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<?xml version="1.0"?>
<archimedes xmlns:xlink="http://www.w3.org/1999/xlink" >
<info>
    <author>Pseudo-Aristotle</author>
    <title>Problemata Mechanica</title> 
    <date>1831</date>
    <place>Berlin</place>
    <translator></translator>
    <lang>el</lang>
    <cvs_file>arist_mecha_080_el_1831.xml</cvs_file>
    <cvs_version></cvs_version>
    <locator>080.xml</locator>
</info>
<text>
<front></front>
<body>
<chap>
<p n="1">

<s id="g0110101"><pb xlink:href="080/01/001.jpg" ed="Bekker" n="847a"></pb>Θαυμάζεται τῶν μὲν κατὰ φύσιν συμβαινόντων, ὅσων  <lb></lb>ἀγνοεῖται τὸ αἴτιον, τῶν δὲ παρὰ φύσιν, ὅσα γίνεται διὰ <lb></lb>τέχνην πρὸς τὸ συμφέρον τοῖς ἀνθρώποις.</s>
<s id="g0110102">ἐν πολλοῖς γὰρ <lb></lb>ἡ φύσις ὑπεναντίον πρὸς τὸ χρήσιμον ἡμῖν ποιεῖ· </s>
<s id="g0110103">ἡ μὲν <lb></lb>γὰρ φύσις ἀεὶ τὸν αὐτὸν ἔχει τρόπον καὶ ἁπλῶς, τὸ δὲ <lb></lb>χρήσιμον μεταβάλλει πολλαχῶς.</s>
</p><p n="2">
<s id="g0110201">ὅταν οὖν δέῃ τι παρὰ <lb></lb>φύσιν πρᾶξαι, διὰ τὸ χαλεπὸν ἀπορίαν παρέχει καὶ δεῖται <lb></lb>τέχνης. </s>
<s id="g0110201a">διὸ καὶ καλοῦμεν τῆς τέχνης τὸ πρὸς τὰς τοιαύτας <lb></lb>ἀπορίας βοηθοῦν μέρος μηχανήν.</s>
<s id="g0110202">καθάπερ γὰρ ἐποίησεν <lb></lb>Ἀντιφῶν ὁ ποιητής, οὕτω καὶ ἔχει· τέχνῃ γὰρ κρατοῦμεν, <lb></lb>ὧν φύσει νικώμεθα.</s>
<s id="g0110203">τοιαῦτα δέ ἐστιν ἐν οἷς τά τε ἐλάττονα <lb></lb>κρατεῖ τῶν μειζόνων, καὶ τὰ ῥοπὴν ἔχοντα μικρὰν κινεῖ <lb></lb>βάρη μεγάλα, καὶ πάντα σχεδὸν ὅσα τῶν προβλημάτων <lb></lb>μηχανικὰ προσαγορεύομεν.</s>
</p><p n="3">
<s id="g0110301">ἔστι δὲ ταῦτα τοῖς φυσικοῖς <lb></lb>προβλήμασιν οὔτε ταὐτὰ πάμπαν οὔτε κεχωρισμένα λίαν, <lb></lb>ἀλλὰ κοινὰ τῶν τε μαθηματικῶν θεωρημάτων καὶ τῶν <lb></lb>φυσικῶν· τὸ μὲν γὰρ ὣς διὰ τῶν μαθηματικῶν δῆλον, τὸ <lb></lb>δὲ περὶ ὃ διὰ τῶν φυσικῶν.</s>
</p><p n="4">
<s id="g0120101">περιέχεται δὲ τῶν ἀπορουμένων<pb xlink:href="080/01/002.jpg" ed="Bekker" n="847b"></pb><lb></lb> ἐν τῷ γένει τούτῳ τὰ περὶ τὸν μοχλόν.</s>
<s id="g0120102">ἄτοπον γὰρ <lb></lb>εἶναι δοκεῖ τὸ κινεῖσθαι μέγα βάρος ὑπὸ μικρᾶς ἰσχύος, <lb></lb>καὶ ταῦτα μετὰ βάρους πλείονος· ὃ γὰρ ἄνευ μοχλοῦ κινεῖν <lb></lb>οὐ δύναταί τις, τοῦτο ταὐτὸ βάρος, προσλαβὼν ἔτι τὸ <lb></lb>τοῦ μοχλοῦ βάρος, κινεῖ θᾶττον.</s>
<s id="g0120103">πάντων δὲ τῶν τοιούτων <lb></lb>ἔχει τῆς αἰτίας τὴν ἀρχὴν ὁ κύκλος.</s>
<s id="g0120104">καὶ τοῦτο εὐλόγως <lb></lb>συμβέβηκεν· ἐκ μὲν γὰρ θαυμασιωτέρου συμβαίνειν τι <lb></lb>θαυμαστὸν οὐδὲν ἄτοπον, </s>
</p><p n="5">
<s id="g0120201">θαυμασιώτατον δὲ τὸ τἀναντία <lb></lb>γίνεσθαι μετ&#039; ἀλλήλων.</s>
<s id="g0120202">ὁ δὲ κύκλος συνέστηκεν ἐκ τοιούτων· </s>
<s id="g0120203"><lb></lb>εὐθὺς γὰρ ἐκ κινουμένου τε γεγένηται καὶ μένοντος, ὧν ἡ <lb></lb>φύσις ἐστὶν ὑπεναντία ἀλλήλοις. ὥστ&#039; ἐνταῦθα ἔστιν ἐπιβλέψασιν <lb></lb>ἧττον θαυμάζειν τὰς συμβαινούσας ὑπεναντιώσεις <lb></lb>περὶ αὐτόν.</s>
<s id="g0120204">πρῶτον μὲν γὰρ τῇ περιεχούσῃ γραμμῇ τὸν <lb></lb>κύκλον, πλάτος οὐθὲν ἐχούσῃ, τἀναντία πως προσεμφαίνεται, <lb></lb>τὸ κοῖλον καὶ τὸ κυρτόν.</s>
<s id="g0120205">ταῦτα δὲ διέστηκεν ἀλλήλων <lb></lb>ὃν τρόπον τὸ μέγα καὶ τὸ μικρόν· ἐκείνων τε γὰρ <lb></lb>μέσον τὸ ἴσον καὶ τούτων τὸ εὐθύ. διὸ μεταβάλλοντα εἰς <lb></lb>ἄλληλα τὰ μὲν ἀναγκαῖον ἴσα γενέσθαι πρότερον ἢ τῶν<pb xlink:href="080/01/003.jpg" ed="Bekker" n="848a"></pb><lb></lb> ἄκρων ὁποτερονοῦν, τὴν δὲ γραμμὴν εὐθεῖαν, ὅταν ἐκ κυρτῆς <lb></lb>εἰς κοῖλον ἢ πάλιν ἐκ ταύτης γίνηται κυρτὴ καὶ περιφερής. <lb></lb>ἓν μὲν οὖν τοῦτο τῶν ἀτόπων ὑπάρχει περὶ τὸν κύκλον,</s>
</p><p n="6">
<s id="g0120301"><lb></lb>δεύτερον δὲ ὅτι ἅμα κινεῖται τὰς ἐναντίας κινήσεις· <lb></lb>ἅμα γὰρ εἰς τὸν ἔμπροσθεν κινεῖται τόπον καὶ τὸν ὄπισθεν.</s>
<s id="g0120302"><lb></lb>ἥ τε γράφουσα γραμμὴ τὸν κύκλον ὡσαύτως ἔχει· ἐξ <lb></lb>οὗ γὰρ ἄρχεται τόπου τὸ πέρας αὐτῆς, εἰς τὸν αὐτὸν τοῦτον τόπον <lb></lb>ἔρχεται πάλιν· συνεχῶς γὰρ κινουμένης αὐτῆς τὸ ἔσχατον <lb></lb>πάλιν ἀπῆλθε πρῶτον, ὥστε καὶ φανερὸν ὅτι μετέβαλεν <lb></lb>ἐντεῦθεν.</s>
<s id="g0120303">διό, καθάπερ εἴρηται πρότερον, οὐδὲν ἄτοπον τὸ <lb></lb>πάντων εἶναι τῶν θαυμάτων αὐτὸν ἀρχήν.</s></p><p n="7">
<s id="g0120401">τὰ μὲν οὖν περὶ <lb></lb>τὸν ζυγὸν γινόμενα εἰς τὸν κύκλον ἀνάγεται, τὰ δὲ περὶ <lb></lb>τὸν μοχλὸν εἰς τὸν ζυγόν, τὰ δ&#039; ἄλλα πάντα σχεδὸν τὰ <lb></lb>περὶ τὰς κινήσεις τὰς μηχανικὰς εἰς τὸν μοχλόν.</s>
<s id="g0120402">ἔτι δὲ <lb></lb>διὰ τὸ μιᾶς οὔσης τῆς ἐκ τοῦ κέντρου γραμμῆς μηθὲν ἕτερον <lb></lb>ἑτέρῳ φέρεσθαι τῶν σημείων τῶν ἐν αὐτῇ ἰσοταχῶς, ἀλλ&#039; ἀεὶ <lb></lb>τὸ τοῦ μένοντος πέρατος πορρώτερον ὂν θᾶττον, πολλὰ τῶν θαυμαζομένων <lb></lb>συμβαίνει περὶ τὰς κινήσεις τῶν κύκλων· περὶ <lb></lb>ὧν ἐν τοῖς ἑπομένοις προβλήμασιν ἔσται δῆλον.</s></p><p n="8">
<s id="g0120501">διὰ δὲ τὸ <lb></lb>τὰς ἐναντίας κινήσεις ἅμα κινεῖσθαι τὸν κύκλον, καὶ τὸ <lb></lb>μὲν ἕτερον τῆς διαμέτρου τῶν ἄκρων, ἐφ&#039; οὗ τὸ Α, εἰς τοὔμπροσθεν <lb></lb>κινεῖσθαι, θάτερον δέ, ἐφ&#039; οὗ τὸ Β, εἰς τοὔπισθεν, <lb></lb>κατασκευάζουσί τινες ὥστ&#039; ἀπὸ μιᾶς κινήσεως πολλοὺς ὑπεναντίους <lb></lb>ἅμα κινεῖσθαι κύκλους, ὥσπερ οὓς ἀνατιθέασιν ἐν <lb></lb>τοῖς ἱεροῖς ποιήσαντες τροχίσκους χαλκοῦς τε καὶ σιδηροῦς.</s>
<s id="g0120502"><lb></lb>εἰ γὰρ εἴη τοῦ ΑΒ κύκλου ἁπτόμενος ἕτερος κύκλος ἐφ&#039; οὗ <lb></lb>ΓΔ, τοῦ κύκλου τοῦ ἐφ&#039; οὗ ΑΒ κινουμένης τῆς διαμέτρου <lb></lb>εἰς τοὔμπροσθεν, κινηθήσεται ἡ ΓΔ εἰς τοὔπισθεν τοῦ κύκλου <lb></lb>τοῦ ἐφ&#039; οὗ Α, κινουμένης τῆς διαμέτρου περὶ τὸ αὐτό.</s>
<s id="g0120503">εἰς <lb></lb>τοὐναντίον ἄρα κινηθήσεται ὁ ἐφ&#039; οὗ ΓΔ κύκλος τῷ ἐφ&#039; <lb></lb>οὗ τὸ ΑΒ· καὶ πάλιν αὐτὸς τὸν ἐφεξῆς, ἐφ&#039; οὗ ΕΖ, εἰς <lb></lb>τοὐναντίον αὑτῷ κινήσει διὰ τὴν αὐτὴν αἰτίαν.</s>
<figure id="id.080.01.003.1.jpg" xlink:href="080/01/003/1.jpg"></figure></p><p n="9">
<s id="g0120601">τὸν αὐτὸν δὲ <lb></lb>τρόπον κἂν πλείους ὦσι, τοῦτο ποιήσουσιν ἑνὸς μόνου κινηθέντος.</s>
<s id="g0120602"><lb></lb>ταύτην οὖν λαβόντες ὑπάρχουσαν ἐν τῷ κύκλῳ τὴν <lb></lb>φύσιν οἱ δημιουργοὶ κατασκευάζουσιν ὄργανον κρύπτοντες <lb></lb>τὴν ἀρχήν, ὅπως ᾖ τοῦ μηχανήματος φανερὸν μόνον τὸ <lb></lb>θαυμαστόν, τὸ δ&#039; αἴτιον ἄδηλον. <pb xlink:href="080/01/004.jpg" ed="Bekker" n="848b"></pb><lb></lb></s></p><p n="10">
<s id="g0120701prop01">Πρῶτον μὲν οὖν τὰ συμβαίνοντα περὶ τὸν ζυγὸν ἀπορεῖται, <lb></lb>διὰ τίνα αἰτίαν ἀκριβέστερά ἐστι τὰ ζυγὰ τὰ μείζω <lb></lb>τῶν ἐλαττόνων.</s>
<s id="g0120702">τούτου δὲ ἀρχή, διὰ τί ποτε ἐν τῷ κύκλῳ <lb></lb>ἡ πλεῖον ἀφεστηκυῖα γραμμὴ τοῦ κέντρου τῆς ἐγγὺς τῇ <lb></lb>αὐτῇ ἰσχύι κινουμένης θᾶττον φέρεται τῆς ἐλάττονοσ</s>
<s id="g0120703">τὸ <lb></lb>γὰρ θᾶττον λέγεται διχῶς·</s>
<s id="g0120704">ἄν τε γὰρ ἐν ἐλάττονι χρόνῳ <lb></lb>ἴσον τόπον διεξέλθῃ, θᾶττον εἶναι λέγομεν, καὶ ἐὰν ἐν ἴσῳ <lb></lb>πλείω.</s>
<s id="g0120705">ἡ δὲ μείζων ἐν ἴσῳ χρόνῳ γράφει μείζονα κύκλον· <lb></lb>ὁ γὰρ ἐκτὸς μείζων τοῦ ἐντός.</s>
<s id="g0120706">αἴτιον δὲ τούτων ὅτι φέρεται <lb></lb>δύο φορὰς ἡ γράφουσα τὸν κύκλον.</s>
<s id="g0120707">ὅταν μὲν οὖν ἐν λόγῳ <lb></lb>τινὶ φέρηται, ἐπ&#039; εὐθείας ἀνάγκη φέρεσθαι τὸ φερόμενον, <lb></lb>καὶ γίνεται διάμετρος αὐτὴ τοῦ σχήματος ὃ ποιοῦσιν αἱ <lb></lb>ἐν τούτῳ τῷ λόγῳ συντεθεῖσαι γραμμαί.</s>
<s id="g0120708">ἔστω γὰρ ὁ λόγος <lb></lb>ὃν φέρεται τὸ φερόμενον, ὃν ἔχει ἡ ΑΒ πρὸς τὴν ΑΓ· <lb></lb>καὶ τὸ μὲν ΑΓ φερέσθω πρὸς τὸ Β, ἡ δὲ ΑΒ ὑποφερέσθω <lb></lb>πρὸς τὴν ΗΓ· ἐνηνέχθω δὲ τὸ μὲν Α πρὸς τὸ Δ, ἡ δὲ ἐφ&#039; <lb></lb>ᾗ ΑΒ πρὸς τὸ Ε. εἰ οὖν ἐπὶ τῆς φορᾶς ὁ λόγος ἦν ὃν ἡ <lb></lb>ΑΒ ἔχει πρὸς τὴν ΑΓ, ἀνάγκη καὶ τὴν ΑΔ πρὸς τὴν <lb></lb>ΑΕ τοῦτον ἔχειν τὸν λόγον.</s>
<s id="g0120709">ὅμοιον ἄρα ἐστὶ τῷ λόγῳ τὸ <lb></lb>μικρὸν τετράπλευρον τῷ μείζονι, ὥστε καὶ ἡ αὐτὴ διάμετρος <lb></lb>αὐτῶν, καὶ τὸ Α ἔσται πρὸς Ζ.</s>
<figure id="id.080.01.004.1.jpg" xlink:href="080/01/004/1.jpg"></figure></p><p n="11">
<s id="g0120801">τὸν αὐτὸν δὴ τρόπον <lb></lb>δειχθήσεται κἂν ὁπουοῦν διαληφθῇ ἡ φορά· αἰεὶ γὰρ <lb></lb>ἔσται ἐπὶ τῆς διαμέτρου.</s>
<s id="g0120802">φανερὸν οὖν ὅτι τὸ κατὰ τὴν διάμετρον <lb></lb>φερόμενον ἐν δύο φοραῖς ἀνάγκη τὸν τῶν πλευρῶν <lb></lb>φέρεσθαι λόγον.</s>
<s id="g0120803">εἰ γὰρ ἄλλον τινά, οὐκ οἰσθήσεται κατὰ <lb></lb>τὴν διάμετρον.</s>
<s id="g0120804">ἐὰν δὲ ἐν μηδενὶ λόγῳ φέρηται δύο φορὰς <lb></lb>κατὰ μηδένα χρόνον, ἀδύνατον εὐθεῖαν εἶναι τὴν φοράν.</s>
<s id="g0120805"><lb></lb>ἔστω γὰρ εὐθεῖα.</s>
<s id="g0120806">τεθείσης οὖν ταύτης διαμέτρου, καὶ παραπληρωθεισῶν <lb></lb>τῶν πλευρῶν, ἀνάγκη τὸν τῶν πλευρῶν λόγον <lb></lb>φέρεσθαι τὸ φερόμενον· τοῦτο γὰρ δέδεικται πρότερον.</s>
<s id="g0120807">οὐκ <lb></lb>ἄρα ποιήσει εὐθεῖαν τὸ ἐν μηδενὶ λόγῳ φερόμενον μηδένα <lb></lb>χρόνον.</s>
<s id="g0120808">ἐὰν γάρ τινα λόγον ἐνεχθῇ ἐν χρόνῳ τινί, τοῦτον <lb></lb>ἀνάγκη τὸν χρόνον εὐθεῖαν εἶναι φορὰν διὰ τὰ προειρημένα.</s>
<s id="g0120809"><lb></lb>ὥστε περιφερὲς γίνεται, δύο φερόμενον φορὰς ἐν μηθενὶ <lb></lb>λόγῳ μηθένα χρόνον.</s></p><p n="12">
<s id="g0120901">ὅτι μὲν τοίνυν ἡ τὸν κύκλον γράφουσα <lb></lb>φέρεται δύο φορὰς ἅμα, φανερὸν ἔκ τε τούτων, <lb></lb>καὶ ὅτι τὸ φερόμενον κατ&#039; εὐθεῖαν ἐπὶ τὴν κάθετον ἀφι-<pb xlink:href="080/01/005.jpg" ed="Bekker" n="849a"></pb><lb></lb>κνεῖται, ὥστε εἶναι πάλιν αὐτὴν ἀπὸ τοῦ κέντρου κάθετον.</s></p><p n="13">
<s id="g0121001"><lb></lb>ἔστω κύκλος ὁ ΑΒΓ, τὸ δ&#039; ἄκρον τὸ ἐφ&#039; οὗ Β φερέσθω <lb></lb>ἐπὶ τὸ Δ· ἀφικνεῖται δέ ποτε ἐπὶ τὸ Γ.</s>
<s id="g0121002">εἰ μὲν οὖν ἐν τῷ <lb></lb>λόγῳ ἐφέρετο ὃν ἔχει ἡ ΒΔ πρὸς τὴν ΔΓ, ἐφέρετο ἂν <lb></lb>τὴν διάμετρον τὴν ἐφ&#039; ᾗ ΒΓ.</s>
<s id="g0121003">νῦν δέ, ἐπείπερ ἐν οὐδενὶ <lb></lb>λόγῳ, ἐπὶ τὴν περιφέρειαν φέρεται τὴν ἐφ&#039; ᾗ ΒΕΓ.</s>
<figure id="id.080.01.005.1.jpg" xlink:href="080/01/005/1.jpg"></figure></p><p n="14">
<s id="g0121101">ἐὰν <lb></lb>δὲ δυοῖν φερομένοιν ἀπὸ τῆς αὐτῆς ἰσχύος τὸ μὲν ἐκκρούοιτο <lb></lb>πλεῖον τὸ δὲ ἔλαττον, εὔλογον βραδύτερον κινηθῆναι <lb></lb>τὸ πλεῖον ἐκκρουόμενον τοῦ ἔλαττον ἐκκρουομένου· ὃ δοκεῖ <lb></lb>συμβαίνειν ἐπὶ τῆς μείζονος καὶ ἐλάττονος τῶν ἐκ τοῦ <lb></lb>κέντρου γραφουσῶν τοὺς κύκλους.</s>
<s id="g0121102">διὰ γὰρ τὸ ἐγγύτερον <lb></lb>εἶναι τοῦ μένοντος τῆς ἐλάττονος τὸ ἄκρον ἢ τὸ τῆς μείζονος, <lb></lb>ὥσπερ ἀντισπώμενον εἰς τοὐναντίον, ἐπὶ τὸ μέσον βραδύτερον <lb></lb>φέρεται τὸ τῆς ἐλάττονος ἄκρον.</s></p><p n="15">
<s id="g0121201">πάσῃ μὲν οὖν <lb></lb>κύκλον γραφούσῃ τοῦτο συμβαίνει, καὶ φέρεται τὴν μὲν <lb></lb>κατὰ φύσιν κατὰ τὴν περιφέρειαν, τὴν δὲ παρὰ φύσιν <lb></lb>εἰς τὸ πλάγιον καὶ τὸ κέντρον. μείζω δ&#039; ἀεὶ τὴν παρὰ <lb></lb>φύσιν ἡ ἐλάττων φέρεται· διὰ γὰρ τὸ ἐγγύτερον εἶναι τοῦ <lb></lb>κέντρου τοῦ ἀντισπῶντος κρατεῖται μᾶλλον.</s></p><p n="16">
<s id="g0121301">ὅτι δὲ μεῖζον <lb></lb>τὸ παρὰ φύσιν κινεῖται ἡ ἐλάττων τῆς μείζονος τῶν ἐκ τοῦ <lb></lb>κέντρου γραφουσῶν τοὺς κύκλους, ἐκ τῶνδε δῆλον.</s>
<s id="g0121302">ἔστω <lb></lb>κύκλος ἐφ&#039; οὗ ΒΓΔΕ, καὶ ἄλλος ἐν τούτῳ ἐλάττων, <lb></lb>ἐφ&#039; οὗ ΧΝΜΞ, περὶ τὸ αὐτὸ κέντρον τὸ Α· καὶ ἐκβεβλήσθωσαν <lb></lb>αἱ διάμετροι, ἐν μὲν τῷ μεγάλῳ, ἐφ&#039; ὧν ΓΔ <lb></lb>καὶ ΒΕ, ἐν δὲ τῷ ἐλάττονι αἱ ΜΧ ΝΞ· καὶ τὸ ἑτερόμηκες <lb></lb>παραπεπληρώσθω, τὸ ΔΨΡΓ. εἰ δὴ ἡ ΑΒ γράφουσα <lb></lb>κύκλον ἥξει ἐπὶ τὸ αὐτὸ ὅθεν ὡρμήθη ἐπὶ τὴν ΑΕ, δῆλον <lb></lb>ὅτι φέρεται πρὸς αὑτήν.</s>
<s id="g0121303">ὁμοίως δὲ καὶ ἡ ΑΧ πρὸς τὴν <lb></lb>ΑΧ ἥξει.</s>
<s id="g0121304">βραδύτερον δὲ φέρεται ἡ ΑΧ τῆς ΑΒ, ὥσπερ <lb></lb>εἴρηται, διὰ τὸ γίνεσθαι μείζονα τὴν ἔκκρουσιν καὶ ἀντισπᾶσθαι <lb></lb>μᾶλλον τὴν ΑΧ.</s></p><p n="17">
<s id="g0121401">ἤχθω δὲ ἡ ΑΘΗ, καὶ ἀπὸ <lb></lb>τοῦ Θ κάθετος ἐπὶ τὴν ΑΒ ἡ ΘΖ ἐν τῷ κύκλῳ, καὶ πάλιν <lb></lb>ἀπὸ τοῦ Θ ἤχθω παρὰ τὴν ΑΒ ἡ ΘΩ, καὶ ἡ ΩΥ <lb></lb>ἐπὶ τὴν ΑΒ κάθετον, καὶ ἡ ΗΚ.</s>
<s id="g0121402">αἱ δὴ ἐφ&#039; ὧν ΩΥ καὶ <lb></lb>ΘΖ ἴσαι. ἡ ἄρα ΒΥ ἐλάττων τῆς ΧΖ·</s>
<s id="g0121403">αἱ γὰρ ἴσαι <lb></lb>εὐθεῖαι ἐπ&#039; ἀνίσους κύκλους ἐμβληθεῖσαι πρὸς ὀρθὰς τῇ <lb></lb>διαμέτρῳ ἔλαττον τμῆμα ἀποτέμνουσι τῆς διαμέτρου ἐν <lb></lb>τοῖς μείζοσι κύκλοις, ἔστι δὲ ἡ ΩΥ ἴση τῇ ΘΖ.</s>
<s id="g0121404">ἐν ὅσῳ<pb xlink:href="080/01/006.jpg" ed="Bekker" n="849b"></pb><lb></lb> δὴ χρόνῳ ἡ ΑΘ τὴν ΧΘ ἐνηνέχθη, ἐν τοσούτῳ χρόνῳ ἐν <lb></lb>τῷ κύκλῳ τῷ μείζονι μείζονα τῆς ΒΩ ἐνήνεκται τὸ ἄκρον <lb></lb>τῆς ΒΑ.</s>
<figure id="id.080.01.006.1.jpg" xlink:href="080/01/006/1.jpg"></figure></p><p n="18">
<s id="g0121501">ἡ μὲν γὰρ κατὰ φύσιν φορὰ ἴση, ἡ δὲ παρὰ <lb></lb>φύσιν ἐλάττων· ἡ δὲ ΒΥ τῆς ΖΧ.</s>
<s id="g0121502">δεῖ δὲ ἀνάλογον εἶναι, <lb></lb>ὡς τὸ κατὰ φύσιν πρὸς τὸ κατὰ φύσιν, τὸ παρὰ φύσιν <lb></lb>πρὸς τὸ παρὰ φύσιν.</s>
<s id="g0121503">μείζονα ἄρα περιφέρειαν διελήλυθε <lb></lb>τὴν ΗΒ τῆς ΩΒ.</s>
<s id="g0121504">ἀνάγκη δὲ τὴν ΗΒ ἐν τούτῳ τῷ χρόνῳ <lb></lb>διεληλυθέναι· </s>
<s id="g0121505">ἐνταῦθα γὰρ ἔσται, ὅταν ἀνάλογον ἀμφοτέρως <lb></lb>συμβαίνῃ τὸ παρὰ φύσιν πρὸς τὸ κατὰ φύσιν.</s>
<s id="g0121506">εἰ δὴ <lb></lb>μεῖζόν ἐστι τὸ κατὰ φύσιν ἐν τῇ μείζονι, καὶ τὸ παρὰ φύσιν <lb></lb>μᾶλλον ἂν ἐνταῦθα συμπίπτοι μοναχῶς, </s>
<s id="g0121507">ὥστε τὸ Β ἐνηνέχθαι <lb></lb>ἂν τὴν ΒΗ ἐν τῷ ἐφ&#039; οὗ Χ σημεῖον. ἐνταῦθα γὰρ <lb></lb>κατὰ φύσιν μὲν γίνεται τῷ Β σημείῳ τὸ κέντρον [1ἔστι γὰρ <lb></lb>αὐτὴ ἀπὸ τοῦ Η κάθετοσ]1, παρὰ φύσιν δὲ ἐς τὸ ΚΒ.</s>
<s id="g0121508">ἔστι <lb></lb>δὲ ὡς τὸ ΗΚ πρὸς τὸ ΚΒ, τὸ ΘΖ πρὸς τὸ ΖΧ. φανερὸν <lb></lb>δὲ ἐὰν ἐπιζευχθῶσιν ἀπὸ τῶν ΒΧ ἐπὶ τὰ ΗΘ.</s>
<s id="g0121509">εἰ δὲ <lb></lb>ἐλάττων ἢ μείζων τῆς ΗΒ ἔσται, ἣν ἠνέχθη τὸ Β, οὐχ ὁμοίως <lb></lb>ἔσται οὐδὲ ἀνάλογον ἐν ἀμφοῖν τὸ κατὰ φύσιν πρὸς τὸ <lb></lb>παρὰ φύσιν.</s>
<s id="g0121510">δι&#039; ἣν μὲν τοίνυν αἰτίαν ἀπὸ τῆς αὐτῆς <lb></lb>ἰσχύος φέρεται θᾶττον τὸ πλέον ἀπέχον τοῦ κέντρου σημεῖον, <lb></lb>δῆλον διὰ τῶν εἰρημένων· </s></p><p n="19">
<s id="g0130101">διότι δὲ τὰ μὲν μείζω ζυγὰ <lb></lb>ἀκριβέστερά ἐστι τῶν ἐλαττόνων, φανερὸν ἐκ τούτων.</s>
<s id="g0130102">γίνεται <lb></lb>γὰρ τὸ μὲν σπάρτον κέντρον [1μένει γὰρ τοῦτο]1, τὸ δὲ ἐπὶ <lb></lb>ἑκάτερον μέρος τῆς πλάστιγγος αἱ ἐκ τοῦ κέντρου.</s>
<s id="g0130103">ἀπὸ οὖν <lb></lb>τοῦ αὐτοῦ βάρους ἀνάγκη θᾶττον κινεῖσθαι τὸ ἄκρον τῆς <lb></lb>πλάστιγγος, ὅσῳ ἂν πλεῖον ἀπέχῃ τοῦ σπάρτου, </s>
<s id="g0130104">καὶ ἔνια <lb></lb>μὲν μὴ δῆλα εἶναι ἐν τοῖς μικροῖς ζυγοῖς πρὸς τὴν αἴσθησιν <lb></lb>ἐπιτιθέμενα βάρη, ἐν δὲ τοῖς μεγάλοις δῆλα </s>
<s id="g0130105">οὐθὲν γὰρ <lb></lb>κωλύει ἔλαττον κινηθῆναι μέγεθος ἢ ὥστε εἶναι τῇ ὄψει <lb></lb>φανερόν.</s>
<s id="g0130106">ἐπὶ δὲ τῆς μεγάλης πλάστιγγος ποιεῖ ὁρατὸν τὸ <lb></lb>αὐτὸ βάρος μέγεθος.</s>
<s id="g0130107">ἔνια δὲ δῆλα μὲν ἐπ&#039; ἀμφοῖν ἐστίν, <lb></lb>ἀλλὰ πολλῷ μᾶλλον ἐπὶ τῶν μειζόνων διὰ τὸ πολλῷ <lb></lb>μεῖζον γίνεσθαι τὸ μέγεθος τῆς ῥοπῆς ὑπὸ τοῦ αὐτοῦ βάρους <lb></lb>ἐν τοῖς μείζοσι.</s>
<s id="g0130108">καὶ διὰ τοῦτο τεχνάζουσιν οἱ ἁλουργοπῶλαι <lb></lb>πρὸς τὸ παρακρούεσθαι ἱστάντες, τό τε σπάρτον <lb></lb>οὐκ ἐν μέσῳ τιθέντες, καὶ μόλυβδον τῆς φάλαγγος εἰς <lb></lb>θάτερον μέρος ἐγχέοντες, ἢ τοῦ ξύλου τὸ πρὸς τὴν ῥίζαν <lb></lb>πρὸς ὃ βούλονται ῥέπειν ποιοῦντες, ἢ ἐὰν ἔχῃ ὄζον· βαρύ-<pb xlink:href="080/01/007.jpg" ed="Bekker" n="850a"></pb><lb></lb>τερον γὰρ ἐν ᾧ μέρος ἡ ῥίζα τοῦ ξύλου ἐστίν, ὁ δὲ ὄζος ῥίζα <lb></lb>τίς ἐστιν.</s></p><p n="20">
<s id="g0130201prop02"><lb></lb>Διὰ τί, ἐὰν μὲν ἄνωθεν ᾖ τὸ σπαρτίον, ὅταν κάτωθεν <lb></lb>ῥέψαντος ἀφέλῃ τὸ βάρος, πάλιν ἀναφέρεται τὸ ζυγόν, <lb></lb>ἐὰν δὲ κάτωθεν ὑποστῇ, οὐκ ἀναφέρεται ἀλλὰ μένει; </s>
<s id="g0130202">ἢ <lb></lb>διότι ἄνωθεν μὲν τοῦ σπαρτίου ὄντος πλεῖον τοῦ ζυγοῦ γίνεται <lb></lb>τὸ ἐπέκεινα τῆς καθέτου; τὸ γὰρ σπαρτίον ἐστὶ κάθετος. <lb></lb>ὥστε ἀνάγκη ἐστὶ κάτω ῥέπειν τὸ πλέον, ἕως ἂν ἔλθῃ ἡ <lb></lb>δίχα διαιροῦσα τὸ ζυγὸν ἐπὶ τὴν κάθετον αὐτήν, ἐπικειμένου <lb></lb>τοῦ βάρους ἐν τῷ ἀνεσπασμένῳ μορίῳ τοῦ ζυγοῦ.</s>
<s id="g0130203"><lb></lb>ἔστω ζυγὸν ὀρθὸν ἐφ&#039; οὗ ΒΓ, σπαρτίον δὲ τὸ ΑΔ. ἐκβαλλόμενον <lb></lb>δὴ τοῦτο κάτω κάθετος ἔσται ἐφ&#039; ἧς ἡ ΑΔΜ.</s>
<s id="g0130204"><lb></lb>ἐὰν οὖν ἐπὶ τὸ Β ἡ ῥοπὴ ἐπιτεθῇ, ἔσται τὸ μὲν Β οὗ τὸ Ε, <lb></lb>τὸ δὲ Γ οὗ τὸ Ζ, ὥστε ἡ δίχα διαιροῦσα τὸ ζυγὸν πρῶτον <lb></lb>μὲν ἦν ἡ ΔΜ τῆς καθέτου αὐτῆς, ἐπικειμένης δὲ τῆς ῥοπῆς <lb></lb>ἔσται ἡ ΔΘ· ὥστε τοῦ ζυγοῦ ἐφ&#039; ᾧ ΕΖ τὸ ἔξω τῆς καθέτου <lb></lb>τῆς ἐφ&#039; ἧς ΑΒ, τοῦ ἐν ᾧ ΦΠ, μείζω τοῦ ἡμίσεος.</s>
<s id="g0130205"><lb></lb>ἐὰν οὖν ἀφαιρεθῇ τὸ βάρος ἀπὸ τοῦ Ε, ἀνάγκη κάτω φέρεσθαι <lb></lb>τὸ Ζ· ἔλαττον γάρ ἐστι τὸ Ε.</s>
<s id="g0130206">ἐὰν μὲν οὖν ἄνω τὸ <lb></lb>σπαρτίον ἔχῃ, πάλιν διὰ τοῦτο ἀναφέρεται τὸ ζυγόν.</s>
<figure id="id.080.01.007.1.jpg" xlink:href="080/01/007/1.jpg"></figure>
<s id="g0130207">ἐὰν <lb></lb>δὲ κάτωθεν ᾖ τὸ ὑποκείμενον, τοὐναντίον ποιεῖ· πλεῖον γὰρ <lb></lb>γίνεται τοῦ ἡμίσεος τοῦ ζυγοῦ τὸ κάτω μέρος ἢ ὡς ἡ κάθετος <lb></lb>διαιρεῖ ὥστε οὐκ ἀναφέρεται· κουφότερον γὰρ τὸ ἐπηρτημένον.</s>
<s id="g0130208"><lb></lb>ἔστω ζυγὸν τὸ ἐφ&#039; οὗ ΝΞ, τὸ ὀρθόν, κάθετος δὲ ἡ <lb></lb>ΚΛΜ. δίχα δὴ διαιρεῖται τὸ ΝΞ.</s>
<s id="g0130209">ἐπιτεθέντος δὲ βάρους <lb></lb>ἐπὶ τὸ Ν, ἔσται τὸ μὲν Ν οὗ τὸ Ο, τὸ δὲ Ξ οὗ τὸ Ρ, ἡ δὲ <lb></lb>ΚΛ οὗ τὸ ΛΘ, ὥστε μεῖζόν ἐστι τὸ ΚΟ τοῦ ΛΡ τῷ ΘΚΛ.</s>
<s id="g0130210"><lb></lb>καὶ ἀφαιρεθέντος οὖν τοῦ βάρους ἀνάγκη μένειν· ἐπίκειται <lb></lb>γὰρ ὥσπερ βάρος ἡ ὑπεροχὴ ἡ τοῦ ἡμίσεος τοῦ ἐν ᾧ τὸ Κ.</s>
<figure id="id.080.01.007.2.jpg" xlink:href="080/01/007/2.jpg"></figure></p><p n="21">
<s id="g0130301prop03"><lb></lb>Διὰ τί κινοῦσι μεγάλα βάρη μικραὶ δυνάμεις τῷ μοχλῷ, <lb></lb>ὥσπερ ἐλέχθη καὶ κατ&#039; ἀρχήν, προσλαβόντι βάρος <lb></lb>ἔτι τὸ τοῦ μοχλοῦ; ῥᾷον δὲ τὸ ἔλαττόν ἐστι κινῆσαι βάρος, <lb></lb>ἔλαττον δέ ἐστιν ἄνευ τοῦ μοχλοῦ.</s>
<s id="g0130302">ἢ ὅτι αἴτιόν ἐστιν ὁ μοχλός, <lb></lb>ζυγὸν [ὢν] κάτωθεν ἔχον τὸ σπαρτίον καὶ εἰς ἄνισα διῃρημένον; <lb></lb>τὸ γὰρ ὑπομόχλιον ἀντὶ σπαρτίου γίνεται· μένει <lb></lb>γὰρ ἄμφω ταῦτα, ὥσπερ τὸ κέντρον.</s>
<s id="g0130303">ἐπεὶ δὲ θᾶττον ὑπὸ <lb></lb>τοῦ ἴσου βάρους κινεῖται ἡ μείζων τῶν ἐκ τοῦ κέντρου, ἔστι δὲ <lb></lb>τρία τὰ περὶ τὸν μοχλόν, τὸ μὲν ὑπομόχλιον, σπάρτον <lb></lb>καὶ κέντρον, δύο δὲ βάρη, ὅ τε κινῶν καὶ τὸ κινούμενον· ὃ<pb xlink:href="080/01/008.jpg" ed="Bekker" n="850b"></pb><lb></lb> οὖν τὸ κινούμενον βάρος πρὸς τὸ κινοῦν, τὸ μῆκος πρὸς τὸ μῆκος <lb></lb>ἀντιπέπονθεν.</s>
<s id="g0130304"> αἰεὶ δὲ ὅσῳ ἂν μεῖζον ἀφεστήκῃ τοῦ ὑπομοχλίου, <lb></lb>ῥᾷον κινήσει.</s>
<s id="g0130305">αἰτία δέ ἐστιν ἡ προλεχθεῖσα, ὅτι ἡ <lb></lb>πλεῖον ἀπέχουσα ἐκ τοῦ κέντρου μείζονα κύκλον γράφει.</s>
<s id="g0130306">ὥστε <lb></lb>ἀπὸ τῆς αὐτῆς ἰσχύος πλέον μεταστήσεται τὸ κινοῦν τὸ <lb></lb>πλεῖον τοῦ ὑπομοχλίου ἀπέχον.</s>
<s id="g0130307">ἔστω μοχλὸς ἐφ&#039; οὗ ΑΒ, <lb></lb>βάρος δὲ ἐφ&#039; ᾧ τὸ Γ, τὸ δὲ κινοῦν ἐφ&#039; ᾧ τὸ Δ, ὑπομόχλιον <lb></lb>ἐφ&#039; ᾧ τὸ Ε, </s>
<s id="g0130308">τὸ δὲ ἐφ&#039; ᾧ τὸ Δ κινῆσαν ἐφ&#039; ᾧ τὸ Η, κινούμενον <lb></lb>δὲ τὸ ἐφ&#039; οὗ Γ, βάρος ἐφ&#039; οὗ Κ.</s>
<figure id="id.080.01.008.1.jpg" xlink:href="080/01/008/1.jpg"></figure></p><p n="22">
<s id="g0130401prop04"><lb></lb>Διὰ τί οἱ μεσόνεοι μάλιστα τὴν ναῦν κινοῦσιν; </s>
<s id="g0130402">ἢ διότι <lb></lb>ἡ κώπη μοχλός ἐστιν; ὑπομόχλιον μὲν γὰρ ὁ σκαλμὸς γίνεται <lb></lb>[1μένει γὰρ δὴ τοῦτο]1, τὸ δὲ βάρος ἡ θάλαττα, ἣν <lb></lb>ἀπωθεῖ ἡ κώπη· ὁ δὲ κινῶν τὸν μοχλὸν ὁ ναύτης ἐστίν.</s>
<s id="g0130403"><lb></lb>ἀεὶ δὲ πλέον βάρος κινεῖ, ὅσῳ ἂν πλέον ἀφεστήκῃ τοῦ ὑπομοχλίου <lb></lb>ὁ κινῶν τὸ βάρος· </s>
<s id="g0130404">μείζων γὰρ οὕτω γίνεται ἡ ἐκ <lb></lb>τοῦ κέντρου, ὁ δὲ σκαλμὸς ὑπομόχλιον ὢν κέντρον ἐστίν.</s>
<s id="g0130405">ἐν <lb></lb>μέσῃ δὲ τῇ νηὶ̈ πλεῖστον τῆς κώπης ἐντός ἐστιν· καὶ γὰρ ἡ <lb></lb>ναῦς ταύτῃ εὐρυτάτη ἐστίν, ὥστε πλεῖον ἐπ&#039; ἀμφότερα ἐνδέχεσθαι <lb></lb>μέρος τῆς κώπης ἑκατέρου τοίχου ἐντὸς εἶναι τῆς <lb></lb>νεώς.</s>
<s id="g0130406">κινεῖται μὲν οὖν ἡ ναῦς διὰ τὸ ἀπερειδομένης τῆς κώπης <lb></lb>εἰς τὴν θάλασσαν τὸ ἄκρον τῆς κώπης τὸ ἐντὸς προϊέναι <lb></lb>εἰς τὸ πρόσθεν, </s>
<s id="g0130407">τὴν δὲ ναῦν προσδεδεμένην τῷ σκαλμῷ συμπροϊέναι, <lb></lb>ᾗ τὸ ἄκρον τῆς κώπης.</s>
<s id="g0130408">ᾗ γὰρ πλείστην θάλασσαν <lb></lb>διαιρεῖ ἡ κώπη, ταύτῃ ἀνάγκη μάλιστα προωθεῖσθαι· πλείστην <lb></lb>δὲ διαιρεῖ ᾗ πλεῖστον μέρος ἀπὸ τοῦ σκαλμοῦ τῆς κώπης <lb></lb>ἐστίν.</s>
<s id="g0130409">διὰ τοῦτο οἱ μεσόνεοι μάλιστα κινοῦσιν· μέγιστον γὰρ <lb></lb>ἐν μέσῃ νηὶ̈ τὸ ἀπὸ τοῦ σκαλμοῦ τῆς κώπης τὸ ἐντός ἐστιν.</s></p><p n="23">
<s id="g0130501prop05"><lb></lb>Διὰ τί τὸ πηδάλιον μικρὸν ὄν, καὶ ἐπ&#039; ἐσχάτῳ τῷ <lb></lb>πλοίῳ, τοσαύτην δύναμιν ἔχει ὥστε ὑπὸ μικροῦ οἴακος καὶ <lb></lb>ἑνὸς ἀνθρώπου δυνάμεως, καὶ ταύτης ἠρεμαίας, μεγάλα κινεῖσθαι <lb></lb>μεγέθη πλοίων; </s>
<s id="g0130502">ἢ διότι καὶ τὸ πηδάλιόν ἐστι μοχλός, <lb></lb>καὶ μοχλεύει ὁ κυβερνήτης. ᾗ μὲν οὖν προσήρμοσται <lb></lb>τῷ πλοίῳ, γίνεται ὑπομόχλιον, τὸ δὲ ὅλον πηδάλιον ὁ <lb></lb>μοχλός, τὸ δὲ βάρος ἡ θάλασσα, ὁ δὲ κυβερνήτης ὁ κινῶν.</s>
<s id="g0130503"><lb></lb>οὐ κατὰ πλάτος δὲ λαμβάνει τὴν θάλασσαν, ὥσπερ ἡ κώπη, <lb></lb>τὸ πηδάλιον. οὐ γὰρ εἰς τὸ πρόσθεν κινεῖ τὸ πλοῖον, ἀλλὰ <lb></lb>κινούμενον κλίνει, πλαγίως τὴν θάλατταν δεχόμενον.</s>
<s id="g0130504">ἐπεὶ <lb></lb>γὰρ τὸ βάρος ἦν ἡ θάλασσα, τοὐναντίον ἀπερειδόμενον κλίνει <lb></lb>τὸ πλοῖον. τὸ γὰρ ὑπομόχλιον εἰς τοὐναντίον στρέφεται,<pb xlink:href="080/01/009.jpg" ed="Bekker" n="851a"></pb><lb></lb> ἡ θάλασσα δὲ ἐντός· ἐκεῖνο δὲ εἰς τὸ ἐκτός. τούτῳ δὲ ἀκολουθεῖ <lb></lb>τὸ πλοῖον διὰ τὸ συνδεδέσθαι.</s>
<s id="g0130505">ἡ μὲν οὖν κώπη κατὰ <lb></lb>πλάτος τὸ βάρος ὠθοῦσα καὶ ὑπ&#039; ἐκείνου ἀντωθουμένη εἰς τὸ <lb></lb>εὐθὺ προάγει· τὸ δὲ πηδάλιον, ὥσπερ κάθηται πλάγιον, <lb></lb>τὴν εἰς τὸ πλάγιον, ἢ δεῦρο ἢ ἐκεῖ, ποιεῖ κίνησιν.</s>
<s id="g0130506">ἐπ&#039; ἄκρου <lb></lb>δὲ καὶ οὐκ ἐν μέσῳ κεῖται, ὅτι ῥᾷστον τὸ κινούμενον κινῆσαι <lb></lb>ἀπ&#039; ἄκρου κινοῦν. </s>
<s id="g0130507">τάχιστα γὰρ φέρεται τὸ πρῶτον μέρος <lb></lb>διὰ τὸ ὥσπερ ἐν τοῖς φερομένοις ἐπὶ τέλει λήγειν τὴν φοράν, <lb></lb>οὕτω καὶ τοῦ συνεχοῦς ἐπὶ τέλους ἀσθενεστάτη ἐστὶν ἡ φορά. <lb></lb>εἰ δὲ ἀσθενεστάτη, ῥᾳδία ἐκκρούειν. </s>
<s id="g0130508">διά τε δὴ ταῦτα ἐν τῇ <lb></lb>πρύμνῃ τὸ πηδάλιόν ἐστι, καὶ ὅτι ἐνταῦθα μικρᾶς κινήσεως <lb></lb>γενομένης πολλῷ μεῖζον τὸ διάστημα ἐπὶ τῷ ἐσχάτῳ γίνεται, <lb></lb>διὰ τὸ τὴν ἴσην γωνίαν ἐπὶ μείζονα καθῆσθαι, καὶ ὅσῳ <lb></lb>ἂν μείζους ὦσιν αἱ περιέχουσαι.</s>
<s id="g0130509">δῆλον δὲ ἐκ τούτου καὶ δι&#039; ἣν <lb></lb>αἰτίαν μᾶλλον προέρχεται εἰς τοὐναντίον τὸ πλοῖον ἢ ἡ τῆς <lb></lb>κώπης πλάτη· τὸ αὐτὸ γὰρ μέγεθος τῇ αὐτῇ ἰσχύϊ κινούμενον <lb></lb>ἐν ἀέρι πλέον ἢ ἐν τῷ ὕδατι πρόεισιν.</s>
<s id="g0130510">ἔστω γὰρ ἡ Α <lb></lb>Β κώπη, τὸ δὲ Γ ὁ σκαλμός, τὸ δὲ Α τὸ ἐν τῷ πλοίῳ, ἡ <lb></lb>ἀρχὴ τῆς κώπης, τὸ δὲ Β τὸ ἐν τῇ θαλάττῃ.</s>
<s id="g0130511">εἰ δὴ τὸ Α <lb></lb>οὗ τὸ Δ μετακεκίνηται, τὸ Β οὐκ ἔσται οὗ τὸ Ε· ἴση γὰρ ἡ Β <lb></lb>Ε τῇ ΑΔ. ἴσον οὖν μετακεχωρηκὸς ἔσται.</s>
<s id="g0130512">ἀλλ&#039; ἦν ἔλαττον. <lb></lb>ἔσται δὴ οὗ τὸ Ζ ἢ τὸ Θ. ἄρα τοίνυν τὴν ΑΒ, καὶ οὐχ ἡ τὸ <lb></lb>Γ, καὶ κάτωθεν. ἐλάττων γὰρ ἡ ΒΖ τῆς ΑΔ, ὥστε καὶ <lb></lb>ἡ ΘΖ τῆς ΔΘ· ὅμοια γὰρ τὰ τρίγωνα.</s>
<s id="g0130513">καθεστηκὸς δὲ <lb></lb>ἔσται καὶ τὸ μέσον, τὸ ἐφ&#039; οὗ Γ· εἰς τοὐναντίον γὰρ τῷ ἐν τῇ <lb></lb>θαλάττῃ ἄκρῳ τῷ Β μεταχωρεῖ, ᾗπερ τὸ ἐν τῷ πλοίῳ <lb></lb>ἄκρον τὸ Α </s>
<s id="g0130514">μὴ ἐχώρει οὗ τὸ Δ. ὥστε μετακινηθήσεται τὸ <lb></lb>πλοῖον, καὶ ἐκεῖ οὗ ἡ ἀρχὴ τῆς κώπης μεταφέρεται.</s>
<figure id="id.080.01.009.1.jpg" xlink:href="080/01/009/1.jpg"></figure>
<figure id="id.080.01.009.2.jpg" xlink:href="080/01/009/2.jpg"></figure>
<s id="g0130515">τὸ δ&#039; <lb></lb>αὐτὸ καὶ τὸ πηδάλιον ποιεῖ, πλὴν ὅτι εἰς τὸ πρόσθεν οὐδὲν <lb></lb>συμβάλλεται τῷ πλοίῳ, ὥσπερ ἐλέχθη ἐπὶ ἄνω, ἀλλὰ <lb></lb>μόνον τὴν πρύμναν εἰς τὸ πλάγιον ἀπωθεῖ ἔνθα ἢ ἔνθα· εἰς <lb></lb>τοὐναντίον γὰρ ἡ πρῷρα οὕτω νεύει.</s>
<s id="g0130516">ᾗ μὲν δὴ τὸ πηδάλιον <lb></lb>προσέζευκται, δεῖ οἷόν τι τοῦ κινουμένου μέσον νοεῖν, καὶ ὥσπερ <lb></lb>ὁ σκαλμὸς τῇ κώπῃ· τὸ δὲ μέσον ὑποχωρεῖ, ᾗ ὁ οἴαξ μετακινεῖται.</s>
<s id="g0130517"><lb></lb>ἐὰν μὲν εἴσω ἄγῃ, καὶ ἡ πρύμνα δεῦρο μεθέστηκεν· <lb></lb>ἡ δὲ πρῷρα εἰς τοὐναντίον νεύει· ἐν γὰρ τῷ αὐτῷ <lb></lb>οὔσης τῆς πρῴρας τὸ πλοῖον μεθέστηκεν ὅλον.</s>
<figure id="id.080.01.009.3.jpg" xlink:href="080/01/009/3.jpg"></figure></p><p n="24">
<s id="g0130601prop06"><lb></lb>Διὰ τί, ὅσῳ ἂν ἡ κεραία ἀνωτέρα ᾖ, θᾶττον πλεῖ τὰ <lb></lb>πλοῖα τῷ αὐτῷ ἱστίῳ καὶ τῷ αὐτῷ πνεύματι; </s>
<s id="g0130602">ἢ διότι γίνεται <lb></lb>ὁ μὲν ἱστὸς μοχλός, ὑπομόχλιον δὲ τὸ ἑδώλιον ἐν ᾧ<pb xlink:href="080/01/010.jpg" ed="Bekker" n="851b"></pb><lb></lb> ἐμπέπηγεν, ὃ δὲ δεῖ κινεῖν βάρος, τὸ πλοῖον, τὸ δὲ κινοῦν <lb></lb>τὸ ἐν τῷ ἱστίῳ πνεῦμα.</s>
<s id="g0130603">εἰ δ&#039; ὅσῳ ἂν πορρώτερον ᾖ τὸ ὑπομόχλιον, <lb></lb>ῥᾷον κινεῖ καὶ θᾶττον ἡ αὐτὴ δύναμις τὸ αὐτὸ <lb></lb>βάρος, ἡ οὖν κεραία ἀνώτερον ἀγομένη καὶ τὸ ἱστίον πορρώτερον <lb></lb>ποιεῖ τοῦ ἑδωλίου ὑπομοχλίου ὄντος.</s></p><p n="25">
<s id="g0130701prop07"><lb></lb>Διὰ τί, ὅταν ἐξ οὐρίας βούλωνται διαδραμεῖν μὴ οὐρίου <lb></lb>τοῦ πνεύματος ὄντος, τὸ μὲν πρὸς τὸν κυβερνήτην τοῦ ἱστίου <lb></lb>μέρος στέλλονται, τὸ δὲ πρὸς τὴν πρῷραν ποδιαῖον ποιησάμενοι <lb></lb>ἐφιᾶσιν; </s>
<s id="g0130702">ἢ διότι ἀντισπᾶν τὸ πηδάλιον πολλῷ μὲν <lb></lb>ὄντι τῷ πνεύματι οὐ δύναται, ὀλίγῳ δέ, ὃ ὑποστέλλονται.</s>
<s id="g0130703"><lb></lb>προάγει μὲν οὖν τὸ πνεῦμα, εἰς οὔριον δὲ καθίστησι τὸ <lb></lb>πηδάλιον, ἀντισπῶν καὶ μοχλεῦον τὴν θάλατταν.</s>
<s id="g0130704">ἅμα <lb></lb>δὲ καὶ οἱ ναῦται μάχονται τῷ πνεύματι· ἀνακλίνουσι γὰρ <lb></lb>ἐπὶ τὸ ἐναντίον ἑαυτούς.</s></p><p n="26">
<s id="g0130801prop08"><lb></lb>Διὰ τί τὰ στρογγύλα καὶ περιφερῆ τῶν σχημάτων <lb></lb>εὐκινητότερα; </s>
<s id="g0130802">τριχῶς δὲ ἐνδέχεται τὸν κύκλον κυλισθῆναι· <lb></lb>ἢ γὰρ κατὰ τὴν ἁψῖδα, συμμεταβάλλοντος τοῦ κέντρου, <lb></lb>ὥσπερ ὁ τροχὸς ὁ τῆς ἁμάξης κυλίεται· ἢ περὶ τὸ κέντρον <lb></lb>μόνον, ὥσπερ αἱ τροχιλέαι, τοῦ κέντρου μένοντος· ἢ παρὰ <lb></lb>τὸ ἐπίπεδον, τοῦ κέντρου μένοντος, ὥσπερ ὁ κεραμεικὸς τροχὸς <lb></lb>κυλίνδεται.</s>
<s id="g0130803">εἰ μὲν δὴ τάχιστα τὰ τοιαῦτα, διά τε τὸ <lb></lb>μικρῷ ἅπτεσθαι τοῦ ἐπιπέδου, ὥσπερ ὁ κύκλος κατὰ στιγμήν, <lb></lb>καὶ διὰ τὸ μὴ προσκόπτειν· ἀφέστηκε γὰρ τῆς γῆς <lb></lb>ἡ γωνία.</s>
<s id="g0130804">καὶ ἔτι ᾧ ἂν ἀπαντήσῃ σώματι, πάλιν τούτου <lb></lb>κατὰ μικρὸν ἅπτεται.</s>
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<s id="g0130805">εἰ δ&#039; εὐθύγραμμον ἦν, τῇ εὐθείᾳ <lb></lb>ἐπὶ πολὺ ἥπτετο ἂν τοῦ ἐπιπέδου.</s>
<s id="g0130806">ἔτι ᾗ ῥέπει ἐπὶ τὸ βάρος, <lb></lb>ταύτῃ κινεῖ ὁ κινῶν. ὅταν μὲν γὰρ πρὸς ὄρθιον ἡ διάμετρος <lb></lb>ᾖ τοῦ κύκλου τῷ ἐπιπέδῳ, ἁπτομένου τοῦ κύκλου κατὰ στιγμὴν <lb></lb>τοῦ ἐπιπέδου, ἴσον τὸ βάρος ἐπ&#039; ἀμφότερα διαλαμβάνει <lb></lb>ἡ διάμετρος· ὅταν δὲ κινῆται, εὐθὺς πλέον ἐφ&#039; ᾧ <lb></lb>κινεῖται, ὥσπερ ῥέπον. ἐντεῦθεν εὐκινητότερον τῷ ὠθοῦντι εἰς <lb></lb>τοὔμπροσθεν· ἐφ&#039; ὃ γὰρ ῥέπει ἕκαστον, εὐκίνητόν ἐστιν, <lb></lb>εἴπερ καὶ τὸ ἐπὶ τὸ ἐναντίον τῆς ῥοπῆς δυσκίνητον.</s>
<s id="g0130807">ἔτι λέγουσί <lb></lb>τινες ὅτι καὶ ἡ γραμμὴ ἡ τοῦ κύκλου ἐν φορᾷ ἐστὶν <lb></lb>ἀεί, ὥσπερ τὰ μένοντα, διὰ τὸ ἀντερείδειν, οἷον καὶ τοῖς <lb></lb>μείζοσι κύκλοις ὑπάρχει πρὸς τοὺς ἐλάττονας. </s>
<s id="g0130808">θᾶττον γὰρ <lb></lb>ὑπὸ τῆς ἴσης ἰσχύος κινοῦνται οἱ μείζους καὶ τὰ βάρη κινοῦσι, <lb></lb>διὰ τὸ ῥοπήν τινα ἔχειν τὴν γωνίαν τὴν τοῦ μείζονος <lb></lb>κύκλου πρὸς τὴν τοῦ ἐλάττονος, καὶ εἶναι ὅπερ ἡ διάμετρος <lb></lb>πρὸς τὴν διάμετρον. ἀλλὰ μὴν πᾶς κύκλος μείζων πρὸσ<pb xlink:href="080/01/011.jpg" ed="Bekker" n="852a"></pb><lb></lb> ἐλάττονα· ἄπειροι γὰρ οἱ ἐλάττονες.</s>
<s id="g0130809">εἰ δὲ καὶ πρὸς ἕτερον <lb></lb>ἔχει ῥοπὴν ὁ κύκλος, ὁμοίως δὲ εὐκίνητος, καὶ ἄλλην ἂν <lb></lb>ἔχοι ῥοπὴν ὁ κύκλος </s>
<s id="g0130810">καὶ τὰ ὑπὸ κύκλου κινούμενα, κἂν μὴ <lb></lb>τῇ ἁψῖδι ἅπτηται τοῦ ἐπιπέδου, ἀλλ&#039; ἢ παρὰ τὸ ἐπίπεδον, <lb></lb>ἢ ὡς αἱ τροχιλέαι· καὶ γὰρ οὕτως ἔχοντα ῥᾷστα κινοῦνται <lb></lb>καὶ κινοῦσι τὸ βάρος.</s>
<s id="g0130811">ἢ οὐ τῷ κατὰ μικρὸν ἅπτεσθαι καὶ <lb></lb>προσκρούειν, ἀλλὰ δι&#039; ἄλλην αἰτίαν.</s>
<s id="g0130812">αὕτη δέ ἐστιν ἡ εἰρημένη <lb></lb>πρότερον, ὅτι ἐκ δύο φορῶν γεγένηται ὁ κύκλος, ὥστε <lb></lb>μίαν αὐτῶν αἰεὶ ἔχειν ῥοπήν, καὶ οἷον φερόμενον αὐτὸν <lb></lb>αἰεὶ κινοῦσιν οἱ κινοῦντες, ὅταν κινῶσι κατὰ τὴν περιφέρειαν <lb></lb>ὁπωσοῦν. φερομένην γὰρ αὐτὴν κινοῦσιν· </s>
<s id="g0130813">τὴν μὲν γὰρ εἰς <lb></lb>τὸ πλάγιον αὐτοῦ κίνησιν ὠθεῖ τὸ κινοῦν, τὴν δὲ ἐπὶ τῆς <lb></lb>διαμέτρου αὐτὸς κινεῖται.</s></p><p n="27">
<s id="g0130901prop09"><lb></lb>Διὰ τί τὰ διὰ τῶν μειζόνων κύκλων αἰρόμενα καὶ <lb></lb>ἑλκόμενα ῥᾷον καὶ θᾶττον κινοῦμεν; οἷον καὶ αἱ τροχιλέαι <lb></lb>αἱ μείζους τῶν ἐλαττόνων, καὶ αἱ σκυτάλαι ὁμοίως.</s>
<s id="g0130902">ἢ <lb></lb>διότι ὅσῳ ἂν μείζων ἡ ἐκ τοῦ κέντρου ᾖ, ἐν τῷ ἴσῳ χρόνῳ <lb></lb>πλέον κινεῖται χωρίον, </s>
<s id="g0130903">ὥστε καὶ τοῦ ἴσου βάρους ἐπόντος <lb></lb>ποιήσει τὸ αὐτό, ὥσπερ εἴπομεν καὶ τὰ μείζω ζυγὰ τῶν <lb></lb>ἐλαττόνων ἀκριβέστερα εἶναι.</s>
<s id="g0130904">τὸ μὲν γὰρ σπαρτίον ἐστὶ <lb></lb>κέντρον, τοῦ δὲ ζυγοῦ αἱ ἐπὶ τάδε τοῦ σπαρτίου αἱ ἐκ τοῦ <lb></lb>κέντρου.</s></p><p n="28">
<s id="g0131001prop10"><lb></lb>Διὰ τί ῥᾷον, ὅταν ἄνευ βάρους ᾖ, κινεῖται τὸ ζυγόν, <lb></lb>ἢ ἔχον βάρος; </s>
<s id="g0131002">ὁμοίως δὲ καὶ τροχὸς ἢ ἄλλο τοιοῦτο τὸ <lb></lb>βαρύτερον μὲν μεῖζον δὲ τοῦ ἐλάττονος καὶ κουφοτέρου.</s>
<s id="g0131003">ἢ <lb></lb>ὅτι οὐ μόνον εἰς τοὐναντίον τὸ βαρύ, ἀλλὰ καὶ εἰς τὸ πλάγιον <lb></lb>δυσκίνητόν ἐστιν.</s>
<s id="g0131004">ἐναντίον γὰρ τῇ ῥοπῇ κινῆσαι χαλεπῶς, <lb></lb>ἐφ&#039; ὃ δὲ ῥέπει, ῥᾳδίως· εἰς δὲ τὸ πλάγιον οὐ ῥέπει.</s></p><p n="29">
<s id="g0131101prop11"><lb></lb>Διὰ τί ἐπὶ τῶν σκυτάλων ῥᾷον τὰ φορτία κομίζεται <lb></lb>ἢ ἐπὶ τῶν ἁμαξῶν, ἐχουσῶν τῶν μὲν μεγάλους τροχούς, <lb></lb>τῶν δὲ μικρούς; </s>
<s id="g0131102">ἢ διότι ἐπὶ τῶν σκυτάλων οὐδεμίαν ἔχει <lb></lb>πρόσκοψιν, τὸ δὲ ἐπὶ τῶν ἁμαξῶν τὸν ἄξονα, καὶ προσκόπτει <lb></lb>αὐτῷ· ἔκ τε γὰρ τῶν ἄνωθεν πιέζει αὐτὸν καὶ ἐκ <lb></lb>τῶν πλαγίων.</s>
<s id="g0131103">τὸ δὲ ἐπὶ τῶν σκυτάλων ἐπὶ δύο τούτων κινεῖται, <lb></lb>τῇ τε κάτω χώρᾳ ὑποκειμένῃ καὶ τῷ βάρει τῷ <lb></lb>ἐπικειμένῳ· ἐπ&#039; ἀμφοτέρων γὰρ τούτων κυλίεται τῶν τόπων <lb></lb>ὁ κύκλος καὶ φερόμενος ὠθεῖται.</s></p><p n="30">
<s id="g0131201prop12"><lb></lb>Διὰ τί πορρωτέρω τὰ βέλη φέρεται ἀπὸ τῆς σφενδόνης <lb></lb>ἢ ἀπὸ τῆς χειρός; καίτοι κρατεῖ γε ὁ βάλλων τῇ χειρὶ<pb xlink:href="080/01/012.jpg" ed="Bekker" n="852b"></pb><lb></lb> μᾶλλον ἢ ἀπαρτήσας τὸ βάρος.</s>
<s id="g0131202">καὶ ἔτι οὕτω μὲν δύο βάρη <lb></lb>κινεῖ, τό τε τῆς σφενδόνης καὶ τὸ βέλος, ἐκείνως δὲ τὸ <lb></lb>βέλος μόνον.</s>
<s id="g0131203">πότερον ὅτι ἐν μὲν τῇ σφενδόνῃ κινούμενον τὸ <lb></lb>βέλος ῥίπτει ὁ βάλλων [1περιαγαγὼν γὰρ κύκλῳ πολλάκις <lb></lb>ἀφίησιν]1, </s>
<s id="g0131204">ἐκ δὲ τῆς χειρὸς ἀπὸ τῆς ἠρεμίας ἡ ἀρχή· <lb></lb>πάντα δὲ εὐκινητότερα κινούμενα ἢ ἠρεμοῦντα.</s>
<s id="g0131205">ἢ διά τε <lb></lb>τοῦτο, καὶ διότι ἐν μὲν τῷ σφενδονᾶν ἡ μὲν χεὶρ γίνεται <lb></lb>κέντρον, ἡ δὲ σφενδόνη ἡ ἐκ τοῦ κέντρου· ὅσῳ ἂν ᾖ μείζων <lb></lb>ἡ ἀπὸ τοῦ κέντρου, κινεῖται θᾶττον. ἡ δὲ ἀπὸ τῆς χειρὸς <lb></lb>βολὴ πρὸς τὴν σφενδόνην βραχεῖα ἐστίν.</s></p><p n="31">
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<s id="g0131301prop13"><lb></lb>Διὰ τί ῥᾷον κινοῦνται περὶ τὸ αὐτὸ ζυγὸν οἱ μείζους <lb></lb>τῶν ἐλαττόνων κόλλοπες, καὶ οἱ αὐτοὶ ὄνοι οἱ λεπτότεροι <lb></lb>ὑπὸ τῆς αὐτῆς ἰσχύος τῶν παχυτέρων; </s>
<s id="g0131302">ἢ διότι ὁ μὲν ὄνος <lb></lb>καὶ τὸ ζυγὸν κέντρον ἐστίν, τὰ δὲ ἀπέχοντα μεγέθη αἱ ἐκ <lb></lb>τοῦ κέντρου; θᾶττον δὲ κινοῦνται καὶ πλέον ἀπὸ τῆς αὐτῆς <lb></lb>ἰσχύος αἱ τῶν μειζόνων κύκλων ἢ αἱ τῶν ἐλαττόνων· ὑπὸ <lb></lb>τῆς αὐτῆς γὰρ ἰσχύος θᾶττον μεθίσταται τὸ ἄκρον τὸ πορρώτερον <lb></lb>τοῦ κέντρου.</s>
<s id="g0131303">διὸ πρὸς μὲν τὸ ζυγὸν τοὺς κόλλοπας <lb></lb>ὄργανα ποιοῦνται, οἷς ῥᾷον στρέφουσιν· ἐν δὲ τοῖς λεπτοῖς <lb></lb>ὄνοις πλεῖον γίνεται τὸ ἔξω τοῦ ξύλου, αὕτη δὲ γίνεται <lb></lb>ἡ ἐκ τοῦ κέντρου.</s></p><p n="32">
<s id="g0131401prop14"><lb></lb>Διὰ τί τὸ αὐτὸ μέγεθος ξύλον ῥᾷον κατεάσσεται περὶ <lb></lb>τὸ γόνυ, ἐὰν ἴσον ἀποστήσας τῶν ἄκρων ἐχόμενος καταγνύῃ, <lb></lb>ἢ παρὰ τὸ γόνυ ἐγγὺς ὄντος· καὶ ἐὰν πρὸς τὴν γῆν <lb></lb>ἐρείσας καὶ τῷ ποδὶ προσβὰς πόρρωθεν τῇ χειρὶ καταγνύῃ, <lb></lb>ἢ ἐγγύθεν; ἢ διότι ἔνθα μὲν τὸ γόνυ κέντρον, ἔνθα δὲ ὁ <lb></lb>πούς.</s>
<s id="g0131402">ὅσῳ δ&#039; ἂν πορρώτερον ᾖ τοῦ κέντρου, ῥᾷον κινεῖται <lb></lb>ἅπαν. κινηθῆναι δὲ ἀνάγκη καταγνύμενον.</s></p><p n="33">
<s id="g0131501prop15"><lb></lb>Διὰ τί περὶ τοὺς αἰγιαλοὺς αἱ καλούμεναι κρόκαι στρογγύλαι <lb></lb>εἰσίν, ἐκ μακρῶν τῶν λίθων καὶ ὀστράκων τὸ ἐξ <lb></lb>ὑπαρχῆς ὄντων; </s>
<s id="g0131502">ἢ διότι τὰ πλεῖον ἀπέχοντα τοῦ μέσου ἐν <lb></lb>ταῖς κινήσεσι θᾶττον φέρεται.</s>
<s id="g0131503">τὸ μὲν γὰρ μέσον γίνεται <lb></lb>κέντρον, τὸ δὲ διάστημα ἡ ἐκ τοῦ κέντρου.</s>
<s id="g0131504">ἀεὶ δὲ ἡ μείζων <lb></lb>ἀπὸ τῆς ἴσης κινήσεως μείζω γράφει κύκλον. τὸ δ&#039; ἐν <lb></lb>ἴσῳ χρόνῳ μείζω διεξιὸν θᾶττον φέρεται. τὰ δὲ φερόμενα <lb></lb>θᾶττον ἐκ τοῦ ἴσου ἀποστήματος σφοδρότερον τύπτει. τὰ δὲ <lb></lb>τύπτοντα μᾶλλον καὶ αὐτὰ τύπτεται μᾶλλον.</s>
<s id="g0131505">ὥστε ἀνάγκη <lb></lb>θραύεσθαι αἰεὶ τὰ πλέον ἀπέχοντα τοῦ μέσου. τοῦτο δὲ <lb></lb>πάσχοντα ἀνάγκη γίνεσθαι περιφερῆ.</s>
<s id="g0131506">ταῖς δὲ κρόκαις διὰ<pb xlink:href="080/01/013.jpg" ed="Bekker" n="853a"></pb><lb></lb> τὴν τῆς θαλάττης κίνησιν, διὰ τὸ μετὰ τῆς θαλάττης κινεῖσθαι, <lb></lb>συμβαίνει ἀεὶ ἐν κινήσει εἶναι καὶ κυλιομέναις <lb></lb>προσκόπτειν.</s>
<s id="g0131507">τοῦτο δὲ ἀνάγκη μάλιστα συμβαίνειν αὐτοῖς <lb></lb>τοῖς ἄκροις.</s></p><p n="34">
<s id="g0131601prop16"><lb></lb>Διὰ τί, ὅσῳ ἂν ᾖ μακρότερα τὰ ξύλα, τοσούτῳ ἀσθενέστερα <lb></lb>γίνεται, καὶ κάμπτεται αἰρόμενα μᾶλλον, κἂν ᾖ <lb></lb>τὸ μὲν βραχύ, ὅσον δίπηχυ, λεπτόν, τὸ δὲ ἑκατὸν πηχῶν <lb></lb>παχύ; </s>
<s id="g0131602">ἢ διότι μοχλὸς γίνεται καὶ βάρος καὶ ὑπομόχλιον <lb></lb>ἐν τῷ αἴρεσθαι τοῦ ξύλου τὸ μῆκος; </s>
<s id="g0131603">τὸ μὲν γὰρ πρῶτον μέρος <lb></lb>αὐτοῦ, ὃ ἡ χεὶρ αἴρει, οἷον ὑπομόχλιον γίνεται, τὸ δ&#039; <lb></lb>ἐπὶ τῷ ἄκρῳ βάρος.</s>
<s id="g0131604">ὥστε ὅσῳ ἂν ᾖ μακρότερον τὸ ἀπὸ τοῦ <lb></lb>ὑπομοχλίου, τοσούτῳ ἀνάγκη κάμπτεσθαι μᾶλλον· ὅσῳ <lb></lb>γὰρ ἂν πλέον ἀπέχῃ τοῦ ὑπομοχλίου, τοσούτῳ ἀνάγκη <lb></lb>κάμπτεσθαι μεῖζον.</s>
<s id="g0131605">ἀνάγκη οὖν αἴρεσθαι τὰ ἄκρα τοῦ <lb></lb>μοχλοῦ.</s>
<s id="g0131606">ἐὰν οὖν ᾖ καμπτόμενος ὁ μοχλός, ἀνάγκη αὐτὸν <lb></lb>κάμπτεσθαι μᾶλλον αἰρόμενον. ὅπερ συμβαίνει ἐπὶ τῶν <lb></lb>ξύλων τῶν μακρῶν· ἐν δὲ τοῖς βραχέσιν ἐγγὺς τὸ ἔσχατον <lb></lb>τοῦ ὑπομοχλίου γίνεται τοῦ ἠρεμοῦντος.</s></p><p n="35">
<s id="g0131701prop17"><lb></lb>Διὰ τί τῷ σφηνὶ ὄντι μικρῷ μεγάλα βάρη διίσταται <lb></lb>καὶ μεγέθη σωμάτων, καὶ θλῖψις ἰσχυρὰ γίνεται; </s>
<s id="g0131702">ἢ διότι <lb></lb>ὁ σφὴν δύο μοχλοί εἰσιν ἐναντίοι ἀλλήλοις, ἔχει δὲ ἑκάτερος <lb></lb>τὸ μὲν βάρος τὸ δὲ ὑπομόχλιον, ὃ καὶ ἀνασπᾷ ἢ <lb></lb>πιέζει.</s>
<s id="g0131703">ἔτι δὲ ἡ τῆς πληγῆς φορὰ τὸ βάρος, ὃ τύπτει καὶ <lb></lb>κινεῖ, ποιεῖ μέγα· </s>
<s id="g0131704">καὶ διὰ τὸ κινούμενον κινεῖν τῇ ταχυτῆτι <lb></lb>ἰσχύει ἔτι πλέον. μικρῷ δὲ ὄντι μεγάλαι δυνάμεις <lb></lb>ἀκολουθοῦσι· διὸ λανθάνει κινῶν παρὰ τὴν ἀξίαν τοῦ μεγέθους.</s>
<s id="g0131705"><lb></lb>ἔστω σφὴν ἐφ&#039; ᾧ ΑΒΓ, τὸ δὲ σφηνούμενον ΔΕΗΖ. <lb></lb>μοχλὸς δὴ γίνεται ἡ ΑΒ, βάρος δὲ τὸ τοῦ Β κάτωθεν, <lb></lb>ὑπομόχλιον δὲ τὸ ΖΔ. ἐναντίος δὲ τούτῳ μοχλὸς τὸ ΒΓ.</s>
<s id="g0131706"><lb></lb>ἡ δὲ ΑΓ κοπτομένη ἑκατέρᾳ τούτων χρῆται μοχλῷ· ἀνασπᾷ <lb></lb>γὰρ τὸ Β.</s>
<figure id="id.080.01.013.1.jpg" xlink:href="080/01/013/1.jpg"></figure></p><p n="36">
<figure id="id.080.01.013.2.jpg" xlink:href="080/01/013/2.jpg"></figure>
<s id="g0131801prop18"><lb></lb>Διὰ τί, ἐάν τις δύο τροχιλέας ποιήσας ἐπὶ δυσὶ ξύλοις <lb></lb>συμβάλλουσιν ἑαυτοῖς ἐναντίως αὑταῖς κύκλῳ περιβάλῃ <lb></lb>καλώδιον, ἔχον τὸ ἄρτημα ἐκ θατέρου τῶν ξύλων, <lb></lb>θάτερον δὲ ᾖ προσερηρεισμένον ἢ προστεθειμένον κατὰ τὰς <lb></lb>τροχαλίας, ἐὰν ἕλκῃ τις τῇ ἀρχῇ τοῦ καλωδίου, μεγάλα <lb></lb>βάρη προσάγει, κἂν ᾖ μικρὰ ἡ ἕλκουσα ἰσχύς; </s>
<s id="g0131802">ἢ διότι τὸ <lb></lb>αὐτὸ βάρος ἀπὸ ἐλάττονος ἰσχύος, εἰ μοχλεύεται, ἐγείρεται, <lb></lb>ἢ ἀπὸ χειρός; ἡ δὲ τροχιλέα τὸ αὐτὸ ποιεῖ τῷ μο-<pb xlink:href="080/01/014.jpg" ed="Bekker" n="853b"></pb><lb></lb>χλῷ, ὥστε ἡ μία ῥᾷον ἕλξει, καὶ ἀπὸ μιᾶς ὁλκῆς τοῦ <lb></lb>κατὰ χεῖρα πολὺ ἕλξει βαρύτερον. τοῦτο δ&#039; αἱ δύο τροχαλίαι <lb></lb>πλέον ἢ διπλασίῳ τάχει αἴρουσαι.</s>
<s id="g0131803">ἔλαττον γὰρ <lb></lb>ἔτι ἡ ἑτέρα ἕλκει ἢ εἰ αὐτὴ καθ&#039; ἑαυτὴν εἷλκεν, ὅταν <lb></lb>παρὰ τῆς ἑτέρας ἐπιβληθῇ τὸ σχοινίον· ἐκείνη γὰρ ἔτι <lb></lb>ἔλαττον ἐποίησε τὸ βάρος.</s>
<s id="g0131804">καὶ οὕτως ἐὰν εἰς πλείους ἐπιβάλληται <lb></lb>τὸ καλώδιον, ἐν ὀλίγαις τροχιλέαις πολλὴ γίνεται <lb></lb>διαφορά, ἢ ὥστε ὑπὸ τῆς πρώτης τοῦ βάρους ἕλκοντος <lb></lb>τέτταρας μνᾶς, ὑπὸ τῆς τελευταίας ἕλκεσθαι πολλῷ <lb></lb>ἐλάττω.</s>
<s id="g0131805">καὶ ἐν τοῖς οἰκοδομικοῖς ἔργοις ῥᾳδίως κινοῦσι μεγάλα <lb></lb>βάρη· μεταφέρουσι γὰρ ἀπὸ τῆς αὐτῆς τροχιλέας <lb></lb>ἐφ&#039; ἑτέραν, καὶ πάλιν ἀπ&#039; ἐκείνης εἰς ὄνους καὶ μοχλούς· <lb></lb>τοῦτο δὲ ταὐτόν ἐστι τῷ ποιεῖν πολλὰς τροχιλέας.</s></p><p n="37">
<s id="g0131901prop19"><lb></lb>Διὰ τί, ἐὰν μέν τις ἐπιθῇ ἐπὶ τὸ ξύλον πέλεκυν μέγαν <lb></lb>καὶ φορτίον μέγα ἐπ&#039; αὐτῷ, οὐ διαιρεῖ τὸ ξύλον, ὅ τι καὶ <lb></lb>λόγου ἄξιον· ἐὰν δὲ ἄρας τὸν πέλεκύν τις πατάξῃ αὐτῷ, <lb></lb>διασχίζει, ἔλαττον βάρος ἔχοντος τοῦ τύπτοντος πολὺ μᾶλλον <lb></lb>ἢ τοῦ ἐπικειμένου καὶ πιεζοῦντος; </s>
<s id="g0131902">ἢ διότι πάντα τῇ κινήσει <lb></lb>ἐργάζεται, καὶ τὸ βαρὺ τὴν τοῦ βάρους κίνησιν λαμβάνει <lb></lb>μᾶλλον κινούμενον ἢ ἠρεμοῦν; </s>
<s id="g0131903">ἐπικείμενον οὖν οὐ κινεῖται τὴν <lb></lb>τοῦ βάρους κίνησιν, φερόμενον δὲ ταύτην τε καὶ τὴν τοῦ <lb></lb>τύπτοντος.</s>
<s id="g0131904">ἔτι δὲ καὶ γίνεται σφὴν ὁ πέλεκυς· ὁ δὲ σφὴν <lb></lb>μικρὸς ὢν μεγάλα διίστησι διὰ τὸ εἶναι ἐκ δύο μοχλῶν <lb></lb>ἐναντίως συγκειμένων.</s></p><p n="38">
<figure id="id.080.01.014.1.jpg" xlink:href="080/01/014/1.jpg"></figure>
<s id="g0132001prop20"><lb></lb>Διὰ τί αἱ φάλαγγες τὰ κρέα ἱστᾶσιν ἀπὸ μικροῦ ἀρτήματος <lb></lb>μεγάλα βάρη, τοῦ ὅλου ἡμιζυγίου ὄντος; οὗ μὲν γὰρ <lb></lb>τὸ βάρος ἐντίθεται, κατήρτηται μόνον ἡ πλάστιγξ, ἐπὶ θάτερον <lb></lb>δὲ ἡ φάλαγξ ἐστὶ μόνον.</s>
<s id="g0132002">ἢ ὅτι ἅμα συμβαίνει ζυγὸν <lb></lb>καὶ μοχλὸν εἶναι τὴν φάλαγγα; ζυγὸν μὲν γὰρ, ᾗ <lb></lb>τῶν σπαρτίων ἕκαστον γίνεται τὸ κέντρον τῆς φάλαγγος. τὸ <lb></lb>μὲν οὖν ἐπὶ θάτερα ἔχει πλάστιγγα, τὸ δὲ ἐπὶ θάτερα ἀντὶ <lb></lb>τῆς πλάστιγγος τὸ σφαίρωμα, ὃ τῷ ζυγῷ ἔγκειται, ὥσπερ <lb></lb>εἴ τις τὴν ἑτέραν πλάστιγγα καὶ τὸν σταθμὸν ἐπιθείη ἐπὶ τὸ <lb></lb>ἄκρον τῆς πλάστιγγος· </s>
<s id="g0132003">δῆλον γὰρ ὅτι ἕλκει τοσοῦτον βάρος <lb></lb>ἐν τῇ ἑτέρᾳ κείμενον πλάστιγγι.</s>
<s id="g0132004">ὅπως δὲ τὸ ἓν ζυγὸν πολλὰ <lb></lb>ᾖ ζυγά, τοιαῦτα τὰ σπαρτία πολλὰ ἔγκειται ἐν τῷ τοιούτῳ <lb></lb>ζυγῷ, ὧν ἑκάστου τὸ ἐπὶ τάδε ἐπὶ τὸ σφαίρωμα τὸ ἥμισυ <lb></lb>τῆς φάλαγγός ἐστι, καὶ ὁ σταθμὸς δι&#039; ἴσου τῶν ἀπ&#039; ἀλλήλων <lb></lb>τῶν σπαρτίων κινουμένων, ὥστε συμμετρεῖσθαι πόσον βάροσ<pb xlink:href="080/01/015.jpg" ed="Bekker" n="854a"></pb><lb></lb> ἕλκει τὸ ἐν τῇ πλάστιγγι κείμενον· ὥστε γινώσκειν, ὅταν <lb></lb>ὀρθὴ ἡ φάλαγξ ᾖ, ἀπὸ ποίου σπάρτου πόσον βάρος ἔχει ἡ <lb></lb>πλάστιγξ, καθάπερ εἴρηται.</s>
<s id="g0132005">ὅλως μέν ἐστι τοῦτο ζυγόν, ἔχον <lb></lb>μίαν μὲν πλάστιγγα, ἐν ᾗ ἵσταται τὸ βάρος, τὴν δ&#039; ἑτέραν, <lb></lb>ἐν ᾗ τὸ σταθμὸν ἐν τῇ φάλαγγι.</s>
<s id="g0132006">διὸ σφαίρωμά ἐστιν ἡ <lb></lb>φάλαγξ ἐπὶ θάτερον. τοιοῦτον δὲ ὂν πολλὰ ζυγά ἐστι, καὶ <lb></lb>τοσαῦτα ὅσαπέρ ἐστι τὰ σπαρτία.</s>
<s id="g0132007">ἀεὶ δὲ τὸ ἐγγύτερον <lb></lb>σπαρτίον τῆς πλάστιγγος καὶ τοῦ ἱσταμένου βάρους μεῖζον ἕλκει <lb></lb>βάρος, διὰ τὸ γίνεσθαι τὴν μὲν φάλαγγα πᾶσαν μοχλὸν <lb></lb>ἀνεστραμμένον [1ὑπομόχλιον μὲν γὰρ τὸ σπαρτίον <lb></lb>ἕκαστον ἄνωθεν ὄν, τὸ δὲ βάρος τὸ ἐνὸν ἐν τῇ πλάστιγγι]1, </s>
<s id="g0132008"><lb></lb>ὅσῳ δ&#039; ἂν μακρότερον ᾖ τὸ μῆκος τοῦ μοχλοῦ τοῦ ἀπὸ τοῦ <lb></lb>ὑπομοχλίου, τοσούτῳ ἐκεῖ μὲν ῥᾷον κινεῖ, ἐνταῦθα δὲ σήκωμα <lb></lb>ποιεῖ, καὶ ἵστησι τὸ πρὸς τὸ σφαίρωμα βάρος τῆς <lb></lb>φάλαγγος.</s></p><p n="39">
<s id="g0132101prop21"><lb></lb>Διὰ τί οἱ ἰατροὶ ῥᾷον ἐξαιροῦσι τοὺς ὀδόντας προσλαμβάνοντες <lb></lb>βάρος τὴν ὀδοντάγραν ἢ τῇ χειρὶ μόνῃ ψιλῇ; </s>
<s id="g0132102"><lb></lb>πότερον διὰ τὸ μᾶλλον ἐξολισθαίνειν διὰ τῆς χειρὸς τὸν <lb></lb>ὀδόντα ἢ ἐκ τῆς ὀδοντάγρας; ἢ μᾶλλον ὀλισθαίνει τῆς <lb></lb>χειρὸς ὁ σίδηρος, καὶ οὐ περιλαμβάνει αὐτὸν κύκλῳ· μαλθακὴ <lb></lb>γὰρ οὖσα ἡ σὰρξ τῶν δακτύλων καὶ προσμένει μᾶλλον <lb></lb>καὶ περιαρμόττει.</s>
<s id="g0132103">ἀλλ&#039; ὅτι ἡ ὀδοντάγρα δύο μοχλοί <lb></lb>εἰσιν ἀντικείμενοι, ἓν τὸ ὑπομόχλιον ἔχοντες τὴν σύναψιν <lb></lb>τῆς θερμαστρίδος· </s>
<s id="g0132104">τοῦ ῥᾷον οὖν κινῆσαι χρῶνται τῷ ὀργάνῳ <lb></lb>πρὸς τὴν ἐξαίρεσιν.</s>
<s id="g0132105">ἔστω γὰρ τῆς ὀδοντάγρας τὸ μὲν ἕτερον <lb></lb>ἄκρον ἐφ&#039; ᾧ τὸ Α, τὸ δὲ ἕτερον, τὸ Β, ὃ ἐξαιρεῖ· ὁ <lb></lb>δὲ μοχλὸς ἐφ&#039; ᾧ ΑΔΖ, ὁ δὲ ἄλλος μοχλὸς ἐφ&#039; ᾧ Β <lb></lb>ΓΕ, ὑπομόχλιον δὲ τὸ ΓΘΔ· ὁ δὲ ὀδοὺς ἐφ&#039; οὗ Ι σύναψις· <lb></lb>ὁ δὲ τὸ βάρος.</s>
<figure id="id.080.01.015.1.jpg" xlink:href="080/01/015/1.jpg"></figure>
<s id="g0132106">ἑκατέρῳ οὖν τῶν ΒΖ καὶ ἅμα λαβὼν <lb></lb>κινεῖ. ὅταν δὲ κινήσῃ, ἐξεῖλε ῥᾷον τῇ χειρὶ ἢ τῷ <lb></lb>ὀργάνῳ.</s></p><p n="40">
<s id="g0132201prop22"><lb></lb>Διὰ τί τὰ κάρυα ῥᾳδίως καταγνύουσιν ἄνευ πληγῆς ἐν <lb></lb>τοῖς ὀργάνοις ἃ ποιοῦσι πρὸς τὸ καταγνύναι αὐτά; πολλὴ <lb></lb>γὰρ ἀφαιρεῖται ἰσχὺς ἡ τῆς φορᾶς καὶ βίας. ἔτι δὲ σκληρῷ <lb></lb>καὶ βαρεῖ συνθλίβων θᾶττον ἂν κατάξαι ἢ ξυλίνῳ καὶ κούφῳ <lb></lb>τῷ ὀργάνῳ. ἢ διότι οὕτως ἐπ&#039; ἀμφότερα θλίβεται ὑπὸ δύο <lb></lb>μοχλῶν τὸ κάρυον, τῷ δὲ μοχλῷ ῥᾳδίως διαιρεῖται τὰ <lb></lb>βάρη; </s>
<s id="g0132202">τὸ γὰρ ὄργανον ἐκ δύο σύγκειται μοχλῶν, ὑπομόχλιον <lb></lb>ἐχόντων τὸ αὐτό, τὴν συναφὴν ἐφ&#039; ἧς τὸ Α.</s>
<s id="g0132203">ὥσπερ<pb xlink:href="080/01/016.jpg" ed="Bekker" n="854b"></pb><lb></lb> οὖν εἰ ἦσαν ἐκβεβλημέναι, ὑφ&#039; ὧν κινουμένων εἰς τὰ τῶν <lb></lb>ΓΔ ἄκρα αἱ ΕΖ συνήγοντο ῥᾳδίως ἀπὸ μικρᾶς ἰσχύος· </s>
<figure id="id.080.01.016.1.jpg" xlink:href="080/01/016/1.jpg"></figure>
<s id="g0132204"><lb></lb>ἣν οὖν ἐν τῇ πληγῇ τὸ βάρος ἐποίει, ταύτην ἡ κρείττων ταύτης, <lb></lb>ἡ τὸ ΕΓ καὶ ΖΔ, μοχλοὶ ὄντες ποιοῦσι· τῇ ἄρσει γὰρ <lb></lb>εἰς τοὐναντίον αἴρονται, καὶ θλίβοντες καταγνύουσι τὸ ἐφ&#039; ᾧ Κ.</s>
<s id="g0132205"><lb></lb>δι&#039; αὐτὸ δὲ τοῦτο καὶ ὅσῳ ἂν ἐγγύτερον ᾖ τῆς Α τὸ Κ, συντρίβεται <lb></lb>θᾶττον· ὅσῳ γὰρ ἂν πλεῖον ἀπέχῃ τοῦ ὑπομοχλίου <lb></lb>ὁ μοχλός, ῥᾷον κινεῖ καὶ πλεῖον ἀπὸ τῆς ἰσχύος τῆς αὐτῆς.</s>
<s id="g0132206"><lb></lb>ἔστιν οὖν τὸ μὲν Α ὑπομόχλιον, ἡ δὲ ΔΑΖ μοχλός, καὶ ἡ <lb></lb>ΓΑΕ.</s>
<s id="g0132207">ὅσῳ ἂν οὖν τὸ Κ ἐγγυτέρω ᾖ τῆς γωνίας τῶν Α, <lb></lb>τοσούτῳ ἐγγύτερον γίνεται τῆς συναφῆς τῶν Α· τοῦτο δέ ἐστι <lb></lb>τὸ ὑπομόχλιον.</s>
<s id="g0132208">ἀνάγκη τοίνυν ἀπὸ τῆς αὐτῆς ἰσχύος συναγούσης <lb></lb>τὸ ΖΕ αἴρεσθαι πλέον.</s>
<s id="g0132209">ὥστε ἐπεί ἐστιν ἐξ ἐναντίας <lb></lb>ἡ ἄρσις, ἀνάγκη θλίβεσθαι μᾶλλον· τὸ δὲ μᾶλλον θλιβόμενον <lb></lb>κατάγνυται θᾶττον.</s></p><p n="41">
<s id="g0132301prop23"><lb></lb>Διὰ τί φερομένων δύο φορὰς ἐν τῷ ῥόμβῳ τῶν ἄκρων <lb></lb>σημείων ἀμφοτέρων, οὐ τὴν ἴσην ἑκάτερον αὐτῶν εὐθεῖαν διέρχεται, <lb></lb>ἀλλὰ πολλαπλασίαν θάτερον; </s>
<s id="g0132302">ὁ αὐτὸς δὲ λόγος καὶ <lb></lb>διὰ τί τὸ ἐπὶ τῆς πλευρᾶς φερόμενον ἐλάττω διέρχεται τῆς <lb></lb>πλευρᾶς. τὸ μὲν γὰρ τὴν διάμετρον τὴν ἐλάττω, ἡ δὲ τὴν <lb></lb>πλευρὰν τὴν μείζω, καὶ ἡ μὲν μίαν, τὸ δὲ δύο φέρεται <lb></lb>φοράς.</s>
<s id="g0132303">φερέσθω γὰρ ἐπὶ τῆς ΑΒ τὸ μὲν Α πρὸς τὸ Β, τὸ <lb></lb>δὲ Β πρὸς τὸ Δ τῷ αὐτῷ τάχει· φερέσθω δὲ καὶ ἡ ΑΒ <lb></lb>ἐπὶ τῆς ΑΓ παρὰ τὴν ΓΔ τῷ αὐτῷ τάχει τούτοις.</s>
<figure id="id.080.01.016.2.jpg" xlink:href="080/01/016/2.jpg"></figure>
<s id="g0132304">ἀνάγκη <lb></lb>δὴ τὸ μὲν Α ἐπὶ τῆς ΑΔ διαμέτρου φέρεσθαι, τὸ δὲ Β ἐπὶ <lb></lb>τῆς ΒΓ, καὶ ἅμα διεληλυθέναι ἑκατέραν, καὶ τὴν ΑΒ τὴν <lb></lb>ΑΓ πλευράν.</s>
<s id="g0132305">ἐνηνέχθω γὰρ τὸ μὲν Α τὴν ΑΕ, ἡ δὲ Α <lb></lb>Β τὴν ΑΖ, καὶ ἔστω ἐκβεβλημένη ἡ ΖΗ παρὰ τὴν ΑΒ, <lb></lb>καὶ ἀπὸ τοῦ Ε πεπληρώσθω.</s>
<s id="g0132306">ὅμοιον οὖν γίνεται τὸ παραπληρωθὲν <lb></lb>τῷ ὅλῳ.</s>
<s id="g0132307">ἴση ἄρα ἡ ΑΖ τῇ ΑΕ, ὥστε τὸ Α <lb></lb>ἐπὶ τῆς πλευρᾶς ἐνήνεκται τῆς ΑΕ. ἡ δὲ ΑΒ τὴν ΑΖ <lb></lb>εἴη ἂν ἐνηνεγμένη. ἔσται ἄρα ἐπὶ τῆς διαμέτρου κατὰ τὸ Θ.</s>
<s id="g0132308"><lb></lb>καὶ αἰεὶ δὲ ἀνάγκη αὐτὸ φέρεσθαι κατὰ τὴν διάμετρον. <lb></lb>καὶ ἅμα ἡ πλευρὰ ἡ ΑΒ τὴν πλευρὰν τὴν ΑΓ δίεισι, <lb></lb>καὶ τὸ Α τὴν διάμετρον δίεισι τὴν ΑΔ.</s>
<s id="g0132309">ὁμοίως δὲ δειχθήσεται <lb></lb>καὶ τὸ Β ἐπὶ τῆς ΑΓ διαμέτρου φερόμενον. ἴση <lb></lb>γάρ ἐστιν ἡ ΒΕ τῇ ΒΗ.</s>
<s id="g0132310">παραπληρωθέντος οὖν ἀπὸ τοῦ Η, <lb></lb>ὅμοιόν ἐστι τῷ ὅλῳ τὸ ἐντός. καὶ τὸ Β ἐπὶ τῆς διαμέτρου <lb></lb>ἔσται κατὰ τὴν σύναψιν τῶν πλευρῶν, καὶ ἅμα δίεισιν ἥ<pb xlink:href="080/01/017.jpg" ed="Bekker" n="855a"></pb><lb></lb> τε πλευρὰ τὴν πλευρὰν καὶ τὸ Β τὴν ΒΓ διάμετρον.</s>
<s id="g0132311"><lb></lb>ἅμα ἄρα καὶ τὸ Β τὴν πολλαπλασίαν τῆς ΑΒ δίεισι <lb></lb>καὶ ἡ πλευρὰ τὴν ἐλάττονα πλευράν, τῷ αὐτῷ τάχει φερόμενα, <lb></lb>καὶ ἡ πλευρὰ μείζω τοῦ Α διελήλυθε μίαν φορὰν <lb></lb>φερομένη.</s>
<s id="g0132312">ὅσῳ γὰρ ἂν ὀξύτερος γένηται ὁ ῥόμβος, ἡ <lb></lb>μὲν διάμετρος ἡ ἐλάττων γίνεται, ἡ δὲ ΒΓ μείζων, ἡ δὲ <lb></lb>πλευρὰ τῆς ΒΓ ἐλάττων.</s>
<s id="g0132313">ἄτοπον γάρ, ὥσπερ ἐλέχθη, τὸ <lb></lb>δύο φορὰς φερόμενον ἐνίοτε βραδύτερον φέρεσθαι τοῦ μίαν, <lb></lb>καὶ ἀμφοτέρων ἰσοταχῶν σημείων δοθέντων μείζω διεξιέναι <lb></lb>θάτερον.</s>
<s id="g0132314">αἴτιον δὲ ὅτι τοῦ μὲν ἀπὸ τῆς ἀμβλείας φερομένου <lb></lb>σχεδὸν ἐναντίαι ἀμφότεραι γίνονται, ἥν τε αὐτὴ <lb></lb>φέρεται καὶ ἣν ὑπὸ τῆς πλευρᾶς ὑποφέρεται, </s>
<s id="g0132315">τοῦ δὲ ἀπὸ <lb></lb>τῆς ὀξείας συμβαίνει φέρεσθαι ἐπὶ τὸ αὐτό. συνεπουρίζει <lb></lb>γὰρ ἡ τῆς πλευρᾶς τὴν ἐπὶ τῆς διαμέτρου· καὶ ὅσῳ ἂν <lb></lb>τὴν μὲν ὀξυτέραν ποιήσῃ, τὴν δὲ ἀμβλυτέραν, ἡ μὲν βραδυτέρα <lb></lb>ἔσται, ἡ δὲ θάττων.</s>
<s id="g0132316">αἱ μὲν γὰρ ἐναντιώτεραι γίνονται <lb></lb>διὰ τὸ ἀμβλυτέραν γίνεσθαι τὴν γωνίαν, αἱ δὲ <lb></lb>μᾶλλον ἐπὶ τὰ αὐτὰ διὰ τὸ συνάγεσθαι τὰς γραμμάς. <lb></lb>τὸ μὲν γὰρ Β σχεδὸν ἐπὶ τὸ αὐτὸ φέρεται κατ&#039; ἀμφοτέρας <lb></lb>τὰς φοράς· </s>
<s id="g0132317">συνεπουρίζεται οὖν ἡ ἑτέρα, καὶ ὅσῳ ἂν <lb></lb>ὀξυτέρα γίνηται ἡ γωνία, τοσούτῳ μᾶλλον. τὸ Α δὲ ἐπὶ <lb></lb>τοὐναντίον· αὐτὸ μὲν γὰρ πρὸς τὸ Β φέρεται, ἡ δὲ πλευρὰ <lb></lb>ὑποφέρει αὐτὸ πρὸς τὸ Δ.</s>
<s id="g0132318">καὶ ὅσῳ ἂν ἀμβλυτέρα ἡ γωνία <lb></lb>ᾖ, ἐναντιώτεραι αἱ φοραὶ γίνονται· εὐθυτέρα γὰρ ἡ <lb></lb>γραμμὴ γίνεται.</s>
<s id="g0132319">εἰ δ&#039; ὅλως εὐθεῖα γένοιτο, παντελῶς ἂν <lb></lb>εἴησαν ἐναντίαι. ἡ δὲ πλευρὰ ὑπ&#039; οὐθενὸς κωλύεται μίαν <lb></lb>φερομένη φοράν. εὐλόγως οὖν τὴν μείζω διέρχεται.</s></p><p n="42">
<s id="g0132401prop24"><lb></lb>Ἀπορεῖται διὰ τί ποτε ὁ μείζων κύκλος τῷ ἐλάττονι <lb></lb>κύκλῳ ἴσην ἐξελίττεται γραμμήν, ὅταν περὶ τὸ αὐτὸ κέντρον <lb></lb>τεθῶσι; χωρὶς δὲ ἐκκυλιόμενοι, ὥσπερ τὸ μέγεθος αὐτῶν <lb></lb>πρὸς τὸ μέγεθος ἔχει, οὕτως καὶ αἱ γραμμαὶ αὐτῶν <lb></lb>γίνονται πρὸς ἀλλήλας.</s>
<s id="g0132402">ἔτι δὲ ἑνὸς καὶ τοῦ αὐτοῦ κέντρου <lb></lb>ὄντος ἀμφοῖν, ὁτὲ μὲν τηλικαύτη γίνεται ἡ γραμμὴ ἣν <lb></lb>ἐκκυλίονται, ἡλίκην ὁ ἐλάττων κύκλος καθ&#039; αὑτὸν ἐκκυλίεται, <lb></lb>ὁτὲ δὲ ὅσην ὁ μείζων.</s>
<s id="g0132403">ὅτι μὲν οὖν μείζω ἐκκυλίεται <lb></lb>ὁ μείζων, φανερόν. γωνία μὲν γὰρ δοκεῖ κατὰ τὴν <lb></lb>αἴσθησιν εἶναι ἡ περιφέρεια ἑκάστου τῆς οἰκείας διαμέτρου, <lb></lb>ἡ τοῦ μείζονος κύκλου μείζων, ἡ δὲ τοῦ ἐλάττονος ἐλάττων, <lb></lb>ὥστε τὸν αὐτὸν τοῦτον ἕξουσι λόγον, καθ&#039; ἃς ἐξεκυλίσθησαν<pb xlink:href="080/01/018.jpg" ed="Bekker" n="855b"></pb><lb></lb> αἱ γραμμαὶ πρὸς ἀλλήλας κατὰ τὴν αἴσθησιν.</s>
<s id="g0132404">ἀλλὰ μὴν <lb></lb>καὶ ὅτι τὴν ἴσην ἐκκυλίονται, ὅταν περὶ τὸ αὐτὸ κέντρον <lb></lb>κείμενοι ὦσι, δῆλον· καὶ οὕτως γίνεται ὁτὲ μὲν ἴση τῇ <lb></lb>γραμμῇ ἣν ὁ μείζων κύκλος ἐκκυλίεται, ὁτὲ δὲ ἐλάττων.</s>
<figure id="id.080.01.018.1.jpg" xlink:href="080/01/018/1.jpg"></figure></p><p n="43">
<s id="g0132405"><lb></lb>ἔστω γὰρ κύκλος ὁ μείζων μὲν ἐφ&#039; οὗ τὰ ΔΖΓ, ὁ δὲ <lb></lb>ἐλάττων ἐφ&#039; οὗ τὰ ΕΗΒ, κέντρον δὲ ἀμφοῖν τὸ Α· καὶ <lb></lb>ἣν μὲν ἐξελίττεται καθ&#039; αὑτὸν ὁ μέγας, ἡ ἐφ&#039; ἧς ΖΙ ἔστω, <lb></lb>ἣν δὲ ὁ ἐλάττων καθ&#039; αὑτόν, ἡ ἐφ&#039; ἧς ΗΚ, ἴση τῇ ΑΖ.</s>
<s id="g0132406"><lb></lb>ἐὰν δὴ κινῶ τὸν ἐλάττονα, τὸ αὐτὸ κέντρον κινῶ, ἐφ&#039; οὗ <lb></lb>τὸ Α· ὁ δὲ μέγας προσηρμόσθω. ὅταν οὖν ἡ ΑΒ ὀρθὴ γένηται <lb></lb>πρὸς τὴν ΗΚ, ἅμα καὶ ἡ ΑΓ γίνεται ὀρθὴ πρὸς τὴν <lb></lb>ΖΛ, ὥστε ἔσται ἴσην ἀεὶ διεληλυθυῖα, τὴν μὲν ΗΚ, ἐφ&#039; <lb></lb>ᾧ ΗΒ περιφέρεια, τὴν δὲ ΖΛ ἡ ἐφ&#039; ἧς ΖΓ.</s>
<s id="g0132407">εἰ δὲ τὸ <lb></lb>τέταρτον μέρος ἴσην ἐξελίττεται, δῆλον ὅτι καὶ ὁ ὅλος κύκλος <lb></lb>τῷ ὅλῳ κύκλῳ ἴσην ἐξελιχθήσεται, ὥστε ὅταν ἡ ΒΗ <lb></lb>γραμμὴ ἔλθῃ ἐπὶ τὸ Κ, καὶ ἡ ΖΓ ἔσται περιφέρεια ἐπὶ <lb></lb>τῆς ΖΛ, καὶ ὁ κύκλος ὅλος ἐξειλιγμένος.</s>
<s id="g0132408">ὁμοίως δὲ καὶ <lb></lb>ἐὰν τὸν μέγαν κινῶ, ἐναρμόσας τὸν μικρόν, τοῦ αὐτοῦ κέντρου <lb></lb>ὄντος, ἅμα τῇ ΑΓ ἡ ΑΒ κάθετος καὶ ὀρθὴ ἔσται, ἡ <lb></lb>μὲν πρὸς τὴν ΖΙ, ἡ δὲ πρὸς τὴν ΗΘ.</s>
<s id="g0132409">ὥστε ὅταν ἴσην ἡ <lb></lb>μὲν τῇ ΗΘ ἔσται διεληλυθυῖα, ἡ δὲ τῇ ΖΙ, καὶ γένηται <lb></lb>ὀρθὴ πάλιν ἡ ΖΑ πρὸς τὴν ΖΛ, καὶ ἡ ΑΓ ὀρθὴ πάλιν, <lb></lb>ὡς τὸ ἐξ ἀρχῆς ἔσονται ἐπὶ τῶν ΘΙ.</s>
<s id="g0132410">τὸ δὲ μήτε στάσεως <lb></lb>γινομένης τὸ μεῖζον τῷ ἐλάττονι, ὥστε μένειν τινὰ χρόνον <lb></lb>ἐπὶ τοῦ αὐτοῦ σημείου· κινοῦνται γὰρ συνεχῶς ἄμφω ἀμφοτεράκις. <lb></lb>μὴ ὑπερπηδῶντος τοῦ ἐλάττονος μηθὲν σημεῖον, <lb></lb>τὸν μὲν μείζω τῷ ἐλάττονι ἴσην διεξιέναι, τὸν δὲ τῷ μείζονι, <lb></lb>ἄτοπον.</s>
<s id="g0132411">ἔτι δὲ μιᾶς κινήσεως οὔσης ἀεὶ τὸ κέντρον <lb></lb>τὸ κινούμενον ὁτὲ μὲν τὴν μεγάλην ὁτὲ δὲ τὴν ἐλάττονα <lb></lb>ἐκκυλίεσθαι θαυμαστόν.</s>
<s id="g0132412">τὸ γὰρ αὐτὸ τῷ αὐτῷ τάχει φερόμενον <lb></lb>ἴσην πέφυκε διεξιέναι· τῷ αὐτῷ δὲ τάχει ἴσην ἐστὶ <lb></lb>κινεῖν ἀμφοτεράκις.</s>
<s id="g0132413">ἀρχὴ δὲ ληπτέα ἥδε περὶ τῆς αἰτίας <lb></lb>αὐτῶν, ὅτι ἡ αὐτὴ δύναμις καὶ ἴση τὸ μὲν βραδύτερον <lb></lb>κινεῖ μέγεθος, τὸ δὲ ταχύτερον.</s>
<s id="g0132414">εἰ δή τι εἴη ὃ μὴ πέφυκεν <lb></lb>ὑφ&#039; ἑαυτοῦ κινεῖσθαι, ἐὰν τοῦτο ἅμα καὶ αὐτὸ κινῇ τὸ πεφυκὸς <lb></lb>κινεῖσθαι, βραδύτερον κινηθήσεται ἢ εἰ αὐτὴ καθ&#039; <lb></lb>αὑτὴν ἐκινεῖτο.</s>
<s id="g0132415">καὶ ἐὰν μὲν πεφυκὸς ᾖ κινεῖσθαι, μὴ συγκινῆται <lb></lb>δὲ μηθέν, ὡσαύτως ἕξει.</s>
<s id="g0132416">καὶ ἀδύνατον δὴ κινεῖσθαι <lb></lb>πλέον ἢ τὸ κινοῦν· οὐ γὰρ τὴν αὑτοῦ κινεῖται κίνησιν, ἀλλὰ<pb xlink:href="080/01/019.jpg" ed="Bekker" n="856a"></pb><lb></lb> τὴν τοῦ κινοῦντος.</s>
<s id="g0132417">εἴη δὴ κύκλος ὁ μὲν μείζων τὸ Α, ὁ δὲ <lb></lb>ἐλάττων ἐφ&#039; ᾧ Β. εἰ ὠθοίη δ&#039; ὁ ἐλάττων τὸν μείζω, μὴ <lb></lb>κυλιομένου αὐτοῦ, φανερὸν ὅτι τοσοῦτον δίεισι τῆς εὐθείας <lb></lb>ὁ μείζων, ὅσον ἐώσθη ὑπὸ τοῦ ἐλάττονος. τοσοῦτον δέ γε <lb></lb>ἐώσθη ὅσον ὁ μικρὸς ἐκινήθη. ἴσην ἄρα τῆς εὐθείας διεληλύθασιν.</s>
<s id="g0132418"><lb></lb>ἀνάγκη τοίνυν καὶ εἰ κυλιόμενος ὁ ἐλάττων τὸν <lb></lb>μείζω ὠθοίη, κυλισθῆναι μὲν ἅμα τῇ ὤσει, τοσοῦτον δ&#039; ὅσον <lb></lb>ὁ ἐλάττων ἐκυλίσθη, εἰ μηθὲν αὐτὸς τῇ αὐτῇ κινήσει κινεῖται.</s>
<s id="g0132419"><lb></lb>ὡς γὰρ καὶ ὅσον ἐκίνει, τοσοῦτον κεκινῆσθαι ἀνάγκη <lb></lb>τὸ κινούμενον ὑπ&#039; ἐκείνου. ἀλλὰ μὴν ὅ τε κύκλος τοσοῦτον <lb></lb>ἐκίνησε τὸ αὐτό, κύκλῳ τε καὶ ποδιαίαν [1ἔστω γὰρ τοσοῦτον <lb></lb>ὃ ἐκινήθη]1, καὶ ὁ μέγας ἄρα τοσοῦτον ἐκινήθη.</s>
<s id="g0132420">ὁμοίως <lb></lb>δὲ κἂν ὁ μέγας τὸν μικρὸν κινήσῃ, ἔσται κεκινημένος ὁ μικρὸς <lb></lb>ὡς καὶ ὁ μείζων.</s>
<s id="g0132421">καθ&#039; αὑτὸν μὲν δὴ κινηθεὶς ὁποτεροσοῦν, <lb></lb>ἐάν τε ταχὺ ἐάν τε βραδέως· τῷ αὐτῷ δὲ τάχει <lb></lb>εὐθὺς ὅσην ὁ μείζων πέφυκεν ἐξελιχθῆναι γραμμήν. ὅπερ <lb></lb>καὶ ποιεῖ τὴν ἀπορίαν, ὅτι οὐκέτι ὁμοίως ποιοῦσιν ὅταν συναρμοσθῶσιν. <lb></lb>τὸ δ&#039; ἔστιν, εἰ ὁ ἕτερος ὑπὸ τοῦ ἑτέρου κινεῖται <lb></lb>οὐχ ἣν πέφυκεν, οὐδὲ τὴν αὑτοῦ κίνησιν.</s>
<s id="g0132422">οὐθὲν γὰρ <lb></lb>διαφέρει περιθεῖναι καὶ ἐναρμόσαι ἢ προσθεῖναι ὁποτερονοῦν <lb></lb>ὁποτέρῳ· ὁμοίως γάρ, ὅταν ὁ μὲν κινῇ ὁ δὲ κινῆται ὑπὸ <lb></lb>τούτου, ὅσον ἂν κινῇ ἅτερος, τοσοῦτον κινηθήσεται ἅτερος.</s>
<s id="g0132423"><lb></lb>ὅταν μὲν οὖν προσκείμενον κινῇ ἢ προσκρεμάμενον, οὐκ ἀεὶ <lb></lb>κυλίει τις· ὅταν δὲ περὶ τὸ αὐτὸ κέντρον τεθῶσιν, ἀνάγκη <lb></lb>κυλίεσθαι ἀεὶ τὸν ἕτερον ὑπὸ τοῦ ἑτέρου.</s>
<s id="g0132424">ἀλλ&#039; οὐθὲν ἧττον <lb></lb>οὐ τὴν αὑτοῦ κίνησιν ἅτερος κινεῖται, ἀλλ&#039; ὥσπερ ἂν εἰ μηδεμίαν <lb></lb>εἶχε κίνησιν. κἂν ἔχῃ, μὴ χρῆται δ&#039; αὐτῇ, ταὐτὸ <lb></lb>συμβαίνει.</s>
<s id="g0132425">ὅταν μὲν οὖν ὁ μέγας κινῇ ἐνδεδεμένον τὸν μικρόν, <lb></lb>ὁ μικρὸς κινεῖται ὅσηνπερ οὗτος· ὅταν δὲ ὁ μικρός, <lb></lb>πάλιν ὁ μέγας ὅσην οὗτος. χωριζόμενος δὲ ἑκάτερος αὑτὸν <lb></lb>κινεῖ αὐτός.</s>
<s id="g0132426">ὅτι δὲ τοῦ αὐτοῦ κέντρου ὄντος καὶ κινοῦντος <lb></lb>τῷ αὐτῷ τάχει συμβαίνει ἄνισον διεξιέναι αὐτοὺς γραμμήν, <lb></lb>παραλογίζεται ὁ ἀπορῶν σοφιστικῶς.</s>
<s id="g0132427">τὸ αὐτὸ μὲν <lb></lb>γάρ ἐστι κέντρον ἀμφοῖν, ἀλλὰ κατὰ συμβεβηκός, ὡς <lb></lb>μουσικὸν καὶ λευκόν· τὸ γὰρ εἶναι ἑκατέρου κέντρου τῶν <lb></lb>κύκλων οὐ τῷ αὐτῷ χρῆται.</s>
<s id="g0132428">ὅταν μὲν οὖν ὁ κινῶν ᾖ ὁ <lb></lb>μικρός, ὡς ἐκείνου κέντρον καὶ ἀρχή, ὅταν δὲ ὁ μέγας, ὡς <lb></lb>ἐκείνου.</s>
<s id="g0132429">οὔκουν τὸ αὐτὸ κινεῖ ἁπλῶς, ἀλλ&#039; ἔστιν ὥς.</s></p><p n="44">
<figure id="id.080.01.019.1.jpg" xlink:href="080/01/019/1.jpg"></figure>
<s id="g0132501prop25"><lb></lb>Διὰ τί τὰς κλίνας ποιοῦσι διπλασιοπλεύρους, τὴν μὲν<pb xlink:href="080/01/020.jpg" ed="Bekker" n="856b"></pb><lb></lb> ἓξ ποδῶν καὶ μικρῷ μείζω πλευράν, τὴν δὲ τριῶν; καὶ <lb></lb>διὰ τί ἐντείνουσιν οὐ κατὰ διάμετρον; </s>
<s id="g0132502">ἢ τὸ μὲν μέγεθος τηλικαύτας, <lb></lb>ὅπως τοῖς σώμασιν ὦσι σύμμετροι; γίνονται <lb></lb>γὰρ οὕτω διπλασιόπλευροι, τετραπήχεις μὲν τὸ μῆκος, διπήχεις <lb></lb>δὲ τὸ πλάτος.</s>
<s id="g0132503">ἐντείνουσι δὲ οὐ κατὰ διάμετρον ἀλλ&#039; <lb></lb>ἀπ&#039; ἐναντίας, ὅπως τά τε ξύλα ἧττον διασπᾶται· τάχιστα <lb></lb>γὰρ σχίζεται κατὰ φύσιν διαιρούμενα ταύτῃ, καὶ ἑλκόμενα <lb></lb>πονεῖ μάλιστα.</s>
<s id="g0132504">ἔτι ἐπειδὴ δεῖ βάρος δύνασθαι τὰ <lb></lb>σπαρτία φέρειν, οὕτως ἧττον πονέσει λοξοῖς τοῖς σπαρτίοις <lb></lb>ἐπιτιθεμένου τοῦ βάρους ἢ πλαγίοις.</s>
<s id="g0132505">ἔτι δὲ ἔλαττον οὕτω <lb></lb>σπαρτίον ἀναλίσκεται.</s>
<s id="g0132506">ἔστω γὰρ κλίνη ἡ ΑΖΗΙ, καὶ δίχα <lb></lb>διῃρήσθω ἡ ΖΗ κατὰ τὸ Β. ἴσα δὴ τρυπήματά ἐστιν <lb></lb>ἐν τῇ ΖΒ καὶ ἐν τῇ ΖΑ. καὶ γὰρ αἱ πλευραὶ ἴσαι εἰσίν· <lb></lb>ἡ γὰρ ὅλη ΖΗ διπλασία ἐστίν.</s>
<s id="g0132507">ἐντείνουσι δ&#039; ὡς γέγραπται, <lb></lb>ἀπὸ τοῦ Α ἐπὶ τὸ Β, εἶτα οὗ τὸ Γ, εἶτα οὗ τὸ Δ, εἶτα οὗ <lb></lb>τὸ Θ, εἶτα οὗ τὸ Ε. καὶ οὕτως ἀεί, ἕως ἂν εἰς γωνίαν <lb></lb>καταστρέψωσιν ἄλλην· </s>
<figure id="id.080.01.020.1.jpg" xlink:href="080/01/020/1.jpg"></figure>
<s id="g0132508">δύο γὰρ ἔχουσι γωνίαι τὰς ἀρχὰς <lb></lb>τοῦ σπαρτίου.</s>
<s id="g0132509">ἴσα δέ ἐστι τὰ σπαρτία κατὰ τὰς κάμψεις, <lb></lb>τό τε ΑΒ καὶ ΒΓ τῷ ΓΔ καὶ ΔΘ. καὶ τὰ ἄλλα δὲ <lb></lb>τὰ τοιαῦτά ἐστιν, ὅτι οὕτως ἔχει ἡ αὐτὴ ἀπόδειξις.</s>
<s id="g0132510">ἡ μὲν <lb></lb>γὰρ ΑΒ τῇ ΕΘ ἴση· ἴσαι γάρ εἰσιν αἱ πλευραὶ τοῦ ΒΗΚ <lb></lb>Α χωρίου, καὶ τὰ τρυπήματα ἴσα διέστηκεν.</s>
<s id="g0132511">ἡ δὲ ΒΗ ἴση <lb></lb>τῇ ΚΑ· ἡ γὰρ Β γωνία ἴση τῇ Η. ἐν ἴσοις γὰρ ἡ μὲν <lb></lb>ἐκτός, ἡ δὲ ἐντός· καὶ ἡ μὲν Β ἐστὶν ἡμίσεια ὀρθῆς· ἡ <lb></lb>γὰρ ΖΒ ἴση τῇ ΖΑ· καὶ γωνία δὲ ἡ κατὰ τὸ Ζ ὀρθή.</s>
<s id="g0132512">ἡ <lb></lb>δὲ Β γωνία ἴση τῇ κατὰ τὸ Η· ἡ γὰρ κατὰ τὸ Ζ ὀρθή, <lb></lb>ἐπειδὴ διπλασιόπλευρον τὸ ἑτερόμηκες καὶ πρὸς μέσον κέκλασται. <lb></lb>ὥστε ἡ ΑΓ τῇ ΕΗ ἴση. ταύτῃ δὲ ἡ ΚΘ· παράλληλος <lb></lb>γάρ. ὥστε ἡ ΒΓ ἴση τῇ ΚΘ. ἡ δὲ ΓΕ τῇ ΔΘ.</s>
<s id="g0132513"><lb></lb>ὁμοίως δὲ καὶ αἱ ἄλλαι δείκνυνται ὅτι ἴσαι εἰσὶν αἱ κατὰ <lb></lb>τὰς κάμψεις δύο ταῖς δυσίν.</s>
<s id="g0132514">ὥστε δῆλον ὅτι τὰ τηλικαῦτα <lb></lb>σπαρτία ὅσον τὸ ΑΒ, τέσσαρα τοσαῦτ&#039; ἔνεστιν ἐν τῇ κλίνῃ· </s>
<s id="g0132515"><lb></lb>ὅσον δ&#039; ἐστὶ τὸ πλῆθος τῶν ἐν τῇ ΖΗ πλευρᾷ τρυπημάτων, <lb></lb>καὶ ἐν τῷ ἡμίσει τῷ ΖΒ τὰ ἡμίση.</s>
<s id="g0132516">ὥστε ἐν τῇ ἡμισείᾳ <lb></lb>κλίνῃ τηλικαῦτα μεγέθη σπαρτίων ἐστὶν ὅσον τῷ ΒΑ ἔνεστι, <lb></lb>τοσαῦτα δὲ τὸ πλῆθος ὅσαπερ ἐν τῷ ΒΗ τρυπήματα.</s>
<s id="g0132517"><lb></lb>ταῦτα δὲ οὐδὲν διαφέρει λέγειν ἢ ὅσα ἐν τῇ ΑΖ καὶ ΒΖ <lb></lb>τὰ συνάμφω.</s>
<figure id="id.080.01.020.2.jpg" xlink:href="080/01/020/2.jpg"></figure>
<s id="g0132518">εἰ δὲ κατὰ διάμετρον ἐνταθῇ τὰ σπαρτία, <lb></lb>ὡς ἐν τῇ ΑΒΓΔ κλίνῃ ἔχει, τὰ ἡμίσεά εἰσιν οὐ τοσαῦτα<pb xlink:href="080/01/021.jpg" ed="Bekker" n="857a"></pb><lb></lb> ὅσα αἱ πλευραὶ ἀμφοῖν, αἱ ΑΖ ΖΗ· τὰ ἴσα δέ, ὅσα <lb></lb>ἐν τῷ ΖΒΖΑ τρυπήματα ἔνεστιν.</s>
<s id="g0132519">μείζονες δέ εἰσιν αἱ ΑΖ <lb></lb>ΒΖ δύο οὖσαι τῆς ΑΒ. ὥστε καὶ τὸ σπαρτίον μεῖζον τοσούτῳ <lb></lb>ὅσον αἱ πλευραὶ ἄμφω μείζους εἰσὶ τῆς διαμέτρου.</s>
<figure id="id.080.01.021.1.jpg" xlink:href="080/01/021/1.jpg"></figure></p><p n="45">
<s id="g0132601prop26"><lb></lb>Διὰ τί χαλεπώτερον τὰ μακρὰ ξύλα ἀπ&#039; ἄκρου <lb></lb>φέρειν ἐπὶ τῷ ὤμῳ ἢ κατὰ τὸ μέσον, ἴσου τοῦ βάρους ὄντος; </s>
<s id="g0132602"><lb></lb>πότερον ὅτι σαλευομένου τοῦ ξύλου τὸ ἄκρον κωλύει φέρειν, <lb></lb>μᾶλλον ἀντισπῶν τῇ σαλεύσει τὴν φοράν; ἢ κἂν <lb></lb>μηθὲν κάμπτηται μηδ&#039; ἔχῃ πολὺ μῆκος, ὅμως χαλεπώτερον <lb></lb>φέρειν ἀπ&#039; ἄκρου; ἀλλ&#039; ὅτι καὶ ῥᾷον αἴρεται ἀπ&#039; <lb></lb>ἄκρου ἢ ἐκ μέσου, διὰ τὸ αὐτὸ καὶ φέρειν οὕτω ῥᾴδιον.</s>
<s id="g0132603"><lb></lb>αἴτιον δὲ ὅτι ἐκ μέσου μὲν αἰρόμενον ἀεὶ ἐπικουφίζει ἄλληλα <lb></lb>τὰ ἄκρα, καὶ θάτερον μέρος τὸ ἐπὶ θάτερον εὖ αἴρει. <lb></lb>ὥσπερ γὰρ κέντρον γίνεται τὸ μέσον, ᾗ ἔχει τὸ αἶρον ἢ <lb></lb>φέρον.</s>
<s id="g0132604">εἰς τὸ ἄνω οὖν κουφίζεται ἑκάτερον τῶν ἄκρων εἰς <lb></lb>τὸ κάτω ῥέπον. ἀπὸ δὲ τοῦ ἄκρου αἰρόμενον ἢ φερόμενον οὐ <lb></lb>ποιεῖ τοῦτο, ἀλλ&#039; ἅπαν τὸ βάρος ῥέπει ἐφ&#039; ἓν μέσον, εἰς <lb></lb>ὅπερ αἴρεται ἢ φέρεται.</s>
<s id="g0132605">ἔστω μέσον ἐφ&#039; οὗ Α, ἄκρα ΒΓ.</s>
<s id="g0132606"><lb></lb>αἰρομένου οὖν ἢ φερομένου κατὰ τὸ Α, τὸ μὲν Β κάτω <lb></lb>ῥέπον ἄνω αἴρει τὸ Γ, τὸ δὲ Γ κάτω ῥέπον τὸ Β ἄνω αἴρει· <lb></lb>ἅμα δὲ αἰρόμενα ἄνω ποιεῖ ταῦτα.</s>
<figure id="id.080.01.021.2.jpg" xlink:href="080/01/021/2.jpg"></figure></p><p n="46">
<s id="g0132701prop27"><lb></lb>Διὰ τί, ἐὰν ᾖ λίαν μακρὸν τὸ αὐτὸ βάρος, χαλεπώτερον <lb></lb>φέρειν ἐπὶ τοῦ ὤμου, κἂν μέσον φέρῃ τις, ἢ ἐὰν <lb></lb>ἔλαττον ᾖ; </s>
<s id="g0132702">πάλαι ἐλέχθη ὡς οὐκ ἔστιν αἴτιον ἡ σάλευσις· <lb></lb>ἀλλ&#039; ἡ σάλευσις νῦν αἴτιόν ἐστιν.</s>
<s id="g0132703">ὅταν γὰρ ᾖ μακρότερον, <lb></lb>τὰ ἄκρα σαλεύεται, ὥστε εἴη ἂν καὶ τὸν φέροντα χαλεπώτερον <lb></lb>φέρειν μᾶλλον.</s>
<s id="g0132704">αἴτιον δὲ τοῦ σαλεύεσθαι μᾶλλον, <lb></lb>ὅτι τῆς αὐτῆς κινήσεως οὔσης μεθίσταται τὰ ἄκρα, ὅσῳπερ <lb></lb>ἂν ᾖ μακρότερον τὸ ξύλον.</s>
<s id="g0132705">ὁ μὲν γὰρ ὦμος κέντρον, ἐφ&#039; <lb></lb>οὗ τὸ Α [1μένει γὰρ τοῦτο]1, αἱ δὲ ΑΒ καὶ ΑΓ αἱ ἐκ τοῦ <lb></lb>κέντρου. ὅσῳ δ&#039; ἂν ᾖ μεῖζον τὸ ἐκ τοῦ κέντρου ἢ τὸ ΑΒ <lb></lb>ἢ καὶ τὸ ΑΓ, πλέον μεθίσταται μέγεθος. δέδεικται δὲ <lb></lb>τοῦτο πρότερον.</s>
<figure id="id.080.01.021.3.jpg" xlink:href="080/01/021/3.jpg"></figure></p><p n="47">
<figure id="id.080.01.021.4.jpg" xlink:href="080/01/021/4.jpg"></figure>
<s id="g0132801prop28"><lb></lb>Διὰ τί ἐπὶ τοῖς φρέασι τὰ κηλώνεια ποιοῦσι τοῦτον τὸν <lb></lb>τρόπον; προστιθέασι γὰρ βάρος ἐν τῷ ξύλῳ τὸν μόλιβδον, <lb></lb>ὄντος βάρους τοῦ κάδου αὐτοῦ, καὶ κενοῦ καὶ πλήρους ὄντος.</s>
<s id="g0132802"><lb></lb>ἢ ὅτι ἐν δυσὶ χρόνοις διῃρημένου τοῦ ἔργου [1βάψαι γὰρ δεῖ, <lb></lb>καὶ τοῦτ&#039; ἄνω ἑλκύσαι]1 συμβαίνει καθιέναι μὲν κενὸν ῥᾳ-<pb xlink:href="080/01/022.jpg" ed="Bekker" n="857b"></pb><lb></lb>δίως, αἴρειν δὲ πλήρη χαλεπῶς; </s>
<s id="g0132803">λυσιτελεῖ οὖν μικρῷ βραδύτερον <lb></lb>εἶναι τὸ καταγαγεῖν πρὸς τὸ πολὺ κουφίσαι τὸ <lb></lb>βάρος ἀνάγοντι. τοῦτο οὖν ποιεῖ ἐπ&#039; ἄκρῳ τῷ κηλωνείῳ ὁ <lb></lb>μόλιβδος προσκείμενος ἢ ὁ λίθος.</s>
<s id="g0132804">καθιμῶντι μὲν γὰρ γίνεται <lb></lb>βάρος μεῖζον ἢ εἰ μόνον κενὸν δεῖ κατάγειν τὸν κάδον· <lb></lb>ὅταν δὲ πλήρης ᾖ, ἀνάγει ὁ μόλιβδος, ἢ ὅ τι ἂν ᾖ <lb></lb>τὸ προσκείμενον βάρος.</s>
<s id="g0132805">ὥστ&#039; ἐστὶ ῥᾷον αὐτῷ τὰ ἄμφω <lb></lb>ἢ ἐκείνῳ.</s></p><p n="48">
<figure id="id.080.01.022.1.jpg" xlink:href="080/01/022/1.jpg"></figure>
<s id="g0132901prop29"><lb></lb>Διὰ τί, ὅταν φέρωσιν ἐπὶ ξύλου ἤ τινος τοιούτου δύο <lb></lb>ἄνθρωποι ἴσον βάρος, οὐχ ὁμοίως θλίβονται, ἐὰν μὴ ἐπὶ <lb></lb>τῷ μέσῳ ᾖ τὸ βάρος, ἀλλὰ μᾶλλον ὅσῳ ἂν ἐγγύτερον ᾖ <lb></lb>τῶν φερόντων; </s>
<s id="g0132902">ἢ διότι μοχλὸς μὲν γίνεται οὕτως ἐχόντων <lb></lb>τὸ ξύλον, τὸ δὲ βάρος ὑπομόχλιον, </s>
<s id="g0132903">ὁ δὲ ἐγγύτερος τοῦ <lb></lb>βάρους τῶν φερόντων τὸ βάρος τὸ κινούμενον, ἅτερος δὲ <lb></lb>τῶν φερόντων τὸ βάρος ὁ κινῶν.</s>
<s id="g0132904">ὅσῳ γὰρ πλέον ἀπέχει τοῦ <lb></lb>βάρους, τοσούτῳ ῥᾷον κινεῖ, καὶ θλίβει μᾶλλον τὸν ἕτερον <lb></lb>εἰς τὸ κάτω, ὥσπερ ἀντερείδοντος τοῦ βάρους τοῦ ἐπικειμένου <lb></lb>καὶ γινομένου ὑπομοχλίου.</s>
<s id="g0132905">ἐν μέσῳ δὲ ὑποκειμένου τοῦ <lb></lb>βάρους, οὐδὲν μᾶλλον ἅτερος θατέρῳ γίνεται βάρος, οὐδὲ <lb></lb>κινεῖ, ἀλλ&#039; ὁμοίως ἑκάτερος ἑκατέρῳ γίνεται βάρος.</s></p><p n="49">
<s id="g0133001prop30"><lb></lb>Διὰ τί οἱ ἀνιστάμενοι πάντες πρὸς ὀξεῖαν γωνίαν τῷ <lb></lb>μηρῷ ποιήσαντες τὴν κνήμην ἀνίστανται, καὶ τῷ θώρακι <lb></lb>πρὸς τὸν μηρόν; εἰ δὲ μή, οὐκ ἂν δύναιντο ἀναστῆναι.</s>
<s id="g0133002">πότερον <lb></lb>ὅτι τὸ ἴσον ἠρεμίας πανταχοῦ αἴτιον, ἡ δὲ ὀρθὴ γωνία <lb></lb>τοῦ ἴσου, καὶ ποιεῖ στάσιν· διὸ καὶ φέρεται πρὸς ὁμοίας <lb></lb>γωνίας τῇ περιφερείᾳ τῆς γῆς. οὐ γὰρ ὅτι καὶ πρὸς ὀρθὴν <lb></lb>ἔσται τῷ ἐπιπέδῳ.</s>
<s id="g0133003">ἢ ὅτι ἀνιστάμενος γίνεται ὀρθός, ἀνάγκη <lb></lb>δὲ τὸν ἑστῶτα κάθετον εἶναι πρὸς τὴν γῆν.</s>
<s id="g0133004">εἰ οὖν μέλλει <lb></lb>ἔσεσθαι πρὸς ὀρθήν, τοῦτο δέ ἐστι τὸ τὴν κεφαλὴν ἔχειν <lb></lb>κατὰ τοὺς πόδας, καὶ γίνεσθαι δὴ ὅτε ἀνίσταται.</s>
<s id="g0133005">ὅταν μὲν <lb></lb>οὖν καθήμενος ᾖ, παράλληλον ἔχει τὴν κεφαλὴν καὶ τοὺς <lb></lb>πόδας, καὶ οὐκ ἐπὶ μιᾶς εὐθείας.</s>
<figure id="id.080.01.022.2.jpg" xlink:href="080/01/022/2.jpg"></figure>
<s id="g0133006">ἡ κεφαλὴ Α ἔστω, θώραξ <lb></lb>ΑΒ, μηρὸς ΒΓ, κνήμη ΓΔ.</s>
<s id="g0133007">πρὸς ὀρθὴν δὲ γίνεται <lb></lb>ὅ τε θώραξ [ἐφ&#039; ὧν ΑΒ] τῷ μηρῷ καὶ ὁ μηρὸς τῇ κνήμῃ <lb></lb>οὕτως καθημένῳ. ὥστε οὕτως ἔχοντα ἀδύνατον ἀναστῆναι.</s>
<s id="g0133008"><lb></lb>ἀνάγκη δὲ ἐγκλῖναι τὴν κνήμην καὶ ποιεῖν τοὺς πόδας ὑπὸ <lb></lb>τὴν κεφαλήν.</s>
<s id="g0133009">τοῦτο δὲ ἔσται, ἐὰν ἡ ΓΔ ἐφ&#039; ἧς τὰ ΓΖ <lb></lb>γένηται, καὶ ἅμα ἀναστῆναι συμβήσεται, καὶ ἔχειν ἐπὶ<pb xlink:href="080/01/023.jpg" ed="Bekker" n="858a"></pb><lb></lb> τῆς αὐτῆς ἴσης τὴν κεφαλήν τε καὶ τοὺς πόδας. ἡ δὲ ΓΖ <lb></lb>ὀξεῖαν ποιεῖ γωνίαν πρὸς τὴν ΒΓ.</s></p><p n="50">
<s id="g0133101prop31"><lb></lb>Διὰ τί ῥᾷον κινεῖται τὸ κινούμενον ἢ τὸ μένον, οἷον <lb></lb>τὰς ἁμάξας θᾶττον κινουμένας ὑπάγουσιν ἢ ἀρχομένας; </s>
<s id="g0133102"><lb></lb>ἢ ὅτι χαλεπώτατον μὲν τὸ εἰς τοὐναντίον κινούμενον κινῆσαι <lb></lb>βάρος; ἀφαιρεῖται γάρ τι τῆς τοῦ κινοῦντος δυνάμεως, κἂν <lb></lb>πολὺ θᾶττον ᾖ· ἀνάγκη γὰρ βραδυτέραν γίνεσθαι τὴν ὦσιν <lb></lb>τοῦ ἀντωθουμένου.</s>
<s id="g0133103">δεύτερον δέ, ἐὰν ἠρεμῇ· ἀντιτείνει γὰρ καὶ <lb></lb>τὸ ἠρεμοῦν.</s>
<s id="g0133104">τὸ δὲ κινούμενον ἐπὶ τὸ αὐτὸ τῷ ὠθοῦντι ὅμοιον <lb></lb>ποιεῖ ὥσπερ ἂν εἰ αὐξήσειέ τις τὴν τοῦ κινοῦντος δύναμιν <lb></lb>καὶ ταχυτῆτα· ὃ γὰρ ὑπ&#039; ἐκείνου ἂν ἔπασχε, τοῦτο αὐτὸ <lb></lb>ποιεῖ εἰς τὸ πρὸ ὁδοῦ κινούμενον.</s></p><p n="51">
<s id="g0133201prop32"><lb></lb>Διὰ τί παύεται φερόμενα τὰ ῥιφέντα; </s>
<s id="g0133202">πότερον ὅταν <lb></lb>λήγῃ ἡ ἰσχὺς ἡ ἀφεῖσα, ἢ διὰ τὸ ἀντισπᾶσθαι, ἢ διὰ <lb></lb>τὴν ῥοπήν, ἐὰν κρείττων ᾖ τῆς ἰσχύος τῆς ῥιψάσης; </s>
<s id="g0133203">ἢ ἄτοπον <lb></lb>τὸ ταῦτ&#039; ἀπορεῖν, ἀφέντα τὴν ἀρχήν.</s></p><p n="52">
<s id="g0133301prop33"><lb></lb>Διὰ τί φέρεταί τι οὐ τὴν αὑτοῦ φοράν, μὴ ἀκολουθοῦντος <lb></lb>καὶ ὠθοῦντος τοῦ ἀφέντος; </s>
<s id="g0133302">ἢ δῆλον ὅτι ἐποίησε τοιοῦτον <lb></lb>τὸ πρῶτον ὡς θάτερον ὠθεῖν, καὶ τοῦθ&#039; ἕτερον· </s>
<s id="g0133303">παύεται δέ, <lb></lb>ὅταν μηκέτι δύνηται ποιεῖν τὸ προωθοῦν τὸ φερόμενον ὥστε <lb></lb>ὠθεῖν, καὶ ὅταν τὸ τοῦ φερομένου βάρος ῥέπῃ μᾶλλον τῆς <lb></lb>εἰς τὸ πρόσθεν δυνάμεως τοῦ ὠθοῦντος.</s></p><p n="53">
<s id="g0133401prop34"><lb></lb>Διὰ τί οὔτε τὰ ἐλάττονα οὔτε τὰ μεγάλα πόρρω φέρεται <lb></lb>ῥιπτούμενα, ἀλλὰ δεῖ συμμετρίαν τινὰ ἔχειν πρὸς <lb></lb>τὸν ῥιπτοῦντα; </s>
<s id="g0133402">πότερον ὅτι ἀνάγκη τὸ ῥιπτούμενον καὶ <lb></lb>ὠθούμενον ἀντερείδειν ὅθεν ὠθεῖται; </s>
<s id="g0133403">τὸ δὲ μηθὲν ὑπεῖκον διὰ <lb></lb>μέγεθος ἢ μηδὲν ἀντερεῖσαν δι&#039; ἀσθένειαν οὐ ποιεῖ ῥῖψιν <lb></lb>οὐδὲ ὦσιν.</s>
<s id="g0133404">τὸ μὲν οὖν πολὺ ὑπερβάλλον τῆς ἰσχύος τῆς <lb></lb>ὠθούσης οὐθὲν ὑπείκει, τὸ δὲ πολὺ ἀσθενέστερον οὐδὲν ἀνερείδει.</s>
<s id="g0133405"><lb></lb>ἢ ὅτι τοσοῦτον φέρεται τὸ φερόμενον, ὅσον ἂν <lb></lb>ἀέρα κινήσῃ εἰς βάθος; τὸ δὲ μηδὲν κινούμενον οὐδ&#039; ἂν <lb></lb>κινήσειεν οὐδέν. συμβαίνει δὴ ἀμφότερα τούτοις ἔχειν.</s>
<s id="g0133406"><pb xlink:href="080/01/024.jpg" ed="Bekker" n="858b"></pb><lb></lb>τό τε γὰρ σφόδρα μέγα καὶ τὸ σφόδρα μικρὸν ὥσπερ οὐθὲν <lb></lb>κινούμενά ἐστι· τὸ μὲν γὰρ αὐτὸ καθ&#039; ἓν κινεῖ, τὸ δ&#039; <lb></lb>οὐθὲν κινεῖται.</s></p><p n="54">
<figure id="id.080.01.024.1.jpg" xlink:href="080/01/024/1.jpg"></figure>
<s id="g0133501prop35"><lb></lb>Διὰ τί τὰ φερόμενα ἐν τῷ δινουμένῳ ὕδατι εἰς τὸ <lb></lb>μέσον τελευτῶντα φέρονται ἅπαντα; </s>
<s id="g0133502">πότερον ὅτι μέγεθος <lb></lb>ἔχει τὸ φερόμενον, ὥστε ἐν δυσὶ κύκλοις εἶναι, τῷ μὲν <lb></lb>ἐλάττονι τῷ δὲ μείζονι, ἑκάτερον αὐτοῦ τῶν ἄκρων. ὥστε <lb></lb>περισπᾷ ὁ μείζων διὰ τὸ φέρεσθαι θᾶττον, καὶ πλάγιον <lb></lb>ἀπωθεῖ αὐτὸ εἰς τὸν ἐλάττω. ἐπεὶ δὲ πλάτος ἔχει τὸ <lb></lb>φερόμενον, καὶ οὗτος πάλιν τὸ αὐτὸ ποιεῖ, καὶ ἀπωθεῖ εἰς <lb></lb>τὸν ἐντός, ἕως ἂν εἰς τὸ μέσον ἔλθῃ.</s>
<s id="g0133503">καὶ τότε μένει διὰ <lb></lb>τὸ ὁμοίως ἔχειν πρὸς ἅπαντας τοὺς κύκλους τὸ φερόμενον, <lb></lb>διὰ τὸ μέσον· καὶ γὰρ τὸ μέσον ἴσον ἀπέχει ἐν ἑκάστῳ <lb></lb>τῶν κύκλων.</s>
<s id="g0133504">ἢ ὅτι ὅσων μὲν μὴ κρατεῖ ἡ φορὰ τοῦ δινουμένου <lb></lb>ὕδατος διὰ τὸ μέγεθος, ἀλλ&#039; ὑπερέχει τῇ βαρύτητι <lb></lb>τῆς τοῦ κύκλου ταχυτῆτος, ἀνάγκη ὑπολείπεσθαι καὶ βραδύτερον <lb></lb>φέρεσθαι.</s>
<s id="g0133505">βραδύτερον δὲ ὁ ἐλάττων κύκλος φέρεται· <lb></lb>τὸ αὐτὸ γὰρ ἐν ἴσῳ χρόνῳ ὁ μέγας τῷ μικρῷ στρέφεται <lb></lb>κύκλῳ, ὅταν ὦσι περὶ τὸ αὐτὸ μέσον.</s>
<s id="g0133506">ὥστε εἰς τὸν <lb></lb>ἐλάττονα κύκλον ἀναγκαῖον ἀπολείπεσθαι, ἕως ἂν ἐπὶ τὸ <lb></lb>μέσον ἔλθῃ.</s>
<s id="g0133507">ὅσων δὲ πρότερον κρατεῖ ἡ φορά, λήγουσα <lb></lb>ταὐτὸ ποιήσει. δεῖ γὰρ τὸν μὲν εὐθύ, τὸν δὲ ἕτερον κρατεῖν <lb></lb>τῇ ταχυτῆτι τοῦ βάρους, ὥστε εἰς τὸν ἐντὸς ἀεὶ κύκλον <lb></lb>ὑπολείπεσθαι πᾶν.</s>
<s id="g0133508">ἀνάγκη γὰρ αὐτὸ ἐντὸς ἢ ἐκτὸς κινεῖσθαι <lb></lb>τὸ μὴ κρατούμενον.</s>
<s id="g0133509">ἐν αὐτῷ δὴ τοίνυν ἐν ᾧ ἐστίν, <lb></lb>ἀδύνατον φέρεσθαι τὸ μὴ κρατούμενον. ἔτι δὲ ἧττον ἐν τῷ <lb></lb>ἐκτός· θάττων γὰρ ἡ φορὰ τοῦ ἐκτὸς κύκλου.</s>
<s id="g0133510">λείπεται δὲ <lb></lb>εἰς τὸν ἐντὸς τὸ μὴ κρατούμενον μεθίστασθαι. ἀεὶ δὲ ἕκαστον <lb></lb>ἐπιδίδωσιν εἰς τὸ μὴ κρατεῖσθαι.</s>
<s id="g0133511">ἐπεὶ δὲ πέρας τοῦ μὴ κινεῖσθαι <lb></lb>ποιεῖ τὸ εἰς μέσον ἐλθεῖν, μένει δὲ τὸ κέντρον μόνον, <lb></lb>ἅπαντα ἀνάγκη εἰς τοῦτο δὴ ἀθροίζεσθαι.</s>
</p>
</chap>
</body>
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</text>
</archimedes>