena observed in the balance can be referred to the circle, and those 15observed in the lever to the balance; while practically all the other phenomena of mechanical motion are connected with the lever. Furthermore, since no two points on one and the same radius travel with the same rapidity, but of two points that which is further from the fixed centre travels more quickly, many marvellous phenomena occur in the motions of 20circles, which will be demonstrated in the following problems. ; Because a circle moves in two contrary forms of motion at the same time, and because one extremity of the diameter, A, moves forwards and the other, B, moves backwards, some people contrive so that as the result of a single movement a number of circles move simultaneously in contrary directions, 25like the wheels of brass and iron which they make and dedicate in the temples. Let AB be a circle and ΓΔ another circle in contact with it; then if Pye; τ; the diameter of the circle AB moves forward, the diameter ΓΔ will move in a backward direction as compared with the circle AB, as long as the diameter moves round the same point, The circle TA therefore 30will move in the. opposite _ direction to the circle ΑΒ, Again, the circle TA will itself make the adjoining circle EZ move in an opposite direction to itself for the same reason, The same thing will happen in the case of a larger number of circles, only one of them being set in motion. Mechanicians seizing on this inherent peculiarity of the circle, 35and hiding the principle, construct an instrument so as to exhibit the marvellous character _ of the device, while they obscure the cause of it.
Chapter 2 (848b1–850a2)
848b
1 Πρῶτον μὲν οὖν τὰ συμβαίνοντα περὶ τὸν ζυγὸν ἀπορεῖται,
διὰ τίνα αἰτίαν ἀκριβέστερά ἐστι τὰ ζυγὰ τὰ μείζω
τῶν ἐλαττόνων. τούτου δὲ ἀρχή, διὰ τί ποτε ἐν τῷ κύκλῳ
ἡ πλεῖον ἀφεστηκυῖα γραμμὴ τοῦ κέντρου τῆς ἐγγὺς τῇ
5 αὐτῇ ἰσχύϊ κινουμένης θᾶττον φέρεται τῆς ἐλάττονος; τὸ
γὰρ θᾶττον λέγεται διχῶς· ἄν τε γὰρ ἐν ἐλάττονι χρόνῳ
ἴσον τόπον διεξέλθῃ, θᾶττον εἶναι λέγομεν, καὶ ἐὰν ἐν ἴσῳ
πλείω. ἡ δὲ μείζων ἐν ἴσῳ χρόνῳ γράφει μείζονα κύκλον·
ὁ γὰρ ἐκτὸς μείζων τοῦ ἐντός. αἴτιον δὲ τούτων ὅτι φέρεται
10 δύο φορὰς ἡ γράφουσα τὸν κύκλον. ὅταν μὲν οὖν ἐν λόγῳ
τινὶ φέρηται, ἐπ' εὐθείας ἀνάγκη φέρεσθαι τὸ φερόμενον,
καὶ γίνεται διάμετρος αὐτὴ τοῦ σχήματος ὃ ποιοῦσιν αἱ
ἐν τούτῳ τῷ λόγῳ συντεθεῖσαι γραμμαί. ἔστω γὰρ ὁ λόγος
ὃν φέρεται τὸ φερόμενον, ὃν ἔχει ἡ ΑΒ πρὸς τὴν ΑΓ·
15 καὶ τὸ μὲν ΑΓ φερέσθω πρὸς τὸ Β, ἡ δὲ ΑΒ ὑποφερέσθω
πρὸς τὴν ΗΓ· ἐνηνέχθω δὲ τὸ μὲν Α πρὸς τὸ Δ, ἡ δὲ ἐφ'
ᾗ ΑΒ πρὸς τὸ Ε. εἰ οὖν ἐπὶ τῆς φορᾶς ὁ λόγος ἦν ὃν ἡ
ΑΒ ἔχει πρὸς τὴν ΑΓ, ἀνάγκη καὶ τὴν ΑΔ πρὸς τὴν
ΑΕ τοῦτον ἔχειν τὸν λόγον. ὅμοιον ἄρα ἐστὶ τῷ λόγῳ τὸ
20 μικρὸν τετράπλευρον τῷ μείζονι, ὥστε καὶ ἡ αὐτὴ διάμετρος
αὐτῶν, καὶ τὸ Α ἔσται πρὸς Ζ. τὸν αὐτὸν δὴ τρόπον
δειχθήσεται κἂν ὁπουοῦν διαληφθῇ ἡ φορά· αἰεὶ γὰρ
ἔσται ἐπὶ τῆς διαμέτρου. φανερὸν οὖν ὅτι τὸ κατὰ τὴν διάμετρον
φερόμενον ἐν δύο φοραῖς ἀνάγκη τὸν τῶν πλευρῶν
25 φέρεσθαι λόγον. εἰ γὰρ ἄλλον τινά, οὐκ οἰσθήσεται κατὰ
τὴν διάμετρον. ἐὰν δὲ ἐν μηδενὶ λόγῳ φέρηται δύο φορὰς
κατὰ μηδένα χρόνον, ἀδύνατον εὐθεῖαν εἶναι τὴν φοράν.
ἔστω γὰρ εὐθεῖα. τεθείσης οὖν ταύτης διαμέτρου, καὶ παραπληρωθεισῶν
τῶν πλευρῶν, ἀνάγκη τὸν τῶν πλευρῶν λόγον
30 φέρεσθαι τὸ φερόμενον· τοῦτο γὰρ δέδεικται πρότερον. οὐκ
ἄρα ποιήσει εὐθεῖαν τὸ ἐν μηδενὶ λόγῳ φερόμενον μηδένα
χρόνον. ἐὰν γάρ τινα λόγον ἐνεχθῇ ἐν χρόνῳ τινί, τοῦτον
ἀνάγκη τὸν χρόνον εὐθεῖαν εἶναι φορὰν διὰ τὰ προειρημένα.
ὥστε περιφερὲς γίνεται, δύο φερόμενον φορὰς ἐν μηθενὶ
35 λόγῳ μηθένα χρόνον. ὅτι μὲν τοίνυν ἡ τὸν κύκλον γράφουσα
φέρεται δύο φορὰς ἅμα, φανερὸν ἔκ τε τούτων,
καὶ ὅτι τὸ φερόμενον κατ' εὐθεῖαν ἐπὶ τὴν κάθετον ἀφικνεῖται,
1I First, then, a question arises as to what takes place 848? in the case of the balance. Why are larger balances more accurate than smaller? And the fundamental principle of this is, why is it that the radius which extends further from the centre is displaced quicker 5than the smaller radius, ; when the near radius is moved by the same force? Now we use the word ‘ quicker’ in two senses; if an object traverses an equal distance in less time, we call it quicker, and also if it traverses a greater distance in equal time. j Now the greater radius describes a greater circle in equal time; for the outer 10circumference is greater than the inner. The reason of this is that the radius undergoes two displacements. Now if the two displacements of a body are in any fixed proportion, the resulting displacement must necessarily be a straight line, and this? line is the diagonal of the figure, made by the lines drawn in this proportion. Let the 15proportion of the two displacements be as AB to Β Z r H AT,? and let A be brought ® to B, and the line AB brought down to ΗΓ, Again, let A be brought to A and the line AB to E; then if the proportion of the two displacements be maintained, AA must necessarily have the same pro- . portion to AE as AB to AT. Therefore the small parallelogram 20is similar to the greater, and their diagonal is the same, so that A will be at Z. In the same way it can be shown, at whatever points the displacement.be arrested, that the point A will in all cases be on the diagonal. And the converse is also true. It is plain that, if a point be moved along the diagonal by two displacements, it 25is necessarily moved according to the proportion of the sides of the parallelogram ; for otherwise it will not be moved along the diagonal. If it be moved in two displacements in no fixed ratio for any time, its displacement cannot be in a straight line. For let it be a straight line. This then being drawn as a diagonal, and the sides of 30the parallelogram filled in, the point must necessarily be moved according to the proportion of the sides; for this has already been proved. Therefore, if the same proportion be not maintained during any interval of time, the point will not describe a straight line; for, if the proportion were maintained during any interval, the point 35must necessarily describe a straight line, by the reasoning above. So that, if the two displacements do not maintain any proportion A.
849a
1 ὥστε εἶναι πάλιν αὐτὴν ἀπὸ τοῦ κέντρου κάθετον.
ἔστω κύκλος ὁ ΑΒΓ, τὸ δ' ἄκρον τὸ ἐφ' οὗ Β φερέσθω
ἐπὶ τὸ Δ· ἀφικνεῖται δέ ποτε ἐπὶ τὸ Γ. εἰ μὲν οὖν ἐν τῷ
λόγῳ ἐφέρετο ὃν ἔχει ἡ ΒΔ πρὸς τὴν ΔΓ, ἐφέρετο ἂν
5 τὴν διάμετρον τὴν ἐφ' ᾗ ΒΓ. νῦν δέ, ἐπείπερ ἐν οὐδενὶ
λόγῳ, ἐπὶ τὴν περιφέρειαν φέρεται τὴν ἐφ' ᾗ ΒΕΓ. ἐὰν
δὲ δυοῖν φερομένοιν ἀπὸ τῆς αὐτῆς ἰσχύος τὸ μὲν ἐκκρούοιτο
πλεῖον τὸ δὲ ἔλαττον, εὔλογον βραδύτερον κινηθῆναι
τὸ πλεῖον ἐκκρουόμενον τοῦ ἔλαττον ἐκκρουομένου· ὃ δοκεῖ
10 συμβαίνειν ἐπὶ τῆς μείζονος καὶ ἐλάττονος τῶν ἐκ τοῦ
κέντρου γραφουσῶν τοὺς κύκλους. διὰ γὰρ τὸ ἐγγύτερον
εἶναι τοῦ μένοντος τῆς ἐλάττονος τὸ ἄκρον ἢ τὸ τῆς μείζονος,
ὥσπερ ἀντισπώμενον εἰς τοὐναντίον, ἐπὶ τὸ μέσον βραδύτερον
φέρεται τὸ τῆς ἐλάττονος ἄκρον. πάσῃ μὲν οὖν
15 κύκλον γραφούσῃ τοῦτο συμβαίνει, καὶ φέρεται τὴν μὲν
κατὰ φύσιν κατὰ τὴν περιφέρειαν, τὴν δὲ παρὰ φύσιν
εἰς τὸ πλάγιον καὶ τὸ κέντρον. μείζω δ' ἀεὶ τὴν παρὰ
φύσιν ἡ ἐλάττων φέρεται· διὰ γὰρ τὸ ἐγγύτερον εἶναι τοῦ
κέντρου τοῦ ἀντισπῶντος κρατεῖται μᾶλλον. ὅτι δὲ μεῖζον
20 τὸ παρὰ φύσιν κινεῖται ἡ ἐλάττων τῆς μείζονος τῶν ἐκ τοῦ
κέντρου γραφουσῶν τοὺς κύκλους, ἐκ τῶνδε δῆλον. ἔστω
κύκλος ἐφ' οὗ ΒΓΔΕ, καὶ ἄλλος ἐν τούτῳ ἐλάττων,
ἐφ' οὗ ΧΝΜΞ, περὶ τὸ αὐτὸ κέντρον τὸ Α· καὶ ἐκβεβλήσθωσαν
αἱ διάμετροι, ἐν μὲν τῷ μεγάλῳ, ἐφ' ὧν ΓΔ
25 καὶ ΒΕ, ἐν δὲ τῷ ἐλάττονι αἱ ΜΧ ΝΞ· καὶ τὸ ἑτερόμηκες
παραπεπληρώσθω, τὸ ΔΨΡΓ. εἰ δὴ ἡ ΑΒ γράφουσα
κύκλον ἥξει ἐπὶ τὸ αὐτὸ ὅθεν ὡρμήθη ἐπὶ τὴν ΑΕ, δῆλον
ὅτι φέρεται πρὸς αὑτήν. ὁμοίως δὲ καὶ ἡ ΑΧ πρὸς τὴν
ΑΧ ἥξει. βραδύτερον δὲ φέρεται ἡ ΑΧ τῆς ΑΒ, ὥσπερ
30 εἴρηται, διὰ τὸ γίνεσθαι μείζονα τὴν ἔκκρουσιν καὶ ἀντισπᾶσθαι
μᾶλλον τὴν ΑΧ. ἤχθω δὲ ἡ ΑΘΗ, καὶ ἀπὸ
τοῦ Θ κάθετος ἐπὶ τὴν ΑΒ ἡ ΘΖ ἐν τῷ κύκλῳ, καὶ πάλιν
ἀπὸ τοῦ Θ ἤχθω παρὰ τὴν ΑΒ ἡ ΘΩ, καὶ ἡ ΩΥ
ἐπὶ τὴν ΑΒ κάθετον, καὶ ἡ ΗΚ. αἱ δὴ ἐφ' ὧν ΩΥ καὶ
35 ΘΖ ἴσαι. ἡ ἄρα ΒΥ ἐλάττων τῆς ΧΖ· αἱ γὰρ ἴσαι
εὐθεῖαι ἐπ' ἀνίσους κύκλους ἐμβληθεῖσαι πρὸς ὀρθὰς τῇ
διαμέτρῳ ἔλαττον τμῆμα ἀποτέμνουσι τῆς διαμέτρου ἐν
τοῖς μείζοσι κύκλοις, ἔστι δὲ ἡ ΩΥ ἴση τῇ ΘΖ. ἐν ὅσῳ
1during any interval, a curve is produced. Now that the radius of a circle has two simultaneous displacements is plain from these considerations, and because the point! from being vertically above the 8495 centre comes back to the perpendicular, so as to be A again perpendicularly 5above the centre. be displaced to A by one force, and come eventually to Γ by the other force. If then it were moved in the proportion of BA to AT, it would move along the diagonal Br. But in the present case, as it is moved in no such proportion, it moves along the curve BET. And, if one of two displacements caused by the same forces is 10more interfered with and the other less, it is reasonable to suppose that the motion more interfered with will be slower than the motion less interfered with; which seems to happen in the case of the greater and less of the radii of circles. For on account ο of the extremity of the lesser radius being nearer the stationary centre than that of 15the greater, being as it were pulled in a contrary direction, towards the middle,? the extremity of the lesser moves more slowly. This is the case with every radius, and it moves in a curve, naturally along the tangent, and unnaturally towards the centre. And the lesser radius is always moved more in respect of its unnatural motion ; for 20being nearer to the retarding centre it is more constrained. And that the less of two radii having the same centre is moved more than the greater in respect of the unnatural motion is plain from what diameters be drawn, ΓΔ ww σι follows. Let BEA be a circle, and XNM& another smaller circle same centre A, and let the and BE in the large circle, 25and MX and ΝΞ in the small; and let the rect- ; angle A¥PT be completed. Δ If the radius AB comes back to the same position eG ma < from which it started, i.e. ᾿ to AB, it is plain that it moved towards itself; and & likewise AX will come to AX. But AX moves more slowly than AB, as has been stated, because the interference is greater and AX is 30more retarded. Now let AOH be drawn, and from © a perpendicular upon AB within the circle, ΘΖ ; and, further, from © let OQ be drawn parallel to AB, and 2Y and HK perpendiculars on AB; then QY and ΘΖ are equal. Therefore BY is less than XZ; for in unequal circles equal straight lines drawn perpendicular to the diameter cut off smaller portions 35of the diameter in the greater circles; QY and ΘΖ being equal.” 2. 8405 38. According to the parallelogram of distances, the result equal, but BY and XZ unequal; so that the theory of the parallelogram , Ὁ.
849b
1 δὴ χρόνῳ ἡ ΑΘ τὴν ΧΘ ἐνηνέχθη, ἐν τοσούτῳ χρόνῳ ἐν
τῷ κύκλῳ τῷ μείζονι μείζονα τῆς ΒΩ ἐνήνεκται τὸ ἄκρον
τῆς ΒΑ. ἡ μὲν γὰρ κατὰ φύσιν φορὰ ἴση, ἡ δὲ παρὰ
φύσιν ἐλάττων· ἡ δὲ ΒΥ τῆς ΖΧ. δεῖ δὲ ἀνάλογον εἶναι,
5 ὡς τὸ κατὰ φύσιν πρὸς τὸ κατὰ φύσιν, τὸ παρὰ φύσιν
πρὸς τὸ παρὰ φύσιν. μείζονα ἄρα περιφέρειαν διελήλυθε
τὴν ΗΒ τῆς ΩΒ. ἀνάγκη δὲ τὴν ΗΒ ἐν τούτῳ τῷ χρόνῳ
διεληλυθέναι· ἐνταῦθα γὰρ ἔσται, ὅταν ἀνάλογον ἀμφοτέρως
συμβαίνῃ τὸ παρὰ φύσιν πρὸς τὸ κατὰ φύσιν. εἰ δὴ
10 μεῖζόν ἐστι τὸ κατὰ φύσιν ἐν τῇ μείζονι, καὶ τὸ παρὰ φύσιν
μᾶλλον ἂν ἐνταῦθα συμπίπτοι μοναχῶς, ὥστε τὸ Β ἐνηνέχθαι
ἂν τὴν ΒΗ ἐν τῷ ἐφ' οὗ Χ σημεῖον. ἐνταῦθα γὰρ
κατὰ φύσιν μὲν γίνεται τῷ Β σημείῳ τὸ κέντρον (ἔστι γὰρ
αὐτὴ ἀπὸ τοῦ Η κάθετος), παρὰ φύσιν δὲ ἐς τὸ ΚΒ. ἔστι
15 δὲ ὡς τὸ ΗΚ πρὸς τὸ ΚΒ, τὸ ΘΖ πρὸς τὸ ΖΧ. φανερὸν
δὲ ἐὰν ἐπιζευχθῶσιν ἀπὸ τῶν ΒΧ ἐπὶ τὰ ΗΘ. εἰ δὲ
ἐλάττων ἢ μείζων τῆς ΗΒ ἔσται, ἣν ἠνέχθη τὸ Β, οὐχ ὁμοίως
ἔσται οὐδὲ ἀνάλογον ἐν ἀμφοῖν τὸ κατὰ φύσιν πρὸς τὸ
παρὰ φύσιν. δι' ἣν μὲν τοίνυν αἰτίαν ἀπὸ τῆς αὐτῆς
20 ἰσχύος φέρεται θᾶττον τὸ πλέον ἀπέχον τοῦ κέντρου σημεῖον,
δῆλον διὰ τῶν εἰρημένων· διότι δὲ τὰ μὲν μείζω ζυγὰ
ἀκριβέστερά ἐστι τῶν ἐλαττόνων, φανερὸν ἐκ τούτων. γίνεται
γὰρ τὸ μὲν σπάρτον κέντρον (μένει γὰρ τοῦτο), τὸ δὲ ἐπὶ
ἑκάτερον μέρος τῆς πλάστιγγος αἱ ἐκ τοῦ κέντρου. ἀπὸ οὖν
25 τοῦ αὐτοῦ βάρους ἀνάγκη θᾶττον κινεῖσθαι τὸ ἄκρον τῆς
πλάστιγγος, ὅσῳ ἂν πλεῖον ἀπέχῃ τοῦ σπάρτου, καὶ ἔνια
μὲν μὴ δῆλα εἶναι ἐν τοῖς μικροῖς ζυγοῖς πρὸς τὴν αἴσθησιν
ἐπιτιθέμενα βάρη, ἐν δὲ τοῖς μεγάλοις δῆλα οὐθὲν γὰρ
κωλύει ἔλαττον κινηθῆναι μέγεθος ἢ ὥστε εἶναι τῇ ὄψει
30 φανερόν. ἐπὶ δὲ τῆς μεγάλης πλάστιγγος ποιεῖ ὁρατὸν τὸ
αὐτὸ βάρος μέγεθος. ἔνια δὲ δῆλα μὲν ἐπ' ἀμφοῖν ἐστίν,
ἀλλὰ πολλῷ μᾶλλον ἐπὶ τῶν μειζόνων διὰ τὸ πολλῷ
μεῖζον γίνεσθαι τὸ μέγεθος τῆς ῥοπῆς ὑπὸ τοῦ αὐτοῦ βάρους
ἐν τοῖς μείζοσι. καὶ διὰ τοῦτο τεχνάζουσιν οἱ ἁλουργοπῶλαι
35 πρὸς τὸ παρακρούεσθαι ἱστάντες, τό τε σπάρτον
οὐκ ἐν μέσῳ τιθέντες, καὶ μόλυβδον τῆς φάλαγγος εἰς
θάτερον μέρος ἐγχέοντες, ἢ τοῦ ξύλου τὸ πρὸς τὴν ῥίζαν
πρὸς ὃ βούλονται ῥέπειν ποιοῦντες, ἢ ἐὰν ἔχῃ ὄζον· βαρύτερον
1as the extremity of the radius BA has described an arc greater than BQ in the greater circle; for the natural displacement is equal and the unnatural less, BY being less than XZ. Whereas they ought to be in proportion, the two natural motions in the same ratio to each other as the 5two unnatural motions. Now the radius AB has described an arc BH greater than BQ. It must necessarily have described BH in the time in which X describes ΧΘ ; for that will be its position when in the two circles the proportion between the unnatural and natural movements holds good. If, then, the natural movement is greater in the greater circle, the unnatural 10movement, too, would agree in being propor: tionally greater + in that case only, where B is moved along BH while X is moved along X®. For in that case the point B comes by its natural movement to H, and by its unnatural movement to K, HK being perpendicular from H. if B and X be joined to H and ©? But, if the arc described by B be less or greater 15than HB, the result will not be the same, nor will the natural movement be proportional to the unnatural in the two circles. So that the reason why the point further from the centre is moved quicker by the same force, and the greater radius describes the greater circle, is plain from what has been said ; and hence the reason is also clear why larger balances 20are more accurate than smaller. For the cord by which a balance is suspended acts as the centre, for it is at rest, and the parts of the balance on either side form the radii, Therefore by the same weight the end of the balance must necessarily be moved quicker in proportion as it is more distant from the cord, and some weight must be imperceptible 25to the senses in small balances, but perceptible in large balances; for there is nothing to prevent the fails. Why is this? The answer is that the same force moves longer radii quicker than shorter. 645+8 K οι [Ὁ δ movement being so small as to be invisible to the eye. Whereas in the large balance the same load makes the movement visible. In some cases 30the effect is clearly seen in both balances, but much more in the larger on account of the amplitude of the displacement caused by the same load being much greater in the larger balance. And thus dealers in purple, in weighing it, use contrivances with intent to deceive, putting the cord out of centre and pouring lead into one arm of the balance, or using 35the wood towards the root of a tree for the end towards which they want it to incline, or a knot, if there be one in the wood; for the part of the wood where the root is is heavier, and a knot is a kind of root.
850a
1 γὰρ ἐν ᾧ μέρος ἡ ῥίζα τοῦ ξύλου ἐστίν, ὁ δὲ ὄζος ῥίζα
τίς ἐστιν.
Chapter 3 (850a3–29)
Διὰ τί, ἐὰν μὲν ἄνωθεν ᾖ τὸ σπαρτίον, ὅταν κάτωθεν
ῥέψαντος ἀφέλῃ τὸ βάρος, πάλιν ἀναφέρεται τὸ ζυγόν,
5 ἐὰν δὲ κάτωθεν ὑποστῇ, οὐκ ἀναφέρεται ἀλλὰ μένει; ἢ
διότι ἄνωθεν μὲν τοῦ σπαρτίου ὄντος πλεῖον τοῦ ζυγοῦ γίνεται
τὸ ἐπέκεινα τῆς καθέτου; τὸ γὰρ σπαρτίον ἐστὶ κάθετος.
ὥστε ἀνάγκη ἐστὶ κάτω ῥέπειν τὸ πλέον, ἕως ἂν ἔλθῃ ἡ
δίχα διαιροῦσα τὸ ζυγὸν ἐπὶ τὴν κάθετον αὐτήν, ἐπικειμένου
10 τοῦ βάρους ἐν τῷ ἀνεσπασμένῳ μορίῳ τοῦ ζυγοῦ.
ἔστω ζυγὸν ὀρθὸν ἐφ' οὗ ΒΓ, σπαρτίον δὲ τὸ ΑΔ. ἐκβαλλόμενον
δὴ τοῦτο κάτω κάθετος ἔσται ἐφ' ἧς ἡ ΑΔΜ.
ἐὰν οὖν ἐπὶ τὸ Β ἡ ῥοπὴ ἐπιτεθῇ, ἔσται τὸ μὲν Β οὗ τὸ Ε,
τὸ δὲ Γ οὗ τὸ Ζ, ὥστε ἡ δίχα διαιροῦσα τὸ ζυγὸν πρῶτον
15 μὲν ἦν ἡ ΔΜ τῆς καθέτου αὐτῆς, ἐπικειμένης δὲ τῆς ῥοπῆς
ἔσται ἡ ΔΘ· ὥστε τοῦ ζυγοῦ ἐφ' ᾧ ΕΖ τὸ ἔξω τῆς καθέτου
τῆς ἐφ' ἧς ΑΒ, τοῦ ἐν ᾧ ΦΠ, μείζω τοῦ ἡμίσεος.
ἐὰν οὖν ἀφαιρεθῇ τὸ βάρος ἀπὸ τοῦ Ε, ἀνάγκη κάτω φέρεσθαι
τὸ Ζ· ἔλαττον γάρ ἐστι τὸ Ε. ἐὰν μὲν οὖν ἄνω τὸ
20 σπαρτίον ἔχῃ, πάλιν διὰ τοῦτο ἀναφέρεται τὸ ζυγόν. ἐὰν
δὲ κάτωθεν ᾖ τὸ ὑποκείμενον, τοὐναντίον ποιεῖ· πλεῖον γὰρ
γίνεται τοῦ ἡμίσεος τοῦ ζυγοῦ τὸ κάτω μέρος ἢ ὡς ἡ κάθετος
διαιρεῖ ὥστε οὐκ ἀναφέρεται· κουφότερον γὰρ τὸ ἐπηρτημένον.
ἔστω ζυγὸν τὸ ἐφ' οὗ ΝΞ, τὸ ὀρθόν, κάθετος δὲ ἡ
25 ΚΛΜ. δίχα δὴ διαιρεῖται τὸ ΝΞ. ἐπιτεθέντος δὲ βάρους
ἐπὶ τὸ Ν, ἔσται τὸ μὲν Ν οὗ τὸ Ο, τὸ δὲ Ξ οὗ τὸ Ρ, ἡ δὲ
ΚΛ οὗ τὸ ΛΘ, ὥστε μεῖζόν ἐστι τὸ ΚΟ τοῦ ΛΡ τῷ ΘΚΛ.
καὶ ἀφαιρεθέντος οὖν τοῦ βάρους ἀνάγκη μένειν· ἐπίκειται
γὰρ ὥσπερ βάρος ἡ ὑπεροχὴ ἡ τοῦ ἡμίσεος τοῦ ἐν ᾧ τὸ Κ.
1How is it that if the cord is attached to the upper surface of the beam of a balance, if A one takes away the weight when the balance is depressed on one side, the beam rises again ; whereas, if the cord is attached to the lower surface of the beam, it does not rise but BL δ ἢ remains in the same posi- Π tion, Is it because, when the 5cord is attached above, there is more of the beam on one side of the perpendicular than on the other, the cord being the perpendicular ? In that case the side on which the greater part of the beam -is must necessarily sink until the line which divides the το beam into two equal parts reaches the actual perpendicular, since the weight now presses on the side of the beam which is elevated, Let ΒΓ be a straight beam, 10and AA acord, If AA be produced it will form the perpendicular AAM. If the portion of the beam towards B be depressed, B will be displaced to E and [to Z; and so the line dividing the beam into two halves, which was originally AM, part οἵ. TOPS Se RIS see a UNE cine we tings 2b A ον O02 ne + ᾿ the perpendicular, will become ΔΘ when the beam is depressed ; so that the part of the beam EZ which is outside the perpendicular 15AM will be greater by ΘΠ than half the beam. If therefore the weight at E be taken away, Z must sink, because the side towards E is shorter, It has been proved then that when the cord is attached above, if the weight be removed the beam rises again. But if the support be from below, the contrary takes place. For then the part which is depressed is more than half of the beam, or in other words, more than the part 20marked off by the original perpendicular; it does not therefore rise, when the weight is removed, for the NE = part that is elevated is A lighter. Let NE be the beam when horizontal, and KAM the _ perpendicular ὁ dividing NE intotwo halves. When the weight is placed M at N, N will be displaced to O and & to P, and KA to ΛΘ, so that KO! is greater than AP by OAK. If the weight, therefore, is removed the beam must 25necessarily remain in the same position; for the excess of the part in which OK ? is over _half the beam acts as a weight and remains depressed. of this treatise,? the exercise of little force raises great weights with the help of a lever, in spite of the added weight of the lever; whereas the less heavy a weight is, the easier it is to move, and the weight is less without the lever? Does the reason lie in the fact 30that the lever acts like the beam of a balance with the cord attached below and
Chapter 4 (850a30–850b9)
30 Διὰ τί κινοῦσι μεγάλα βάρη μικραὶ δυνάμεις τῷ μοχλῷ,
ὥσπερ ἐλέχθη καὶ κατ' ἀρχήν, προσλαβόντι βάρος
ἔτι τὸ τοῦ μοχλοῦ; ῥᾷον δὲ τὸ ἔλαττόν ἐστι κινῆσαι βάρος,
ἔλαττον δέ ἐστιν ἄνευ τοῦ μοχλοῦ. ἢ ὅτι αἴτιόν ἐστιν ὁ μοχλός,
ζυγὸν [ὢν] κάτωθεν ἔχον τὸ σπαρτίον καὶ εἰς ἄνισα διῃρημένον;
35 τὸ γὰρ ὑπομόχλιον ἀντὶ σπαρτίου γίνεται· μένει
γὰρ ἄμφω ταῦτα, ὥσπερ τὸ κέντρον. ἐπεὶ δὲ θᾶττον ὑπὸ
τοῦ ἴσου βάρους κινεῖται ἡ μείζων τῶν ἐκ τοῦ κέντρου, ἔστι δὲ
τρία τὰ περὶ τὸν μοχλόν, τὸ μὲν ὑπομόχλιον, σπάρτον
καὶ κέντρον, δύο δὲ βάρη, ὅ τε κινῶν καὶ τὸ κινούμενον· ὃ
Why is it that, as has been remarked at the beginning ra EE divided into two unequal parts? The fulcrum, then, takes the place of the cord, for both remain at rest and act as the centre. Now since a longer radius moves more quickly than a shorter one under pressure of an equal weight ; and since the lever requires three elements, viz. the 35fulcrum —corresponding to the cord of a balance and forming the centre—and two weights, that exerted by the person using the lever and the weight which is to be moved; this being so, as the weight moved is to the weight moving it, so, inversely, is the length of the arm bearing the weight to the length of the arm nearer to the power.
850b
1 οὖν τὸ κινούμενον βάρος πρὸς τὸ κινοῦν, τὸ μῆκος πρὸς τὸ μῆκος
ἀντιπέπονθεν. αἰεὶ δὲ ὅσῳ ἂν μεῖζον ἀφεστήκῃ τοῦ ὑπομοχλίου,
ῥᾷον κινήσει. αἰτία δέ ἐστιν ἡ προλεχθεῖσα, ὅτι ἡ
πλεῖον ἀπέχουσα ἐκ τοῦ κέντρου μείζονα κύκλον γράφει. ὥστε
5 ἀπὸ τῆς αὐτῆς ἰσχύος πλέον μεταστήσεται τὸ κινοῦν τὸ
πλεῖον τοῦ ὑπομοχλίου ἀπέχον. ἔστω μοχλὸς ἐφ' οὗ ΑΒ,
βάρος δὲ ἐφ' ᾧ τὸ Γ, τὸ δὲ κινοῦν ἐφ' ᾧ τὸ Δ, ὑπομόχλιον
ἐφ' ᾧ τὸ Ε, τὸ δὲ ἐφ' ᾧ τὸ Δ κινῆσαν ἐφ' ᾧ τὸ Η, κινούμενον
δὲ τὸ ἐφ' οὗ Γ, βάρος ἐφ' οὗ Κ.
1The further i one is from the fulcrum, the more easily will one raise the weight; the reason being that which has already been stated,! namely, that a longer radius describes a larger circle. So with the exertion of the same force themotive weight will change its position more than 5the weight which it moves, because it is further from the fulcrum. Let AB be a lever, Γ the weight to be lifted, A the motive weight, and E the fulcrum; the position of A after it has raised the weight will be H, and that of I’, the ΕΝ raised, will be K.
Chapter 5 (850b10–27)
10 Διὰ τί οἱ μεσόνεοι μάλιστα τὴν ναῦν κινοῦσιν; ἢ διότι
ἡ κώπη μοχλός ἐστιν; ὑπομόχλιον μὲν γὰρ ὁ σκαλμὸς γίνεται
(μένει γὰρ δὴ τοῦτο), τὸ δὲ βάρος ἡ θάλαττα, ἣν
ἀπωθεῖ ἡ κώπη· ὁ δὲ κινῶν τὸν μοχλὸν ὁ ναύτης ἐστίν.
ἀεὶ δὲ πλέον βάρος κινεῖ, ὅσῳ ἂν πλέον ἀφεστήκῃ τοῦ ὑπομοχλίου
15 ὁ κινῶν τὸ βάρος· μείζων γὰρ οὕτω γίνεται ἡ ἐκ
τοῦ κέντρου, ὁ δὲ σκαλμὸς ὑπομόχλιον ὢν κέντρον ἐστίν. ἐν
μέσῃ δὲ τῇ νηῒ πλεῖστον τῆς κώπης ἐντός ἐστιν· καὶ γὰρ ἡ
ναῦς ταύτῃ εὐρυτάτη ἐστίν, ὥστε πλεῖον ἐπ' ἀμφότερα ἐνδέχεσθαι
μέρος τῆς κώπης ἑκατέρου τοίχου ἐντὸς εἶναι τῆς
20 νεώς. κινεῖται μὲν οὖν ἡ ναῦς διὰ τὸ ἀπερειδομένης τῆς κώπης
εἰς τὴν θάλασσαν τὸ ἄκρον τῆς κώπης τὸ ἐντὸς προϊέναι
εἰς τὸ πρόσθεν, τὴν δὲ ναῦν προσδεδεμένην τῷ σκαλμῷ συμπροϊέναι,
ᾗ τὸ ἄκρον τῆς κώπης. ᾗ γὰρ πλείστην θάλασσαν
διαιρεῖ ἡ κώπη, ταύτῃ ἀνάγκη μάλιστα προωθεῖσθαι· πλείστην
25 δὲ διαιρεῖ ᾗ πλεῖστον μέρος ἀπὸ τοῦ σκαλμοῦ τῆς κώπης
ἐστίν. διὰ τοῦτο οἱ μεσόνεοι μάλιστα κινοῦσιν· μέγιστον γὰρ
ἐν μέσῃ νηῒ τὸ ἀπὸ τοῦ σκαλμοῦ τῆς κώπης τὸ ἐντός ἐστιν.
oa Why is it that those rowers who are amidships move the ship most? Is it because the oar acts as 10a lever? The fulcrum then is the thole-pin (for it remains in the same place) ; and the weight is the sea which the oar displaces; and the power that moves the lever is the rower. The further he who moves a weight is from the fulcrum, the greater is the weight which he moves; for then the radius becomes greater, and the thole-pin acting as the fulcrum 15is the centre. Now amidships there is more of the oar inside the ship than elsewhere ; for there the ship is widest, so that on both sides a longer portion of the oar can be fh AG ἃς inside the two walls of the vessel.+.The ship then moves zo because, as the blade presses against the sea, the handle of the oar, which is inside the ship, advances 20forward, and the ship, being firmly attached to the thole-pin, advances with it in the same direction as the handle of the oar. For where the blade displaces most water, there necessarily must the ship be propelled most; and it displaces as most water where the handle is furthest from the thole-pin. This is why the rowers who are amidships move the ship 25most; for it is in the middle of the ship that the length of the oar from the thole-pin inside the ship is greatest,
Chapter 6 (850b28–851a37)
Διὰ τί τὸ πηδάλιον μικρὸν ὄν, καὶ ἐπ' ἐσχάτῳ τῷ
πλοίῳ, τοσαύτην δύναμιν ἔχει ὥστε ὑπὸ μικροῦ οἴακος καὶ
30 ἑνὸς ἀνθρώπου δυνάμεως, καὶ ταύτης ἠρεμαίας, μεγάλα κινεῖσθαι
μεγέθη πλοίων; ἢ διότι καὶ τὸ πηδάλιόν ἐστι μοχλός,
καὶ μοχλεύει ὁ κυβερνήτης. ᾗ μὲν οὖν προσήρμοσται
τῷ πλοίῳ, γίνεται ὑπομόχλιον, τὸ δὲ ὅλον πηδάλιον ὁ
μοχλός, τὸ δὲ βάρος ἡ θάλασσα, ὁ δὲ κυβερνήτης ὁ κινῶν.
35 οὐ κατὰ πλάτος δὲ λαμβάνει τὴν θάλασσαν, ὥσπερ ἡ κώπη,
τὸ πηδάλιον. οὐ γὰρ εἰς τὸ πρόσθεν κινεῖ τὸ πλοῖον, ἀλλὰ
κινούμενον κλίνει, πλαγίως τὴν θάλατταν δεχόμενον. ἐπεὶ
γὰρ τὸ βάρος ἦν ἡ θάλασσα, τοὐναντίον ἀπερειδόμενον κλίνει
τὸ πλοῖον. τὸ γὰρ ὑπομόχλιον εἰς τοὐναντίον στρέφεται,
Why is it that the rudder, being small and at the extreme end of the ship, has such power that vessels of great. burden can be moved by a small tiller and the strength of one man only gently exerted? Is it because the rudder, too, is a 30lever and the steersman works it? The fulcrum then is the point at which the rudder is attached to the ship, and the whole rudder is the lever, and the sea is the weight, and the steersman the moving force. The rudder does not ; take the sea squarely, as the oar does; for it does not move the ship forward, but diverts it as it moves, taking the sea 35obliquely. For since, as we saw, the sea is the weight, the rudder pressing in a contrary direction diverts the ship. For the fulcrum turns in a contrary direction to the sea; when the sea turns inwards, the fulcrum turns outwards; and the ship follows it because it is attached to it.
851a
1 ἡ θάλασσα δὲ ἐντός· ἐκεῖνο δὲ εἰς τὸ ἐκτός. τούτῳ δὲ ἀκολουθεῖ
τὸ πλοῖον διὰ τὸ συνδεδέσθαι. ἡ μὲν οὖν κώπη κατὰ
πλάτος τὸ βάρος ὠθοῦσα καὶ ὑπ' ἐκείνου ἀντωθουμένη εἰς τὸ
εὐθὺ προάγει· τὸ δὲ πηδάλιον, ὥσπερ κάθηται πλάγιον,
5 τὴν εἰς τὸ πλάγιον, ἢ δεῦρο ἢ ἐκεῖ, ποιεῖ κίνησιν. ἐπ' ἄκρου
δὲ καὶ οὐκ ἐν μέσῳ κεῖται, ὅτι ῥᾷστον τὸ κινούμενον κινῆσαι
ἀπ' ἄκρου κινοῦν. τάχιστα γὰρ φέρεται τὸ πρῶτον μέρος
διὰ τὸ ὥσπερ ἐν τοῖς φερομένοις ἐπὶ τέλει λήγειν τὴν φοράν,
οὕτω καὶ τοῦ συνεχοῦς ἐπὶ τέλους ἀσθενεστάτη ἐστὶν ἡ φορά.
10 εἰ δὲ ἀσθενεστάτη, ῥᾳδία ἐκκρούειν. διά τε δὴ ταῦτα ἐν τῇ
πρύμνῃ τὸ πηδάλιόν ἐστι, καὶ ὅτι ἐνταῦθα μικρᾶς κινήσεως
γενομένης πολλῷ μεῖζον τὸ διάστημα ἐπὶ τῷ ἐσχάτῳ γίνεται,
διὰ τὸ τὴν ἴσην γωνίαν ἐπὶ μείζονα καθῆσθαι, καὶ ὅσῳ
ἂν μείζους ὦσιν αἱ περιέχουσαι. δῆλον δὲ ἐκ τούτου καὶ δι' ἣν
15 αἰτίαν μᾶλλον προέρχεται εἰς τοὐναντίον τὸ πλοῖον ἢ ἡ τῆς
κώπης πλάτη· τὸ αὐτὸ γὰρ μέγεθος τῇ αὐτῇ ἰσχύϊ κινούμενον
ἐν ἀέρι πλέον ἢ ἐν τῷ ὕδατι πρόεισιν. ἔστω γὰρ ἡ Α
Β κώπη, τὸ δὲ Γ ὁ σκαλμός, τὸ δὲ Α τὸ ἐν τῷ πλοίῳ, ἡ
ἀρχὴ τῆς κώπης, τὸ δὲ Β τὸ ἐν τῇ θαλάττῃ. εἰ δὴ τὸ Α
20 οὗ τὸ Δ μετακεκίνηται, τὸ Β οὐκ ἔσται οὗ τὸ Ε· ἴση γὰρ ἡ Β
Ε τῇ ΑΔ. ἴσον οὖν μετακεχωρηκὸς ἔσται. ἀλλ' ἦν ἔλαττον.
ἔσται δὴ οὗ τὸ Ζ ἢ τὸ Θ. ἄρα τοίνυν τὴν ΑΒ, καὶ οὐχ ἡ τὸ
Γ, καὶ κάτωθεν. ἐλάττων γὰρ ἡ ΒΖ τῆς ΑΔ, ὥστε καὶ
ἡ ΘΖ τῆς ΔΘ· ὅμοια γὰρ τὰ τρίγωνα. καθεστηκὸς δὲ
25 ἔσται καὶ τὸ μέσον, τὸ ἐφ' οὗ Γ· εἰς τοὐναντίον γὰρ τῷ ἐν τῇ
θαλάττῃ ἄκρῳ τῷ Β μεταχωρεῖ, ᾗπερ τὸ ἐν τῷ πλοίῳ
ἄκρον τὸ Α μὴ ἐχώρει οὗ τὸ Δ. ὥστε μετακινηθήσεται τὸ
πλοῖον, καὶ ἐκεῖ οὗ ἡ ἀρχὴ τῆς κώπης μεταφέρεται. τὸ δ'
αὐτὸ καὶ τὸ πηδάλιον ποιεῖ, πλὴν ὅτι εἰς τὸ πρόσθεν οὐδὲν
30 συμβάλλεται τῷ πλοίῳ, ὥσπερ ἐλέχθη ἐπὶ ἄνω, ἀλλὰ
μόνον τὴν πρύμναν εἰς τὸ πλάγιον ἀπωθεῖ ἔνθα ἢ ἔνθα· εἰς
τοὐναντίον γὰρ ἡ πρῷρα οὕτω νεύει. ᾗ μὲν δὴ τὸ πηδάλιον
προσέζευκται, δεῖ οἷόν τι τοῦ κινουμένου μέσον νοεῖν, καὶ ὥσπερ
ὁ σκαλμὸς τῇ κώπῃ· τὸ δὲ μέσον ὑποχωρεῖ, ᾗ ὁ οἴαξ μετακινεῖται.
35 ἐὰν μὲν εἴσω ἄγῃ, καὶ ἡ πρύμνα δεῦρο μεθέστηκεν·
ἡ δὲ πρῷρα εἰς τοὐναντίον νεύει· ἐν γὰρ τῷ αὐτῷ
οὔσης τῆς πρῴρας τὸ πλοῖον μεθέστηκεν ὅλον.
1The oar pushing the weight squarely, and being itself thrust in turn by it, impels the ship straight forward; but the rudder, as it has an oblique position, causes an oblique motion one way or the other. It is placed at the stern and not amidships, because it is easiest to move a mass which 5has to be moved, if it is moved from one extremity. For the fore part travels quickest, because, just as in objects that are travelling along, the movement ceases at the end ; so, too, in any object which is continuous the movement is weakest towards the end,! and if it is weakest in that part τὸ σι it is easy to check it.- For this reason, then, the rudder is 10placed at the stern, and also because, as there is little motion there, the displacement is much greater at the extremity, since the equal angle stands on a longer base in proportion as the enclosing lines are longer.1 From this it is also plain why the ship advances in the opposite direction more than does the oar-blade; for the same bulk moved by the same 15force progresses more in air than in water. For let AB be the oar and I the thole-pin, and A the end of the oar inside the : ship, and B, that in the sea. Εἰ Then if A be moved to Δ, ee: B will not be at E; for Γ, Β BE is equal to AA, and ᾿ so B, if it were δὲ E, Γ would have changed its Δ position as much as A, whereas it has really, as we saw, traversed a shorter 20distance. B will therefore be at Z. © then cuts AB not at T but below it. For BZ is less than AA, so 25The centre I will also have been displaced; for it moves in a contrary direction to B, the end of the oar in the sea, and in the same direction as A, the end in the ship, and A changes its position to A. So the ship will also change its position, and it 25advances in the same direction as the handle of the oar. The rudder also acts in the same way, except that, as we saw above, 3° it contributes nothing to the forward motion of the ship, but merely thrusts the stern sideways one way or the other ; object the fore part has more motion than the hinder part ; it is perhaps due to a false generalization from S the fact 30that in the case of a horse Z age ὃ and cart, the motive power is in front. for then the bow inclines in the contrary direction. The pomt where the rudder is attached must be considered, as it were, the centre of the mass which is moved, corresponding to the thole-pim in the case of the oar; but the middle of the ship moves in the direction to which the tiller 35is put over. If the steersman puts it inwards, the stern alters its position in that direction, but the bow inclines in the contrary direction; for while the bow remains in the same place, the position of the ship as a whole is altered.
Chapter 7 (851a38–851b5)
Διὰ τί, ὅσῳ ἂν ἡ κεραία ἀνωτέρα ᾖ, θᾶττον πλεῖ τὰ
πλοῖα τῷ αὐτῷ ἱστίῳ καὶ τῷ αὐτῷ πνεύματι; ἢ διότι γίνεται
40 ὁ μὲν ἱστὸς μοχλός, ὑπομόχλιον δὲ τὸ ἑδώλιον ἐν ᾧ
Why is it that the higher the yard-arm is raised, the quicker does a vessel travel with the same sail and in the same breeze?!
851b
1 ἐμπέπηγεν, ὃ δὲ δεῖ κινεῖν βάρος, τὸ πλοῖον, τὸ δὲ κινοῦν
τὸ ἐν τῷ ἱστίῳ πνεῦμα. εἰ δ' ὅσῳ ἂν πορρώτερον ᾖ τὸ ὑπομόχλιον,
ῥᾷον κινεῖ καὶ θᾶττον ἡ αὐτὴ δύναμις τὸ αὐτὸ
βάρος, ἡ οὖν κεραία ἀνώτερον ἀγομένη καὶ τὸ ἱστίον πορρώτερον
5 ποιεῖ τοῦ ἑδωλίου ὑπομοχλίου ὄντος.
1Is it because the mast is a lever, and the socket in which it is fixed, the fulcrum, and the weight which it has to move is the boat, and the motive power is the wind in the sail? If the same power moves the same weight more easily and quickly the further away the fulcrum is, then the yard-arm, being raised higher, brings the sail also further away 5from the mast-socket, which is the fulcrum. οι
Chapter 8 (851b6–852a13)
Διὰ τί, ὅταν ἐξ οὐρίας βούλωνται διαδραμεῖν μὴ οὐρίου
τοῦ πνεύματος ὄντος, τὸ μὲν πρὸς τὸν κυβερνήτην τοῦ ἱστίου
μέρος στέλλονται, τὸ δὲ πρὸς τὴν πρῷραν ποδιαῖον ποιησάμενοι
ἐφιᾶσιν; ἢ διότι ἀντισπᾶν τὸ πηδάλιον πολλῷ μὲν
10 ὄντι τῷ πνεύματι οὐ δύναται, ὀλίγῳ δέ, ὃ ὑποστέλλονται.
προάγει μὲν οὖν τὸ πνεῦμα, εἰς οὔριον δὲ καθίστησι τὸ
πηδάλιον, ἀντισπῶν καὶ μοχλεῦον τὴν θάλατταν. ἅμα
δὲ καὶ οἱ ναῦται μάχονται τῷ πνεύματι· ἀνακλίνουσι γὰρ
ἐπὶ τὸ ἐναντίον ἑαυτούς.
15 Διὰ τί τὰ στρογγύλα καὶ περιφερῆ τῶν σχημάτων
εὐκινητότερα; τριχῶς δὲ ἐνδέχεται τὸν κύκλον κυλισθῆναι·
ἢ γὰρ κατὰ τὴν ἁψῖδα, συμμεταβάλλοντος τοῦ κέντρου,
ὥσπερ ὁ τροχὸς ὁ τῆς ἁμάξης κυλίεται· ἢ περὶ τὸ κέντρον
μόνον, ὥσπερ αἱ τροχιλέαι, τοῦ κέντρου μένοντος· ἢ παρὰ
20 τὸ ἐπίπεδον, τοῦ κέντρου μένοντος, ὥσπερ ὁ κεραμεικὸς τροχὸς
κυλίνδεται. εἰ μὲν δὴ τάχιστα τὰ τοιαῦτα, διά τε τὸ
μικρῷ ἅπτεσθαι τοῦ ἐπιπέδου, ὥσπερ ὁ κύκλος κατὰ στιγμήν,
καὶ διὰ τὸ μὴ προσκόπτειν· ἀφέστηκε γὰρ τῆς γῆς
ἡ γωνία. καὶ ἔτι ᾧ ἂν ἀπαντήσῃ σώματι, πάλιν τούτου
25 κατὰ μικρὸν ἅπτεται. εἰ δ' εὐθύγραμμον ἦν, τῇ εὐθείᾳ
ἐπὶ πολὺ ἥπτετο ἂν τοῦ ἐπιπέδου. ἔτι ᾗ ῥέπει ἐπὶ τὸ βάρος,
ταύτῃ κινεῖ ὁ κινῶν. ὅταν μὲν γὰρ πρὸς ὄρθιον ἡ διάμετρος
ᾖ τοῦ κύκλου τῷ ἐπιπέδῳ, ἁπτομένου τοῦ κύκλου κατὰ στιγμὴν
τοῦ ἐπιπέδου, ἴσον τὸ βάρος ἐπ' ἀμφότερα διαλαμβάνει
30 ἡ διάμετρος· ὅταν δὲ κινῆται, εὐθὺς πλέον ἐφ' ᾧ
κινεῖται, ὥσπερ ῥέπον. ἐντεῦθεν εὐκινητότερον τῷ ὠθοῦντι εἰς
τοὔμπροσθεν· ἐφ' ὃ γὰρ ῥέπει ἕκαστον, εὐκίνητόν ἐστιν,
εἴπερ καὶ τὸ ἐπὶ τὸ ἐναντίον τῆς ῥοπῆς δυσκίνητον. ἔτι λέγουσί
τινες ὅτι καὶ ἡ γραμμὴ ἡ τοῦ κύκλου ἐν φορᾷ ἐστὶν
35 ἀεί, ὥσπερ τὰ μένοντα, διὰ τὸ ἀντερείδειν, οἷον καὶ τοῖς
μείζοσι κύκλοις ὑπάρχει πρὸς τοὺς ἐλάττονας. θᾶττον γὰρ
ὑπὸ τῆς ἴσης ἰσχύος κινοῦνται οἱ μείζους καὶ τὰ βάρη κινοῦσι,
διὰ τὸ ῥοπήν τινα ἔχειν τὴν γωνίαν τὴν τοῦ μείζονος
κύκλου πρὸς τὴν τοῦ ἐλάττονος, καὶ εἶναι ὅπερ ἡ διάμετρος
40 πρὸς τὴν διάμετρον. ἀλλὰ μὴν πᾶς κύκλος μείζων πρὸς
Why is it that, when sailors wish to keep their course in an unfavourable wind, they draw in the part of the sail which is nearer to the steersman, and, working the sheet, let out the part towards the bows? Is it because the rudder cannot counteract the wind when it is strong, but can do so when there is only a little wind, and so? they draw in sail? The wind then bears the ship along, 10while the rudder turns the wind into a favouring breeze, counteracting it and serving as a lever against the sea. The sailors also at the same time contend with the wind by leaning their weight in the opposite direction. is Why is it that spherical and circular forms are easier to move? A circle can revolve in three different ways : either along its circumference, the centre correspondingly changing its position, as a carriage 15wheel revolves; or ᾿ round the centre only, as pulleys move, the centre being at rest; or it can turn, as does the potter’s wheel, parallel to the ground, the centre being at rest. Do not circular forms move quickest, firstly because they have a very slight contact with the ground (like a circle in contact at a single point), and secondly, because there is no friction, for the angle is well away from the ground? Further, if. they come 20into collision with another body, they only are in contact with it again to a very small extent, (If it { were a question of a rectilinear body, owing to its sides' being straight, it would have a considerable contact with the ground.) Further, he who moves circular objects moves them in a direction to which they have an inclination as regards weight. For when the diameter of the circle is perpendicular to the ground, the circle 25being in contact with the ground only at one point, the diameter divides the weight equally on either side of it; but as soon as it is set in motion, there is more weight on the side to which } it is moved, as though it had an inclination in that direction.2 Hence, it is easier for one who pushes it forward to move it; for it is easier to move any body inadirection. to which it inclines, just as it is difficult to move it contrary to 30its inclination. Some people further assert that the cir- i cumference of a circle keeps up a continual motion, just as bodies which are at rest remain so owing to their resistance.® includes both (1) the tendency of bodies at rest to remain at rest, and (2) the tendency of bodies in motion to continue in motion, TRS AEA AS ie RS -This~can be illustrated by a comparison of larger with smaller circles; larger circles can be moved more 35readily with an exertion of the same amount of force and move other weights with them, because the angle! of the larger circle as compared with that of the smaller has an inclination which is in the same proportion as the diameter of the one is to the diameter of the other. Now if any circle be taken, there is always a lesser circle than 852* which it is greater; for the lesser circles which can be described are infinite in number.
852a
1 ἐλάττονα· ἄπειροι γὰρ οἱ ἐλάττονες. εἰ δὲ καὶ πρὸς ἕτερον
ἔχει ῥοπὴν ὁ κύκλος, ὁμοίως δὲ εὐκίνητος, καὶ ἄλλην ἂν
ἔχοι ῥοπὴν ὁ κύκλος καὶ τὰ ὑπὸ κύκλου κινούμενα, κἂν μὴ
τῇ ἁψῖδι ἅπτηται τοῦ ἐπιπέδου, ἀλλ' ἢ παρὰ τὸ ἐπίπεδον,
5 ἢ ὡς αἱ τροχιλέαι· καὶ γὰρ οὕτως ἔχοντα ῥᾷστα κινοῦνται
καὶ κινοῦσι τὸ βάρος. ἢ οὐ τῷ κατὰ μικρὸν ἅπτεσθαι καὶ
προσκρούειν, ἀλλὰ δι' ἄλλην αἰτίαν. αὕτη δέ ἐστιν ἡ εἰρημένη
πρότερον, ὅτι ἐκ δύο φορῶν γεγένηται ὁ κύκλος, ὥστε
μίαν αὐτῶν αἰεὶ ἔχειν ῥοπήν, καὶ οἷον φερόμενον αὐτὸν
10 αἰεὶ κινοῦσιν οἱ κινοῦντες, ὅταν κινῶσι κατὰ τὴν περιφέρειαν
ὁπωσοῦν. φερομένην γὰρ αὐτὴν κινοῦσιν· τὴν μὲν γὰρ εἰς
τὸ πλάγιον αὐτοῦ κίνησιν ὠθεῖ τὸ κινοῦν, τὴν δὲ ἐπὶ τῆς
διαμέτρου αὐτὸς κινεῖται.
1Now if it is the case that one circle has a greater inclina- tion as compared with another circle, and is correspondingly easy to move, then it is also the case that if a circle does not touch the ground with its circumference, but moves — either parallel to the ground ? or with the motion of a pulley, the 5circle and the bodies moved by the circle will have a further cause of inclination ; for circular objects of this kind move most easily and move weights*® with them. Can it be that this is due to a reason other than that they have only a very slight contact with the ground, and consequently encounter little friction? This reason is that which we have already mentioned,* namely, that 10the circle is made up of two forms of motion—and so one of them always has an inclination—and those who move a circle move it when it has, as it were, a motion of its own, when they move it at any point on its circumference. They are moving the circumference when it is already in motion ; for the motive force pushes it in a tangential direction, while the circle itself moves in the 15motion which takes place along the diameter. σι " ο a * Ch. ft.
Chapter 9 (852a14–22)
Διὰ τί τὰ διὰ τῶν μειζόνων κύκλων αἰρόμενα καὶ
15 ἑλκόμενα ῥᾷον καὶ θᾶττον κινοῦμεν; οἷον καὶ αἱ τροχιλέαι
αἱ μείζους τῶν ἐλαττόνων, καὶ αἱ σκυτάλαι ὁμοίως. ἢ
διότι ὅσῳ ἂν μείζων ἡ ἐκ τοῦ κέντρου ᾖ, ἐν τῷ ἴσῳ χρόνῳ
πλέον κινεῖται χωρίον, ὥστε καὶ τοῦ ἴσου βάρους ἐπόντος
ποιήσει τὸ αὐτό, ὥσπερ εἴπομεν καὶ τὰ μείζω ζυγὰ τῶν
20 ἐλαττόνων ἀκριβέστερα εἶναι. τὸ μὲν γὰρ σπαρτίον ἐστὶ
κέντρον, τοῦ δὲ ζυγοῦ αἱ ἐπὶ τάδε τοῦ σπαρτίου αἱ ἐκ τοῦ
κέντρου.
How is it that we can move objects more easily and quickly when they are lifted or drawn along by circles of large circumference? Why, for example, are large pulleys more effective than small, and similarly large rollers? Is it because the longer the radius is the further the object is moved in the same time, and so it 20will do the same” also with an equal weight upon it?1 Just as we said? that large balances are more accurate than small; for the cord is the centre and the parts of the beam on either side of the cord are the radii.
Chapter 10 (852a23–28)
Διὰ τί ῥᾷον, ὅταν ἄνευ βάρους ᾖ, κινεῖται τὸ ζυγόν,
ἢ ἔχον βάρος; ὁμοίως δὲ καὶ τροχὸς ἢ ἄλλο τοιοῦτο τὸ
25 βαρύτερον μὲν μεῖζον δὲ τοῦ ἐλάττονος καὶ κουφοτέρου. ἢ
ὅτι οὐ μόνον εἰς τοὐναντίον τὸ βαρύ, ἀλλὰ καὶ εἰς τὸ πλάγιον
δυσκίνητόν ἐστιν. ἐναντίον γὰρ τῇ ῥοπῇ κινῆσαι χαλεπῶς,
ἐφ' ὃ δὲ ῥέπει, ῥᾳδίως· εἰς δὲ τὸ πλάγιον οὐ ῥέπει.
Why is it that a balance moves more easily without Io a weight upon it than with one? So too with a wheel or anything of that nature, the smaller and lighter is easier to 25move than the heavier and larger.2 Is it because that which is heavy is difficult to move not only vertically, but also horizontally? For one can move‘ a weight with difficulty contrary to its inclination, but easily in the direction of its inclination ; and it does not incline in a horizontal direction. ©
Chapter 11 (852a29–37)
Διὰ τί ἐπὶ τῶν σκυτάλων ῥᾷον τὰ φορτία κομίζεται
30 ἢ ἐπὶ τῶν ἁμαξῶν, ἐχουσῶν τῶν μὲν μεγάλους τροχούς,
τῶν δὲ μικρούς; ἢ διότι ἐπὶ τῶν σκυτάλων οὐδεμίαν ἔχει
πρόσκοψιν, τὸ δὲ ἐπὶ τῶν ἁμαξῶν τὸν ἄξονα, καὶ προσκόπτει
αὐτῷ· ἔκ τε γὰρ τῶν ἄνωθεν πιέζει αὐτὸν καὶ ἐκ
τῶν πλαγίων. τὸ δὲ ἐπὶ τῶν σκυτάλων ἐπὶ δύο τούτων κινεῖται,
35 τῇ τε κάτω χώρᾳ ὑποκειμένῃ καὶ τῷ βάρει τῷ
ἐπικειμένῳ· ἐπ' ἀμφοτέρων γὰρ τούτων κυλίεται τῶν τόπων
ὁ κύκλος καὶ φερόμενος ὠθεῖται.
Why is it that it is easier to convey heavy weights on rollers than on carts, 30though the latter have large wheels and the former a small circumference? Is it because a weight placed upon rollers encounters no friction, whereas when placed upon a cart it has the axle at which it encounters friction? For it presses on the axle from above in addition to the horizontal pressure. But an object on rollers is moved at two points on them, where the ground supports 35them below and where the weight is imposed above; the circle revolves at both these points and is thrust along as it moves.
Chapter 12 (852a38–852b10)
Διὰ τί πορρωτέρω τὰ βέλη φέρεται ἀπὸ τῆς σφενδόνης
ἢ ἀπὸ τῆς χειρός; καίτοι κρατεῖ γε ὁ βάλλων τῇ χειρὶ
Why is it that a missile travels further from a sling than from the hand, although he who casts it has more control over the missile in his hand than when he holds the weight suspended?
852b
1 μᾶλλον ἢ ἀπαρτήσας τὸ βάρος. καὶ ἔτι οὕτω μὲν δύο βάρη
κινεῖ, τό τε τῆς σφενδόνης καὶ τὸ βέλος, ἐκείνως δὲ τὸ
βέλος μόνον. πότερον ὅτι ἐν μὲν τῇ σφενδόνῃ κινούμενον τὸ
βέλος ῥίπτει ὁ βάλλων (περιαγαγὼν γὰρ κύκλῳ πολλάκις
5 ἀφίησιν), ἐκ δὲ τῆς χειρὸς ἀπὸ τῆς ἠρεμίας ἡ ἀρχή·
πάντα δὲ εὐκινητότερα κινούμενα ἢ ἠρεμοῦντα. ἢ διά τε
τοῦτο, καὶ διότι ἐν μὲν τῷ σφενδονᾶν ἡ μὲν χεὶρ γίνεται
κέντρον, ἡ δὲ σφενδόνη ἡ ἐκ τοῦ κέντρου· ὅσῳ ἂν ᾖ μείζων
ἡ ἀπὸ τοῦ κέντρου, κινεῖται θᾶττον. ἡ δὲ ἀπὸ τῆς χειρὸς
10 βολὴ πρὸς τὴν σφενδόνην βραχεῖα ἐστίν.
1Further, in the latter case he moves two weights, that of the sling and the missile, while in the former case he moves only the missile. Is it because he who casts’ the missile does so when it is already in motion in the sling (for he swings it round many times before he lets it go), whereas when cast from the hand it starts 5from a state οἵ. rest? Now any object is easier to move when it is already in motion than when it is at rest. Or, while this is one reason, is there a further reason, namely, that in using a sling the hand becomes the centre and the sling the radius, and the longer the radius is the more quickly it moves, and so a cast from the hand is short as compared with a cast from a sling? το
Chapter 13 (852b11–21)
Διὰ τί ῥᾷον κινοῦνται περὶ τὸ αὐτὸ ζυγὸν οἱ μείζους
τῶν ἐλαττόνων κόλλοπες, καὶ οἱ αὐτοὶ ὄνοι οἱ λεπτότεροι
ὑπὸ τῆς αὐτῆς ἰσχύος τῶν παχυτέρων; ἢ διότι ὁ μὲν ὄνος
καὶ τὸ ζυγὸν κέντρον ἐστίν, τὰ δὲ ἀπέχοντα μεγέθη αἱ ἐκ
15 τοῦ κέντρου; θᾶττον δὲ κινοῦνται καὶ πλέον ἀπὸ τῆς αὐτῆς
ἰσχύος αἱ τῶν μειζόνων κύκλων ἢ αἱ τῶν ἐλαττόνων· ὑπὸ
τῆς αὐτῆς γὰρ ἰσχύος θᾶττον μεθίσταται τὸ ἄκρον τὸ πορρώτερον
τοῦ κέντρου. διὸ πρὸς μὲν τὸ ζυγὸν τοὺς κόλλοπας
ὄργανα ποιοῦνται, οἷς ῥᾷον στρέφουσιν· ἐν δὲ τοῖς λεπτοῖς
20 ὄνοις πλεῖον γίνεται τὸ ἔξω τοῦ ξύλου, αὕτη δὲ γίνεται
ἡ ἐκ τοῦ κέντρου.
Why is it that 10longer bars are moved more easily than shorter ones round the same capstan, and similarly lighter? windlasses are moved more easily by the same force than stouter * windlasses ?_ Is it because the windlass and the capstan form a centre and the outer masses‘ the radii? For the radii of greater circles are moved more readily and further by the same force than those of lesser circles ; for the extremity 15further from the centre is moved more readily by the same force. Therefore in the case of the capstan they use the bars as a means whereby they ° turn it more easily ; and in the case of the lighter ® windlasses the part outside the central cylinder is more extended, and this portion forms the radius of the circle. is) ο i.e. lighter and broader. i.e. stouter and narrower. i,e. lighter and broader.
Chapter 14 (852b22–28)
Διὰ τί τὸ αὐτὸ μέγεθος ξύλον ῥᾷον κατεάσσεται περὶ
τὸ γόνυ, ἐὰν ἴσον ἀποστήσας τῶν ἄκρων ἐχόμενος καταγνύῃ,
ἢ παρὰ τὸ γόνυ ἐγγὺς ὄντος· καὶ ἐὰν πρὸς τὴν γῆν
25 ἐρείσας καὶ τῷ ποδὶ προσβὰς πόρρωθεν τῇ χειρὶ καταγνύῃ,
ἢ ἐγγύθεν; ἢ διότι ἔνθα μὲν τὸ γόνυ κέντρον, ἔνθα δὲ ὁ
πούς. ὅσῳ δ' ἂν πορρώτερον ᾖ τοῦ κέντρου, ῥᾷον κινεῖται
ἅπαν. κινηθῆναι δὲ ἀνάγκη καταγνύμενον.
Why is 20it that a piece of wood? of the same size is more easily broken against the knee, if one breaks it holding the ends at equal distance from the knee, than if it is held close to the knee? And if one leans a piece of wood upon the ground and places one’s foot on it, why does one break it more easily if one grasps it at a distance from the foot rather than near it? Is it because in the former case the 25knee, and in the latter the foot is the centre, and the further an object is from the centre the more easily is it always moved, and that which is to be broken must be moved ?
Chapter 15 (852b29–853a4)
Διὰ τί περὶ τοὺς αἰγιαλοὺς αἱ καλούμεναι κρόκαι στρογγύλαι
30 εἰσίν, ἐκ μακρῶν τῶν λίθων καὶ ὀστράκων τὸ ἐξ
ὑπαρχῆς ὄντων; ἢ διότι τὰ πλεῖον ἀπέχοντα τοῦ μέσου ἐν
ταῖς κινήσεσι θᾶττον φέρεται. τὸ μὲν γὰρ μέσον γίνεται
κέντρον, τὸ δὲ διάστημα ἡ ἐκ τοῦ κέντρου. ἀεὶ δὲ ἡ μείζων
ἀπὸ τῆς ἴσης κινήσεως μείζω γράφει κύκλον. τὸ δ' ἐν
35 ἴσῳ χρόνῳ μείζω διεξιὸν θᾶττον φέρεται. τὰ δὲ φερόμενα
θᾶττον ἐκ τοῦ ἴσου ἀποστήματος σφοδρότερον τύπτει. τὰ δὲ
τύπτοντα μᾶλλον καὶ αὐτὰ τύπτεται μᾶλλον. ὥστε ἀνάγκη
θραύεσθαι αἰεὶ τὰ πλέον ἀπέχοντα τοῦ μέσου. τοῦτο δὲ
πάσχοντα ἀνάγκη γίνεσθαι περιφερῆ. ταῖς δὲ κρόκαις διὰ
Why is it that the so-called pebbles found on beaches go are round, though they are originally formed from stones and shells which are elongated in shape? Is it because objects whose outer surfaces are far removed from their middle 30point are borne along more quickly by the move- © ments to which they are subjected? The middle of such objects acts as the centre and the distance thence to the exterior becomes the radius, and a longer radius always describes a greater circle than a shorter radius when the . force which moves them is equal. An object which traverses a greater space in the same time travels more quickly, and objects 35which travel more quickly from an equal distance strike harder against other objects, and the more they strike the more they are themselves struck. It follows, therefore, that objects in which the distance from the middle to the exterior is greater always become broken, and in this process they must necessarily become round.
853a
1 τὴν τῆς θαλάττης κίνησιν, διὰ τὸ μετὰ τῆς θαλάττης κινεῖσθαι,
συμβαίνει ἀεὶ ἐν κινήσει εἶναι καὶ κυλιομέναις
προσκόπτειν. τοῦτο δὲ ἀνάγκη μάλιστα συμβαίνειν αὐτοῖς
τοῖς ἄκροις.
1So in the case of pebbles, because the sea moves and they move with it, the result is that they are always in motion, and, as they roll about, they come into collision with other objects; and it is their extremities which are necessarily most affected. a ae
Chapter 16 (853a5–18)
5 Διὰ τί, ὅσῳ ἂν ᾖ μακρότερα τὰ ξύλα, τοσούτῳ ἀσθενέστερα
γίνεται, καὶ κάμπτεται αἰρόμενα μᾶλλον, κἂν ᾖ
τὸ μὲν βραχύ, ὅσον δίπηχυ, λεπτόν, τὸ δὲ ἑκατὸν πηχῶν
παχύ; ἢ διότι μοχλὸς γίνεται καὶ βάρος καὶ ὑπομόχλιον
ἐν τῷ αἴρεσθαι τοῦ ξύλου τὸ μῆκος; τὸ μὲν γὰρ πρῶτον μέρος
10 αὐτοῦ, ὃ ἡ χεὶρ αἴρει, οἷον ὑπομόχλιον γίνεται, τὸ δ'
ἐπὶ τῷ ἄκρῳ βάρος. ὥστε ὅσῳ ἂν ᾖ μακρότερον τὸ ἀπὸ τοῦ
ὑπομοχλίου, τοσούτῳ ἀνάγκη κάμπτεσθαι μᾶλλον· ὅσῳ
γὰρ ἂν πλέον ἀπέχῃ τοῦ ὑπομοχλίου, τοσούτῳ ἀνάγκη
κάμπτεσθαι μεῖζον. ἀνάγκη οὖν αἴρεσθαι τὰ ἄκρα τοῦ
15 μοχλοῦ. ἐὰν οὖν ᾖ καμπτόμενος ὁ μοχλός, ἀνάγκη αὐτὸν
κάμπτεσθαι μᾶλλον αἰρόμενον. ὅπερ συμβαίνει ἐπὶ τῶν
ξύλων τῶν μακρῶν· ἐν δὲ τοῖς βραχέσιν ἐγγὺς τὸ ἔσχατον
τοῦ ὑπομοχλίου γίνεται τοῦ ἠρεμοῦντος.
Why is it that the longera plank 5of wood is, the weaker16 _ it is, and the more it bends when lifted up? Why, for example, does a short thin plank about two cubits long bend less than a thick plank a hundred cubits long? [5 it because the length of the plank when it is lifted forms a lever, a weight, and afulcrum? The first part of it, then, which the hand raises becomes as it were, a fulcrum, 10and : the part towards the end becomes the weight; and so the longer the space is from the fulcrum to the end, the more the plank must bend; for it must necessarily bend more the further away it is from the fulcrum. Therefore the ends of the lever must be subject to pressure.! If, then, the lever is bent, it must bend more when it is lifted up. This is exactly 15what happens in the case of long planks of wood ; whereas in the case of shorter planks, the extremity is near the fulcrum which is at rest.
Chapter 17 (853a19–31)
Διὰ τί τῷ σφηνὶ ὄντι μικρῷ μεγάλα βάρη διίσταται
20 καὶ μεγέθη σωμάτων, καὶ θλῖψις ἰσχυρὰ γίνεται; ἢ διότι
ὁ σφὴν δύο μοχλοί εἰσιν ἐναντίοι ἀλλήλοις, ἔχει δὲ ἑκάτερος
τὸ μὲν βάρος τὸ δὲ ὑπομόχλιον, ὃ καὶ ἀνασπᾷ ἢ
πιέζει. ἔτι δὲ ἡ τῆς πληγῆς φορὰ τὸ βάρος, ὃ τύπτει καὶ
κινεῖ, ποιεῖ μέγα· καὶ διὰ τὸ κινούμενον κινεῖν τῇ ταχυτῆτι
25 ἰσχύει ἔτι πλέον. μικρῷ δὲ ὄντι μεγάλαι δυνάμεις
ἀκολουθοῦσι· διὸ λανθάνει κινῶν παρὰ τὴν ἀξίαν τοῦ μεγέθους.
ἔστω σφὴν ἐφ' ᾧ ΑΒΓ, τὸ δὲ σφηνούμενον ΔΕΗΖ.
μοχλὸς δὴ γίνεται ἡ ΑΒ, βάρος δὲ τὸ τοῦ Β κάτωθεν,
ὑπομόχλιον δὲ τὸ ΖΔ. ἐναντίος δὲ τούτῳ μοχλὸς τὸ ΒΓ.
30 ἡ δὲ ΑΓ κοπτομένη ἑκατέρᾳ τούτων χρῆται μοχλῷ· ἀνασπᾷ
γὰρ τὸ Β.
How is it that great weights and masses can be split and violent pressure be exerted with a wedge, which is Δ Ε a small thing? Is it because the wedge forms two levers working in opposite directions, and each has a weight and 20fulcrum which presses upwards or downwards? Further, the impetus of the blow causes the weight which strikes the wedge and moves it to be very considerable; and it has all the more force because by reason of its speed it is moving what is already moving. Although the lever is short, great force accompanies it, and so it causes a much more violent movement than 25we should expect from an ΓΔ 8538 15, ‘Tenendum scilicet Aristoteli proprium esse verbum caeteroquin nulla e/evaéio, sed potius contrarius motus, locum habeat (Capelle). Cf. 854” 6. yee the object which is acted upon by it; then AB is a lever and the weight is below at B, and the fulcrum is ZA. On the opposite side is the lever ΒΓ, When AT is struck it brings both 30of these into use as levers; for it presses upwards at the point B.
Chapter 18 (853a32–853b13)
Διὰ τί, ἐάν τις δύο τροχιλέας ποιήσας ἐπὶ δυσὶ ξύλοις
συμβάλλουσιν ἑαυτοῖς ἐναντίως αὑταῖς κύκλῳ περιβάλῃ
καλώδιον, ἔχον τὸ ἄρτημα ἐκ θατέρου τῶν ξύλων,
35 θάτερον δὲ ᾖ προσερηρεισμένον ἢ προστεθειμένον κατὰ τὰς
τροχαλίας, ἐὰν ἕλκῃ τις τῇ ἀρχῇ τοῦ καλωδίου, μεγάλα
βάρη προσάγει, κἂν ᾖ μικρὰ ἡ ἕλκουσα ἰσχύς; ἢ διότι τὸ
αὐτὸ βάρος ἀπὸ ἐλάττονος ἰσχύος, εἰ μοχλεύεται, ἐγείρεται,
ἢ ἀπὸ χειρός; ἡ δὲ τροχιλέα τὸ αὐτὸ ποιεῖ τῷ μοχλῷ,
Why is it that if one puts two pulleys on two blocks which are in opposite positions, and places round them a cord with one end attached to one of the blocks and the other supported by or passed over the pulleys, if one pulls at the end of the cord, one can move great weights, even if the force 35which draws them is small? Is it because the same weight is raised by less force, if a lever is employed, than by the hand, and the pulley acts in the same way as a lever, so that a single pulley will draw more easily and draw a far heavier weight with a slight pull than the hand alone can?
853b
1 ὥστε ἡ μία ῥᾷον ἕλξει, καὶ ἀπὸ μιᾶς ὁλκῆς τοῦ
κατὰ χεῖρα πολὺ ἕλξει βαρύτερον. τοῦτο δ' αἱ δύο τροχαλίαι
πλέον ἢ διπλασίῳ τάχει αἴρουσαι. ἔλαττον γὰρ
ἔτι ἡ ἑτέρα ἕλκει ἢ εἰ αὐτὴ καθ' ἑαυτὴν εἷλκεν, ὅταν
5 παρὰ τῆς ἑτέρας ἐπιβληθῇ τὸ σχοινίον· ἐκείνη γὰρ ἔτι
ἔλαττον ἐποίησε τὸ βάρος. καὶ οὕτως ἐὰν εἰς πλείους ἐπιβάλληται
τὸ καλώδιον, ἐν ὀλίγαις τροχιλέαις πολλὴ γίνεται
διαφορά, ἢ ὥστε ὑπὸ τῆς πρώτης τοῦ βάρους ἕλκοντος
τέτταρας μνᾶς, ὑπὸ τῆς τελευταίας ἕλκεσθαι πολλῷ
10 ἐλάττω. καὶ ἐν τοῖς οἰκοδομικοῖς ἔργοις ῥᾳδίως κινοῦσι μεγάλα
βάρη· μεταφέρουσι γὰρ ἀπὸ τῆς αὐτῆς τροχιλέας
ἐφ' ἑτέραν, καὶ πάλιν ἀπ' ἐκείνης εἰς ὄνους καὶ μοχλούς·
τοῦτο δὲ ταὐτόν ἐστι τῷ ποιεῖν πολλὰς τροχιλέας.
1Two pulleys raise this weight with more than double the velocity; for the second pulley draws a still less weight than if it drew alone by itself, when the rope is passed on to it from the other pulley : for the other pulley makes the weight still less. Thus if the cord is passed through a greater 5number, the difference is great, even when there are only a few pulleys, so that, if the load under the first weighs four minae, much less is drawn by the last. το In building operations they easily move great weights ; for they transfer them from one? pulley to another and thence again to windlasses and levers, and this is equivalent to constructing a number of pulleys.
Chapter 19 (853b14–24)
Διὰ τί, ἐὰν μέν τις ἐπιθῇ ἐπὶ τὸ ξύλον πέλεκυν μέγαν
15 καὶ φορτίον μέγα ἐπ' αὐτῷ, οὐ διαιρεῖ τὸ ξύλον, ὅ τι καὶ
λόγου ἄξιον· ἐὰν δὲ ἄρας τὸν πέλεκύν τις πατάξῃ αὐτῷ,
διασχίζει, ἔλαττον βάρος ἔχοντος τοῦ τύπτοντος πολὺ μᾶλλον
ἢ τοῦ ἐπικειμένου καὶ πιεζοῦντος; ἢ διότι πάντα τῇ κινήσει
ἐργάζεται, καὶ τὸ βαρὺ τὴν τοῦ βάρους κίνησιν λαμβάνει
20 μᾶλλον κινούμενον ἢ ἠρεμοῦν; ἐπικείμενον οὖν οὐ κινεῖται τὴν
τοῦ βάρους κίνησιν, φερόμενον δὲ ταύτην τε καὶ τὴν τοῦ
τύπτοντος. ἔτι δὲ καὶ γίνεται σφὴν ὁ πέλεκυς· ὁ δὲ σφὴν
μικρὸς ὢν μεγάλα διίστησι διὰ τὸ εἶναι ἐκ δύο μοχλῶν
ἐναντίως συγκειμένων.
10How is it that, if you place a heavy axe on a piece of I9 rg wood and put a heavy weight on the top of it, it does not cleave the wood to any considerable extent, whereas, if you lift the axe and strike the wood with it, it does split it, although the axe when it strikes the blow has much less : weight upon it than when it is placed on the wood and . pressing on it? Is 15it because the effect is produced entirely by movement, and that which is heavy gets more movement from its weight when it is in motion than when it is RA Sit gic licen apn ideas ~ at rest? So when it is merely placed on the wood, it does not move with the movement derived from its weight ; but when it is put into motion, it moves with the movement derived from its weight 20and also with that imparted by the striker. Furthermore, the axe works like a wedge; and a wedge, though small, can split large masses because it is made up of two levers working in opposite directions.
Chapter 20 (853b25–854a15)
25 Διὰ τί αἱ φάλαγγες τὰ κρέα ἱστᾶσιν ἀπὸ μικροῦ ἀρτήματος
μεγάλα βάρη, τοῦ ὅλου ἡμιζυγίου ὄντος; οὗ μὲν γὰρ
τὸ βάρος ἐντίθεται, κατήρτηται μόνον ἡ πλάστιγξ, ἐπὶ θάτερον
δὲ ἡ φάλαγξ ἐστὶ μόνον. ἢ ὅτι ἅμα συμβαίνει ζυγὸν
καὶ μοχλὸν εἶναι τὴν φάλαγγα; ζυγὸν μὲν γὰρ, ᾗ
30 τῶν σπαρτίων ἕκαστον γίνεται τὸ κέντρον τῆς φάλαγγος. τὸ
μὲν οὖν ἐπὶ θάτερα ἔχει πλάστιγγα, τὸ δὲ ἐπὶ θάτερα ἀντὶ
τῆς πλάστιγγος τὸ σφαίρωμα, ὃ τῷ ζυγῷ ἔγκειται, ὥσπερ
εἴ τις τὴν ἑτέραν πλάστιγγα καὶ τὸν σταθμὸν ἐπιθείη ἐπὶ τὸ
ἄκρον τῆς πλάστιγγος· δῆλον γὰρ ὅτι ἕλκει τοσοῦτον βάρος
35 ἐν τῇ ἑτέρᾳ κείμενον πλάστιγγι. ὅπως δὲ τὸ ἓν ζυγὸν πολλὰ
ᾖ ζυγά, τοιαῦτα τὰ σπαρτία πολλὰ ἔγκειται ἐν τῷ τοιούτῳ
ζυγῷ, ὧν ἑκάστου τὸ ἐπὶ τάδε ἐπὶ τὸ σφαίρωμα τὸ ἥμισυ
τῆς φάλαγγός ἐστι, καὶ ὁ σταθμὸς δι' ἴσου τῶν ἀπ' ἀλλήλων
τῶν σπαρτίων κινουμένων, ὥστε συμμετρεῖσθαι πόσον βάρος
«Wuvyisit that steelyards! weigh great weights of meat with a small counterpoise, the whole forming only a half balance ὃ For a pan is fixed only at the end where the 25object weighed is placed, and at the other end there is nothing but the steelyard. Is it because the steelyard is at once a beam and a lever? For it is a beam, inasmuch as each position of the cord becomes the centre of the steelyard. Now at one end it has a pan, and at the other instead of a pan the counterpoise which is fixed in the beam, just as if one were to place the 30other pan with the counterpoise in it at the end of the steelyard; for it is clear that it draws the same weight when it lies in this second pan. But in order that the single beam may act as many beams, many such positions for the cord are situated along a beam of this kind, in each of which the part on the side of the counterpoise forms half the steelyard and acts as the 35weight,” the _ positions of the cord being moved through equal intervals, so that one can calculate how much weight is drawn by what lies in the pan, and thus know, when the steelyard is horizontal, how much weight the pan holds for each of the several positions of the cord, as has been explained.
854a
1 ἕλκει τὸ ἐν τῇ πλάστιγγι κείμενον· ὥστε γινώσκειν, ὅταν
ὀρθὴ ἡ φάλαγξ ᾖ, ἀπὸ ποίου σπάρτου πόσον βάρος ἔχει ἡ
πλάστιγξ, καθάπερ εἴρηται. ὅλως μέν ἐστι τοῦτο ζυγόν, ἔχον
μίαν μὲν πλάστιγγα, ἐν ᾗ ἵσταται τὸ βάρος, τὴν δ' ἑτέραν,
5 ἐν ᾗ τὸ σταθμὸν ἐν τῇ φάλαγγι. διὸ σφαίρωμά ἐστιν ἡ
φάλαγξ ἐπὶ θάτερον. τοιοῦτον δὲ ὂν πολλὰ ζυγά ἐστι, καὶ
τοσαῦτα ὅσαπέρ ἐστι τὰ σπαρτία. ἀεὶ δὲ τὸ ἐγγύτερον
σπαρτίον τῆς πλάστιγγος καὶ τοῦ ἱσταμένου βάρους μεῖζον ἕλκει
βάρος, διὰ τὸ γίνεσθαι τὴν μὲν φάλαγγα πᾶσαν μοχλὸν
10 ἀνεστραμμένον (ὑπομόχλιον μὲν γὰρ τὸ σπαρτίον
ἕκαστον ἄνωθεν ὄν, τὸ δὲ βάρος τὸ ἐνὸν ἐν τῇ πλάστιγγι),
ὅσῳ δ' ἂν μακρότερον ᾖ τὸ μῆκος τοῦ μοχλοῦ τοῦ ἀπὸ τοῦ
ὑπομοχλίου, τοσούτῳ ἐκεῖ μὲν ῥᾷον κινεῖ, ἐνταῦθα δὲ σήκωμα
ποιεῖ, καὶ ἵστησι τὸ πρὸς τὸ σφαίρωμα βάρος τῆς
15 φάλαγγος.
1In short, this may be regarded as a balance, having one pan in which the object weighed is placed, and the other in which is the weight? of the steelyard, and so the steel- 5. yard at the other end is the counterpoise. And, since it is as described, it acts as an adjustable balance 5beam, with as many forms as there are positions of the cord. And in all cases, when the cord is nearer the pan and the weight upon it, it draws a greater weight, on account of the whole steelyard being an inverted! lever (for the cord in each position is a fulcrum, although it is above, and the weight is what is in the pan), and the greater the length of 10the lever from the fulcrum, the more easily it produces motion in the case of the lever, and in the case of the balance causes equilibrium and counterbalances the weight of the steelyard near the counterpoise. ἢ - “ον ttl:
Chapter 21 (854a16–31)
Διὰ τί οἱ ἰατροὶ ῥᾷον ἐξαιροῦσι τοὺς ὀδόντας προσλαμβάνοντες
βάρος τὴν ὀδοντάγραν ἢ τῇ χειρὶ μόνῃ ψιλῇ;
πότερον διὰ τὸ μᾶλλον ἐξολισθαίνειν διὰ τῆς χειρὸς τὸν
ὀδόντα ἢ ἐκ τῆς ὀδοντάγρας; ἢ μᾶλλον ὀλισθαίνει τῆς
20 χειρὸς ὁ σίδηρος, καὶ οὐ περιλαμβάνει αὐτὸν κύκλῳ· μαλθακὴ
γὰρ οὖσα ἡ σὰρξ τῶν δακτύλων καὶ προσμένει μᾶλλον
καὶ περιαρμόττει. ἀλλ' ὅτι ἡ ὀδοντάγρα δύο μοχλοί
εἰσιν ἀντικείμενοι, ἓν τὸ ὑπομόχλιον ἔχοντες τὴν σύναψιν
τῆς θερμαστρίδος· τοῦ ῥᾷον οὖν κινῆσαι χρῶνται τῷ ὀργάνῳ
25 πρὸς τὴν ἐξαίρεσιν. ἔστω γὰρ τῆς ὀδοντάγρας τὸ μὲν ἕτερον
ἄκρον ἐφ' ᾧ τὸ Α, τὸ δὲ ἕτερον, τὸ Β, ὃ ἐξαιρεῖ· ὁ
δὲ μοχλὸς ἐφ' ᾧ ΑΔΖ, ὁ δὲ ἄλλος μοχλὸς ἐφ' ᾧ Β
ΓΕ, ὑπομόχλιον δὲ τὸ ΓΘΔ· ὁ δὲ ὀδοὺς ἐφ' οὗ Ι σύναψις·
ὁ δὲ τὸ βάρος. ἑκατέρῳ οὖν τῶν ΒΖ καὶ ἅμα λαβὼν
30 κινεῖ. ὅταν δὲ κινήσῃ, ἐξεῖλε ῥᾷον τῇ χειρὶ ἢ τῷ
ὀργάνῳ.
How is it that dentists extract teeth more easily by applying the additional weight of a tooth-extractor than with the bare hand only? 15Is it because the tooth is more inclined to slip in the fingers = _A ΕΒ: than from the _ tooth-ex- — Θ ao tractor ? or does not the iron slip more than the hand and fail to grasp the tooth all A round, since the flesh of the E bees fingers being soft both ad- ΕἸα. 10. heres to and fits round the tooth better? The truth is that the tooth-extractor consists 20of two levers opposed to one another, with the same fulcrum at the point where the pincers join; so they use the instrument to draw teeth, in order to move them more easily. _ Let A be one extremity of the tooth-extractor and B the other extremity which draws the tooth, and AAZ one lever and BIE the other, and P@A the fulcrum, and let the tooth, which 25is the weight to be lifted, be at the point I, where the two levers meet. The dentist holds and moves the tooth at the same time with B and Z; and when he has moved it, he can take it out more easily with his fingers than with the instrument.
Chapter 22 (854a32–854b15)
Διὰ τί τὰ κάρυα ῥᾳδίως καταγνύουσιν ἄνευ πληγῆς ἐν
τοῖς ὀργάνοις ἃ ποιοῦσι πρὸς τὸ καταγνύναι αὐτά; πολλὴ
γὰρ ἀφαιρεῖται ἰσχὺς ἡ τῆς φορᾶς καὶ βίας. ἔτι δὲ σκληρῷ
35 καὶ βαρεῖ συνθλίβων θᾶττον ἂν κατάξαι ἢ ξυλίνῳ καὶ κούφῳ
τῷ ὀργάνῳ. ἢ διότι οὕτως ἐπ' ἀμφότερα θλίβεται ὑπὸ δύο
μοχλῶν τὸ κάρυον, τῷ δὲ μοχλῷ ῥᾳδίως διαιρεῖται τὰ
βάρη; τὸ γὰρ ὄργανον ἐκ δύο σύγκειται μοχλῶν, ὑπομόχλιον
ἐχόντων τὸ αὐτό, τὴν συναφὴν ἐφ' ἧς τὸ Α. ὥςπερ
Why is it that men easily crack nuts, without striking ' 8545 10. ‘Inverted’ because the cord is regarded as 30supporting from above, whereas the fulcrum supports from below; but the cord really supports below, the beam resting on the loop. : a blow upon them, in the instruments made for this purpose? For with nut-crackers much power is lost, namely, that of motion and violent impetus.! Further, if one crushes them with a hard and heavy instrument, one can crack 35them much more quickly than with a light wooden instrument, Is it because the nut is crushed on two of its sides by two " levers, and bodies can easily be rent asunder with a lever ? For the nut-cracker consists of two levers, with the same fulcrum, namely, A, their point of connexion.
854b
1 οὖν εἰ ἦσαν ἐκβεβλημέναι, ὑφ' ὧν κινουμένων εἰς τὰ τῶν
ΓΔ ἄκρα αἱ ΕΖ συνήγοντο ῥᾳδίως ἀπὸ μικρᾶς ἰσχύος·
ἣν οὖν ἐν τῇ πληγῇ τὸ βάρος ἐποίει, ταύτην ἡ κρείττων ταύτης,
ἡ τὸ ΕΓ καὶ ΖΔ, μοχλοὶ ὄντες ποιοῦσι· τῇ ἄρσει γὰρ
5 εἰς τοὐναντίον αἴρονται, καὶ θλίβοντες καταγνύουσι τὸ ἐφ' ᾧ Κ.
δι' αὐτὸ δὲ τοῦτο καὶ ὅσῳ ἂν ἐγγύτερον ᾖ τῆς Α τὸ Κ, συντρίβεται
θᾶττον· ὅσῳ γὰρ ἂν πλεῖον ἀπέχῃ τοῦ ὑπομοχλίου
ὁ μοχλός, ῥᾷον κινεῖ καὶ πλεῖον ἀπὸ τῆς ἰσχύος τῆς αὐτῆς.
ἔστιν οὖν τὸ μὲν Α ὑπομόχλιον, ἡ δὲ ΔΑΖ μοχλός, καὶ ἡ
10 ΓΑΕ. ὅσῳ ἂν οὖν τὸ Κ ἐγγυτέρω ᾖ τῆς γωνίας τῶν Α,
τοσούτῳ ἐγγύτερον γίνεται τῆς συναφῆς τῶν Α· τοῦτο δέ ἐστι
τὸ ὑπομόχλιον. ἀνάγκη τοίνυν ἀπὸ τῆς αὐτῆς ἰσχύος συναγούσης
τὸ ΖΕ αἴρεσθαι πλέον. ὥστε ἐπεί ἐστιν ἐξ ἐναντίας
ἡ ἄρσις, ἀνάγκη θλίβεσθαι μᾶλλον· τὸ δὲ μᾶλλον θλιβόμενον
15 κατάγνυται θᾶττον.
1As, therefore, E and Z would have been easily moved by a small force if they had been pushed apart, so they are easily brought together, the levers being moved at the points A andT.? So EF and ZA being levers exert the same or even greater force than that which the weight exerted Γ Ζ when 5the nut was cracked by a blow; for when weight is put? upon the levers they move in opposite directions and compress and break the object at K. For this very reason, too, the nearer K is to A, the sooner it is subjected to pressure; for the further the lever extends from the fulcrum, the more easily and more powerfully does it move an object with the exercise of 10the same force. A, then, is the fulcrum, and AAZ and TAE are the levers. The nearer, therefore, K is to the angle at A, the nearer it is to to the point where the levers are connected,’ and this is the fulcrum. So with the same force bringing them together, Z and E must be subjected to more weight ; -1 854234. i.e. the force which would be brought into use if the 15nut were broken by a blow. : 8s4>2, Omitting ὑφ᾽ ὧν, which is clearly corrupt, and placing 645-8 L and so, when weight is exerted from two contrary directions, more compression must take place, and the more an object is compressed, the sooner it breaks.
Chapter 23 (854b16–855a27)
Διὰ τί φερομένων δύο φορὰς ἐν τῷ ῥόμβῳ τῶν ἄκρων
σημείων ἀμφοτέρων, οὐ τὴν ἴσην ἑκάτερον αὐτῶν εὐθεῖαν διέρχεται,
ἀλλὰ πολλαπλασίαν θάτερον; ὁ αὐτὸς δὲ λόγος καὶ
διὰ τί τὸ ἐπὶ τῆς πλευρᾶς φερόμενον ἐλάττω διέρχεται τῆς
20 πλευρᾶς. τὸ μὲν γὰρ τὴν διάμετρον τὴν ἐλάττω, ἡ δὲ τὴν
πλευρὰν τὴν μείζω, καὶ ἡ μὲν μίαν, τὸ δὲ δύο φέρεται
φοράς. φερέσθω γὰρ ἐπὶ τῆς ΑΒ τὸ μὲν Α πρὸς τὸ Β, τὸ
δὲ Β πρὸς τὸ Δ τῷ αὐτῷ τάχει· φερέσθω δὲ καὶ ἡ ΑΒ
ἐπὶ τῆς ΑΓ παρὰ τὴν ΓΔ τῷ αὐτῷ τάχει τούτοις. ἀνάγκη
25 δὴ τὸ μὲν Α ἐπὶ τῆς ΑΔ διαμέτρου φέρεσθαι, τὸ δὲ Β ἐπὶ
τῆς ΒΓ, καὶ ἅμα διεληλυθέναι ἑκατέραν, καὶ τὴν ΑΒ τὴν
ΑΓ πλευράν. ἐνηνέχθω γὰρ τὸ μὲν Α τὴν ΑΕ, ἡ δὲ Α
Β τὴν ΑΖ, καὶ ἔστω ἐκβεβλημένη ἡ ΖΗ παρὰ τὴν ΑΒ,
καὶ ἀπὸ τοῦ Ε πεπληρώσθω. ὅμοιον οὖν γίνεται τὸ παραπληρωθὲν
30 τῷ ὅλῳ. ἴση ἄρα ἡ ΑΖ τῇ ΑΕ, ὥστε τὸ Α
ἐπὶ τῆς πλευρᾶς ἐνήνεκται τῆς ΑΕ. ἡ δὲ ΑΒ τὴν ΑΖ
εἴη ἂν ἐνηνεγμένη. ἔσται ἄρα ἐπὶ τῆς διαμέτρου κατὰ τὸ Θ.
καὶ αἰεὶ δὲ ἀνάγκη αὐτὸ φέρεσθαι κατὰ τὴν διάμετρον.
καὶ ἅμα ἡ πλευρὰ ἡ ΑΒ τὴν πλευρὰν τὴν ΑΓ δίεισι,
35 καὶ τὸ Α τὴν διάμετρον δίεισι τὴν ΑΔ. ὁμοίως δὲ δειχθήσεται
καὶ τὸ Β ἐπὶ τῆς ΑΓ διαμέτρου φερόμενον. ἴση
γάρ ἐστιν ἡ ΒΕ τῇ ΒΗ. παραπληρωθέντος οὖν ἀπὸ τοῦ Η,
ὅμοιόν ἐστι τῷ ὅλῳ τὸ ἐντός. καὶ τὸ Β ἐπὶ τῆς διαμέτρου
ἔσται κατὰ τὴν σύναψιν τῶν πλευρῶν, καὶ ἅμα δίεισιν ἥ
Why is it that in a rhombus, when the points at the 23 _ . extremities are moved in two movements, they do not 20describe equal straight lines, but one of them amuch longer line than the other? Further (and this is the same question), why does the point [A] moving along the side [AB] describe a resultant line [AA] less than the side? Forthe — point describes the diagonal, the shorter distance, and the line [AB] moves along the side [AT], the longer distance ; and yet the line 25has but one movement, and the point two movements. For let A move along AB to B, and B to A with the same velocity ; and let the line AB move along AT parallel to Ε Β Θ r Ἢ i glee ΓΔ with the same velocity. Then the point A must move along the diagonal AA, and B along Br; and both must describe these diagonals simultaneously, while AB moves along the side AT. 30For let A be moved the distance AE, and the line AB the distance AZ, and let ZH be drawn parallel to AB, and a line drawn from E to complete the parallelogram [AZO@E]. The small parallelogram then thus formed is similar to the Whole parallelogram. Thus AZ equals AE, so that A has been moved along the side AE [to E], while the line AB would be moved the distance AZ. 35Thus A will be on the diagonal at ©, and so must always move along the diagonal; and [in the whole parallelogram} the side AB will describe the side AT, and the point A the diagonal AA simultaneously. In the same way it may be proved that B moves along the diagonal BI, BE being equal to BH.
855a
1 τε πλευρὰ τὴν πλευρὰν καὶ τὸ Β τὴν ΒΓ διάμετρον.
ἅμα ἄρα καὶ τὸ Β τὴν πολλαπλασίαν τῆς ΑΒ δίεισι
καὶ ἡ πλευρὰ τὴν ἐλάττονα πλευράν, τῷ αὐτῷ τάχει φερόμενα,
καὶ ἡ πλευρὰ μείζω τοῦ Α διελήλυθε μίαν φορὰν
5 φερομένη. ὅσῳ γὰρ ἂν ὀξύτερος γένηται ὁ ῥόμβος, ἡ
μὲν διάμετρος ἡ ἐλάττων γίνεται, ἡ δὲ ΒΓ μείζων, ἡ δὲ
πλευρὰ τῆς ΒΓ ἐλάττων. ἄτοπον γάρ, ὥσπερ ἐλέχθη, τὸ
δύο φορὰς φερόμενον ἐνίοτε βραδύτερον φέρεσθαι τοῦ μίαν,
καὶ ἀμφοτέρων ἰσοταχῶν σημείων δοθέντων μείζω διεξιέναι
10 θάτερον. αἴτιον δὲ ὅτι τοῦ μὲν ἀπὸ τῆς ἀμβλείας φερομένου
σχεδὸν ἐναντίαι ἀμφότεραι γίνονται, ἥν τε αὐτὴ
φέρεται καὶ ἣν ὑπὸ τῆς πλευρᾶς ὑποφέρεται, τοῦ δὲ ἀπὸ
τῆς ὀξείας συμβαίνει φέρεσθαι ἐπὶ τὸ αὐτό. συνεπουρίζει
γὰρ ἡ τῆς πλευρᾶς τὴν ἐπὶ τῆς διαμέτρου· καὶ ὅσῳ ἂν
15 τὴν μὲν ὀξυτέραν ποιήσῃ, τὴν δὲ ἀμβλυτέραν, ἡ μὲν βραδυτέρα
ἔσται, ἡ δὲ θάττων. αἱ μὲν γὰρ ἐναντιώτεραι γίνονται
διὰ τὸ ἀμβλυτέραν γίνεσθαι τὴν γωνίαν, αἱ δὲ
μᾶλλον ἐπὶ τὰ αὐτὰ διὰ τὸ συνάγεσθαι τὰς γραμμάς.
τὸ μὲν γὰρ Β σχεδὸν ἐπὶ τὸ αὐτὸ φέρεται κατ' ἀμφοτέρας
20 τὰς φοράς· συνεπουρίζεται οὖν ἡ ἑτέρα, καὶ ὅσῳ ἂν
ὀξυτέρα γίνηται ἡ γωνία, τοσούτῳ μᾶλλον. τὸ Α δὲ ἐπὶ
τοὐναντίον· αὐτὸ μὲν γὰρ πρὸς τὸ Β φέρεται, ἡ δὲ πλευρὰ
ὑποφέρει αὐτὸ πρὸς τὸ Δ. καὶ ὅσῳ ἂν ἀμβλυτέρα ἡ γωνία
ᾖ, ἐναντιώτεραι αἱ φοραὶ γίνονται· εὐθυτέρα γὰρ ἡ
25 γραμμὴ γίνεται. εἰ δ' ὅλως εὐθεῖα γένοιτο, παντελῶς ἂν
εἴησαν ἐναντίαι. ἡ δὲ πλευρὰ ὑπ' οὐθενὸς κωλύεται μίαν
φερομένη φοράν. εὐλόγως οὖν τὴν μείζω διέρχεται.
1For, if the parallelogram be completed by drawing a line from H, the interior parallelogram [EOHB] will be similar to the whole parallelogram ; and B will be on the diagonal at the point where the sides meet; and the side [BA] will describe At the same time then B will describe a line [ΒΓ] which is much longer than AB, and the side 5[AB] will pass along the side [ΑΓ] which is shorter [than the diagonal], though the velocity is the same, in the same time (and the side [AB] has moved further than A, though it is moved by only one movement). For as the rhombus becomes more.acute [at B the side [AB] less than ΒΓ, For it is strange, as has been remarked, that in some cases a point moved by two movements travels more slowly than a point moved by one, 10and that, while both the given points have equal velocity, either one of them describes a greater line. The reason is that, when a point moves from an obtuse angle, the sides are in almost opposite directions, namely, that in which the point itself! is moved and that in which it is moved down? by the side; but when it moves from an acute angle, it moves, as it were, in actual fact towards the same position. For the 15angle of the sides contributes to increase the speed of the diagonal; and in proportion as one makes the one angle more acute and the other more obtuse, the movement is slower or quicker. For the sides are brought into more opposite direction by the angle becoming more obtuse; but they are brought into the same direction by the sides being brought nearer together. For B moves in practically the same direction in virtue 20of both its movements; thus one contributes to. assist the other, and more so, the more acute the angle becomes. And the reverse is the case with A; " fe) ° τὸ L2 for it itself moves towards B, while the movement of the Side [AB] brings it down to Δ; and the more obtuse the angle is, the more opposite will the movements be; for the two sides become more like a straight line. If they became actually a straight line, 25the components would be absolutely in opposite directions. But the side, being moved in one direction only, is interfered with by nothing. In that case it naturally moves through a longer distance.
Chapter 24 (855a28–856a38)
Ἀπορεῖται διὰ τί ποτε ὁ μείζων κύκλος τῷ ἐλάττονι
κύκλῳ ἴσην ἐξελίττεται γραμμήν, ὅταν περὶ τὸ αὐτὸ κέντρον
30 τεθῶσι; χωρὶς δὲ ἐκκυλιόμενοι, ὥσπερ τὸ μέγεθος αὐτῶν
πρὸς τὸ μέγεθος ἔχει, οὕτως καὶ αἱ γραμμαὶ αὐτῶν
γίνονται πρὸς ἀλλήλας. ἔτι δὲ ἑνὸς καὶ τοῦ αὐτοῦ κέντρου
ὄντος ἀμφοῖν, ὁτὲ μὲν τηλικαύτη γίνεται ἡ γραμμὴ ἣν
ἐκκυλίονται, ἡλίκην ὁ ἐλάττων κύκλος καθ' αὑτὸν ἐκκυλίεται,
35 ὁτὲ δὲ ὅσην ὁ μείζων. ὅτι μὲν οὖν μείζω ἐκκυλίεται
ὁ μείζων, φανερόν. γωνία μὲν γὰρ δοκεῖ κατὰ τὴν
αἴσθησιν εἶναι ἡ περιφέρεια ἑκάστου τῆς οἰκείας διαμέτρου,
ἡ τοῦ μείζονος κύκλου μείζων, ἡ δὲ τοῦ ἐλάττονος ἐλάττων,
ὥστε τὸν αὐτὸν τοῦτον ἕξουσι λόγον, καθ' ἃς ἐξεκυλίσθησαν
THERE is a question why a large circle traces out a path equal to that of a smaller circle, when they are placed about the same centre,? but when they are rolled separately,® their paths are to one another in the proportion 30of their dimensions. And, further, the centre of both being one and the same, at one time the path which they trace is of the same length as the smaller traces out alone, and at another time of the length which the larger circle traces.* Now it is manifest that the larger circle traces out the longer path. For by mere observation it is plain that the angle which the circumference of each makes with its own diameter 35is greater in the case of the larger circle than in the smaller;® so that, by observation, the paths along which they roll will have this same proportion to one <—___ «-------.a> a Bt a’s final position will be that from which he started, 8 will have moved from f! to β3, Ὁ 855° 36. i.e. the angle AZI is greater than the angle AHB, cf.
855b
1 αἱ γραμμαὶ πρὸς ἀλλήλας κατὰ τὴν αἴσθησιν. ἀλλὰ μὴν
καὶ ὅτι τὴν ἴσην ἐκκυλίονται, ὅταν περὶ τὸ αὐτὸ κέντρον
κείμενοι ὦσι, δῆλον· καὶ οὕτως γίνεται ὁτὲ μὲν ἴση τῇ
γραμμῇ ἣν ὁ μείζων κύκλος ἐκκυλίεται, ὁτὲ δὲ ἐλάττων.
5 ἔστω γὰρ κύκλος ὁ μείζων μὲν ἐφ' οὗ τὰ ΔΖΓ, ὁ δὲ
ἐλάττων ἐφ' οὗ τὰ ΕΗΒ, κέντρον δὲ ἀμφοῖν τὸ Α· καὶ
ἣν μὲν ἐξελίττεται καθ' αὑτὸν ὁ μέγας, ἡ ἐφ' ἧς ΖΙ ἔστω,
ἣν δὲ ὁ ἐλάττων καθ' αὑτόν, ἡ ἐφ' ἧς ΗΚ, ἴση τῇ ΑΖ.
ἐὰν δὴ κινῶ τὸν ἐλάττονα, τὸ αὐτὸ κέντρον κινῶ, ἐφ' οὗ
10 τὸ Α· ὁ δὲ μέγας προσηρμόσθω. ὅταν οὖν ἡ ΑΒ ὀρθὴ γένηται
πρὸς τὴν ΗΚ, ἅμα καὶ ἡ ΑΓ γίνεται ὀρθὴ πρὸς τὴν
ΖΛ, ὥστε ἔσται ἴσην ἀεὶ διεληλυθυῖα, τὴν μὲν ΗΚ, ἐφ'
ᾧ ΗΒ περιφέρεια, τὴν δὲ ΖΛ ἡ ἐφ' ἧς ΖΓ. εἰ δὲ τὸ
τέταρτον μέρος ἴσην ἐξελίττεται, δῆλον ὅτι καὶ ὁ ὅλος κύκλος
15 τῷ ὅλῳ κύκλῳ ἴσην ἐξελιχθήσεται, ὥστε ὅταν ἡ ΒΗ
γραμμὴ ἔλθῃ ἐπὶ τὸ Κ, καὶ ἡ ΖΓ ἔσται περιφέρεια ἐπὶ
τῆς ΖΛ, καὶ ὁ κύκλος ὅλος ἐξειλιγμένος. ὁμοίως δὲ καὶ
ἐὰν τὸν μέγαν κινῶ, ἐναρμόσας τὸν μικρόν, τοῦ αὐτοῦ κέντρου
ὄντος, ἅμα τῇ ΑΓ ἡ ΑΒ κάθετος καὶ ὀρθὴ ἔσται, ἡ
20 μὲν πρὸς τὴν ΖΙ, ἡ δὲ πρὸς τὴν ΗΘ. ὥστε ὅταν ἴσην ἡ
μὲν τῇ ΗΘ ἔσται διεληλυθυῖα, ἡ δὲ τῇ ΖΙ, καὶ γένηται
ὀρθὴ πάλιν ἡ ΖΑ πρὸς τὴν ΖΛ, καὶ ἡ ΑΓ ὀρθὴ πάλιν,
ὡς τὸ ἐξ ἀρχῆς ἔσονται ἐπὶ τῶν ΘΙ. τὸ δὲ μήτε στάσεως
γινομένης τὸ μεῖζον τῷ ἐλάττονι, ὥστε μένειν τινὰ χρόνον
25 ἐπὶ τοῦ αὐτοῦ σημείου· κινοῦνται γὰρ συνεχῶς ἄμφω ἀμφοτεράκις.
μὴ ὑπερπηδῶντος τοῦ ἐλάττονος μηθὲν σημεῖον,
τὸν μὲν μείζω τῷ ἐλάττονι ἴσην διεξιέναι, τὸν δὲ τῷ μείζονι,
ἄτοπον. ἔτι δὲ μιᾶς κινήσεως οὔσης ἀεὶ τὸ κέντρον
τὸ κινούμενον ὁτὲ μὲν τὴν μεγάλην ὁτὲ δὲ τὴν ἐλάττονα
30 ἐκκυλίεσθαι θαυμαστόν. τὸ γὰρ αὐτὸ τῷ αὐτῷ τάχει φερόμενον
ἴσην πέφυκε διεξιέναι· τῷ αὐτῷ δὲ τάχει ἴσην ἐστὶ
κινεῖν ἀμφοτεράκις. ἀρχὴ δὲ ληπτέα ἥδε περὶ τῆς αἰτίας
αὐτῶν, ὅτι ἡ αὐτὴ δύναμις καὶ ἴση τὸ μὲν βραδύτερον
κινεῖ μέγεθος, τὸ δὲ ταχύτερον. εἰ δή τι εἴη ὃ μὴ πέφυκεν
35 ὑφ' ἑαυτοῦ κινεῖσθαι, ἐὰν τοῦτο ἅμα καὶ αὐτὸ κινῇ τὸ πεφυκὸς
κινεῖσθαι, βραδύτερον κινηθήσεται ἢ εἰ αὐτὴ καθ'
αὑτὴν ἐκινεῖτο. καὶ ἐὰν μὲν πεφυκὸς ᾖ κινεῖσθαι, μὴ συγκινῆται
δὲ μηθέν, ὡσαύτως ἕξει. καὶ ἀδύνατον δὴ κινεῖσθαι
πλέον ἢ τὸ κινοῦν· οὐ γὰρ τὴν αὑτοῦ κινεῖται κίνησιν, ἀλλὰ
1and note. Dc gern Gy hg mH or ὁ another. But, in fact, it is manifest that, when they are situated about the same centre, this is not so, but they trace out an equal path; so that it comes to this, that in the one case the path is equal to that traced by the larger circle, in the other to that traced by the smaller. Let 5AZI be the greater circle, EHB the lesser, A the common centre, ZI the path along which the greater circle moves by its own motion,! and HK the path of the smaller circle by its own motion, equal to ZA. r = cae K Θ Η Ζ Λ When, then, I move the smaller circle,? I move the same centre A; and now let the large circle be fixed to it. Whenever, therefore, AB becomes perpendicular to HK [at ΚΙ, AT at the same 10time becomes perpendicular to ZA [at A] ; so that they will always have traversed an equal distance, HK representing the arc HB, and ZA representing the arc ZT. And if one quadrant traces an equal path, it is plain that the whole circle will trace out a path equal to that of the other whole circle; so that whenever the line HB comes to K, the arc ΖΓ will move along ZA; and the same is the case with 15the whole circle after one revolution. . In like manner if I roll the large circle,’ fastening th smaller circle to it, about the same centre, AB. will be perpendicular and vertical at the same time as AT, the latter to ZI [at I], the former to ΗΘ [at ©]. - So that, whenever the one [HB] shall have traversed a distance equal to HO and the other [ZI] a distance equal to ZI, and ZA again becomes perpendicular 20to ZA and AH to HK, they will be in their original position at the points © and 1, And, since there is no halting of the greater for the lesser, so as to be at rest during an interval at the same point (for in both cases both are moved continuously), nor does the lesser skip any point, it is strange that in one case the greater should traverse a distance equal to that traversed by the lesser, and 25in the other case the lesser a distance equal to that traversed by the greater. And, further, it is wonderful that, though there is always only one movement, the centre that is moved should be rolled forward in one case a great and in another a less distance. For the same thing moved at the same velocity naturally traverses an equal distance; and to move a thing at the same velocity is to move it an 30equal distance in both cases. As to the reason, this may be taken as a principle, that the same, or an equal force, moves one mass more slowly and the other more quickly. Suppose that there is a body which is not naturally in motion of itself; if another body which is naturally in motion move it and itself as well, it will be moved more slowly than if it were being moved by its own motion alone; and if 35it be naturally in motion and nothing is moved with it, the same is the case. So it is quite impossible for any body to be moved more than that which moves it ; for it is not moved according to any rate of motion of its 8562 Own, but at the rate of that which moves it. Let there be two circles, a greater A and a lesser B.
856a
1 τὴν τοῦ κινοῦντος. εἴη δὴ κύκλος ὁ μὲν μείζων τὸ Α, ὁ δὲ
ἐλάττων ἐφ' ᾧ Β. εἰ ὠθοίη δ' ὁ ἐλάττων τὸν μείζω, μὴ
κυλιομένου αὐτοῦ, φανερὸν ὅτι τοσοῦτον δίεισι τῆς εὐθείας
ὁ μείζων, ὅσον ἐώσθη ὑπὸ τοῦ ἐλάττονος. τοσοῦτον δέ γε
5 ἐώσθη ὅσον ὁ μικρὸς ἐκινήθη. ἴσην ἄρα τῆς εὐθείας διεληλύθασιν.
ἀνάγκη τοίνυν καὶ εἰ κυλιόμενος ὁ ἐλάττων τὸν
μείζω ὠθοίη, κυλισθῆναι μὲν ἅμα τῇ ὤσει, τοσοῦτον δ' ὅσον
ὁ ἐλάττων ἐκυλίσθη, εἰ μηθὲν αὐτὸς τῇ αὐτῇ κινήσει κινεῖται.
ὡς γὰρ καὶ ὅσον ἐκίνει, τοσοῦτον κεκινῆσθαι ἀνάγκη
10 τὸ κινούμενον ὑπ' ἐκείνου. ἀλλὰ μὴν ὅ τε κύκλος τοσοῦτον
ἐκίνησε τὸ αὐτό, κύκλῳ τε καὶ ποδιαίαν (ἔστω γὰρ τοσοῦτον
ὃ ἐκινήθη), καὶ ὁ μέγας ἄρα τοσοῦτον ἐκινήθη. ὁμοίως
δὲ κἂν ὁ μέγας τὸν μικρὸν κινήσῃ, ἔσται κεκινημένος ὁ μικρὸς
ὡς καὶ ὁ μείζων. καθ' αὑτὸν μὲν δὴ κινηθεὶς ὁποτεροσοῦν,
15 ἐάν τε ταχὺ ἐάν τε βραδέως· τῷ αὐτῷ δὲ τάχει
εὐθὺς ὅσην ὁ μείζων πέφυκεν ἐξελιχθῆναι γραμμήν. ὅπερ
καὶ ποιεῖ τὴν ἀπορίαν, ὅτι οὐκέτι ὁμοίως ποιοῦσιν ὅταν συναρμοσθῶσιν.
τὸ δ' ἔστιν, εἰ ὁ ἕτερος ὑπὸ τοῦ ἑτέρου κινεῖται
οὐχ ἣν πέφυκεν, οὐδὲ τὴν αὑτοῦ κίνησιν. οὐθὲν γὰρ
20 διαφέρει περιθεῖναι καὶ ἐναρμόσαι ἢ προσθεῖναι ὁποτερονοῦν
ὁποτέρῳ· ὁμοίως γάρ, ὅταν ὁ μὲν κινῇ ὁ δὲ κινῆται ὑπὸ
τούτου, ὅσον ἂν κινῇ ἅτερος, τοσοῦτον κινηθήσεται ἅτερος.
ὅταν μὲν οὖν προσκείμενον κινῇ ἢ προσκρεμάμενον, οὐκ ἀεὶ
κυλίει τις· ὅταν δὲ περὶ τὸ αὐτὸ κέντρον τεθῶσιν, ἀνάγκη
25 κυλίεσθαι ἀεὶ τὸν ἕτερον ὑπὸ τοῦ ἑτέρου. ἀλλ' οὐθὲν ἧττον
οὐ τὴν αὑτοῦ κίνησιν ἅτερος κινεῖται, ἀλλ' ὥσπερ ἂν εἰ μηδεμίαν
εἶχε κίνησιν. κἂν ἔχῃ, μὴ χρῆται δ' αὐτῇ, ταὐτὸ
συμβαίνει. ὅταν μὲν οὖν ὁ μέγας κινῇ ἐνδεδεμένον τὸν μικρόν,
ὁ μικρὸς κινεῖται ὅσηνπερ οὗτος· ὅταν δὲ ὁ μικρός,
30 πάλιν ὁ μέγας ὅσην οὗτος. χωριζόμενος δὲ ἑκάτερος αὑτὸν
κινεῖ αὐτός. ὅτι δὲ τοῦ αὐτοῦ κέντρου ὄντος καὶ κινοῦντος
τῷ αὐτῷ τάχει συμβαίνει ἄνισον διεξιέναι αὐτοὺς γραμμήν,
παραλογίζεται ὁ ἀπορῶν σοφιστικῶς. τὸ αὐτὸ μὲν
γάρ ἐστι κέντρον ἀμφοῖν, ἀλλὰ κατὰ συμβεβηκός, ὡς
35 μουσικὸν καὶ λευκόν· τὸ γὰρ εἶναι ἑκατέρου κέντρου τῶν
κύκλων οὐ τῷ αὐτῷ χρῆται. ὅταν μὲν οὖν ὁ κινῶν ᾖ ὁ
μικρός, ὡς ἐκείνου κέντρον καὶ ἀρχή, ὅταν δὲ ὁ μέγας, ὡς
ἐκείνου. οὔκουν τὸ αὐτὸ κινεῖ ἁπλῶς, ἀλλ' ἔστιν ὥς.
1If the lesser were to push along the greater, when the greater is not rolling along, it is plain that the greater will traverse so much distance as it has been pushed by the lesser. And it has been pushed the same distance as the small circle has moved ; so that they have both traversed an equal straight line. Necessarily, therefore, if the lesser 5be rolling while it ee 1... pushes the greater, the latter will be rolled, as well as pushed, just so far as the lesser has been rolled, if the greater have no motion of its own; for in the same way and so far as the moving body moves it, so far must the body which is moved be moved thereby. So, indeed, the lesser? circle has moved the greater so far and in such a way,? viz., in a circle—say one foot, for let that be the extent 10of the movement—and consequently the larger circle has moved that distance. So too, if the large circle move the lesser, the lesser circle will have been moved just as far as the large circle, in whatever way® the latter be moved, whether quickly or slowly, by its own motion; and the lesser circle will trace out a line at the same velocity and of the same length as the greater traced out by its natural movement. And this is just 15what causes the difficulty, that they do not act any longer when they are joined together in the same way as they acted when they were not connected; that is to say, when one is moved by the other not according to its natural motion, nor according to its own motion. For it makes no difference whether one is fixed round the other or fitted inside it, or placed in contact with it; for in all these cases, when one moves and the other 20is moved by it, the one will be moved just so far as the other moves it. Now when one moves a circle by means of another circle in contact with it, or suspended from it, one. does not revolve it continuously ;* but if one places them about the same centre, the one must be continuously revolved by the other. But nevertheless, the former is not moved in accordance with its own motion, but just as if it had no proper motion; and 25if it has a proper motion, but does not make use of it, it comes to the same thing. Whenever, therefore, the large circle moves the small circle affixed to it, the small circle moves the same distance 8567 sh μι as the large, and vice versa. But when they are separate each has its own motion.? If any one raises the difficulty that, when the centre is the same and is moving the two circles with equal velocity, they trace out unequal 30paths, he is reasoning falsely and sophistically. For the centre is, indeed, the ἧ same for both, but only accidentally, just as the same ἷ thing may chance to be ‘ musical’ and ‘ white’; for to q be the centre of each of the circles is not the same for it in the two cases. In conclusion, when it is the smaller circle that moves the greater, the centre and source of motion isto be regarded as belonging to the smaller circle; but 35when the greater circle moves the lesser, it is to be regarded as belonging { to the greater circle. Thus the source of motion is not the same absolutely, though it is in a sense the same. oe AS tie ai
Chapter 25 (856a39–857a4)
Διὰ τί τὰς κλίνας ποιοῦσι διπλασιοπλεύρους, τὴν μὲν
Why do they construct beds so that one dimension is 25 i double the other, one side being six feet long or a little more, the other three feet ?
856b
1 ἓξ ποδῶν καὶ μικρῷ μείζω πλευράν, τὴν δὲ τριῶν; καὶ
διὰ τί ἐντείνουσιν οὐ κατὰ διάμετρον; ἢ τὸ μὲν μέγεθος τηλικαύτας,
ὅπως τοῖς σώμασιν ὦσι σύμμετροι; γίνονται
γὰρ οὕτω διπλασιόπλευροι, τετραπήχεις μὲν τὸ μῆκος, διπήχεις
5 δὲ τὸ πλάτος. ἐντείνουσι δὲ οὐ κατὰ διάμετρον ἀλλ'
ἀπ' ἐναντίας, ὅπως τά τε ξύλα ἧττον διασπᾶται· τάχιστα
γὰρ σχίζεται κατὰ φύσιν διαιρούμενα ταύτῃ, καὶ ἑλκόμενα
πονεῖ μάλιστα. ἔτι ἐπειδὴ δεῖ βάρος δύνασθαι τὰ
σπαρτία φέρειν, οὕτως ἧττον πονέσει λοξοῖς τοῖς σπαρτίοις
10 ἐπιτιθεμένου τοῦ βάρους ἢ πλαγίοις. ἔτι δὲ ἔλαττον οὕτω
σπαρτίον ἀναλίσκεται. ἔστω γὰρ κλίνη ἡ ΑΖΗΙ, καὶ δίχα
διῃρήσθω ἡ ΖΗ κατὰ τὸ Β. ἴσα δὴ τρυπήματά ἐστιν
ἐν τῇ ΖΒ καὶ ἐν τῇ ΖΑ. καὶ γὰρ αἱ πλευραὶ ἴσαι εἰσίν·
ἡ γὰρ ὅλη ΖΗ διπλασία ἐστίν. ἐντείνουσι δ' ὡς γέγραπται,
15 ἀπὸ τοῦ Α ἐπὶ τὸ Β, εἶτα οὗ τὸ Γ, εἶτα οὗ τὸ Δ, εἶτα οὗ
τὸ Θ, εἶτα οὗ τὸ Ε. καὶ οὕτως ἀεί, ἕως ἂν εἰς γωνίαν
καταστρέψωσιν ἄλλην· δύο γὰρ ἔχουσι γωνίαι τὰς ἀρχὰς
τοῦ σπαρτίου. ἴσα δέ ἐστι τὰ σπαρτία κατὰ τὰς κάμψεις,
τό τε ΑΒ καὶ ΒΓ τῷ ΓΔ καὶ ΔΘ. καὶ τὰ ἄλλα δὲ
20 τὰ τοιαῦτά ἐστιν, ὅτι οὕτως ἔχει ἡ αὐτὴ ἀπόδειξις. ἡ μὲν
γὰρ ΑΒ τῇ ΕΘ ἴση· ἴσαι γάρ εἰσιν αἱ πλευραὶ τοῦ ΒΗΚ
Α χωρίου, καὶ τὰ τρυπήματα ἴσα διέστηκεν. ἡ δὲ ΒΗ ἴση
τῇ ΚΑ· ἡ γὰρ Β γωνία ἴση τῇ Η. ἐν ἴσοις γὰρ ἡ μὲν
ἐκτός, ἡ δὲ ἐντός· καὶ ἡ μὲν Β ἐστὶν ἡμίσεια ὀρθῆς· ἡ
25 γὰρ ΖΒ ἴση τῇ ΖΑ· καὶ γωνία δὲ ἡ κατὰ τὸ Ζ ὀρθή. ἡ
δὲ Β γωνία ἴση τῇ κατὰ τὸ Η· ἡ γὰρ κατὰ τὸ Ζ ὀρθή,
ἐπειδὴ διπλασιόπλευρον τὸ ἑτερόμηκες καὶ πρὸς μέσον κέκλασται.
ὥστε ἡ ΑΓ τῇ ΕΗ ἴση. ταύτῃ δὲ ἡ ΚΘ· παράλληλος
γάρ. ὥστε ἡ ΒΓ ἴση τῇ ΚΘ. ἡ δὲ ΓΕ τῇ ΔΘ.
30 ὁμοίως δὲ καὶ αἱ ἄλλαι δείκνυνται ὅτι ἴσαι εἰσὶν αἱ κατὰ
τὰς κάμψεις δύο ταῖς δυσίν. ὥστε δῆλον ὅτι τὰ τηλικαῦτα
σπαρτία ὅσον τὸ ΑΒ, τέσσαρα τοσαῦτ' ἔνεστιν ἐν τῇ κλίνῃ·
ὅσον δ' ἐστὶ τὸ πλῆθος τῶν ἐν τῇ ΖΗ πλευρᾷ τρυπημάτων,
καὶ ἐν τῷ ἡμίσει τῷ ΖΒ τὰ ἡμίση. ὥστε ἐν τῇ ἡμισείᾳ
35 κλίνῃ τηλικαῦτα μεγέθη σπαρτίων ἐστὶν ὅσον τῷ ΒΑ ἔνεστι,
τοσαῦτα δὲ τὸ πλῆθος ὅσαπερ ἐν τῷ ΒΗ τρυπήματα.
ταῦτα δὲ οὐδὲν διαφέρει λέγειν ἢ ὅσα ἐν τῇ ΑΖ καὶ ΒΖ
τὰ συνάμφω. εἰ δὲ κατὰ διάμετρον ἐνταθῇ τὰ σπαρτία,
ὡς ἐν τῇ ΑΒΓΔ κλίνῃ ἔχει, τὰ ἡμίσεά εἰσιν οὐ τοσαῦτα
1And why do they not stretch bed-ropes diagonally? Do they make them of this size so as to fit the body? Thus they have one side twice the length of the other, being four cubits long and two cubits wide. The ropes are not stretched diagonally but from side to side, so that the 5wooden frame may be less likely to break ; for wood can be cleft most easily if split thus in the natural way,” and when there is a pull upon it, it is subject to a considerable strain. Further, since the ropes have to be able to bear a weight, there will be less of a strain when io the weight is put upon them if they are strung crosswise 10rather than diagonally. Again, less rope® is used up by this method. Let AZHI be a bed, and let ZH be divided into two equal parts at B. There is an equal number of holes in ZB and ZA; for the sides are equal,! each to each, for the whole side ZH is double the side ZA. They stretch the rope on the method already mentioned from A to B, then toT, 15A, ©, and E, and so on until they turn back and reach another angle; for the two ends of the rope? come at two different angles.® Now the parts of the rope which form the bends are equal, e.g. AB, ΒΓ are equal to AN BFE TA, AO—and so with other Ζ Η similar pairs of sides, for the As MM same demonstration holds good M P inallcases.4 [For AB is 20equal to A ΕΘ; for the opposite sides of Δ Be ἢ Σ the parallelogram BHKA are equal, and the holes are an equal distance apart from one another. And BH is equal to KA; for the angle at B is equal to the angle at H (for the exterior angle of a parallelogram is equal to the interior opposite angle) ; and the angle at B is half a right angle, for 25ZB is equal to ZA, and the angle at Z is a right angle. And the angle at B is equal to the angle at H; for the angle at Z is a right angle, since the bed is a rectangular figure, one side of which is double the other, and 5. 85618. omdprov should probably be read from the Leid. MS.: see last note. divided into two equal parts; so that BI is 30equal to EH, as also is KO; for it is parallel. So that BI is equal to KO, and TE to ΔΘ. In like manner it can be demonstrated : that all the other pairs of sides which form the bends of the rope are equal to one another. So that clearly there ; are four such lengths of rope as AB in the bed; and there Ἢ is half the number of holes in the half 35ZB that there is in the whole ΖΗ.: So that in the half of the bed there are lengths of rope, such as AB, and they are of the same number as there are holes in BH, or, what comes to the same thing, in AZ, ZB together. But if the rope be strung diagonally, as in the bed ABIA,?
857a
1 ὅσα αἱ πλευραὶ ἀμφοῖν, αἱ ΑΖ ΖΗ· τὰ ἴσα δέ, ὅσα
ἐν τῷ ΖΒΖΑ τρυπήματα ἔνεστιν. μείζονες δέ εἰσιν αἱ ΑΖ
ΒΖ δύο οὖσαι τῆς ΑΒ. ὥστε καὶ τὸ σπαρτίον μεῖζον τοσούτῳ
ὅσον αἱ πλευραὶ ἄμφω μείζους εἰσὶ τῆς διαμέτρου.
1the halves are not of the same length as the sides of both, AZ and ZH; but they are of the same number? as the holes in ZB, ZA. But AZ, ZB, being two, are greater than AB, so that the rope is longer by the amount by which the two sides taken together are greater than the diagonal. νἀ δ
Chapter 26 (857a5–21)
5 Διὰ τί χαλεπώτερον τὰ μακρὰ ξύλα ἀπ' ἄκρου
φέρειν ἐπὶ τῷ ὤμῳ ἢ κατὰ τὸ μέσον, ἴσου τοῦ βάρους ὄντος;
πότερον ὅτι σαλευομένου τοῦ ξύλου τὸ ἄκρον κωλύει φέρειν,
μᾶλλον ἀντισπῶν τῇ σαλεύσει τὴν φοράν; ἢ κἂν
μηθὲν κάμπτηται μηδ' ἔχῃ πολὺ μῆκος, ὅμως χαλεπώτερον
10 φέρειν ἀπ' ἄκρου; ἀλλ' ὅτι καὶ ῥᾷον αἴρεται ἀπ'
ἄκρου ἢ ἐκ μέσου, διὰ τὸ αὐτὸ καὶ φέρειν οὕτω ῥᾴδιον.
αἴτιον δὲ ὅτι ἐκ μέσου μὲν αἰρόμενον ἀεὶ ἐπικουφίζει ἄλληλα
τὰ ἄκρα, καὶ θάτερον μέρος τὸ ἐπὶ θάτερον εὖ αἴρει.
ὥσπερ γὰρ κέντρον γίνεται τὸ μέσον, ᾗ ἔχει τὸ αἶρον ἢ
15 φέρον. εἰς τὸ ἄνω οὖν κουφίζεται ἑκάτερον τῶν ἄκρων εἰς
τὸ κάτω ῥέπον. ἀπὸ δὲ τοῦ ἄκρου αἰρόμενον ἢ φερόμενον οὐ
ποιεῖ τοῦτο, ἀλλ' ἅπαν τὸ βάρος ῥέπει ἐφ' ἓν μέσον, εἰς
ὅπερ αἴρεται ἢ φέρεται. ἔστω μέσον ἐφ' οὗ Α, ἄκρα ΒΓ.
αἰρομένου οὖν ἢ φερομένου κατὰ τὸ Α, τὸ μὲν Β κάτω
20 ῥέπον ἄνω αἴρει τὸ Γ, τὸ δὲ Γ κάτω ῥέπον τὸ Β ἄνω αἴρει·
ἅμα δὲ αἰρόμενα ἄνω ποιεῖ ταῦτα.
5. Why is it more difficult to carry 5a long plank of wood on the shoulder if one holds it at the end than if it is held in the middle, though the weight is the same? Is it because, as the plank vibrates, the end prevents one from carrying it, because it tends to interrupt one’s progress by its vibration? No, for if it does not bend at all and is not very long, it is nevertheless more difficult to carry if it is held at the end. It is easier 10to carry if one holds it in the middle rather than at the end, for the same reason for which it is easier to lift in that way. The reason is that, if one lifts it in the middle, the two ends always lighten one another, and one side lifts the other side up. For the middle, where the lifter or carrier holds it, forms, as it were, the centre, and each of the two ends inclining downwards raises up and 15lightens A B more hopelessly corrupt and unintelligible than those preceding them. 2.85639. A figure, apparently, in which the rope is strung along the diagonals AI and BA and parallel to them on either side. Γ 8 85792, Reading with Capelle the other end; whereas if it is lifted or carried from one end, this effect is not produced, but all the weight inclines in one direction. Let A be the middle of a 20plank which is raised or carried, and let B and I be the extremities. When the go OA [plank is lifted or carried at the B point A, B inclines downwards and raises Γ up, and I inclines downwards and raises B up; the effect is produced by their being raised up at the same moment.
Chapter 27 (857a22–33)
Διὰ τί, ἐὰν ᾖ λίαν μακρὸν τὸ αὐτὸ βάρος, χαλεπώτερον
φέρειν ἐπὶ τοῦ ὤμου, κἂν μέσον φέρῃ τις, ἢ ἐὰν
ἔλαττον ᾖ; πάλαι ἐλέχθη ὡς οὐκ ἔστιν αἴτιον ἡ σάλευσις·
25 ἀλλ' ἡ σάλευσις νῦν αἴτιόν ἐστιν. ὅταν γὰρ ᾖ μακρότερον,
τὰ ἄκρα σαλεύεται, ὥστε εἴη ἂν καὶ τὸν φέροντα χαλεπώτερον
φέρειν μᾶλλον. αἴτιον δὲ τοῦ σαλεύεσθαι μᾶλλον,
ὅτι τῆς αὐτῆς κινήσεως οὔσης μεθίσταται τὰ ἄκρα, ὅσῳπερ
ἂν ᾖ μακρότερον τὸ ξύλον. ὁ μὲν γὰρ ὦμος κέντρον, ἐφ'
30 οὗ τὸ Α (μένει γὰρ τοῦτο), αἱ δὲ ΑΒ καὶ ΑΓ αἱ ἐκ τοῦ
κέντρου. ὅσῳ δ' ἂν ᾖ μεῖζον τὸ ἐκ τοῦ κέντρου ἢ τὸ ΑΒ
ἢ καὶ τὸ ΑΓ, πλέον μεθίσταται μέγεθος. δέδεικται δὲ
τοῦτο πρότερον.
Why is a very long object more difficult to carry on the shoulder, even if one carries it in the middle, than a shorter object 25of the same weight? In the last case we said that the vibration was not the reason; in this case it is the reason. For the longer an object is, the more its extremities vibrate, and so it would be more difficult for the man to carry it. The reason of the ee ee increased vibration 185 that, B - Γ though the movement is the ἘΠῚ τ. same, the extremities change their position more the longer the piece of 30wood is. Let the shoulder, which is the centre (for it is at rest), be at A, and let AB and AT be the radii; then the longer the radius AB or AT is, the greater is the amplitude of movement. This point has already been demonstrated.+ [Ὁ
Chapter 28 (857a34–857b8)
Διὰ τί ἐπὶ τοῖς φρέασι τὰ κηλώνεια ποιοῦσι τοῦτον τὸν
35 τρόπον; προστιθέασι γὰρ βάρος ἐν τῷ ξύλῳ τὸν μόλιβδον,
ὄντος βάρους τοῦ κάδου αὐτοῦ, καὶ κενοῦ καὶ πλήρους ὄντος.
ἢ ὅτι ἐν δυσὶ χρόνοις διῃρημένου τοῦ ἔργου (βάψαι γὰρ δεῖ,
καὶ τοῦτ' ἄνω ἑλκύσαι) συμβαίνει καθιέναι μὲν κενὸν ῥᾳδίως,
Why do they construct ‘ swipes’ by the side of wells by attaching the lead as a weight at the end of the bar, the bucket being itself a weight, whether it is empty or 35full? Is the reason that, the drawing of water being divided into two operations distinct in time (for the bucket has to be dipped and then drawn up), it is an easy task to let it down when it is empty, but difficult to raise it when it is full?
857b
1 αἴρειν δὲ πλήρη χαλεπῶς; λυσιτελεῖ οὖν μικρῷ βραδύτερον
εἶναι τὸ καταγαγεῖν πρὸς τὸ πολὺ κουφίσαι τὸ
βάρος ἀνάγοντι. τοῦτο οὖν ποιεῖ ἐπ' ἄκρῳ τῷ κηλωνείῳ ὁ
μόλιβδος προσκείμενος ἢ ὁ λίθος. καθιμῶντι μὲν γὰρ γίνεται
5 βάρος μεῖζον ἢ εἰ μόνον κενὸν δεῖ κατάγειν τὸν κάδον·
ὅταν δὲ πλήρης ᾖ, ἀνάγει ὁ μόλιβδος, ἢ ὅ τι ἂν ᾖ
τὸ προσκείμενον βάρος. ὥστ' ἐστὶ ῥᾷον αὐτῷ τὰ ἄμφω
ἢ ἐκείνῳ.
1It is therefore of advantage to lower it rather more 857» slowly with a view to lightening the weight considerably when it is drawn up again. This effect is produced by the lead or stone attached to the end of the swipe. In letting it down there is a heavier weight to lift than if one 5has merely to lower the empty bucket; but when it is full, the lead, or whatever the weight attached is, helps to draw it up; and so the two operations taken together are easier than on the other method.
Chapter 29 (857b9–20)
Διὰ τί, ὅταν φέρωσιν ἐπὶ ξύλου ἤ τινος τοιούτου δύο
10 ἄνθρωποι ἴσον βάρος, οὐχ ὁμοίως θλίβονται, ἐὰν μὴ ἐπὶ
τῷ μέσῳ ᾖ τὸ βάρος, ἀλλὰ μᾶλλον ὅσῳ ἂν ἐγγύτερον ᾖ
τῶν φερόντων; ἢ διότι μοχλὸς μὲν γίνεται οὕτως ἐχόντων
τὸ ξύλον, τὸ δὲ βάρος ὑπομόχλιον, ὁ δὲ ἐγγύτερος τοῦ
βάρους τῶν φερόντων τὸ βάρος τὸ κινούμενον, ἅτερος δὲ
15 τῶν φερόντων τὸ βάρος ὁ κινῶν. ὅσῳ γὰρ πλέον ἀπέχει τοῦ
βάρους, τοσούτῳ ῥᾷον κινεῖ, καὶ θλίβει μᾶλλον τὸν ἕτερον
εἰς τὸ κάτω, ὥσπερ ἀντερείδοντος τοῦ βάρους τοῦ ἐπικειμένου
καὶ γινομένου ὑπομοχλίου. ἐν μέσῳ δὲ ὑποκειμένου τοῦ
βάρους, οὐδὲν μᾶλλον ἅτερος θατέρῳ γίνεται βάρος, οὐδὲ
20 κινεῖ, ἀλλ' ὁμοίως ἑκάτερος ἑκατέρῳ γίνεται βάρος.
Why is it that when two men are carrying an equal το weight on a piece of wood -or something of the kind, the pressure on them is not equal unless the 10weight is in the middle, but it presses more on the person carrying it to. whom it is nearest? Is it because the wood, when they hold it in this way, becomes a lever, and the load forms the fulcrum, and the carrier nearer to the load becomes the weight which is to be moved, while the other carrier ἐ becomes the mover of the weight? The further the latter is 15from the weight, the more easily he moves it, and the ᾿ more he presses down the other man, since the load placed on the wood and acting as a fulcrum, as it were, offers ἦ resistance. But if the load is placed in the middle, one carrier does not act as a weight on the other any more than the other on him, or exercise any motive forcé upon him, but each 20is equally a weight upon the other.
Chapter 30 (857b21–858a2)
Διὰ τί οἱ ἀνιστάμενοι πάντες πρὸς ὀξεῖαν γωνίαν τῷ
μηρῷ ποιήσαντες τὴν κνήμην ἀνίστανται, καὶ τῷ θώρακι
πρὸς τὸν μηρόν; εἰ δὲ μή, οὐκ ἂν δύναιντο ἀναστῆναι. πότερον
ὅτι τὸ ἴσον ἠρεμίας πανταχοῦ αἴτιον, ἡ δὲ ὀρθὴ γωνία
25 τοῦ ἴσου, καὶ ποιεῖ στάσιν· διὸ καὶ φέρεται πρὸς ὁμοίας
γωνίας τῇ περιφερείᾳ τῆς γῆς. οὐ γὰρ ὅτι καὶ πρὸς ὀρθὴν
ἔσται τῷ ἐπιπέδῳ. ἢ ὅτι ἀνιστάμενος γίνεται ὀρθός, ἀνάγκη
δὲ τὸν ἑστῶτα κάθετον εἶναι πρὸς τὴν γῆν. εἰ οὖν μέλλει
ἔσεσθαι πρὸς ὀρθήν, τοῦτο δέ ἐστι τὸ τὴν κεφαλὴν ἔχειν
30 κατὰ τοὺς πόδας, καὶ γίνεσθαι δὴ ὅτε ἀνίσταται. ὅταν μὲν
οὖν καθήμενος ᾖ, παράλληλον ἔχει τὴν κεφαλὴν καὶ τοὺς
πόδας, καὶ οὐκ ἐπὶ μιᾶς εὐθείας. ἡ κεφαλὴ Α ἔστω, θώραξ
ΑΒ, μηρὸς ΒΓ, κνήμη ΓΔ. πρὸς ὀρθὴν δὲ γίνεται
ὅ τε θώραξ [ἐφ' ὧν ΑΒ] τῷ μηρῷ καὶ ὁ μηρὸς τῇ κνήμῃ
35 οὕτως καθημένῳ. ὥστε οὕτως ἔχοντα ἀδύνατον ἀναστῆναι.
ἀνάγκη δὲ ἐγκλῖναι τὴν κνήμην καὶ ποιεῖν τοὺς πόδας ὑπὸ
τὴν κεφαλήν. τοῦτο δὲ ἔσται, ἐὰν ἡ ΓΔ ἐφ' ἧς τὰ ΓΖ
γένηται, καὶ ἅμα ἀναστῆναι συμβήσεται, καὶ ἔχειν ἐπὶ
Why is it that when people rise from a sitting position, they always do so by making an acute angle between the thigh and the lower leg and between the chest and the thigh, otherwise they cannot rise? Is it because equality is always a cause of rest, and a right angle causes an equality! and so causes equilibrium? So in 25rising a man moves towards a position at equal angles to the earth’s circumference; for it is not the case that he will actually be at right angles to the ground. Or is it because when a man rises he tends to become upright, and a man who is standing must be perpendicular to the ground? If, then, he is to be at right angles to the ground, that means that 30he must have his head in the same line as his feet, and this occurs when he is rising. As long, then, as he is sitting, he keeps his feet and head parallel to one another and not in the same straight line. Let A be the head, A AB the line of the chest, ΒΓ the thigh, and ΓΔ the lower leg. Then AB, the line of the chest, is at right angles to the thigh, and 35the thigh at right angles to the lower. B a leg, when a man is seated in this way. In this position, then, a man cannot rise; but to do so he must bend the leg and Z A place the feet at a point under the head. Bic.
858a
1 τῆς αὐτῆς ἴσης τὴν κεφαλήν τε καὶ τοὺς πόδας. ἡ δὲ ΓΖ
ὀξεῖαν ποιεῖ γωνίαν πρὸς τὴν ΒΓ.
1iy: This will be the case if ΓΔ be moved to ΓΖ, and the result will be that he can rise immediately, and he will have his head and his feet in 8585 the same straight line;+ and ΓΖ will form an acute angle with ΒΓ,
Chapter 31 (858a3–12)
Διὰ τί ῥᾷον κινεῖται τὸ κινούμενον ἢ τὸ μένον, οἷον
τὰς ἁμάξας θᾶττον κινουμένας ὑπάγουσιν ἢ ἀρχομένας;
5 ἢ ὅτι χαλεπώτατον μὲν τὸ εἰς τοὐναντίον κινούμενον κινῆσαι
βάρος; ἀφαιρεῖται γάρ τι τῆς τοῦ κινοῦντος δυνάμεως, κἂν
πολὺ θᾶττον ᾖ· ἀνάγκη γὰρ βραδυτέραν γίνεσθαι τὴν ὦσιν
τοῦ ἀντωθουμένου. δεύτερον δέ, ἐὰν ἠρεμῇ· ἀντιτείνει γὰρ καὶ
τὸ ἠρεμοῦν. τὸ δὲ κινούμενον ἐπὶ τὸ αὐτὸ τῷ ὠθοῦντι ὅμοιον
10 ποιεῖ ὥσπερ ἂν εἰ αὐξήσειέ τις τὴν τοῦ κινοῦντος δύναμιν
καὶ ταχυτῆτα· ὃ γὰρ ὑπ' ἐκείνου ἂν ἔπασχε, τοῦτο αὐτὸ
ποιεῖ εἰς τὸ πρὸ ὁδοῦ κινούμενον.
Why is it that a body which is already in motion is easier to move than one which is at rest? For example, a wagon which is in 5motion can be propelled more quickly than one which has to be started. Is it because, in the first place, it is very difficult to move in one direction a weight which is already moving in the opposite direction? For though the motive force may be much quicker, yet some of it is lost; for the propulsion exerted by that which is being pushed in the opposite direction must necessarily become slower. And so, secondly, the propulsion 10must be slower if the body is at rest; for even that which is at rest offers resistance. When a body is moving in the same direction as that which pushes it, the effect is just τὸ as if one increased the force and speed of the motive power ; for by moving forward it produces of itself exactly the effect which that power would have upon it.
Chapter 32 (858a13–16)
Διὰ τί παύεται φερόμενα τὰ ῥιφέντα; πότερον ὅταν
λήγῃ ἡ ἰσχὺς ἡ ἀφεῖσα, ἢ διὰ τὸ ἀντισπᾶσθαι, ἢ διὰ
15 τὴν ῥοπήν, ἐὰν κρείττων ᾖ τῆς ἰσχύος τῆς ῥιψάσης; ἢ ἄτοπον
τὸ ταῦτ' ἀπορεῖν, ἀφέντα τὴν ἀρχήν.
Why is it that an object which is thrown eventually comes to a standstill? Does it stop 15when the force which 8585 at started it fails, or because the object is drawn in a contrary direction, or is it due to its downward tendency, which is stronger than the force which threw it? Or is it absurd to discuss such questions, while the principle escapes us?
Chapter 33 (858a17–22)
Διὰ τί φέρεταί τι οὐ τὴν αὑτοῦ φοράν, μὴ ἀκολουθοῦντος
καὶ ὠθοῦντος τοῦ ἀφέντος; ἢ δῆλον ὅτι ἐποίησε τοιοῦτον
τὸ πρῶτον ὡς θάτερον ὠθεῖν, καὶ τοῦθ' ἕτερον· παύεται δέ,
20 ὅταν μηκέτι δύνηται ποιεῖν τὸ προωθοῦν τὸ φερόμενον ὥστε
ὠθεῖν, καὶ ὅταν τὸ τοῦ φερομένου βάρος ῥέπῃ μᾶλλον τῆς
εἰς τὸ πρόσθεν δυνάμεως τοῦ ὠθοῦντος.
How is it that a body is carried along by a motion not its own, if that which started it does not keep following and pushing it along? Is it not clear that in 20the beginning the impelling force so acted as to push one thing along, and this in its turn pushes along something else? 4+ The moving body comes to a standstill when the force which pushes it along can no longer so act as to push it, and when the weight of the moving object has a stronger inclination downwards than the forward force of that which pushes it.
Chapter 34 (858a23–858b3)
Διὰ τί οὔτε τὰ ἐλάττονα οὔτε τὰ μεγάλα πόρρω φέρεται
ῥιπτούμενα, ἀλλὰ δεῖ συμμετρίαν τινὰ ἔχειν πρὸς
25 τὸν ῥιπτοῦντα; πότερον ὅτι ἀνάγκη τὸ ῥιπτούμενον καὶ
ὠθούμενον ἀντερείδειν ὅθεν ὠθεῖται; τὸ δὲ μηθὲν ὑπεῖκον διὰ
μέγεθος ἢ μηδὲν ἀντερεῖσαν δι' ἀσθένειαν οὐ ποιεῖ ῥῖψιν
οὐδὲ ὦσιν. τὸ μὲν οὖν πολὺ ὑπερβάλλον τῆς ἰσχύος τῆς
ὠθούσης οὐθὲν ὑπείκει, τὸ δὲ πολὺ ἀσθενέστερον οὐδὲν ἀνερείδει.
30 ἢ ὅτι τοσοῦτον φέρεται τὸ φερόμενον, ὅσον ἂν
ἀέρα κινήσῃ εἰς βάθος; τὸ δὲ μηδὲν κινούμενον οὐδ' ἂν
κινήσειεν οὐδέν. συμβαίνει δὴ ἀμφότερα τούτοις ἔχειν.
Why is it that neither small nor large bodies travel far when thrown, 25but they must have due relation to the person who throws them? Is it because that which is thrown or pushed must offer resistance to that from which it is pushed, and whatever does not yield owing to its mass, or does not resist owing to its weakness, does not admit of being thrown or pushed? A body, then, which is far beyond the force which tries to push it, does not yield at all; while that which is far weaker offers no 30resistance. 3o Or is it because that which travels along does so only as far as it moves the air to its depths, and that which is not moved cannot itself move anything either?
858b
1 τό τε γὰρ σφόδρα μέγα καὶ τὸ σφόδρα μικρὸν ὥσπερ οὐθὲν
κινούμενά ἐστι· τὸ μὲν γὰρ αὐτὸ καθ' ἓν κινεῖ, τὸ δ'
οὐθὲν κινεῖται.
1Both these things are the case here; that which is very large and that which is very small must be looked upon as not moving at all; for the latter does not move anything, while the former is not itself at all moved.
Chapter 35 (858b4–31)
Διὰ τί τὰ φερόμενα ἐν τῷ δινουμένῳ ὕδατι εἰς τὸ
5 μέσον τελευτῶντα φέρονται ἅπαντα; πότερον ὅτι μέγεθος
ἔχει τὸ φερόμενον, ὥστε ἐν δυσὶ κύκλοις εἶναι, τῷ μὲν
ἐλάττονι τῷ δὲ μείζονι, ἑκάτερον αὐτοῦ τῶν ἄκρων. ὥστε
περισπᾷ ὁ μείζων διὰ τὸ φέρεσθαι θᾶττον, καὶ πλάγιον
ἀπωθεῖ αὐτὸ εἰς τὸν ἐλάττω. ἐπεὶ δὲ πλάτος ἔχει τὸ
10 φερόμενον, καὶ οὗτος πάλιν τὸ αὐτὸ ποιεῖ, καὶ ἀπωθεῖ εἰς
τὸν ἐντός, ἕως ἂν εἰς τὸ μέσον ἔλθῃ. καὶ τότε μένει διὰ
τὸ ὁμοίως ἔχειν πρὸς ἅπαντας τοὺς κύκλους τὸ φερόμενον,
διὰ τὸ μέσον· καὶ γὰρ τὸ μέσον ἴσον ἀπέχει ἐν ἑκάστῳ
τῶν κύκλων. ἢ ὅτι ὅσων μὲν μὴ κρατεῖ ἡ φορὰ τοῦ δινουμένου
15 ὕδατος διὰ τὸ μέγεθος, ἀλλ' ὑπερέχει τῇ βαρύτητι
τῆς τοῦ κύκλου ταχυτῆτος, ἀνάγκη ὑπολείπεσθαι καὶ βραδύτερον
φέρεσθαι. βραδύτερον δὲ ὁ ἐλάττων κύκλος φέρεται·
τὸ αὐτὸ γὰρ ἐν ἴσῳ χρόνῳ ὁ μέγας τῷ μικρῷ στρέφεται
κύκλῳ, ὅταν ὦσι περὶ τὸ αὐτὸ μέσον. ὥστε εἰς τὸν
20 ἐλάττονα κύκλον ἀναγκαῖον ἀπολείπεσθαι, ἕως ἂν ἐπὶ τὸ
μέσον ἔλθῃ. ὅσων δὲ πρότερον κρατεῖ ἡ φορά, λήγουσα
ταὐτὸ ποιήσει. δεῖ γὰρ τὸν μὲν εὐθύ, τὸν δὲ ἕτερον κρατεῖν
τῇ ταχυτῆτι τοῦ βάρους, ὥστε εἰς τὸν ἐντὸς ἀεὶ κύκλον
ὑπολείπεσθαι πᾶν. ἀνάγκη γὰρ αὐτὸ ἐντὸς ἢ ἐκτὸς κινεῖσθαι
25 τὸ μὴ κρατούμενον. ἐν αὐτῷ δὴ τοίνυν ἐν ᾧ ἐστίν,
ἀδύνατον φέρεσθαι τὸ μὴ κρατούμενον. ἔτι δὲ ἧττον ἐν τῷ
ἐκτός· θάττων γὰρ ἡ φορὰ τοῦ ἐκτὸς κύκλου. λείπεται δὲ
εἰς τὸν ἐντὸς τὸ μὴ κρατούμενον μεθίστασθαι. ἀεὶ δὲ ἕκαστον
ἐπιδίδωσιν εἰς τὸ μὴ κρατεῖσθαι. ἐπεὶ δὲ πέρας τοῦ μὴ κινεῖσθαι
30 ποιεῖ τὸ εἰς μέσον ἐλθεῖν, μένει δὲ τὸ κέντρον μόνον,
ἅπαντα ἀνάγκη εἰς τοῦτο δὴ ἀθροίζεσθαι.
Why is it that an object which is carried round in whirling water is always eventually carried into the middle? Is it because the object has magnitude, so that it has position in two circles, one of its extremities revolving in a greater and the other in a lesser circle? The greater circle, then, on account of its greater velocity, draws it round 5and thrusts it sideways into the lesser circle; but since the object has breadth, the lesser circle in its turn does the same thing and thrusts it into the next interior circle, until it reaches the centre. Here the object remains because it stands in the same relation to all the circles, being in the middle; for the middle is equidistant from the circumference in the case of each of the circles. Or is it because an object which, owing to its magnitude, the motion of the whirling water cannot overcome, but which by its weight prevails over the velocity of the revolving circle, must necessarily be left behind and travel along more slowly? Now the lesser circle travels more slowly—for the greater 10and the lesser circle do not! revolve over the same space in an equal time when they move round the same centre—and so the object must be left revolving in a lesser and lesser circle until it reaches the middle. If the force of the whirling water prevails at first, it will go on doing so to the end; for one circle must prevail and then the next over the weight of the object owing to their velocity, so that the whole object is continually being left behind in the next circle towards the centre. For an object over which the water does not prevail must be carried either inwards or outwards. Such an object cannot then be carried along in its original position; still less can it be carried along in the 15outer circle, for the velocity of the outer circle is greater. The only alternative is that the object over which the water does not prevail is transferred to the inner circle. Now every object has a tendency to resist force; but since the arrival at the middle puts an end to motion, and the centre alone is at rest, all objects must necessarily collect there. Io + ο bs b uo 1908 HENRY FROWDE, M.A. PUBLISHER TO THE UNIVERSITY OF OXFORD LONDON, EDINBURGH Text of Aristotle, is to a large extent unintelligible. But M. Hayduck, in the valuable paper which he contributed to the Veue Fahrbiicher fiir Philologie und Paedagogtk (vol. 109, part I, Teubner, 1874), prepared the way; and Otto Apelt, profiting by 20Hayduck’s labours and by a fresh collation of the manuscripts, published a more satisfactory text in his volume Aristotelis quae feruntur de Plantis, &c. (Teubner, 1888). Many of the most difficult passages are discussed and elucidated in the prolegomena to this volume. Finally, Apelt included a German translation of the treatise in his Beitrage zur Geschichte der griechischen Philosophie (Teubner, 1891). In the following paraphrase, I have endeavoured to make a full use of the work of Hayduck and Apelt, with a view to reproducing the subtle and somewhat intricate thought of the author, whoever he may have been. Though the treatise is published amongst the works of Aristotle, there are grounds for 25ascribing it to Theophrastus: whilst, for all we can tell, it may have been written neither by Aristotle nor by Theophrastus, but by Strato, or possibly by some one otherwise unknown. But the work—no matter who wrote it—is interesting for the close texture of its reasoning, and for the light which it throws on certain obscure places in Plato and Aristotle. Its value for the student of the History of Mathematics is no doubt considerable: but my own ignorance of this subject makes me hesitate to express an opinion. I take this opportunity of thanking three of my friends, E. I. Carlyle (Fellow of Lincoln College) and A. L. Dixon (Fellow of Merton College) for their help in several of the mathematical 30passages, and W. D. Ross (Fellow of Oriel College) for his valuable suggestions, most of which I have _ adopted. Hi HJ. January, 1908, a ‘ee