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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:38:23 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes> <info> <author>Pseudo-Aristotle</author> <title>Problemata Mechanica</title> <date>1982</date> <place>Padova</place> <translator></translator> <lang>el</lang> <cvs_file>arist_mecha_108_el_1982.xml</cvs_file> <cvs_version></cvs_version> <locator>108.xml</locator> </info> <text> <front/> <body> <chap> <p n="1"> <s id="g0110101"> <pb xlink:href="080/01/001.jpg" ed="Bekker" n="847a"/> Θαυμάζεται τῶν μὲν κατὰ φύσιν συμβαινόντων, ὅσων <lb/> ἀγνοεῖται τὸ αἴτιον, τῶν δὲ παρὰ φύσιν, ὅσα γίνεται διὰ <lb/> τέχνην πρὸς τὸ συμφέρον τοῖς ἀνθρώποις. </s> <s id="g0110102"> ἐν πολλοῖς γὰρ <lb/> ἡ φύσις ὑπεναντίον πρὸς τὸ χρήσιμον ἡμῖν ποιεῖ· </s> <s id="g0110103"> ἡ μὲν <lb/> γὰρ φύσις ἀεὶ τὸν αὐτὸν ἔχει τρόπον καὶ ἁπλῶς, τὸ δὲ <lb/> χρήσιμον μεταβάλλει πολλαχῶς. </s> </p> <p n="2"> <s id="g0110201"> ὅταν οὖν δέῃ τι παρὰ <lb/> φύσιν πρᾶξαι, διὰ τὸ χαλεπὸν ἀπορίαν παρέχει καὶ δεῖται <lb/> τέχνης. </s> <s> διὸ καὶ καλοῦμεν τῆς τέχνης τὸ πρὸς τὰς τοιαύτας <lb/> ἀπορίας βοηθοῦν μέρος μηχανήν. </s> <s id="g0110202"> καθάπερ γὰρ ἐποίησεν <lb/> Ἀντιφῶν ὁ ποιητής, οὕτω καὶ ἔχει· τέχνῃ γὰρ κρατοῦμεν, <lb/> ὧν φύσει νικώμεθα. </s> <s id="g0110203"> τοιαῦτα δέ ἐστιν ἐν οἷς τά τε ἐλάττονα <lb/> κρατεῖ τῶν μειζόνων, καὶ τὰ ῥοπὴν ἔχοντα μικρὰν κινεῖ <lb/> βάρη μεγάλα, καὶ πάντα σχεδὸν ὅσα τῶν προβλημάτων <lb/> μηχανικὰ προσαγορεύομεν. </s> </p> <p n="3"> <s id="g0110301"> ἔστι δὲ ταῦτα τοῖς φυσικοῖς <lb/> προβλήμασιν οὔτε ταὐτὰ πάμπαν οὔτε κεχωρισμένα λίαν, <lb/> ἀλλὰ κοινὰ τῶν τε μαθηματικῶν θεωρημάτων καὶ τῶν <lb/> φυσικῶν· τὸ μὲν γὰρ ὣς διὰ τῶν μαθηματικῶν δῆλον, τὸ <lb/> δὲ περὶ ὃ διὰ τῶν φυσικῶν. </s> </p> <p n="4"> <s id="g0120101"> περιέχεται δὲ τῶν <pb xlink:href="080/01/002.jpg" ed="Bekker" n="847b"/> <lb/> ἀπορουμένων ἐν τῷ γένει τούτῳ τὰ περὶ τὸν μοχλόν. </s> <s id="g0120102"> ἄτοπον γὰρ <lb/> εἶναι δοκεῖ τὸ κινεῖσθαι μέγα βάρος ὑπὸ μικρᾶς ἰσχύος, <lb/> καὶ ταῦτα μετὰ βάρους πλείονος· ὃ γὰρ ἄνευ μοχλοῦ κινεῖν <lb/> οὐ δύναταί τις, τοῦτο ταὐτὸ βάρος, προσλαβὼν ἔτι τὸ <lb/> τοῦ μοχλοῦ βάρος, κινεῖ θᾶττον. </s> <s id="g0120103"> πάντων δὲ τῶν τοιούτων <lb/> ἔχει τῆς αἰτίας τὴν ἀρχὴν ὁ κύκλος. </s> <s id="g0120104"> καὶ τοῦτο εὐλόγως <lb/> συμβέβηκεν· ἐκ μὲν γὰρ θαυμασιωτέρου συμβαίνειν τι <lb/> θαυμαστὸν οὐδὲν ἄτοπον, </s> </p> <p n="5"> <s id="g0120201"> θαυμασιώτατον δὲ τὸ τἀναντία <lb/> γίνεσθαι μετ' ἀλλήλων. </s> <s id="g0120202"> ὁ δὲ κύκλος συνέστηκεν ἐκ τοιούτων· </s> <s id="g0120203"> <lb/> εὐθὺς γὰρ ἐκ κινουμένου τε γεγένηται καὶ μένοντος, ὧν ἡ <lb/> φύσις ἐστὶν ὑπεναντία ἀλλήλοις. ὥστ' ἐνταῦθα ἔστιν ἐπιβλέψασιν <lb/> ἧττον θαυμάζειν τὰς συμβαινούσας ὑπεναντιώσεις <lb/> περὶ αὐτόν. </s> <s id="g0120204"> πρῶτον μὲν γὰρ τῇ περιεχούσῃ γραμμῇ τὸν <lb/> κύκλον πλάτος οὐθὲν ἐχούσῃ, τἀναντία πως προσεμφαίνεται, <lb/> τὸ κοῖλον καὶ τὸ κυρτόν. </s> <s id="g0120205"> ταῦτα δὲ διέστηκεν ἀλλήλων <lb/> ὃν τρόπον τὸ μέγα καὶ τὸ μικρόν· ἐκείνων τε γὰρ <lb/> μέσον τὸ ἴσον καὶ τούτων τὸ εὐθύ. διὸ μεταβάλλοντα εἰς <lb/> ἄλληλα τὰ μὲν ἀναγκαῖον ἴσα γενέσθαι πρότερον ἢ <pb xlink:href="080/01/003.jpg" ed="Bekker" n="848a"/> <lb/> τῶν ἄκρων ὁποτερονοῦν, τὴν δὲ γραμμὴν εὐθεῖαν, ὅταν ἐκ κυρτῆς <lb/> εἰς κοῖλον ἢ πάλιν ἐκ ταύτης γίνηται κυρτὴ καὶ περιφερής. <lb/> ἓν μὲν οὖν τοῦτο τῶν ἀτόπων ὑπάρχει περὶ τὸν κύκλον, </s> </p> <p n="6"> <s id="g0120301"> <lb/> δεύτερον δὲ ὅτι ἅμα κινεῖται τὰς ἐναντίας κινήσεις· <lb/> ἅμα γὰρ εἰς τὸν ἔμπροσθεν κινεῖται τόπον καὶ τὸν ὄπισθεν. </s> <s id="g0120302"> <lb/> ἥ τε γράφουσα γραμμὴ τὸν κύκλον ὡσαύτως ἔχει· ἐξ <lb/> οὗ γὰρ ἄρχεται τόπου τὸ πέρας αὐτῆς, εἰς τὸν αὐτὸν τοῦτον τόπον <lb/> ἔρχεται πάλιν· συνεχῶς γὰρ κινουμένης αὐτῆς τὸ ἔσχατον <lb/> πάλιν ἀπῆλθε πρῶτον, ὥστε καὶ φανερὸν ὅτι μετέβαλεν <lb/> ἐντεῦθεν. </s> <s id="g0120303"> διό, καθάπερ εἴρηται πρότερον, οὐδὲν ἄτοπον τὸ <lb/> πάντων εἶναι τῶν θαυμάτων αὐτὸν ἀρχήν. </s> </p> <p n="7"> <s id="g0120401"> τὰ μὲν οὖν περὶ <lb/> τὸν ζυγὸν γινόμενα εἰς τὸν κύκλον ἀνάγεται, τὰ δὲ περὶ <lb/> τὸν μοχλὸν εἰς τὸν ζυγόν, τὰ δ' ἄλλα πάντα σχεδὸν τὰ <lb/> περὶ τὰς κινήσεις τὰς μηχανικὰς εἰς τὸν μοχλόν. </s> <s id="g0120402"> ἔτι δὲ <lb/> διὰ τὸ μιᾶς οὔσης τῆς ἐκ τοῦ κέντρου γραμμῆς μηθὲν ἕτερον <lb/> ἑτέρῳ φέρεσθαι τῶν σημείων τῶν ἐν αὐτῇ ἰσοταχῶς, ἀλλ' ἀεὶ <lb/> τὸ τοῦ μένοντος πέρατος πορρώτερον ὂν θᾶττον, πολλὰ τῶν θαυμαζομένων <lb/> συμβαίνει περὶ τὰς κινήσεις τῶν κύκλων· περὶ <lb/> ὧν ἐν τοῖς ἑπομένοις προβλήμασιν ἔσται δῆλον. </s> </p> <p n="8"> <s id="g0120501"> διὰ δὲ τὸ <lb/> τὰς ἐναντίας κινήσεις ἅμα κινεῖσθαι τὸν κύκλον, καὶ τὸ <lb/> μὲν ἕτερον τῆς διαμέτρου τῶν ἄκρων, ἐφ' οὗ τὸ Α, εἰς τοὔμπροσθεν <lb/> κινεῖσθαι, θάτερον δέ, ἐφ' οὗ τὸ Β, εἰς τοὔπισθεν, <lb/> κατασκευάζουσί τινες ὥστ' ἀπὸ μιᾶς κινήσεως πολλοὺς ὑπεναντίους <lb/> ἅμα κινεῖσθαι κύκλους, ὥσπερ οὓς ἀνατιθέασιν ἐν <lb/> τοῖς ἱεροῖς ποιήσαντες τροχίσκους χαλκοῦς τε καὶ σιδηροῦς. </s> <s id="g0120502"> <lb/> εἰ γὰρ εἴη τοῦ ΑΒ κύκλου ἁπτόμενος ἕτερος κύκλος ἐφ' οὗ <lb/> ΓΔ, τοῦ κύκλου τοῦ ἐφ' οὗ ΑΒ κινουμένης τῆς διαμέτρου <lb/> εἰς τοὔμπροσθεν, κινηθήσεται ἡ ΓΔ εἰς τοὔπισθεν τοῦ κύκλου <lb/> τοῦ ἐφ' οὗ Α, κινουμένης τῆς διαμέτρου περὶ τὸ αὐτό. </s> <s id="g0120503"> εἰς <lb/> τοὐναντίον ἄρα κινηθήσεται ὁ ἐφ' οὗ ὁ ΓΔ κύκλος τῷ ἐφ' <lb/> οὗ τὸ ΑΒ· καὶ πάλιν αὐτὸς τὸν ἐφεξῆς, ἐφ' οὗ ΕΖ, εἰς <lb/> τοὐναντίον αὑτῷ κινήσει διὰ τὴν αὐτὴν αἰτίαν. </s> <figure id="id.080.01.003.1.jpg" xlink:href="080/01/003/1.jpg"/> </p> <p n="9"> <s id="g0120601"> τὸν αὐτὸν δὲ <lb/> τρόπον κἂν πλείους ὦσι, τοῦτο ποιήσουσιν ἑνὸς μόνου κινηθέντος. </s> <s id="g0120602"> <lb/> ταύτην οὖν λαβόντες ὑπάρχουσαν ἐν τῷ κύκλῳ τὴν <lb/> φύσιν οἱ δημιουργοὶ κατασκευάζουσιν ὄργανον κρύπτοντες <lb/> τὴν ἀρχήν, ὅπως ᾖ τοῦ μηχανήματος φανερὸν μόνον τὸ <lb/> θαυμαστόν, τὸ δ' αἴτιον ἄδηλον. <pb xlink:href="080/01/004.jpg" ed="Bekker" n="848b"/> <lb/> </s> </p> <p n="10"> <s id="g0120701prop01"> Πρῶτον μὲν οὖν τὰ συμβαίνοντα περὶ τὸν ζυγὸν ἀπορεῖται, <lb/> διὰ τίνα αἰτίαν ἀκριβέστερά ἐστι τὰ ζυγὰ τὰ μείζω <lb/> τῶν ἐλαττόνων. </s> <s id="g0120702"> τούτου δὲ ἀρχή, διὰ τί ποτε ἐν τῷ κύκλῳ <lb/> ἡ πλεῖον ἀφεστηκυῖα γραμμὴ τοῦ κέντρου τῆς ἐγγὺς τῇ <lb/> αὐτῇ ἰσχύι κινουμένης θᾶττον φέρεται τῆς ἐλάττονος; </s> <s id="g0120703"> τὸ <lb/> γὰρ θᾶττον λέγεται διχῶς· </s> <s id="g0120704"> ἄν τε γὰρ ἐν ἐλάττονι χρόνῳ <lb/> ἴσον τόπον διεξέλθῃ, θᾶττον εἶναι λέγομεν, καὶ ἐὰν ἐν ἴσῳ <lb/> πλείω. </s> <s id="g0120705"> ἡ δὲ μείζων ἐν ἴσῳ χρόνῳ γράφει μείζονα κύκλον· <lb/> ὁ γὰρ ἐκτὸς μείζων τοῦ ἐντός. </s> <s id="g0120706"> αἴτιον δὲ τούτων ὅτι φέρεται <lb/> δύο φορὰς ἡ γράφουσα τὸν κύκλον. </s> <s id="g0120707"> ὅταν μὲν οὖν ἐν λόγῳ <lb/> τινὶ φέρηται, ἐπ' εὐθείας ἀνάγκη φέρεσθαι τὸ φερόμενον, <lb/> καὶ γίνεται διάμετρος αὐτὴ τοῦ σχήματος ὃ ποιοῦσιν αἱ <lb/> ἐν τούτῳ τῷ λόγῳ συντεθεῖσαι γραμμαί. </s> <s id="g0120708"> ἔστω γὰρ ὁ λόγος <lb/> ὃν φέρεται τὸ φερόμενον, ὃν ἔχει ἡ ΑΒ πρὸς τὴν ΑΓ· <lb/> καὶ τὸ μὲν ΑΓ φερέσθω πρὸς τὸ Β, ἡ δὲ ΑΒ ὑποφερέσθω <lb/> πρὸς τὴν ΗΓ· ἐνηνέχθω δὲ τὸ μὲν Α πρὸς τὸ Δ, ἡ δὲ ἐφ' <lb/> ᾗ ΑΒ πρὸς τὸ Ε. εἰ οὖν ἐπὶ τῆς φορᾶς ὁ λόγος ἦν ὃν ἡ <lb/> ΑΒ ἔχει πρὸς τὴν ΑΓ, ἀνάγκη καὶ τὴν ΑΔ πρὸς τὴν <lb/> ΑΕ τοῦτον ἔχειν τὸν λόγον. </s> <s id="g0120709"> ὅμοιον ἄρα ἐστὶ τῷ λόγῳ τὸ <lb/> μικρὸν τετράπλευρον τῷ μείζονι, ὥστε καὶ ἡ αὐτὴ διάμετρος <lb/> αὐτῶν, καὶ τὸ Α ἔσται πρὸς Ζ. </s> <figure id="id.080.01.004.1.jpg" xlink:href="080/01/004/1.jpg"/> </p> <p n="11"> <s id="g0120801"> τὸν αὐτὸν δὴ τρόπον <lb/> δειχθήσεται κἂν ὁπουοῦν διαληφθῇ ἡ φορά· αἰεὶ γὰρ <lb/> ἔσται ἐπὶ τῆς διαμέτρου. </s> <s id="g0120802"> φανερὸν οὖν ὅτι τὸ κατὰ τὴν διάμετρον <lb/> φερόμενον ἐν δύο φοραῖς ἀνάγκη τὸν τῶν πλευρῶν <lb/> φέρεσθαι λόγον. </s> <s id="g0120803"> εἰ γὰρ ἄλλον τινά, οὐκ οἰσθήσεται κατὰ <lb/> τὴν διάμετρον. </s> <s id="g0120804"> ἐὰν δὲ ἐν μηδενὶ λόγῳ φέρηται δύο φορὰς <lb/> κατὰ μηδένα χρόνον, ἀδύνατον εὐθεῖαν εἶναι τὴν φοράν. </s> <s id="g0120805"> <lb/> ἔστω γὰρ εὐθεῖα. </s> <s id="g0120806"> τεθείσης οὖν ταύτης διαμέτρου, καὶ παραπληρωθεισῶν <lb/> τῶν πλευρῶν, ἀνάγκη τὸν τῶν πλευρῶν λόγον <lb/> φέρεσθαι τὸ φερόμενον· τοῦτο γὰρ δέδεικται πρότερον. </s> <s id="g0120807"> οὐκ <lb/> ἄρα ποιήσει εὐθεῖαν τὸ ἐν μηδενὶ λόγῳ φερόμενον μηδένα <lb/> χρόνον. </s> <s id="g0120808"> ἐὰν γάρ τινα λόγον ἐνεχθῇ ἐν χρόνῳ τινί, τοῦτον <lb/> ἀνάγκη τὸν χρόνον εὐθεῖαν εἶναι φορὰν διὰ τὰ προειρημένα. </s> <s id="g0120809"> <lb/> ὥστε περιφερὲς γίνεται, δύο φερόμενον φορὰς ἐν μηθενὶ <lb/> λόγῳ μηθένα χρόνον. </s> </p> <p n="12"> <s id="g0120901"> ὅτι μὲν τοίνυν ἡ τὸν κύκλον γράφουσα <lb/> φέρεται δύο φορὰς ἅμα, φανερὸν ἔκ τε τούτων, <lb/> καὶ ὅτι τὸ φερόμενον κατ' εὐθεῖαν ἐπὶ τὴν κάθετον <pb xlink:href="080/01/005.jpg" ed="Bekker" n="849a"/> <lb/> ἀφικνεῖται, ὥστε εἶναι πάλιν αὐτὴν ἀπὸ τοῦ κέντρου κάθετον. </s> </p> <p n="13"> <s id="g0121001"> <lb/> ἔστω κύκλος ὁ ΑΒΓ, τὸ δ' ἄκρον τὸ ἐφ' οὗ Β φερέσθω <lb/> ἐπὶ τὸ Δ· ἀφικνεῖται δέ ποτε ἐπὶ τὸ Γ. </s> <s id="g0121002"> εἰ μὲν οὖν ἐν τῷ <lb/> λόγῳ ἐφέρετο ὃν ἔχει ἡ ΒΔ πρὸς τὴν ΔΓ, ἐφέρετο ἂν <lb/> τὴν διάμετρον τὴν ἐφ' ᾗ ΒΓ. </s> <s id="g0121003"> νῦν, δέ ἐπείπερ ἐν οὐδενὶ <lb/> λόγῳ, ἐπὶ τὴν περιφέρειαν φέρεται τὴν ἐφ' ᾗ ΒΕΓ. </s> <figure id="id.080.01.005.1.jpg" xlink:href="080/01/005/1.jpg"/> </p> <p n="14"> <s id="g0121101"> ἐὰν <lb/> δὲ δυοῖν φερομένοιν ἀπὸ τῆς αὐτῆς ἰσχύος τὸ μὲν ἐκκρούοιτο <lb/> πλεῖον τὸ δὲ ἔλαττον, εὔλογον βραδύτερον κινηθῆναι <lb/> τὸ πλεῖον ἐκκρουόμενον τοῦ ἔλαττον ἐκκρουομένου· ὃ δοκεῖ <lb/> συμβαίνειν ἐπὶ τῆς μείζονος καὶ ἐλάττονος τῶν ἐκ τοῦ <lb/> κέντρου γραφουσῶν τοὺς κύκλους. </s> <s id="g0121102"> διὰ γὰρ τὸ ἐγγύτερον <lb/> εἶναι τοῦ μένοντος τῆς ἐλάττονος τὸ ἄκρον ἢ τὸ τῆς μείζονος, <lb/> ὥσπερ ἀντισπώμενον εἰς τοὐναντίον, ἐπὶ τὸ μέσον βραδύτερον <lb/> φέρεται τὸ τῆς ἐλάττονος ἄκρον. </s> </p> <p n="15"> <s id="g0121201"> πάσῃ μὲν οὖν <lb/> κύκλον γραφούσῃ τοῦτο συμβαίνει, καὶ φέρεται τὴν μὲν <lb/> κατὰ φύσιν τὴν δὲ παρὰ φύσιν κατὰ τὴν περιφέρειαν, <lb/> εἰς τὸ πλάγιον καὶ τὸ κέντρον. μείζω δ' ἀεὶ τὴν παρὰ <lb/> φύσιν ἡ ἐλάττων φέρεται· διὰ γὰρ τὸ ἐγγύτερον εἶναι τοῦ <lb/> κέντρου τοῦ ἀντισπῶντος κρατεῖται μᾶλλον. </s> </p> <p n="16"> <s id="g0121301"> ὅτι δὲ μεῖζον <lb/> τὸ παρὰ φύσιν κινεῖται ἡ ἐλάττων τῆς μείζονος τῶν ἐκ τοῦ <lb/> κέντρου γραφουσῶν τοὺς κύκλους, ἐκ τῶνδε δῆλον. </s> <s id="g0121302"> ἔστω <lb/> κύκλος ἐφ' οὗ ΒΓΔΕ, καὶ ἄλλος ἐν τούτῳ ἐλάττων, <lb/> ἐφ' οὗ ΧΝΜΞ, περὶ τὸ αὐτὸ κέντρον τὸ Α· καὶ ἐκβεβλήσθωσαν <lb/> αἱ διάμετροι, ἐν μὲν τῷ μεγάλῳ, ἐφ' ὧν ΓΔ <lb/> καὶ ΒΕ, ἐν δὲ τῷ ἐλάττονι αἱ ΜΧ ΝΞ· καὶ τὸ ἑτερόμηκες <lb/> παραπεπληρώσθω, τὸ ΔΨΡΓ. εἰ δὴ ἡ ΑΒ γράφουσα <lb/> κύκλον ἥξει ἐπὶ τὸ αὐτὸ ὅθεν ὡρμήθη ἐπὶ τὴν ΑΕ, δῆλον <lb/> ὅτι φέρεται πρὸς αὑτήν. </s> <s id="g0121303"> ὁμοίως δὲ καὶ ἡ ΑΧ πρὸς τὴν <lb/> ΑΧ ἥξει. </s> <s id="g0121304"> βραδύτερον δὲ φέρεται ἡ ΑΧ τῆς ΑΒ, ὥσπερ <lb/> εἴρηται, διὰ τὸ γίνεσθαι μείζονα τὴν ἔκκρουσιν καὶ ἀντισπᾶσθαι <lb/> μᾶλλον τὴν ΑΧ. </s> </p> <p n="17"> <s id="g0121401"> ἤχθω δὲ ἡ ΑΘΗ, καὶ ἀπὸ <lb/> τοῦ Θ κάθετος ἐπὶ τὴν ΑΒ ἡ ΘΖ ἐν τῷ κύκλῳ, καὶ πάλιν <lb/> ἀπὸ τοῦ Θ ἤχθω παρὰ τὴν ΑΒ ἡ ΘΩ, καὶ ἡ ΩΥ <lb/> ἐπὶ τὴν ΑΒ κάθετον, καὶ ἡ ΗΚ. </s> <s id="g0121402"> αἱ δὴ ἐφ' ὧν ΩΥ καὶ <lb/> ΘΖ ἴσαι. ἡ ἄρα ΒΥ ἐλάττων τῆς ΧΖ· </s> <s id="g0121403"> αἱ γὰρ ἴσαι <lb/> εὐθεῖαι ἐπ' ἀνίσους κύκλους ἐμβληθεῖσαι πρὸς ὀρθὰς τῇ <lb/> διαμέτρῳ ἔλαττον τμῆμα ἀποτέμνουσι τῆς διαμέτρου ἐν <lb/> τοῖς μείζοσι κύκλοις, ἔστι δὲ ἡ ΩΥ ἴση τῇ ΘΖ. </s> <s id="g0121404"> ἐν <pb xlink:href="080/01/006.jpg" ed="Bekker" n="849b"/> <lb/> ὅσῳ δὴ χρόνῳ ἡ ΑΘ τὴν ΧΘ ἐνηνέχθη, ἐν τοσούτῳ χρόνῳ ἐν <lb/> τῷ κύκλῳ τῷ μείζονι μείζονα τῆς ΒΩ ἐνήνεκται τὸ ἄκρον <lb/> τῆς ΒΑ. </s> <figure id="id.080.01.006.1.jpg" xlink:href="080/01/006/1.jpg"/> </p> <p n="18"> <s id="g0121501"> ἡ μὲν γὰρ κατὰ φύσιν φορὰ ἴση, ἡ δὲ παρὰ <lb/> φύσιν ἐλάττων· ἡ δὲ ΒΥ τῆς ΖΧ ἐλάττων. </s> <s id="g0121502"> δεῖ δὲ ἀνάλογον εἶναι, <lb/> ὡς τὸ κατὰ φύσιν πρὸς τὸ κατὰ φύσιν, τὸ παρὰ φύσιν <lb/> πρὸς τὸ παρὰ φύσιν. </s> <s id="g0121503"> μείζονα ἄρα περιφέρειαν διελήλυθε <lb/> τὴν ΗΒ τῆς ΩΒ. </s> <s id="g0121504"> ἀνάγκη δὲ τὴν ΗΒ ἐν τούτῳ τῷ χρόνῳ <lb/> διεληλυθέναι· </s> <s id="g0121505"> ἐνταῦθα γὰρ ἔσται, ὅταν ἀνάλογον ἀμφοτέρως <lb/> συμβαίνῃ τὸ παρὰ φύσιν πρὸς τὸ κατὰ φύσιν. </s> <s id="g0121506"> εἰ δὴ <lb/> μεῖζόν ἐστι τὸ κατὰ φύσιν ἐν τῇ μείζονι, καὶ τὸ παρὰ φύσιν <lb/> μᾶλλον ἂν ἐνταῦθα συμπίπτοι μοναχῶς, </s> <s id="g0121507"> ὥστε τὸ Β ἐνηνέχθαι <lb/> ἂν τὴν ΒΗ ἐν ῷ τὸ ἐφ' οὗ Χ σημεῖον. ἐνταῦθα γὰρ <lb/> κατὰ φύσιν μὲν γίνεται τῷ Β σημείῳ τὸ ΚΗ. (ἔστι γὰρ <lb/> αὐτὴ ἀπὸ τοῦ Η κάθετος), παρὰ φύσιν δὲ ἐς τὸ ΚΒ. </s> <s id="g0121508"> ἔστι <lb/> δὲ ὡς τὸ ΗΚ πρὸς τὸ ΚΒ, τὸ ΘΖ πρὸς τὸ ΖΧ. φανερὸν <lb/> δὲ ἐὰν ἐπιζευχθῶσιν ἀπὸ τῶν ΒΧ ἐπὶ τὰ ΗΘ. </s> <s id="g0121509"> εἰ δὲ <lb/> ἐλάττων ἢ μείζων τῆς ΗΒ ἔσται, ἣν ἠνέχθη τὸ Β, οὐχ ὁμοίως <lb/> ἔσται οὐδὲ ἀνάλογον ἐν ἀμφοῖν τὸ κατὰ φύσιν πρὸς τὸ <lb/> παρὰ φύσιν. </s> <s id="g0121510"> δι' ἣν μὲν τοίνυν αἰτίαν ἀπὸ τῆς αὐτῆς <lb/> ἰσχύος φέρεται θᾶττον τὸ πλέον ἀπέχον τοῦ κέντρου σημεῖον, καὶ μείζω γράφει ἡ μείζων <lb/> δῆλον διὰ τῶν εἰρημένων· </s> </p> <p n="19"> <s id="g0130101"> διότι δὲ τὰ μὲν μείζω ζυγὰ <lb/> ἀκριβέστερά ἐστι τῶν ἐλαττόνων, φανερὸν ἐκ τούτων. </s> <s id="g0130102"> γίνεται <lb/> γὰρ τὸ μὲν σπάρτον κέντρον (μένει γὰρ τοῦτο), τὸ δὲ ἐπὶ <lb/> ἑκάτερον μέρος τῆς πλάστιγγος αἱ ἐκ τοῦ κέντρου. </s> <s id="g0130103"> ἀπὸ οὖν <lb/> τοῦ αὐτοῦ βάρους ἀνάγκη θᾶττον κινεῖσθαι τὸ ἄκρον τῆς <lb/> πλάστιγγος, ὅσῳ ἂν πλεῖον ἀπέχῃ τοῦ σπάρτου, </s> <s id="g0130104"> καὶ ἔνια <lb/> μὲν μὴ δῆλα εἶναι ἐν τοῖς μικροῖς ζυγοῖς πρὸς τὴν αἴσθησιν <lb/> ἐπιτιθέμενα βάρη, ἐν δὲ τοῖς μεγάλοις δῆλα </s> <s id="g0130105"> οὐθὲν γὰρ <lb/> κωλύει ἔλαττον κινηθῆναι μέγεθος ἢ ὥστε εἶναι τῇ ὄψει <lb/> φανερόν. </s> <s id="g0130106"> ἐπὶ δὲ τῆς μεγάλης πλάστιγγος ποιεῖ ὁρατὸν τὸ <lb/> αὐτὸ βάρος μέγεθος. </s> <s id="g0130107"> ἔνια δὲ δῆλα μὲν ἐπ' ἀμφοῖν ἐστίν, <lb/> ἀλλὰ πολλῷ μᾶλλον ἐπὶ τῶν μειζόνων διὰ τὸ πολλῷ <lb/> μεῖζον γίνεσθαι τὸ μέγεθος τῆς ῥοπῆς ὑπὸ τοῦ αὐτοῦ βάρους <lb/> ἐν τοῖς μείζοσι. </s> <s id="g0130108"> καὶ διὰ τοῦτο τεχνάζουσιν οἱ ἁλουργοπῶλαι <lb/> πρὸς τὸ παρακρούεσθαι ἱστάντες, τό τε σπάρτον <lb/> οὐκ ἐν μέσῳ τιθέντες, καὶ μόλυβδον τῆς φάλαγγος εἰς <lb/> θάτερον μέρος ἐγχέοντες, ἢ τοῦ ξύλου τὸ πρὸς τὴν ῥίζαν <lb/> πρὸς ὃ βούλονται ῥέπειν ποιοῦντες, ἢ ἐὰν ἔχῃ ὄζον· <pb xlink:href="080/01/007.jpg" ed="Bekker" n="850a"/> <lb/> βαρύτερον γὰρ ἐν ᾧ μέρος ἡ ῥίζα τοῦ ξύλου ἐστίν, ὁ δὲ ὄζος ῥίζα <lb/> τίς ἐστιν. </s> </p> <p n="20"> <s id="g0130201prop02"> <lb/> Διὰ τί, ἐὰν μὲν ἄνωθεν ᾖ τὸ σπαρτίον, ὅταν κάτωθεν <lb/> ῥέψαντος ἀφέλῃ τὸ βάρος, πάλιν ἀναφέρεται τὸ ζυγόν, <lb/> ἐὰν δὲ κάτωθεν ὑποστῇ, οὐκ ἀναφέρεται ἀλλὰ μένει; </s> <s id="g0130202"> ἢ <lb/> διότι ἄνωθεν μὲν τοῦ σπαρτίου ὄντος πλεῖον τοῦ ζυγοῦ γίνεται <lb/> τὸ ἐπέκεινα τῆς καθέτου; τὸ γὰρ σπαρτίον ἐστὶ κάθετος. <lb/> ὥστε ἀνάγκη ἐστὶ κάτω ῥέπειν τὸ πλέον, ἕως ἂν ἔλθῃ ἡ <lb/> δίχα διαιροῦσα τὸ ζυγὸν ἐπὶ τὴν κάθετον αὐτήν, ἐπικειμένου <lb/> τοῦ βάρους ἐν τῷ ἀνεσπασμένῳ μορίῳ τοῦ ζυγοῦ. </s> <s id="g0130203"> <lb/> ἔστω ζυγὸν ὀρθὸν τὸ ἐφ' οὗ ΒΓ, σπαρτίον δὲ τὸ ΑΔ. ἐκβαλλόμενον <lb/> δὴ τοῦτο κάτω κάθετος ἔσται ἐφ' ἧς ἡ ΑΔΜ. </s> <s id="g0130204"> <lb/> ἐὰν οὖν ἐπὶ τὸ Β ἡ ῥοπὴ ἐπιτεθῇ ἔσται, τὸ μὲν Β οὗ τὸ Ε, <lb/> τὸ δὲ Γ οὗ τὸ Ζ, ὥστε ἡ δίχα διαιροῦσα τὸ ζυγὸν πρῶτον <lb/> μὲν ἦν ἡ ΔΜ τῆς καθέτου αὐτῆς, ἐπικειμένης δὲ τῆς ῥοπῆς <lb/> ἔσται ἡ ΔΘ· ὥστε τοῦ ζυγοῦ ἐφ' ᾧ ΕΖ τὸ ἔξω τῆς καθέτου <lb/> τῆς ἐφ' ἧς ΑM, τοῦ ἐν ᾧ ΘΠ, μείζω τοῦ ἡμίσεος. </s> <s id="g0130205"> <lb/> ἐὰν οὖν ἀφαιρεθῇ τὸ βάρος ἀπὸ τοῦ Ε, ἀνάγκη κάτω φέρεσθαι <lb/> τὸ Ζ· ἔλαττον γάρ ἐστι τὸ Ε. </s> <s id="g0130206"> ἐὰν μὲν οὖν ἄνω τὸ <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"/> <s id="g0130207"> ἐὰν <lb/> δὲ κάτωθεν ᾖ τὸ ὑποκείμενον, τοὐναντίον ποιεῖ· πλεῖον γὰρ <lb/> γίνεται τοῦ ἡμίσεος τοῦ ζυγοῦ τὸ κάτω μέρος ἢ ὡς ἡ κάθετος <lb/> διαιρεῖ ὥστε οὐκ ἀναφέρεται· κουφότερον γὰρ τὸ ἐπηρτημένον. </s> <s id="g0130208"> <lb/> ἔστω ζυγὸν τὸ ἐφ' οὗ ΝΞ, τὸ ὀρθόν, κάθετος δὲ ἡ <lb/> ΚΛΜ. δίχα δὴ διαιρεῖται τὸ ΝΞ. </s> <s id="g0130209"> ἐπιτεθέντος δὲ βάρους <lb/> ἐπὶ τὸ Ν, ἔσται τὸ μὲν Ν οὗ τὸ Ο, τὸ δὲ Ξ οὗ τὸ Ρ, ἡ δὲ <lb/> ΚΛ οὗ τὸ ΛΘ, ὥστε μεῖζόν ἐστι τὸ ΚΟ τοῦ ΛΡ τῷ ΘΚΛ. </s> <s id="g0130210"> <lb/> καὶ ἀφαιρεθέντος οὖν τοῦ βάρους ἀνάγκη μένειν· ἐπίκειται <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"/> </p> <p n="21"> <s id="g0130301prop03"> <lb/> Διὰ τί κινοῦσι μεγάλα βάρη μικραὶ δυνάμεις τῷ μοχλῷ, <lb/> ὥσπερ ἐλέχθη καὶ κατ' ἀρχήν, προσλαβόντι βάρος <lb/> ἔτι τὸ τοῦ μοχλοῦ; ῥᾷον δὲ τὸ ἔλαττόν ἐστι κινῆσαι βάρος, <lb/> ἔλαττον δέ ἐστιν ἄνευ τοῦ μοχλοῦ. </s> <s id="g0130302"> ἢ ὅτι αἴτιόν ἐστιν ὁ μοχλός, <lb/> ζυγὸν ὢν κάτωθεν ἔχον τὸ σπαρτίον καὶ εἰς ἄνισα διῃρημένον; <lb/> τὸ γὰρ ὑπομόχλιόν ἐστι τὸ σπαρτίον· μένει <lb/> γὰρ ἄμφω ταῦτα, ὥσπερ τὸ κέντρον. </s> <s id="g0130303"> ἐπεὶ δὲ θᾶττον ὑπὸ <lb/> τοῦ ἴσου βάρους κινεῖται ἡ μείζων τῶν ἐκ τοῦ κέντρου, ἔστι δὲ <lb/> τρία τὰ περὶ τὸν μοχλόν, τὸ μὲν ὑπομόχλιον, σπάρτον <lb/> καὶ κέντρον, δύο δὲ βάρη, ὅ τε κινῶν καὶ τὸ κινούμενον· <pb xlink:href="080/01/008.jpg" ed="Bekker" n="850b"/> <lb/> ὃ οὖν τὸ κινούμενον βάρος πρὸς τὸ κινοῦν, τὸ μῆκος πρὸς τὸ μῆκος <lb/> ἀντιπέπονθεν. </s> <s id="g0130304"> αἰεὶ δὲ ὅσῳ ἂν μεῖζον ἀφεστήκῃ τοῦ ὑπομοχλίου, <lb/> ῥᾷον κινήσει. </s> <s id="g0130305"> αἰτία δέ ἐστιν ἡ προλεχθεῖσα, ὅτι ἡ <lb/> πλεῖον ἀπέχουσα ἐκ τοῦ κέντρου μείζονα κύκλον γράφει. </s> <s id="g0130306"> ὥστε <lb/> ἀπὸ τῆς αὐτῆς ἰσχύος πλέον μεταστήσεται τὸ κινοῦν τὸ <lb/> πλεῖον τοῦ ὑπομοχλίου ἀπέχον. </s> <s id="g0130307"> ἔστω μοχλὸς ἐφ' οὗ ΑΒ, <lb/> βάρος δὲ ἐφ' ᾧ τὸ Γ, τὸ δὲ κινοῦν ἐφ' ᾧ τὸ Δ, ὑπομόχλιον <lb/> ἐφ' ᾧ τὸ Ε, </s> <s id="g0130308"> τὸ δὲ ἐφ' ᾧ τὸ Δ κινῆσαν ἐφ' ᾧ τὸ Η, κινούμενον <lb/> δὲ τὸ ἐφ' οὗ Γ, βάρος ἐφ' οὗ Κ. </s> <figure id="id.080.01.008.1.jpg" xlink:href="080/01/008/1.jpg"/> </p> <p n="22"> <s id="g0130401prop04"> <lb/> Διὰ τί οἱ μεσόνεοι μάλιστα τὴν ναῦν κινοῦσιν; </s> <s id="g0130402"> ἢ διότι <lb/> ἡ κώπη μοχλός ἐστιν; ὑπομόχλιον μὲν γὰρ ὁ σκαλμὸς γίνεται <lb/> (μένει γὰρ δὴ τοῦτο), τὸ δὲ βάρος ἡ θάλαττα, ἣν <lb/> ἀπωθεῖ ἡ κώπη· ὁ δὲ κινῶν τὸν μοχλὸν ὁ ναύτης ἐστίν. </s> <s id="g0130403"> <lb/> ἀεὶ δὲ πλέον βάρος κινεῖ, ὅσῳ ἂν πλέον ἀφεστήκῃ τοῦ ὑπομοχλίου <lb/> ὁ κινῶν τὸ βάρος· </s> <s id="g0130404"> μείζων γὰρ οὕτω γίνεται ἡ ἐκ <lb/> τοῦ κέντρου, ὁ δὲ σκαλμὸς ὑπομόχλιον ὢν κέντρον ἐστίν. </s> <s id="g0130405"> ἐν <lb/> μέσῃ δὲ τῇ νηῒ πλεῖστον τῆς κώπης ἐντός ἐστιν· καὶ γὰρ ἡ <lb/> ναῦς ταύτῃ εὐρυτάτη ἐστίν, ὥστε πλεῖον ἐπ' ἀμφότερα ἐνδέχεσθαι <lb/> μέρος τῆς κώπης ἑκατέρου τοίχου ἐντὸς εἶναι τῆς <lb/> νεώς. </s> <s id="g0130406"> κινεῖται μὲν οὖν ἡ ναῦς διὰ τὸ ἀπερειδομένης τῆς κώπης <lb/> εἰς τὴν θάλασσαν τὸ ἄκρον τῆς κώπης τὸ ἐντὸς προϊέναι <lb/> εἰς τὸ πρόσθεν, </s> <s id="g0130407"> τὴν δὲ ναῦν προσδεδεμένην τῷ σκαλμῷ συμπροϊέναι, <lb/> ᾗ τὸ ἄκρον τῆς κώπης. </s> <s id="g0130408"> ᾗ γὰρ πλείστην θάλασσαν <lb/> διαιρεῖ ἡ κώπη, ταύτῃ ἀνάγκη μάλιστα προωθεῖσθαι· πλείστην <lb/> δὲ διαιρεῖ ᾗ πλεῖστον μέρος ἀπὸ τοῦ σκαλμοῦ τῆς κώπης <lb/> ἐστίν. </s> <s id="g0130409"> διὰ τοῦτο οἱ μεσόνεοι μάλιστα κινοῦσιν· μέγιστον γὰρ <lb/> ἐν μέσῃ νηῒ τὸ ἀπὸ τοῦ σκαλμοῦ τῆς κώπης τὸ ἐντός ἐστιν. </s> </p> <p n="23"> <s id="g0130501prop05"> <lb/> Διὰ τί τὸ πηδάλιον μικρὸν ὄν, καὶ ἐπ' ἐσχάτῳ τῷ <lb/> πλοίῳ, τοσαύτην δύναμιν ἔχει ὥστε ὑπὸ μικροῦ οἴακος καὶ <lb/> ἑνὸς ἀνθρώπου δυνάμεως, καὶ ταύτης ἠρεμαίας, μεγάλα κινεῖσθαι <lb/> μεγέθη πλοίων; </s> <s id="g0130502"> ἢ διότι καὶ τὸ πηδάλιόν ἐστι μοχλός, <lb/> καὶ μοχλεύει ὁ κυβερνήτης. ᾗ μὲν οὖν προσήρμοσται <lb/> τῷ πλοίῳ, γίνεται ὑπομόχλιον, τὸ δὲ ὅλον πηδάλιον ὁ <lb/> μοχλός,, τὸ δὲ βάρος ἡ θάλασσα, ὁ δὲ κυβερνήτης ὁ κινῶν. </s> <s id="g0130503"> <lb/> οὐ κατὰ πλάτος δὲ λαμβάνει τὴν θάλασσαν, ὥσπερ ἡ κώπη, <lb/> τὸ πηδάλιον. οὐ γὰρ εἰς τὸ πρόσθεν κινεῖ τὸ πλοῖον, ἀλλὰ <lb/> κινούμενον κλίνει, πλαγίως τὴν θάλατταν δεχόμενον. </s> <s id="g0130504"> ἐπεὶ <lb/> γὰρ τὸ βάρος ἦν ἡ θάλασσα, τοὐναντίον ἀπερειδόμενον κλίνει <lb/> τὸ πλοῖον. τὸ γὰρ ὑπομόχλιον εἰς τοὐναντίον <pb xlink:href="080/01/009.jpg" ed="Bekker" n="851a"/> <lb/> στρέφεται, ἡ θάλασσα δὲ ἐντός· ἐκεῖνο δὲ εἰς τὸ ἐκτός. τούτῳ δὲ ἀκολουθεῖ; <lb/> τὸ πλοῖον διὰ τὸ συνδεδέσθαι. </s> <s id="g0130505"> ἡ μὲν οὖν κώπη κατὰ <lb/> πλάτος τὸ βάρος ὠθοῦσα καὶ ὑπ' ἐκείνου ἀντωθουμένη εἰς τὸ <lb/> εὐθὺ προάγει· τὸ δὲ πηδάλιον, ὥσπερ κάθηται πλάγιον, <lb/> τὴν εἰς τὸ πλάγιον, ἢ δεῦρο ἢ ἐκεῖ, ποιεῖ κίνησιν. </s> <s id="g0130506"> ἐπ' ἄκρου <lb/> δὲ καὶ οὐκ ἐν μέσῳ κεῖται, ὅτι ῥᾷστον τὸ κινούμενον κινῆσαι <lb/> ἀπ' ἄκρου κινοῦν. </s> <s id="g0130507"> τάχιστα γὰρ φέρεται τὸ πρῶτον μέρος <lb/> διὰ τὸ ὥσπερ ἐν τοῖς φερομένοις ἐπὶ τέλει λήγειν τὴν φοράν, <lb/> οὕτω καὶ τοῦ συνεχοῦς ἐπὶ τέλους ἀσθενεστάτη ἐστὶν ἡ φορά. <lb/> εἰ δὲ ἀσθενεστάτη, ῥᾳδία ἐκκρούειν. </s> <s id="g0130508"> διά τε δὴ ταῦτα ἐν τῇ <lb/> πρύμνῃ τὸ πηδάλιόν ἐστι, καὶ ὅτι ἐνταῦθα μικρᾶς κινήσεως <lb/> γενομένης πολλῷ μεῖζον τὸ διάστημα ἐπὶ τῷ ἐσχάτῳ γίνεται, <lb/> διὰ τὸ τὴν ἴσην γωνίαν ἐπὶ μείζονα καθῆσθαι, καὶ ὅσῳ <lb/> ἂν μείζους ὦσιν αἱ περιέχουσαι. </s> <s id="g0130509"> δῆλον δὲ ἐκ τούτου καὶ δι' ἣν <lb/> αἰτίαν μᾶλλον προέρχεται εἰς τοὐναντίον τὸ πλοῖον ἢ ἡ τῆς <lb/> κώπης πλάτη· τὸ αὐτὸ γὰρ μέγεθος τῇ αὐτῇ ἰσχύϊ κινούμενον <lb/> ἐν ἀέρι πλέον ἢ ἐν τῷ ὕδατι πρόεισιν. </s> <s id="g0130510"> ἔστω γὰρ ἡ <lb/> ΑΒ κώπη, τὸ δὲ Γ ὁ σκαλμός, τὸ δὲ Α τὸ ἐν τῷ πλοίῳ, ἡ <lb/> ἀρχὴ τῆς κώπης, τὸ δὲ Β τὸ ἐν τῇ θαλάττῃ. </s> <s id="g0130511"> εἰ δὴ τὸ Α <lb/> οὗ τὸ Δ μετακεκίνηται, τὸ Β οὐκ ἔσται οὗ τὸ Ε· ἴση γὰρ ἡ <lb/> ΒΕ τῇ ΑΔ. ἴσον οὖν μετακεχωρηκὸς ἔσται. </s> <s id="g0130512"> ἀλλ' ἦν ἔλαττον. <lb/> ἔσται δὴ οὗ τὸ Ζ ἢ τὸ Θ. ἄρα τοίνυν τὴν ΑΒ, καὶ οὐχ ἡ τὸ <lb/> Γ, καὶ κάτωθεν. ἐλάττων γὰρ ἡ ΒΖ τῆς ΑΔ, ὥστε καὶ <lb/> ἡ ΘΖ τῆς ΔΘ· ὅμοια γὰρ τὰ τρίγωνα. </s> <s id="g0130513"> καθεστηκὸς δὲ <lb/> ἔσται καὶ τὸ μέσον, τὸ ἐφ' οὗ Γ· εἰς τοὐναντίον γὰρ τῷ ἐν τῇ <lb/> θαλάττῃ ἄκρῳ τῷ Β μεταχωρεῖ, ᾗπερ τὸ ἐν τῷ πλοίῳ <lb/> ἄκρον τὸ Α </s> <s id="g0130514"> μὴ ἐχώρει οὗ τὸ Δ. ὥστε μετακινηθήσεται τὸ <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 id="id.080.01.009.2.jpg" xlink:href="080/01/009/2.jpg"/> <s id="g0130515"> τὸ δ' <lb/> αὐτὸ καὶ τὸ πηδάλιον ποιεῖ, πλὴν ὅτι εἰς τὸ πρόσθεν οὐδὲν <lb/> συμβάλλεται τῷ πλοίῳ, ὥσπερ ἐλέχθη ἐπὶ ἄνω, ἀλλὰ <lb/> μόνον τὴν πρύμναν εἰς τὸ πλάγιον ἀπωθεῖ ἔνθα ἢ ἔνθα· εἰς <lb/> τοὐναντίον γὰρ ἡ πρῷρα οὕτω νεύει. </s> <s id="g0130516"> ᾗ μὲν δὴ τὸ πηδάλιον <lb/> προσέζευκται, δεῖ οἷόν τι τοῦ κινουμένου μέσον νοεῖν, καὶ ὥσπερ <lb/> ὁ σκαλμὸς τῇ κώπῃ· τὸ δὲ μέσον ὑποχωρεῖ, ᾗ ὁ οἴαξ μετακινεῖται. </s> <s id="g0130517"> <lb/> ἐὰν μὲν εἴσω ἄγῃ, καὶ ἡ πρύμνα δεῦρο μεθέστηκεν· <lb/> ἡ δὲ πρῷρα εἰς τοὐναντίον νεύει· ἐν γὰρ τῷ αὐτῷ <lb/> οὔσης τῆς πρῴρας τὸ πλοῖον μεθέστηκεν ὅλον. </s> <figure id="id.080.01.009.3.jpg" xlink:href="080/01/009/3.jpg"/> </p> <p n="24"> <s id="g0130601prop06"> <lb/> Διὰ τί, ὅσῳ ἂν ἡ κεραία ἀνωτέρα ᾖ, θᾶττον πλεῖ τὰ <lb/> πλοῖα τῷ αὐτῷ ἱστίῳ καὶ τῷ αὐτῷ πνεύματι; </s> <s id="g0130602"> ἢ διότι γίνεται <lb/> ὁ μὲν ἱστὸς μοχλός, ὑπομόχλιον δὲ τὸ ἑδώλιον ἐν <pb xlink:href="080/01/010.jpg" ed="Bekker" n="851b"/> <lb/> ᾧ ἐμπέπηγεν, ὃ δὲ δεῖ κινεῖν βάρος, τὸ πλοῖον, τὸ δὲ κινοῦν <lb/> τὸ ἐν τῷ ἱστίῳ πνεῦμα. </s> <s id="g0130603"> εἰ δ' ὅσῳ ἂν πορρώτερον ᾖ τὸ ὑπομόχλιον, <lb/> ῥᾷον κινεῖ καὶ θᾶττον ἡ αὐτὴ δύναμις τὸ αὐτὸ <lb/> βάρος, ἡ οὖν κεραία ἀνώτερον ἀγομένη καὶ τὸ ἱστίον πορρώτερον <lb/> ποιεῖ τοῦ ἑδωλίου ὑπομοχλίου ὄντος. </s> </p> <p n="25"> <s id="g0130701prop07"> <lb/> Διὰ τί, ὅταν ἐξ οὐρίας βούλωνται διαδραμεῖν μὴ οὐρίου <lb/> τοῦ πνεύματος ὄντος, τὸ μὲν πρὸς τὸν κυβερνήτην τοῦ ἱστίου <lb/> μέρος στέλλονται, τὸ δὲ πρὸς τὴν πρῷραν ποδιαῖον ποιησάμενοι <lb/> ἐφιᾶσιν; </s> <s id="g0130702"> ἢ διότι ἀντισπᾶν τὸ πηδάλιον πολλῷ μὲν <lb/> ὄντι τῷ πνεύματι οὐ δύναται, ὀλίγῳ δέ, ὃ ὑποστέλλονται. </s> <s id="g0130703"> <lb/> προάγει μὲν οὖν τὸ πνεῦμα, εἰς οὔριον δὲ καθίστησι τὸ <lb/> πηδάλιον, ἀντισπῶν καὶ μοχλεῦον τὴν θάλατταν. </s> <s id="g0130704"> ἅμα <lb/> δὲ καὶ οἱ ναῦται μάχονται τῷ πνεύματι· ἀνακλίνουσι γὰρ <lb/> ἐπὶ τὸ ἐναντίον ἑαυτούς. </s> </p> <p n="26"> <s id="g0130801prop08"> <lb/> Διὰ τί τὰ στρογγύλα καὶ περιφερῆ τῶν σχημάτων <lb/> εὐκινητότερα; </s> <s id="g0130802"> τριχῶς δὲ ἐνδέχεται τὸν κύκλον κυλισθῆναι· <lb/> ἢ γὰρ κατὰ τὴν ἁψῖδα, συμμεταβάλλοντος τοῦ κέντρου, <lb/> ὥσπερ ὁ τῆς ἁμάξης τροχὸς κυλίεται· ἢ περὶ τὸ κέντρον <lb/> μόνον, ὥσπερ αἱ τροχιλέαι, τοῦ κέντρου μένοντος· ἢ παρὰ <lb/> τὸ ἐπίπεδον, τοῦ κέντρου μένοντος, ὥσπερ ὁ κεραμεικὸς τροχὸς <lb/> κυλίνδεται. </s> <s id="g0130803"> εἰ μὲν δὴ τάχιστα τὰ τοιαῦτα, διά τε τὸ <lb/> μικρῷ ἅπτεσθαι τοῦ ἐπιπέδου, ὥσπερ ὁ κύκλος κατὰ στιγμήν, <lb/> καὶ διὰ τὸ μὴ προσκόπτειν· ἀφέστηκε γὰρ τῆς γῆς <lb/> ἡ γωνία. </s> <s id="g0130804"> καὶ ἔτι ᾧ ἂν ἀπαντήσῃ σώματι, πάλιν τούτου <lb/> κατὰ μικρὸν ἅπτεται. </s> <figure id="id.080.01.010.1.jpg" xlink:href="080/01/010/1.jpg"/> <s id="g0130805"> εἰ δ' εὐθύγραμμον ἦν, τῇ εὐθείᾳ <lb/> ἐπὶ πολὺ ἥπτετο ἂν τοῦ ἐπιπέδου. </s> <s id="g0130806"> ἔτι ᾗ ῥέπει ἐπὶ τὸ βάρος, <lb/> ταύτῃ κινεῖ ὁ κινῶν. ὅταν μὲν γὰρ πρὸς ὄρθιον ἡ διάμετρος <lb/> ᾖ τοῦ κύκλου τῷ ἐπιπέδῳ, ἁπτομένου τοῦ κύκλου κατὰ στιγμὴν <lb/> τοῦ ἐπιπέδου, ἴσον τὸ βάρος ἐπ' ἀμφότερα διαλαμβάνει <lb/> ἡ διάμετρος· ὅταν δὲ κινῆται, εὐθὺς πλέον ἐφ' ᾧ <lb/> κινεῖται, ὥσπερ ῥέπον. ἐντεῦθεν εὐκινητότερον τῷ ὠθοῦντι εἰς <lb/> τοὔμπροσθεν· ἐφ' ὃ γὰρ ῥέπει ἕκαστον, εὐκίνητόν ἐστιν, <lb/> εἴπερ καὶ τὸ ἐπὶ τὸ ἐναντίον τῆς ῥοπῆς δυσκίνητον. </s> <s id="g0130807"> ἔτι λέγουσί <lb/> τινες ὅτι καὶ ἡ γραμμὴ ἡ τοῦ κύκλου ἐν φορᾷ ἐστὶν <lb/> ἀεί, ὥσπερ τὰ μένοντα, διὰ τὸ ἀντερείδειν, οἷον καὶ τοῖς <lb/> μείζοσι κύκλοις ὑπάρχει πρὸς τοὺς ἐλάττονας. </s> <s id="g0130808"> θᾶττον γὰρ <lb/> ὑπὸ τῆς ἴσης ἰσχύος κινοῦνται οἱ μείζους καὶ τὰ βάρη κινοῦσι, <lb/> διὰ τὸ ῥοπήν τινα ἔχειν τὴν γωνίαν τὴν τοῦ μείζονος <lb/> κύκλου πρὸς τὴν τοῦ ἐλάττονος, καὶ εἶναι ὅπερ ἡ διάμετρος <lb/> πρὸς τὴν διάμετρον. ἀλλὰ μὴν πᾶς κύκλος μείζων <pb xlink:href="080/01/011.jpg" ed="Bekker" n="852a"/> <lb/> πρὸς ἐλάττονα· ἄπειροι γὰρ οἱ ἐλάττονες. </s> <s id="g0130809"> εἰ δὲ καὶ πρὸς ἕτερον <lb/> ἔχει ῥοπὴν ὁ κύκλος, ὁμοίως δὲ εὐκίνητος, καὶ ἄλλην ἂν <lb/> ἔχοι ῥοπὴν ὁ κύκλος </s> <s id="g0130810"> καὶ τὰ ὑπὸ κύκλου κινούμενα, κἂν μὴ <lb/> τῇ ἁψῖδι ἅπτηται τοῦ ἐπιπέδου, ἀλλ' ἢ παρὰ τὸ ἐπίπεδον, <lb/> ἢ ὡς αἱ τροχιλέαι· καὶ γὰρ οὕτως ἔχοντα ῥᾷστα κινοῦνται <lb/> καὶ κινοῦσι τὸ βάρος. </s> <s id="g0130811"> ἢ οὐ τῷ κατὰ μικρὸν ἅπτεσθαι καὶ <lb/> προσκρούειν, ἀλλὰ δι' ἄλλην αἰτίαν. </s> <s id="g0130812"> αὕτη δέ ἐστιν ἡ εἰρημένη <lb/> πρότερον, ὅτι ἐκ δύο φορῶν γεγένηται ὁ κύκλος, ὥστε <lb/> μίαν αὐτῶν αἰεὶ ἔχειν ῥοπήν, καὶ οἷον φερόμενον αὐτὸν <lb/> αἰεὶ κινοῦσιν οἱ κινοῦντες, ὅταν κινῶσι κατὰ τὴν περιφέρειαν <lb/> ὁπωσοῦν. φερομένην γὰρ αὐτὴν κινοῦσιν· </s> <s id="g0130813"> τὴν μὲν γὰρ εἰς <lb/> τὸ πλάγιον αὐτοῦ κίνησιν ὠθεῖ τὸ κινοῦν, τὴν δὲ ἐπὶ τῆς <lb/> διαμέτρου αὐτὸς κινεῖται. </s> </p> <p n="27"> <s id="g0130901prop09"> <lb/> Διὰ τί τὰ διὰ τῶν μειζόνων κύκλων αἰρόμενα καὶ <lb/> ἑλκόμενα ῥᾷον καὶ θᾶττον κινοῦμεν; οἷον καὶ αἱ τροχιλέαι <lb/> αἱ μείζους τῶν ἐλαττόνων, καὶ αἱ σκυτάλαι ὁμοίως. </s> <s id="g0130902"> ἢ <lb/> διότι ὅσῳ ἂν μείζων ἡ ἐκ τοῦ κέντρου ᾖ, ἐν τῷ ἴσῳ χρόνῳ <lb/> πλέον κινεῖται χωρίον, </s> <s id="g0130903"> ὥστε καὶ τοῦ ἴσου βάρους ἐπόντος <lb/> ποιήσει τὸ αὐτό, ὥσπερ εἴπομεν καὶ τὰ μείζω ζυγὰ τῶν <lb/> ἐλαττόνων ἀκριβέστερα εἶναι. </s> <s id="g0130904"> τὸ μὲν γὰρ σπαρτίον ἐστὶ <lb/> κέντρον, τοῦ δὲ ζυγοῦ αἱ ἐπὶ τάδε τοῦ σπαρτίου αἱ ἐκ τοῦ <lb/> κέντρου. </s> </p> <p n="28"> <s id="g0131001prop10"> <lb/> Διὰ τί ῥᾷον, ὅταν ἄνευ βάρους ᾖ, κινεῖται τὸ ζυγόν, <lb/> ἢ ἔχον βάρος; </s> <s id="g0131002"> ὁμοίως δὲ καὶ τροχὸς ἢ ἄλλο τοιοῦτο τὸ <lb/> βαρύτερον μὲν μεῖζον δὲ τοῦ ἐλάττονος καὶ κουφοτέρου. </s> <s id="g0131003"> ἢ <lb/> ὅτι οὐ μόνον εἰς τοὐναντίον τὸ βαρύ, ἀλλὰ καὶ εἰς τὸ πλάγιον <lb/> δυσκίνητόν ἐστιν. </s> <s id="g0131004"> ἐναντίον γὰρ τῇ ῥοπῇ κινῆσαι χαλεπῶς, <lb/> ἐφ' ὃ δὲ ῥέπει, ῥᾳδίως· εἰς δὲ τὸ πλάγιον οὐ ῥέπει. </s> </p> <p n="29"> <s id="g0131101prop11"> <lb/> Διὰ τί ἐπὶ τῶν σκυτάλων ῥᾷον τὰ φορτία κομίζεται <lb/> ἢ ἐπὶ τῶν ἁμαξῶν, ἐχουσῶν τῶν μὲν μεγάλους τροχούς, <lb/> τῶν δὲ μικρούς; </s> <s id="g0131102"> ἢ διότι ἐπὶ τῶν σκυτάλων οὐδεμίαν ἔχει <lb/> πρόσκοψιν, τὸ δὲ ἐπὶ τῶν ἁμαξῶν τὸν ἄξονα, καὶ προσκόπτει <lb/> αὐτῷ· ἔκ τε γὰρ τῶν ἄνωθεν πιέζει αὐτὸν καὶ ἐκ <lb/> τῶν πλαγίων. </s> <s id="g0131103"> τὸ δὲ ἐπὶ τῶν σκυτάλων ἐπὶ δύο τούτων κινεῖται, <lb/> τῇ τε κάτω χώρᾳ ὑποκειμένῃ καὶ τῷ βάρει τῷ <lb/> ἐπικειμένῳ· ἐπ' ἀμφοτέρων γὰρ τούτων κυλίεται τῶν τόπων <lb/> ὁ κύκλος καὶ φερόμενος ὠθεῖται. </s> </p> <p n="30"> <s id="g0131201prop12"> <lb/> Διὰ τί πορρωτέρω τὰ βέλη φέρεται ἀπὸ τῆς σφενδόνης <lb/> ἢ ἀπὸ τῆς χειρός; καίτοι κρατεῖ γε ὁ βάλλων τῇ <pb xlink:href="080/01/012.jpg" ed="Bekker" n="852b"/> <lb/> χειρὶ μᾶλλον ἢ ἀπαρτήσας τὸ βάρος. </s> <s id="g0131202"> καὶ ἔτι οὕτω μὲν δύο βάρη <lb/> κινεῖ, τό τε τῆς σφενδόνης καὶ τὸ βέλος, ἐκεῖνως δὲ τὸ <lb/> βέλος μόνον. </s> <s id="g0131203"> πότερον ὅτι ἐν μὲν τῇ σφενδόνῃ κινούμενον τὸ <lb/> βέλος ῥίπτει ὁ βάλλων (περιαγαγὼν γὰρ κύκλῳ πολλάκις <lb/> ἀφίησιν), </s> <s id="g0131204"> ἐκ δὲ τῆς χειρὸς ἀπὸ τῆς ἠρεμίας ἡ ἀρχή· <lb/> πάντα δὲ εὐκινητότερα κινούμενα ἢ ἠρεμοῦντα. </s> <s id="g0131205"> ἢ διά τε <lb/> τοῦτο, καὶ διότι ἐν μὲν τῷ σφενδονᾶν ἡ μὲν χεὶρ γίνεται <lb/> κέντρον, ἡ δὲ σφενδόνη ἡ ἐκ τοῦ κέντρου· ὅσῳ ἂν ᾖ μείζων <lb/> ἡ ἀπὸ τοῦ κέντρου, κινεῖται θᾶττον. ἡ δὲ ἀπὸ τῆς χειρὸς <lb/> βολὴ πρὸς τὴν σφενδόνην βραχεῖα ἐστίν. </s> </p> <p n="31"> <figure id="id.080.01.012.1.jpg" xlink:href="080/01/012/1.jpg"/> <s id="g0131301prop13"> <lb/> Διὰ τί ῥᾷον κινοῦνται περὶ τὸ αὐτὸ ζυγὸν οἱ μείζους <lb/> τῶν ἐλαττόνων κόλλοπες, καὶ οἱ αὐτοὶ ὄνοι οἱ λεπτότεροι <lb/> ὑπὸ τῆς αὐτῆς ἰσχύος τῶν παχυτέρων; </s> <s id="g0131302"> ἢ διότι ὁ μὲν ὄνος <lb/> καὶ τὸ ζυγὸν κέντρον ἐστίν, τὰ δὲ ἀπέχοντα μεγέθη αἱ ἐκ <lb/> τοῦ κέντρου; θᾶττον δὲ κινοῦνται καὶ πλέον ἀπὸ τῆς αὐτῆς <lb/> ἰσχύος αἱ τῶν μειζόνων κύκλων ἢ αἱ τῶν ἐλαττόνων: ὑπὸ <lb/> τῆς αὐτῆς γὰρ ἰσχύος θᾶττον μεθίσταται τὸ ἄκρον τὸ πορρώτερον <lb/> τοῦ κέντρου. </s> <s id="g0131303"> διὸ πρὸς μὲν τὸ ζυγὸν τοὺς κόλλοπας <lb/> ὄργανα ποιοῦνται, οἷς ῥᾷον στρέφουσιν· ἐν δὲ τοῖς λεπτοῖς <lb/> ὄνοις πλεῖον γίνεται τὸ ἔξω τοῦ ξύλου, αὕτη δὲ γίνεται <lb/> ἡ ἐκ τοῦ κέντρου. </s> </p> <p n="32"> <s id="g0131401prop14"> <lb/> Διὰ τί τὸ αὐτὸ μέγεθος ξύλον ῥᾷον κατεάσσεται περὶ <lb/> τὸ γόνυ, ἐὰν ἴσον ἀποστήσας τῶν ἄκρων ἐχόμενος καταγνύῃ, <lb/> ἢ παρὰ τὸ γόνυ ἐγγὺς ὄντος· καὶ ἐὰν πρὸς τὴν γῆν <lb/> ἐρείσας καὶ τῷ ποδὶ προσβὰς πόρρωθεν τῇ χειρὶ καταγνύῃ, <lb/> ἢ ἐγγύθεν; ἢ διότι ἔνθα μὲν τὸ γόνυ κέντρον, ἔνθα δὲ ὁ <lb/> πούς. </s> <s id="g0131402"> ὅσῳ δ' ἂν πορρώτερον ᾖ τοῦ κέντρου, ῥᾷον κινεῖται <lb/> ἅπαν. κινηθῆναι δὲ ἀνάγκη καταγνύμενον. </s> </p> <p n="33"> <s id="g0131501prop15"> <lb/> Διὰ τί περὶ τοὺς αἰγιαλοὺς αἱ καλούμεναι κρόκαι στρογγύλαι <lb/> εἰσίν, ἐκ μακρῶν τῶν λίθων καὶ ὀστράκων τὸ ἐξ <lb/> ὑπαρχῆς ὄντων; </s> <s id="g0131502"> ἢ διότι τὰ πλεῖον ἀπέχοντα τοῦ μέσου ἐν <lb/> ταῖς κινήσεσι θᾶττον φέρεται. </s> <s id="g0131503"> τὸ μὲν γὰρ μέσον γίνεται <lb/> κέντρον, τὸ δὲ διάστημα ἡ ἐκ τοῦ κέντρου. </s> <s id="g0131504"> ἀεὶ δὲ ἡ μείζων <lb/> ἀπὸ τῆς ἴσης κινήσεως μείζω γράφει κύκλον. τὸ δ' ἐν <lb/> ἴσῳ χρόνῳ μείζω διεξιὸν θᾶττον φέρεται. τὰ δὲ φερόμενα <lb/> θᾶττον ἐκ τοῦ ἴσου ἀποστήματος σφοδρότερον τύπτει. τὰ δὲ <lb/> τύπτοντα μᾶλλον καὶ αὐτὰ τύπτεται μᾶλλον. </s> <s id="g0131505"> ὥστε ἀνάγκη <lb/> θραύεσθαι αἰεὶ τὰ πλέον ἀπέχοντα τοῦ μέσου. τοῦτο δὲ <lb/> πάσχοντα ἀνάγκη γίνεσθαι περιφερῆ. </s> <s id="g0131506"> ταῖς δὲ κρόκαις <pb xlink:href="080/01/013.jpg" ed="Bekker" n="853a"/> <lb/> διὰ τὴν τῆς θαλάττης κίνησιν, διὰ τὸ μετὰ τῆς θαλάττης κινεῖσθαι, <lb/> συμβαίνει ἀεὶ ἐν κινήσει εἶναι καὶ κυλιομέναις <lb/> προσκόπτειν. </s> <s id="g0131507"> τοῦτο δὲ ἀνάγκη μάλιστα συμβαίνειν αὐτοῖς <lb/> τοῖς ἄκροις. </s> </p> <p n="34"> <s id="g0131601prop16"> <lb/> Διὰ τί, ὅσῳ ἂν ᾖ μακρότερα τὰ ξύλα, τοσούτῳ ἀσθενέστερα <lb/> γίνεται, καὶ κάμπτεται αἰρόμενα μᾶλλον, κἂν ᾖ <lb/> τὸ μὲν βραχύ, ὅσον δίπηχυ, λεπτόν, τὸ δὲ ἑκατὸν πηχῶν <lb/> παχύ; </s> <s id="g0131602"> ἢ διότι μοχλὸς γίνεται καὶ βάρος καὶ ὑπομόχλιον <lb/> ἐν τῷ αἴρεσθαι τοῦ ξύλου τὸ μῆκος; </s> <s id="g0131603"> τὸ μὲν γὰρ πρῶτον μέρος <lb/> αὐτοῦ, ὃ ἡ χεὶρ αἴρει, οἷον ὑπομόχλιον γίνεται, τὸ δ' <lb/> ἐπὶ τῷ ἄκρῳ βάρος. </s> <s id="g0131604"> ὥστε ὅσῳ ἂν ᾖ μακρότερον τὸ ἀπὸ τοῦ <lb/> ὑπομοχλίου, τοσούτῳ ἀνάγκη κάμπτεσθαι μᾶλλον· ὅσῳ <lb/> γὰρ ἂν πλέον ἀπέχῃ τοῦ ὑπομοχλίου, τοσούτῳ ἀνάγκη <lb/> κάμπτεσθαι μεῖζον. </s> <s id="g0131605"> ἀνάγκη οὖν αἴρεσθαι τὰ ἄκρα τοῦ <lb/> μοχλοῦ. </s> <s id="g0131606"> ἐὰν οὖν ᾖ καμπτόμενος ὁ μοχλός, ἀνάγκη αὐτὸν <lb/> κάμπτεσθαι μᾶλλον αἰρόμενον. ὅπερ συμβαίνει ἐπὶ τῶν <lb/> ξύλων τῶν μακρῶν· ἐν δὲ τοῖς βραχέσιν ἐγγὺς τὸ ἔσχατον <lb/> τοῦ ὑπομοχλίου γίνεται τοῦ ἠρεμοῦντος. </s> </p> <p n="35"> <s id="g0131701prop17"> <lb/> Διὰ τί τῷ σφηνὶ ὄντι μικρῷ μεγάλα βάρη διίσταται <lb/> καὶ μεγέθη σωμάτων, καὶ θλῖψις ἰσχυρὰ γίνεται; </s> <s id="g0131702"> ἢ διότι <lb/> ὁ σφὴν δύο μοχλοί εἰσιν ἐναντίοι ἀλλήλοις, ἔχει δὲ ἑκάτερος <lb/> τὸ μὲν βάρος τὸ δὲ ὑπομόχλιον, ὃ καὶ ἀνασπᾷ ἢ <lb/> πιέζει. </s> <s id="g0131703"> ἔτι δὲ ἡ τῆς πληγῆς φορὰ τὸ βάρος, ὃ τύπτει καὶ <lb/> κινεῖ, ποιεῖ μέγα· </s> <s id="g0131704"> καὶ διὰ τὸ κινούμενον κινεῖν τῇ ταχυτῆτι <lb/> ἰσχύει ἔτι πλέον. μικρῷ δὲ ὄντι τῷ μοχλῷ μεγάλαι δυνάμεις <lb/> ἀκολουθοῦσι· διὸ λανθάνει κινῶν παρὰ τὴν ἀξίαν τοῦ μεγέθους. </s> <s id="g0131705"> <lb/> ἔστω σφὴν ἐφ' ᾧ ΑΒΓ, τὸ δὲ σφηνούμενον ΔΕΗΖ. <lb/> μοχλὸς δὴ γίνεται ἡ ΑΒ, βάρος δὲ τὸ τοῦ Β κάτωθεν, <lb/> ὑπομόχλιον δὲ τὸ ΖΔ. ἐναντίος δὲ τούτῳ μοχλὸς τὸ ΒΓ. </s> <s id="g0131706"> <lb/> ἡ δὲ ΑΓ κοπτομένη ἑκατέρᾳ τούτων χρῆται μοχλῷ· ἀνασπᾷ <lb/> γὰρ τὸ Β. </s> <figure id="id.080.01.013.1.jpg" xlink:href="080/01/013/1.jpg"/> </p> <p n="36"> <figure id="id.080.01.013.2.jpg" xlink:href="080/01/013/2.jpg"/> <s id="g0131801prop18"> <lb/> Διὰ τί, ἐάν τις δύο τροχιλέας ποιήσας ἐπὶ δυσὶ ξύλοις <lb/> συμβάλλουσιν ἑαυτοῖς ἐναντίως αὑταῖς κύκλῳ περιβάλῃ <lb/> καλώδιον, ἔχον τὸ ἄρτημα ἐκ θατέρου τῶν ξύλων, <lb/> θάτερον δὲ ᾖ προσερηρεισμένον ἢ προστεθειμένον κατὰ τὰς <lb/> τροχαλίας, ἐὰν ἕλκῃ τις τῇ ἀρχῇ τοῦ καλωδίου, μεγάλα <lb/> βάρη προσάγει, κἂν ᾖ μικρὰ ἡ ἕλκουσα ἰσχύς; </s> <s id="g0131802"> ἢ διότι τὸ <lb/> αὐτὸ βάρος ἀπὸ ἐλάττονος ἰσχύος, εἰ μοχλεύεται, ἐγείρεται, <lb/> ἢ ἀπὸ χειρός; ἡ δὲ τροχιλέα τὸ αὐτὸ ποιεῖ τῷ <pb xlink:href="080/01/014.jpg" ed="Bekker" n="853b"/> <lb/> μοχλῷ, ὥστε ἡ μία ῥᾷον ἕλξει, καὶ ἀπὸ μιᾶς ὁλκῆς τοῦ <lb/> κατὰ χεῖρα πολὺ ἕλξει βαρύτερον. τοῦτο δ' αἱ δύο τροχιλίαι <lb/> πλέον ἢ διπλασίῳ τάχει αἴρουσαι. </s> <s id="g0131803"> ἔλαττον γὰρ <lb/> ἔτι ἡ ἑτέρα ἕλκει ἢ εἰ αὐτὴ καθ' ἑαυτὴν εἷλκεν, ὅταν <lb/> παρὰ τῆς ἑτέρας ἐπιβληθῇ τὸ σχοινίον· ἐκείνη γὰρ ἔτι <lb/> ἔλαττον ἐποίησε τὸ βάρος. </s> <s id="g0131804"> καὶ οὕτως ἐὰν εἰς πλείους ἐπιβάλληται <lb/> τὸ καλώδιον, ἐν ὀλίγαις τροχιλέαις πολλὴ γίνεται <lb/> διαφορά, ἢ ὥστε ὑπὸ τῆς πρώτης τοῦ βάρους ἕλκοντος <lb/> τέτταρας μνᾶς, ὑπὸ τῆς τελευταίας ἕλκεσθαι πολλῷ <lb/> ἐλάττω. </s> <s id="g0131805"> καὶ ἐν τοῖς οἰκοδομικοῖς ἔργοις ῥᾳδίως κινοῦσι μεγάλα <lb/> βάρη· μεταφέρουσι γὰρ ἀπὸ τῆς αὐτῆς τροχιλέας <lb/> ἐφ' ἑτέραν, καὶ πάλιν ἀπ' ἐκείνης εἰς ὄνους καὶ μοχλούς· <lb/> τοῦτο δὲ ταὐτόν ἐστι τῷ ποιεῖν πολλὰς τροχιλέας. </s> </p> <p n="37"> <s id="g0131901prop19"> <lb/> Διὰ τί, ἐὰν μέν τις ἐπιθῇ ἐπὶ τὸ ξύλον πέλεκυν μέγαν <lb/> καὶ φορτίον μέγα ἐπ' αὐτῷ, οὐ διαιρεῖ τὸ ξύλον, ὅ τι καὶ <lb/> λόγου ἄξιον· ἐὰν δὲ ἄρας τὸν πέλεκύν τις πατάξῃ αὐτῷ, <lb/> διασχίζει, ἔλαττον βάρος ἔχοντος τοῦ τύπτοντος πολὺ μᾶλλον <lb/> ἢ τοῦ ἐπικειμένου καὶ πιεζοῦντος; </s> <s id="g0131902"> ἢ διότι πάντα τῇ κινήσει <lb/> ἐργάζεται, καὶ τὸ βαρὺ τὴν τοῦ βάρους κίνησιν λαμβάνει <lb/> μᾶλλον κινούμενον ἢ ἠρεμοῦν; </s> <s id="g0131903"> ἐπικείμενον οὖν εὐ κινεῖται τὴν <lb/> τοῦ βάρους κίνησιν, φερόμενον δὲ ταύτην τε καὶ τὴν τοῦ <lb/> τύπτοντος. </s> <s id="g0131904"> ἔτι δὲ καὶ γίνεται σφὴν ὁ πέλεκυς· ὁ δὲ σφὴν <lb/> μικρὸς ὢν μεγάλα διίστησι διὰ τὸ εἶναι ἐκ δύο μοχλῶν <lb/> ἐναντίως συγκειμένων. </s> </p> <p n="38"> <figure id="id.080.01.014.1.jpg" xlink:href="080/01/014/1.jpg"/> <s id="g0132001prop20"> <lb/> Διὰ τί αἱ φάλαγγες τὰ κρέα ἱστᾶσιν ἀπὸ μικροῦ ἀρτήματος <lb/> μεγάλα βάρη, τοῦ ὅλου ἡμιζυγίου ὄντος; οὗ μὲν γὰρ <lb/> τὸ βάρος ἐντίθεται, κατήρτηται μόνον ἡ πλάστιγξ, ἐπὶ θάτερον <lb/> δὲ ἡ φάλαγξ ἐστὶ μόνον. </s> <s id="g0132002"> ἢ ὅτι ἅμα συμβαίνει ζυγὸν <lb/> καὶ μοχλὸν εἶναι τὴν φάλαγγα; ζυγὸν μὲν γὰρ, ᾗ <lb/> τῶν σπαρτίων ἕκαστον γίνεται τὸ κέντρον τῆς φάλαγγος. τὸ <lb/> μὲν οὖν ἐπὶ θάτερα ἔχει πλάστιγγα, τὸ δὲ ἐπὶ θάτερα ἀντὶ <lb/> τῆς πλάστιγγος τὸ σφαίρωμα, ὃ τῷ ζυγῷ ἔγκειται, ὥσπερ <lb/> εἴ τις τὴν ἑτέραν πλάστιγγα καὶ τὸν σταθμὸν ἐπιθείη ἐπὶ τὸ <lb/> ἄκρον τῆς πλάστιγγος· </s> <s id="g0132003"> δῆλον γὰρ ὅτι ἕλκει τοσοῦτον βάρος <lb/> ἐν τῇ ἑτέρᾳ κείμενον πλάστιγγι. </s> <s id="g0132004"> ὅπως δὲ τὸ ἓν ζυγὸν πολλὰ <lb/> ᾖ ζυγά, τοιαῦτα τὰ σπαρτία πολλὰ ἔγκειται ἐν τῷ τοιούτῳ <lb/> ζυγῷ, ὧν ἑκάστου τὸ ἐπὶ τάδε ἐπὶ τὸ σφαίρωμα τὸ ἥμισυ <lb/> τῆς φάλαγγός ἐστι, καὶ ὁ σταθμὸς δι' ἴσου τῶν ἀπ' ἀλλήλων <lb/> τῶν σπαρτίων κινουμένων, ὥστε συμμετρεῖσθαι πόσον <pb xlink:href="080/01/015.jpg" ed="Bekker" n="854a"/> <lb/> βάρος ἕλκει τὸ ἐν τῇ πλάστιγγι κείμενον· ὥστε γινώσκειν, ὅταν <lb/> ὀρθὴ ἡ φάλαγξ ᾖ, ἀπὸ ποίου σπάρτου πόσον βάρος ἔχει ἡ <lb/> πλάστιγξ, καθάπερ εἴρηται. </s> <s id="g0132005"> ὅλως μέν ἐστι τοῦτο ζυγόν, ἔχον <lb/> μίαν μὲν πλάστιγγα, ἐν ᾗ ἵσταται τὸ βάρος, τὴν δ' ἑτέραν, <lb/> ἐν ᾗ τὸ σταθμὸν ἐν τῇ φάλαγγι. </s> <s id="g0132006"> διὸ σφαίρωμά ἐστιν ἡ <lb/> φάλαγξ ἐπὶ θάτερον. τοιοῦτον δὲ ὂν πολλὰ ζυγά ἐστι, καὶ <lb/> τοσαῦτα ὅσαπέρ ἐστι τὰ σπαρτία. </s> <s id="g0132007"> ἀεὶ δὲ τὸ ἐγγύτερον <lb/> σπαρτίον τῆς πλάστιγγος καὶ τοῦ ἱσταμένου βάρους μεῖζον ἕλκει <lb/> βάρος, διὰ τὸ γίνεσθαι τὴν μὲν φάλαγγα πᾶσαν μοχλὸν <lb/> ἀνεστραμμένον (ὑπομόχλιον μὲν γὰρ τὸ σπαρτίον <lb/> ἕκαστον ἄνωθεν ὄν, τὸ δὲ βάρος τὸ ἐνὸν ἐν τῇ πλάστιγγι), </s> <s id="g0132008"> <lb/> ὅσῳ δ' ἂν μακρότερον ᾖ τὸ μῆκος τοῦ μοχλοῦ τοῦ ἀπὸ τοῦ <lb/> ὑπομοχλίου, τοσούτῳ ἐκεῖ μὲν ῥᾷον κινεῖ, ἐνταῦθα δὲ σήκωμα <lb/> ποιεῖ, καὶ ἵστησι τὸ πρὸς τὸ σφαίρωμα βάρος τῆς <lb/> φάλαγγος. </s> </p> <p n="39"> <s id="g0132101prop21"> <lb/> Διὰ τί οἱ ἰατροὶ ῥᾷον ἐξαιροῦσι τοὺς ὀδόντας προσλαμβάνοντες <lb/> βάρος τὴν ὀδοντάγραν ἢ τῇ χειρὶ μόνῃ ψιλῇ; </s> <s id="g0132102"> <lb/> πότερον διὰ τὸ μᾶλλον ἐξολισθαίνειν διὰ τῆς χειρὸς τὸν <lb/> ὀδόντα ἢ ἐκ τῆς ὀδοντάγρας; ἢ μᾶλλον ὀλισθαίνει τῆς <lb/> χειρὸς ὁ σίδηρος, καὶ οὐ περιλαμβάνει αὐτὸν κύκλῳ· μαλθακὴ <lb/> γὰρ οὖσα ἡ σὰρξ τῶν δακτύλων καὶ προσμένει μᾶλλον <lb/> καὶ περιαρμόττει. </s> <s id="g0132103"> ἀλλ' ὅτι ἡ ὀδοντάγρα δύο μοχλοί <lb/> εἰσιν ἀντικείμενοι, ἓν τὸ ὑπομόχλιον ἔχοντες τὴν σύναψιν <lb/> τῆς θερμαστρίδος· </s> <s id="g0132104"> τοῦ ῥᾷον οὖν κινῆσαι χρῶνται τῷ ὀργάνῳ <lb/> πρὸς τὴν ἐξαίρεσιν. </s> <s id="g0132105"> ἔστω γὰρ τῆς ὀδοντάγρας τὸ μὲν ἕτερον <lb/> ἄκρον ἐφ' ᾧ τὸ Α, τὸ δὲ ἕτερον, τὸ Β, ὃ ἐξαιρεῖ· ὁ <lb/> δὲ μοχλὸς ἐφ' ᾧ ΑΔΖ, ὁ δὲ ἄλλος μοχλὸς ἐφ' ᾧ <lb/> ΒΓΕ, ὑπομόχλιον δὲ τὸ ΓΘΔ· ὁ δὲ ὀδοὺς ἐφ' οὗ Ι σύναψις· <lb/> ὁ δὲ τὸ βάρος. </s> <figure id="id.080.01.015.1.jpg" xlink:href="080/01/015/1.jpg"/> <s id="g0132106"> ἑκατέρῳ οὖν τῶν ΒΖ καὶ ἅμα λαβὼν <lb/> κινεῖ. ὅταν δὲ κινήσῃ, ἐξεῖλε ῥᾷον τῇ χειρὶ ἢ τῷ <lb/> ὀργάνῳ. </s> </p> <p n="40"> <s id="g0132201prop22"> <lb/> Διὰ τί τὰ κάρυα ῥᾳδίως καταγνύουσιν ἄνευ πληγῆς ἐν <lb/> τοῖς ὀργάνοις ἃ ποιοῦσι πρὸς τὸ καταγνύναι αὐτά; πολλὴ <lb/> γὰρ ἀφαιρεῖται ἰσχὺς ἡ τῆς φορᾶς καὶ βίας. ἔτι δὲ σκληρῷ <lb/> καὶ βαρεῖ συνθλίβων θᾶττον ἂν κατάξαι ἢ ξυλίνῳ καὶ κούφῳ <lb/> τῷ ὀργάνῳ. ἢ διότι οὕτως ἐπ' ἀμφότερα θλίβεται ὑπὸ δύο <lb/> μοχλῶν τὸ κάρυον, τῷ δὲ μοχλῷ ῥᾳδίως διαιρεῖται τὰ <lb/> βάρη; </s> <s id="g0132202"> τὸ γὰρ ὄργανον ἐκ δύο σύγκειται μοχλῶν, ὑπομόχλιον <lb/> ἐχόντων τὸ αὐτό, τὴν συναφὴν ἐφ' ἧς τὸ Α. </s> <s id="g0132203"> <pb xlink:href="080/01/016.jpg" ed="Bekker" n="854b"/> <lb/> ὥσπερ οὖν εἰ ἦσαν ἐκβεβλημέναι, ὑφ' ὧν κινουμένων εἰς τὰ τῶν <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"/> <s id="g0132204"> <lb/> ἣν οὖν ἐν τῇ πληγῇ τὸ βάρος ἐποίει, ταύτην ἡ κρείττων ταύτης, <lb/> ἡ τὸ ΕΓ καὶ ΖΔ, μοχλοὶ ὄντες ποιοῦσι· τῇ ἄρσει γὰρ <lb/> εἰς τοὐναντίον αἴρονται, καὶ θλίβοντες καταγνύουσι τὸ ἐφ' ᾧ Κ. </s> <s id="g0132205"> <lb/> δι' αὐτὸ δὲ τοῦτο καὶ ὅσῳ ἂν ἐγγύτερον ᾖ τῆς Α τὸ Κ, συντρίβεται <lb/> θᾶττον· ὅσῳ γὰρ ἂν πλεῖον ἀπέχῃ τοῦ ὑπομοχλίου <lb/> ὁ μοχλός, ῥᾷον κινεῖ καὶ πλεῖον ἀπὸ τῆς ἰσχύος τῆς αὐτῆς. </s> <s id="g0132206"> <lb/> ἔστιν οὖν τὸ μὲν Α ὑπομόχλιον, ἡ δὲ ΔΑΖ μοχλός, καὶ ἡ <lb/> ΓΑΕ. </s> <s id="g0132207"> ὅσῳ ἂν οὖν τὸ Κ ἐγγυτέρω ᾖ τῆς γωνίας τῶν Α, <lb/> τοσούτῳ ἐγγύτερον γίνεται τῆς συναφῆς τῶν Α· τοῦτο δέ ἐστι <lb/> τὸ ὑπομόχλιον. </s> <s id="g0132208"> ἀνάγκη τοίνυν ἀπὸ τῆς αὐτῆς ἰσχύος συναγούσης <lb/> τὸ ΖΕ αἴρεσθαι πλέον. </s> <s id="g0132209"> ὥστε ἐπεί ἐστιν ἐξ ἐναντίας <lb/> ἡ ἄρσις, ἀνάγκη θλίβεσθαι μᾶλλον· τὸ δὲ μᾶλλον θλιβόμενον <lb/> κατάγνυται θᾶττον. </s> </p> <p n="41"> <s id="g0132301prop23"> <lb/> Διὰ τί φερομένων δύο φορὰς ἐν τῷ ῥόμβῳ τῶν ἄκρων <lb/> σημείων ἀμφοτέρων, οὐ τὴν ἴσην ἑκάτερον αὐτῶν εὐθεῖαν διέρχεται, <lb/> ἀλλὰ πολλαπλασίαν θάτερον; </s> <s id="g0132302"> ὁ αὐτὸς δὲ λόγος καὶ <lb/> διὰ τί τὸ ἐπὶ τῆς πλευρᾶς φερόμενον ἐλάττω διέρχεται τῆς <lb/> πλευρᾶς. τὸ μὲν γὰρ τὴν ἐλάττω διάμετρον, ἡ δὲ τὴν <lb/> μείζω πλευρὰν, καὶ ἡ μὲν μίαν, τὸ δὲ δύο φέρεται <lb/> φοράς. </s> <s id="g0132303"> φερέσθω γὰρ ἐπὶ τῆς ΑΒ τὸ μὲν Α πρὸς τὸ Β, τὸ <lb/> δὲ Β πρὸς τὸ Δ τῷ αὐτῷ τάχει· φερέσθω δὲ καὶ ἡ ΑΒ <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"/> <s id="g0132304"> ἀνάγκη <lb/> δὴ τὸ μὲν Α ἐπὶ τῆς ΑΔ διαμέτρου φέρεσθαι, τὸ δὲ Β ἐπὶ <lb/> τῆς ΒΓ, καὶ ἅμα διεληλυθέναι ἑκατέραν, καὶ τὴν ΑΒ τὴν <lb/> ΑΓ πλευράν. </s> <s id="g0132305"> ἐνηνέχθω γὰρ τὸ μὲν Α τὴν ΑΕ, ἡ δὲ <lb/> ΑΒ τὴν ΑΖ, καὶ ἔστω ἐκβεβλημένη ἡ ΖΗ παρὰ τὴν ΑΒ, <lb/> καὶ ἀπὸ τοῦ Ε πεπληρώσθω. </s> <s id="g0132306"> ὅμοιον οὖν γίνεται τὸ παραπληρωθὲν <lb/> τῷ ὅλῳ. </s> <s id="g0132307"> ἴση ἄρα ἡ ΑΖ τῇ ΑΕ, ὥστε τὸ Α <lb/> ἐπὶ τῆς πλευρᾶς ἐνήνεκται τῆς ΑΕ. ἡ δὲ ΑΒ τὴν ΑΖ <lb/> εἴη ἂν ἐνηνεγμένη. ἔσται ἄρα ἐπὶ τῆς διαμέτρου κατὰ τὸ Θ. </s> <s id="g0132308"> <lb/> καὶ αἰεὶ δὲ ἀνάγκη αὐτὸ φέρεσθαι κατὰ τὴν διάμετρον. <lb/> καὶ ἅμα ἡ πλευρὰ ἡ ΑΒ τὴν πλευρὰν τὴν ΑΓ δίεισι, <lb/> καὶ τὸ Α τὴν διάμετρον δίεισι τὴν ΑΔ. </s> <s id="g0132309"> ὁμοίως δὲ δειχθήσεται <lb/> καὶ τὸ Β ἐπὶ τῆς ΑΓ διαμέτρου φερόμενον. ἴση <lb/> γάρ ἐστιν ἡ ΒΕ τῇ ΒΗ. </s> <s id="g0132310"> παραπληρωθέντος οὖν ἀπὸ τοῦ Η, <lb/> ὅμοιόν ἐστι τῷ ὅλῳ τὸ ἐντός. καὶ τὸ Β ἐπὶ τῆς διαμέτρου <lb/> ἔσται κατὰ τὴν σύναψιν τῶν πλευρῶν, καὶ ἅμα δίεισιν <pb xlink:href="080/01/017.jpg" ed="Bekker" n="855a"/> <lb/> ἥ τε πλευρὰ τὴν πλευρὰν καὶ τὸ Β τὴν ΒΓ διάμετρον. </s> <s id="g0132311"> <lb/> ἅμα ἄρα καὶ τὸ Β τὴν πολλαπλασίαν τῆς ΑΒ δίεισι <lb/> καὶ ἡ πλευρὰ τὴν ἐλάττονα πλευράν, τῷ αὐτῷ τάχει φερόμενα, <lb/> καὶ ἡ πλευρὰ μείζω τοῦ Α διελήλυθε μίαν φορὰν <lb/> φερομένη. </s> <s id="g0132312"> ὅσῳ γὰρ ἂν ὀξύτερος γένηται ὁ ῥόμβος, ἡ <lb/> μὲν διάμετρος ἡ ἐλάττων γίνεται, ἡ δὲ ΒΓ μείζων, ἡ δὲ <lb/> πλευρὰ τῆς ΒΓ ἐλάττων. </s> <s id="g0132313"> ἄτοπον γάρ, ὥσπερ ἐλέχθη, τὸ <lb/> δύο φορὰς φερόμενον ἐνίοτε βραδύτερον φέρεσθαι τοῦ μίαν, <lb/> καὶ ἀμφοτέρων ἰσοταχῶν σημείων δοθέντων μείζω διεξιέναι <lb/> θάτερον. </s> <s id="g0132314"> αἴτιον δὲ ὅτι τοῦ μὲν ἀπὸ τῆς ἀμβλείας φερομένου <lb/> σχεδὸν ἐναντίαι ἀμφότεραι γίνονται, ἥν τε αὐτὴ <lb/> φέρεται καὶ ἣν ὑπὸ τῆς πλευρᾶς ὑποφέρεται, </s> <s id="g0132315"> τοῦ δὲ ἀπὸ <lb/> τῆς ὀξείας συμβαίνει φέρεσθαι ἐπὶ τὸ αὐτό. συνεπουρίζει <lb/> γὰρ ἡ τῆς πλευρᾶς τὴν ἐπὶ τῆς διαμέτρου· καὶ ὅσῳ ἂν <lb/> τὴν μὲν ὀξυτέραν ποιήσῃ, τὴν δὲ ἀμβλυτέραν, ἡ μὲν βραδυτέρα <lb/> ἔσται, ἡ δὲ θάττων. </s> <s id="g0132316"> αἱ μὲν γὰρ ἐναντιώτεραι γίνονται <lb/> διὰ τὸ ἀμβλυτέραν γίνεσθαι τὴν γωνίαν, αἱ δὲ <lb/> μᾶλλον ἐπὶ τὰ αὐτὰ διὰ τὸ συνάγεσθαι τὰς γραμμάς. <lb/> τὸ μὲν γὰρ Β σχεδὸν ἐπὶ τὸ αὐτὸ φέρεται κατ' ἀμφοτέρας <lb/> τὰς φοράς· </s> <s id="g0132317"> συνεπουρίζεται οὖν ἡ ἑτέρα, καὶ ὅσῳ ἂν <lb/> ὀξυτέρα γίνηται ἡ γωνία, τοσούτῳ μᾶλλον. τὸ Α δὲ ἐπὶ <lb/> τοὐναντίον· αὐτὸ μὲν γὰρ πρὸς τὸ Β φέρεται, ἡ δὲ πλευρὰ <lb/> ὑποφέρει αὐτὸ πρὸς τὸ Δ. </s> <s id="g0132318"> καὶ ὅσῳ ἂν ἀμβλυτέρα ἡ γωνία <lb/> ᾖ, ἐναντιώτεραι αἱ φοραὶ γίνονται· εὐθυτέρα γὰρ ἡ <lb/> γραμμὴ γίνεται. </s> <s id="g0132319"> εἰ δ' ὅλως εὐθεῖα γένοιτο, παντελῶς ἂν <lb/> εἴησαν ἐναντίαι. ἡ δὲ πλευρὰ ὑπ' οὐθενὸς κωλύεται μίαν <lb/> φερομένη φοράν. εὐλόγως οὖν τὴν μείζω διέρχεται. </s> </p> <p n="42"> <s id="g0132401prop24"> <lb/> Ἀπορεῖται διὰ τί ποτε ὁ μείζων κύκλος τῷ ἐλάττονι <lb/> κύκλῳ ἴσην ἐξελίττεται γραμμήν, ὅταν περὶ τὸ αὐτὸ κέντρον <lb/> τεθῶσι; χωρὶς δὲ ἐκκυλιόμενοι, ὥσπερ τὸ μέγεθος αὐτῶν <lb/> πρὸς τὸ μέγεθος ἔχει, οὕτως καὶ αἱ γραμμαὶ αὐτῶν <lb/> γίνονται πρὸς ἀλλήλας. </s> <s id="g0132402"> ἔτι δὲ ἑνὸς καὶ τοῦ αὐτοῦ κέντρου <lb/> ὄντος ἀμφοῖν, ὁτὲ μὲν τηλικαύτη γίνεται ἡ γραμμὴ ἣν <lb/> ἐκκυλίονται, ἡλίκην ὁ ἐλάττων κύκλος καθ' αὑτὸν ἐκκυλίεται, <lb/> ὁτὲ δὲ ὅσην ὁ μείζων. </s> <s id="g0132403"> ὅτι μὲν οὖν μείζω ἐκκυλίεται <lb/> ὁ μείζων, φανερόν. γωνία μὲν γὰρ δοκεῖ κατὰ τὴν <lb/> αἴσθησιν εἶναι ἡ περιφέρεια ἑκάστου τῆς οἰκείας διαμέτρου, <lb/> ἡ τοῦ μείζονος κύκλου μείζων, ἡ δὲ τοῦ ἐλάττονος ἐλάττων, <lb/> ὥστε τὸν αὐτὸν τοῦτον ἕξουσι λόγον, καθ' ἃς <pb xlink:href="080/01/018.jpg" ed="Bekker" n="855b"/> <lb/> ἐξεκυλίσθησαν αἱ γραμμαὶ πρὸς ἀλλήλας κατὰ τὴν αἴσθησιν. </s> <s id="g0132404"> ἀλλὰ μὴν <lb/> καὶ ὅτι τὴν ἴσην ἐκκυλίονται, ὅταν περὶ τὸ αὐτὸ κέντρον <lb/> κείμενοι ὦσι, δῆλον· καὶ οὕτως γίνεται ὁτὲ μὲν ἴση τῇ <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"/> </p> <p n="43"> <s id="g0132405"> <lb/> ἔστω γὰρ κύκλος ὁ μείζων μὲν ἐφ' οὗ τὰ ΔΖΓ, ὁ δὲ <lb/> ἐλάττων ἐφ' οὗ τὰ ΕΗΒ, κέντρον δὲ ἀμφοῖν τὸ Α· καὶ <lb/> ἣν μὲν ἐξελίττεται καθ' αὑτὸν ὁ μέγας, ἡ ἐφ' ἧς ΖΙ ἔστω, <lb/> ἣν δὲ ὁ ἐλάττων καθ' αὑτόν, ἡ ἐφ' ἧς ΗΚ, ἴση τῇ ΑΖ. </s> <s id="g0132406"> <lb/> ἐὰν δὴ κινῶ τὸν ἐλάττονα, τὸ αὐτὸ κέντρον κινῶ ἐφ' οὗ <lb/> τὸ Α· ὁ δὲ μέγας προσηρμόσθω. ὅταν οὖν ἡ ΑΒ ὀρθὴ γένηται <lb/> πρὸς τὴν ΗΚ, ἅμα καὶ ἡ ΑΓ γίνεται ὀρθὴ πρὸς τὴν <lb/> ΖΛ, ὥστε ἔσται ἴσην ἀεὶ διεληλυθυῖα, τὴν μὲν ΗΚ, ἐφ' <lb/> ᾧ ΗΒ περιφέρεια, τὴν δὲ ΖΛ ἡ ἐφ' ἧς ΖΓ. </s> <s id="g0132407"> εἰ δὲ τὸ <lb/> τέταρτον μέρος ἴσην ἐξελίττεται, δῆλον ὅτι καὶ ὁ ὅλος κύκλος <lb/> τῷ ὅλῳ κύκλῳ ἴσην ἐξελιχθήσεται, ὥστε ὅταν ἡ ΒΗ <lb/> γραμμὴ ἔλθῃ ἐπὶ τὸ Κ, καὶ ἡ ΖΓ ἔσται περιφέρεια ἐπὶ <lb/> τῆς ΖΛ, καὶ ὁ κύκλος ὅλος ἐξειλιγμένος. </s> <s id="g0132408"> ὁμοίως δὲ καὶ <lb/> ἐὰν τὸν μέγαν κινῶ, ἐναρμόσας τὸν μικρόν, τοῦ αὐτοῦ κέντρου <lb/> ὄντος, ἅμα τῇ ΑΓ ἡ ΑΒ κάθετος καὶ ὀρθὴ ἔσται, ἡ <lb/> μὲν πρὸς τὴν ΖΙ, ἡ δὲ πρὸς τὴν ΗΘ. </s> <s id="g0132409"> ὥστε ὅταν ἴσην ἡ <lb/> μὲν τῇ ΗΘ ἔσται διεληλυθυῖα, ἡ δὲ τῇ ΖΙ, καὶ γένηται <lb/> ὀρθὴ πάλιν ἡ ΖΑ πρὸς τὴν ΖΛ καὶ ἡ ΑΓ ὀρθὴ πάλιν, <lb/> ὡς τὸ ἐξ ἀρχῆς ἔσονται ἐπὶ τῶν ΘΙ. </s> <s id="g0132410"> τὸ δὲ μήτε στάσεως <lb/> γινομένης τὸ μεῖζον τῷ ἐλάττονι, ὥστε μένειν τινὰ χρόνον <lb/> ἐπὶ τοῦ αὐτοῦ σημείου· κινοῦνται γὰρ συνεχῶς ἄμφω ἀμφοτεράκις. <lb/> μὴ ὑπερπηδόντος τοῦ ἐλάττονος μηθὲν σημεῖον, <lb/> τὸν μὲν μείζω τῷ ἐλάττονι ἴσην διεξιέναι, τὸν δὲ τῷ μείζονι, <lb/> ἄτοπον. </s> <s id="g0132411"> ἔτι δὲ μιᾶς κινήσεως οὔσης ἀεὶ τὸ τὸ <lb/> κινούμενον κέντρον ὁτὲ μὲν τὴν μεγάλην ὁτὲ δὲ τὴν ἐλάττονα <lb/> ἐκκυλίεσθαι θαυμαστόν. </s> <s id="g0132412"> τὸ γὰρ αὐτὸ τῷ αὐτῷ τάχει φερόμενον <lb/> ἴσην πέφυκε διεξιέναι· τῷ αὐτῷ δὲ τάχει ἴσην ἐστὶ <lb/> κινεῖν ἀμφοτεράκις. </s> <s id="g0132413"> ἀρχὴ δὲ ληπτέα ἥδε περὶ τῆς αἰτίας <lb/> αὐτῶν, ὅτι ἡ αὐτὴ δύναμις καὶ ἴση τὸ μὲν βραδύτερον <lb/> κινεῖ μέγεθος, τὸ δὲ ταχύτερον. </s> <s id="g0132414"> εἰ δή τι εἴη ὃ μὴ πέφυκεν <lb/> ὑφ' ἑαυτοῦ κινεῖσθαι, ἐὰν τοῦτο ἅμα καὶ αὐτὸ κινῇ τὸ πεφυκὸς <lb/> κινεῖσθαι, βραδύτερον κινηθήσεται ἢ εἰ αὐτὴ καθ' <lb/> αὑτὴν ἐκινεῖτο. </s> <s id="g0132415"> καὶ ἐὰν μὲν πεφυκὸς ᾖ κινεῖσθαι, μὴ συγκινῆται <lb/> δὲ μηθέν, ὡσαύτως ἕξει. </s> <s id="g0132416"> καὶ ἀδύνατον δὴ κινεῖσθαι <lb/> πλέον ἢ τὸ κινοῦν· οὐ γὰρ τὴν αὑτοῦ κινεῖται κίνησιν, <pb xlink:href="080/01/019.jpg" ed="Bekker" n="856a"/> <lb/> ἀλλὰ τὴν τοῦ κινοῦντος. </s> <s id="g0132417"> εἴη δὴ κύκλος ὁ μὲν μείζων τὸ Α, ὁ δὲ <lb/> ἐλάττων ἐφ' ᾧ Β. εἰ ὠθοίη δ' ὁ ἐλάττων τὸν μείζω, μὴ <lb/> κυλιομένου αὐτοῦ, φανερὸν ὅτι τοσοῦτον δίεισι τῆς εὐθείας <lb/> ὁ μείζων, ὅσον ἐώσθη ὑπὸ τοῦ ἐλάττονος. τοσοῦτον δέ γε <lb/> ἐώσθη ὅσον ὁ μικρὸς ἐκινήθη. ἴσην ἄρα τῆς εὐθείας διεληλύθασιν. </s> <s id="g0132418"> <lb/> ἀνάγκη τοίνυν καὶ εἰ κυλιόμενος ὁ ἐλάττων τὸν <lb/> μείζω ὠθοίη, κυλισθῆναι μὲν ἅμα τῇ ὤσει, τοσοῦτον δ' ὅσον <lb/> ὁ ἐλάττων ἐκυλίσθη, εἰ μηθὲν αὐτὸς τῇ αὐτῇ κινήσει κινεῖται. </s> <s id="g0132419"> <lb/> ὡς γὰρ καὶ ὅσον ἐκίνει, τοσοῦτον κεκινῆσθαι ἀνάγκη <lb/> τὸ κινούμενον ὑπ' ἐκείνου. ἀλλὰ μὴν ὅ τε κύκλος τοσοῦτον <lb/> ἐκίνησε τὸ αὐτό κύκλῳ τε καὶ ποδιαίαν (ἔστω γὰρ τοσοῦτον <lb/> ὃ ἐκινήθη), καὶ ὁ μέγας ἄρα τοσοῦτον ἐκινήθη. </s> <s id="g0132420"> ὁμοίως <lb/> δὲ κἂν ὁ μέγας τὸν μικρὸν κινήσῃ, ἔσται κεκινημένος ὁ μικρὸς <lb/> ὡς καὶ ὁ μείζων. </s> <s id="g0132421"> καθ' αὑτὸν μὲν δὴ κινηθεὶς ὁποτεροσοῦν, <lb/> ἐάν τε ταχὺ ἐάν τε βραδέως· τῷ αὐτῷ δὲ τάχει <lb/> εὐθὺς ὅσην ὁ μείζων πέφυκεν ἐξελιχθῆναι γραμμήν. ὅπερ <lb/> καὶ ποιεῖ τὴν ἀπορίαν, ὅτι οὐκέτι ὁμοίως ποιοῦσιν ὅταν συναρμοσθῶσιν. <lb/> τὸ δ' ἔστιν, εἰ ὁ ἕτερος ὑπὸ τοῦ ἑτέρου κινεῖται <lb/> οὐχ ἣν πέφυκεν, οὐδὲ τὴν αὑτοῦ κίνησιν. </s> <s id="g0132422"> οὐθὲν γὰρ <lb/> διαφέρει περιθεῖναι καὶ ἐναρμόσαι ἢ προσθεῖναι ὁποτερονοῦν <lb/> ὁποτέρῳ· ὁμοίως γάρ, ὅταν ὁ μὲν κινῇ ὁ δὲ κινῆται ὑπὸ <lb/> τούτου, ὅσον ἂν κινῇ ἅτερος, τοσοῦτον κινηθήσεται ἅτερος. </s> <s id="g0132423"> <lb/> ὅταν μὲν οὖν προσκείμενον κινῇ ἢ προσκρεμάμενον, οὐκ ἀεὶ <lb/> κυλίει τις· ὅταν δὲ περὶ τὸ αὐτὸ κέντρον τεθῶσιν, ἀνάγκη <lb/> κυλίεσθαι ἀεὶ τὸν ἕτερον ὑπὸ τοῦ ἑτέρου. </s> <s id="g0132424"> ἀλλ' οὐθὲν ἧττον <lb/> οὐ τὴν αὑτοῦ κίνησιν ἅτερος κινεῖται, ἀλλ' ὥσπερ ἂν εἰ μηδεμίαν <lb/> εἶχε κίνησιν. κἂν ἔχῃ, μὴ χρῆται δ' αὐτῇ, ταὐτὸ <lb/> συμβαίνει. </s> <s id="g0132425"> ὅταν μὲν οὖν ὁ μέγας κινῇ ἐνδεδεμένον τὸν μικρόν, <lb/> ὁ μικρὸς κινεῖται ὅσηνπερ οὗτος· ὅταν δὲ ὁ μικρός, <lb/> πάλιν ὁ μέγας ὅσην οὗτος. χωριζόμενος δὲ ἑκάτερος αὑτὸν <lb/> κινεῖ αὐτός. </s> <s id="g0132426"> ὅτι δὲ τοῦ αὐτοῦ κέντρου ὄντος καὶ κινοῦντος <lb/> τῷ αὐτῷ τάχει συμβαίνει ἄνισον διεξιέναι αὐτοὺς γραμμήν, <lb/> παραλογίζεται ὁ ἀπορῶν σοφιστικῶς. </s> <s id="g0132427"> τὸ αὐτὸ μὲν <lb/> γάρ ἐστι κέντρον ἀμφοῖν, ἀλλὰ κατὰ συμβεβηκός, ὡς <lb/> μουσικὸν καὶ λευκόν· τὸ γὰρ εἶναι ἑκατέρου κέντρου τῶν <lb/> κύκλων οὐ τῷ αὐτῷ χρῆται. </s> <s id="g0132428"> ὅταν μὲν οὖν ὁ κινῶν ᾖ ὁ <lb/> μικρός, ὡς ἐκείνου κέντρον καὶ ἀρχή, ὅταν δὲ ὁ μέγας, ὡς <lb/> ἐκείνου. </s> <s id="g0132429"> οὔκουν τὸ αὐτὸ κινεῖ ἁπλῶς, ἀλλ' ἔστιν ὥς. </s> </p> <p n="44"> <figure id="id.080.01.019.1.jpg" xlink:href="080/01/019/1.jpg"/> <s id="g0132501prop25"> <lb/> Διὰ τί τὰς κλίνας ποιοῦσι διπλασιοπλεύρους, τὴν <pb xlink:href="080/01/020.jpg" ed="Bekker" n="856b"/> <lb/> μὲν ἓξ ποδῶν καὶ μικρῷ μείζω πλευράν, τὴν δὲ τριῶν; καὶ <lb/> διὰ τί ἐντείνουσιν οὐ κατὰ διάμετρον; </s> <s id="g0132502"> ἢ τὸ μὲν μέγεθος τηλικαύτας, <lb/> ὅπως τοῖς σώμασιν ὦσι σύμμετροι; γίνονται <lb/> γὰρ οὕτω διπλασιόπλευροι, τετραπήχεις μὲν τὸ μῆκος, διπήχεις <lb/> δὲ τὸ πλάτος. </s> <s id="g0132503"> ἐντείνουσι δὲ οὐ κατὰ διάμετρον ἀλλ' <lb/> ἀπ' ἐναντίας, ὅπως τά τε ξύλα ἧττον διασπᾶται· τάχιστα <lb/> γὰρ σχίζεται κατὰ φύσιν διαιρούμενα ταύτῃ, καὶ ἑλκόμενα <lb/> πονεῖ μάλιστα. </s> <s id="g0132504"> ἔτι ἐπειδὴ δεῖ βάρος δύνασθαι τὰ <lb/> σπαρτία φέρειν, οὕτως ἧττον πονέσει λοξοῖς τοῖς σπάρτοις <lb/> ἐπιτιθεμένου τοῦ βάρους ἢ πλαγίοις. </s> <s id="g0132505"> ἔτι δὲ ἔλαττον οὕτω <lb/> σπαρτίον ἀναλίσκεται. </s> <s id="g0132506"> ἔστω γὰρ κλίνη ἡ ΑΖΗΙ, καὶ δίχα <lb/> διῃρήσθω ἡ ΖΗ κατὰ τὸ Β. ἴσα δὴ τρυπήματα ἐστιν <lb/> ἐν τῇ ΖΒ καὶ ἐν τῇ ΖΑ. καὶ γὰρ αἱ πλευραὶ ἴσαι εἰσίν· <lb/> ἡ γὰρ ὅλη ΖΗ διπλασία ἐστίν. </s> <s id="g0132507"> ἐντείνουσι δ' ὡς γέγραπται, <lb/> ἀπὸ τοῦ Α ἐπὶ τὸ Β, εἶτα οὗ τὸ Γ, εἶτα οὗ τὸ Δ, εἶτα οὗ <lb/> τὸ Θ, εἶτα οὗ τὸ Ε. καὶ οὕτως ἀεί, ἕως ἂν εἰς γωνίαν <lb/> καταστρέψωσιν ἄλλην· </s> <figure id="id.080.01.020.1.jpg" xlink:href="080/01/020/1.jpg"/> <s id="g0132508"> δύο γὰρ ἔχουσι γωνίαι τὰς ἀρχὰς <lb/> τοῦ σπαρτίου. </s> <s id="g0132509"> ἴσα δέ ἐστι τὰ σπαρτία κατὰ τὰς κάμψεις, <lb/> τό τε ΑΒ καὶ ΒΓ τῷ ΓΔ καὶ ΔΘ. καὶ τὰ ἄλλα δὲ <lb/> τὰ τοιαῦτά ἐστιν, ὅτι οὕτως ἔχει ἡ αὐτὴ ἀπόδειξις. </s> <s id="g0132510"> ἡ μὲν <lb/> γὰρ ΑΒ τῇ ΕΘ ἴση· ἴσαι γάρ εἰσιν αἱ πλευραὶ τοῦ <lb/> ΒΗΚΑ χωρίου, καὶ τὰ τρυπήματα ἴσα διέστηκεν. </s> <s id="g0132511"> ἡ δὲ ΒΗ ἴση <lb/> τῇ ΚΑ· ἡ γὰρ Β γωνία ἴση τῇ Η. ἐν ἴσοις γὰρ ἡ μὲν <lb/> ἐκτός, ἡ δὲ ἐντός· καὶ ἡ μὲν Β ἐστὶν ἡμίσεια ὀρθῆς· ἡ <lb/> γὰρ ΖΒ ἴση τῇ ΖΑ· καὶ γωνία δὲ ἡ κατὰ τὸ Ζ ὀρθή. </s> <s id="g0132512"> ἡ <lb/> δὲ Β γωνία ἴση τῇ κατὰ τὸ Η· ἡ γὰρ κατὰ τὸ Ζ ὀρθή, <lb/> ἐπειδὴ διπλασιόπλευρον τὸ ἑτερόμηκες καὶ πρὸς μέσον κέκλασται. <lb/> ὥστε ἡ ΑΓ τῇ ΕΗ ἴση. ταύτῃ δὲ ἡ ΚΘ· παράλληλος <lb/> γάρ. ὥστε ἡ ΒΓ ἴση τῇ ΚΘ. ἡ δὲ ΓΕ τῇ ΔΘ. </s> <s id="g0132513"> <lb/> ὁμοίως δὲ καὶ αἱ ἄλλαι δείκνυνται ὅτι ἴσαι εἰσὶν αἱ κατὰ <lb/> τὰς κάμψεις δύο ταῖς δυσίν. </s> <s id="g0132514"> ὥστε δῆλον ὅτι τὰ τηλικαῦτα <lb/> σπαρτία ὅσον τὸ ΑΒ, τέσσαρα τοσαῦτ' ἔνεστιν ἐν τῇ κλίνῃ· </s> <s id="g0132515"> <lb/> ὅσον δ' ἐστὶ τὸ πλῆθος τῶν ἐν τῇ ΖΗ πλευρᾷ τρυπημάτων, <lb/> καὶ ἐν τῷ ἡμίσει τῷ ΖΒ τὰ ἡμίση. </s> <s id="g0132516"> ὥστε ἐν τῇ ἡμισείᾳ <lb/> κλίνῃ τηλικαῦτα μεγέθη σπαρτίων ἐστὶν ὅσον τῷ ΒΑ ἔνεστι, <lb/> τοσαῦτα δὲ τὸ πλῆθος ὅσαπερ ἐν τῷ ΒΗ τρυπήματα. </s> <s id="g0132517"> <lb/> ταῦτα δὲ οὐδὲν διαφέρει λέγειν ἢ ὅσα ἐν τῇ ΑΖ καὶ ΒΖ <lb/> τὰ συνάμφω. </s> <figure id="id.080.01.020.2.jpg" xlink:href="080/01/020/2.jpg"/> <s id="g0132518"> εἰ δὲ κατὰ διάμετρον ἐνταθῇ τὰ σπαρτία, <lb/> ὡς ἐν τῇ ΑΒΓΔ κλίνῃ ἔχει, τὰ ἡμίσεά εἰσιν οὐ <pb xlink:href="080/01/021.jpg" ed="Bekker" n="857a"/> <lb/> τοσαῦτα ὅσα αἱ αἱ πλευραὶ ἀμφοῖν, αἱ ΑΖ ΖΗ· τὰ ἴσα δέ, <lb/> ὅσα ἐν τῷ ΑΖΒΖ τρυπήματα </s> <s id="g0132519"> ἔνεστιν. μείζονες δέ εἰσιν αἱ <lb/> ΑΖΒΖ δύο οὖσαι τῆς ΑΒ. ὥστε καὶ τὸ σπαρτίον μεῖζον τοσούτῳ <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"/> </p> <p n="45"> <s id="g0132601prop26"> <lb/> Διὰ τί χαλεπώτερον τὰ μακρὰ ξύλα ἀπὸ τοῦ ἄκρου <lb/> φέρειν ἐπὶ τῷ ὤμῳ ἢ κατὰ τὸ μέσον, ἴσου τοῦ βάρους ὄντος; </s> <s id="g0132602"> <lb/> πότερον ὅτι σαλευομένου τοῦ ξύλου τὸ ἄκρον κωλύει φέρειν, <lb/> μᾶλλον ἀντισπῶν τῇ σαλεύσει τὴν φοράν; ἢ κἂν <lb/> μηθὲν κάμπτηται μηδ' ἔχῃ πολὺ μῆκος, ὅμως χαλεπώτερον <lb/> φέρειν ἀπ' ἄκρου; ἀλλ' ὅτι καὶ ῥᾷον αἴρεται ἀπ' <lb/> ἄκρου ἢ ἐκ μέσου, διὰ τὸ αὐτὸ καὶ φέρειν οὕτω ῥᾴδιον. </s> <s id="g0132603"> <lb/> αἴτιον δὲ ὅτι ἐκ μέσου μὲν αἰρόμενον ἀεὶ ἐπικουφίζει ἄλληλα <lb/> τὰ ἄκρα, καὶ θάτερον μέρος τὸ ἐπὶ θάτερον εὖ αἴρει. <lb/> ὥσπερ γὰρ κέντρον γίνεται τὸ μέσον, ᾗ ἔχει τὸ αἶρον ἢ <lb/> φέρον. </s> <s id="g0132604"> εἰς τὸ ἄνω οὖν κουφίζεται ἑκάτερον τῶν ἄκρων εἰς <lb/> τὸ κάτω ῥέπον. ἀπὸ δὲ τοῦ ἄκρου αἰρόμενον ἢ φερόμενον οὐ <lb/> ποιεῖ τοῦτο, ἀλλ' ἅπαν τὸ βάρος ῥέπει ἐφ' ἓν μέσον, εἰς <lb/> ὅπερ αἴρεται ἢ φέρεται. </s> <s id="g0132605"> ἔστω μέσον ἐφ' οὗ Α, ἄκρα ΒΓ. </s> <s id="g0132606"> <lb/> αἰρομένου οὖν ἢ φερομένου κατὰ τὸ Α, τὸ μὲν Β κάτω <lb/> ῥέπον ἄνω αἴρει τὸ Γ, τὸ δὲ Γ κάτω ῥέπον τὸ Β ἄνω αἴρει· <lb/> ἅμα δὲ αἰρόμενα ἄνω ποιεῖ ταῦτα. </s> <figure id="id.080.01.021.2.jpg" xlink:href="080/01/021/2.jpg"/> </p> <p n="46"> <s id="g0132701prop27"> <lb/> Διὰ τί, ἐὰν ᾖ λίαν μακρὸν τὸ αὐτὸ βάρος, χαλεπώτερον <lb/> φέρειν ἐπὶ τοῦ ὤμου, κἂν μέσον φέρῃ τις, ἢ ἐὰν <lb/> ἔλαττον ᾖ; </s> <s id="g0132702"> πάλαι ἐλέχθη ὡς οὐκ ἔστιν αἴτιον ἡ σάλευσις· <lb/> ἀλλ' ἡ σάλευσις νῦν αἴτιόν ἐστιν. </s> <s id="g0132703"> ὅταν γὰρ ᾖ μακρότερον, <lb/> τὰ ἄκρα σαλεύεται, ὥστε εἴη ἂν καὶ τὸν φέροντα χαλεπώτερον <lb/> φέρειν μᾶλλον. </s> <s id="g0132704"> αἴτιον δὲ τοῦ σαλεύεσθαι μᾶλλον, <lb/> ὅτι τῆς αὐτῆς κινήσεως οὔσης μεθίσταται τὰ ἄκρα, ὅσῳπερ <lb/> ἂν ᾖ μακρότερον τὸ ξύλον. </s> <s id="g0132705"> ὁ μὲν γὰρ ὦμος κέντρον, ἐφ' <lb/> οὗ τὸ Α (μένει γὰρ τοῦτο), αἱ δὲ ΑΒ καὶ ΑΓ αἱ ἐκ τοῦ <lb/> κέντρου. ὅσῳ δ' ἂν ᾖ μεῖζον τὸ ἐκ τοῦ κέντρου ἢ τὸ ΑΒ <lb/> ἢ καὶ τὸ ΑΓ, πλέον μεθίσταται μέγεθος. δέδεικται δὲ <lb/> τοῦτο πρότερον. </s> <figure id="id.080.01.021.3.jpg" xlink:href="080/01/021/3.jpg"/> </p> <p n="47"> <figure id="id.080.01.021.4.jpg" xlink:href="080/01/021/4.jpg"/> <s id="g0132801prop28"> <lb/> Διὰ τί ἐπὶ τοῖς φρέασι τὰ κηλώνεια ποιοῦσι τοῦτον τὸν <lb/> τρόπον; προστιθέασι γὰρ βάρος ἐν τῷ ξύλῳ τὸν μόλιβδον, <lb/> ὄντος βάρους τοῦ κάδου αὐτοῦ, καὶ κενοῦ καὶ πλήρους ὄντος. </s> <s id="g0132802"> <lb/> ἢ ὅτι ἐν δυσὶ χρόνοις διῃρημένου τοῦ ἔργου (βάψαι γὰρ δεῖ, <lb/> καὶ τοῦτ' ἄνω ἑλκύσαι) συμβαίνει καθιέναι μὲν κενὸν <pb xlink:href="080/01/022.jpg" ed="Bekker" n="857b"/> <lb/> ῥᾳδίως, αἴρειν δὲ πλήρη χαλεπῶς; </s> <s id="g0132803"> λυσιτελεῖ οὖν μικρῷ βραδύτερον <lb/> εἶναι τὸ καταγαγεῖν πρὸς τὸ πολὺ κουφίσαι τὸ <lb/> βάρος ἀνάγοντι. τοῦτο οὖν ποιεῖ ἐπ' ἄκρῳ τῷ κηλωνείῳ ὁ <lb/> μόλιβδος προσκείμενος ἢ ὁ λίθος. </s> <s id="g0132804"> καθιμῶντι μὲν γὰρ γίνεται <lb/> βάρος μεῖζον ἢ εἰ μόνον κενὸν δεῖ κατάγειν τὸν κάδον· <lb/> ὅταν δὲ πλήρης ᾖ, ἀνάγει ὁ μόλιβδος, ἢ ὅ τι ἂν ᾖ <lb/> τὸ προσκείμενον βάρος. </s> <s id="g0132805"> ὥστ' ἐστὶ ῥᾷον αὐτῷ τὰ ἄμφω <lb/> ἢ ἐκείνῳ. </s> </p> <p n="48"> <figure id="id.080.01.022.1.jpg" xlink:href="080/01/022/1.jpg"/> <s id="g0132901prop29"> <lb/> Διὰ τί, ὅταν φέρωσιν ἐπὶ ξύλου ἤ τινος τοιούτου δύο <lb/> ἄνθρωποι ἴσον βάρος, οὐχ ὁμοίως θλίβονται, ἐὰν μὴ ἐπὶ <lb/> τῷ μέσῳ ᾖ τὸ βάρος, ἀλλὰ μᾶλλον ὅσῳ ἂν ἐγγύτερον ᾖ <lb/> τῶν φερόντων; </s> <s id="g0132902"> ἢ διότι μοχλὸς μὲν γίνεται οὕτως ἐχόντων <lb/> τὸ ξύλον, τὸ δὲ βάρος ὑπομόχλιον, </s> <s id="g0132903"> ὁ δὲ ἐγγύτερος τοῦ <lb/> βάρους τῶν φερόντων τὸ βάρος τὸ κινούμενον, ἅτερος δὲ <lb/> τῶν φερόντων τὸ βάρος ὁ κινῶν. </s> <s id="g0132904"> ὅσῳ γὰρ πλέον ἀπέχει τοῦ <lb/> βάρους, τοσούτῳ ῥᾷον κινεῖ, καὶ θλίβει μᾶλλον τὸν ἕτερον <lb/> εἰς τὸ κάτω, ὥσπερ ἀντερείδοντος τοῦ βάρους τοῦ ἐπικειμένου <lb/> καὶ γινομένου ὑπομοχλίου. </s> <s id="g0132905"> ἐν μέσῳ δὲ ὑποκειμένου τοῦ <lb/> βάρους, οὐδὲν μᾶλλον ἅτερος θατέρῳ γίνεται βάρος, οὐδὲ <lb/> κινεῖ, ἀλλ' ὁμοίως ἑκάτερος ἑκατέρῳ γίνεται βάρος. </s> </p> <p n="49"> <s id="g0133001prop30"> <lb/> Διὰ τί οἱ ἀνιστάμενοι πάντες πρὸς ὀξεῖαν γωνίαν τῷ <lb/> μηρῷ ποιήσαντες τὴν κνήμην ἀνίστανται, καὶ τῷ θώρακι <lb/> πρὸς τὸν μηρόν; εἰ δὲ μή, οὐκ ἂν δύναιντο ἀναστῆναι. </s> <s id="g0133002"> πότερον <lb/> ὅτι τὸ ἴσον ἠρεμίας πανταχοῦ αἴτιον, ἡ δὲ ὀρθὴ γωνία <lb/> τοῦ ἴσου, καὶ ποιεῖ στάσιν· διὸ καὶ φέρεται πρὸς ὁμοίας <lb/> γωνίας τῇ περιφερείᾳ τῆς γῆς. οὐ γὰρ ὅτι καὶ πρὸς ὀρθὴν <lb/> ἔσται τῷ ἐπιπέδῳ. </s> <s id="g0133003"> ἢ ὅτι ἀνιστάμενος γίνεται ὀρθός, ἀνάγκη <lb/> δὲ τὸν ἑστῶτα κάθετον εἶναι πρὸς τὴν γῆν. </s> <s id="g0133004"> εἰ οὖν μέλλει <lb/> ἔσεσθαι πρὸς ὀρθήν, τοῦτο δέ ἐστι τὸ τὴν κεφαλὴν ἔχειν <lb/> κατὰ τοὺς πόδας, καὶ γίνεσθαι δὴ ὅτε ἀνίσταται. </s> <s id="g0133005"> ὅταν μὲν <lb/> οὖν καθήμενος ᾖ, παράλληλον ἔχει τὴν κεφαλὴν καὶ τοὺς <lb/> πόδας, καὶ οὐκ ἐπὶ μιᾶς εὐθείας. </s> <figure id="id.080.01.022.2.jpg" xlink:href="080/01/022/2.jpg"/> <s id="g0133006"> ἡ κεφαλὴ Α ἔστω, θώραξ <lb/> ΑΒ, μηρὸς ΒΓ, κνήμη ΓΔ. </s> <s id="g0133007"> πρὸς ὀρθὴν δὲ γίνεται <lb/> ὅ τε θώραξ ἐφ' ὧν ΑΒ τῷ μηρῷ καὶ ὁ μηρὸς τῇ κνήμῃ <lb/> οὕτως καθημένῳ. ὥστε οὕτως ἔχοντα ἀδύνατον ἀναστῆναι. </s> <s id="g0133008"> <lb/> ἀνάγκη δὲ ἐγκλῖναι τὴν κνήμην καὶ ποιεῖν τοὺς πόδας ὑπὸ <lb/> τὴν κεφαλήν. </s> <s id="g0133009"> τοῦτο δὲ ἔσται, ἐὰν ἡ ΓΔ ἐφ' ἧς τὰ ΓΖ <lb/> γένηται, καὶ ἅμα ἀναστῆναι συμβήσεται, καὶ ἔχειν <pb xlink:href="080/01/023.jpg" ed="Bekker" n="858a"/> <lb/> ἐπὶ τῆς αὐτῆς ἴσης τὴν κεφαλήν τε καὶ τοὺς πόδας. ἡ δὲ ΓΖ <lb/> ὀξεῖαν ποιεῖ γωνίαν πρὸς τὴν ΒΓ. </s> </p> <p n="50"> <s id="g0133101prop31"> <lb/> Διὰ τί ῥᾷον κινεῖται τὸ κινούμενον ἢ τὸ μένον, οἷον <lb/> τὰς ἁμάξας θᾶττον κινουμένας ὑπάγουσιν ἢ ἀρχομένας; </s> <s id="g0133102"> <lb/> ἢ ὅτι χαλεπώτατον μὲν τὸ εἰς τοὐναντίον κινούμενον κινῆσαι <lb/> βάρος; ἀφαιρεῖται γάρ τι τῆς τοῦ κινοῦντος δυνάμεως, κἂν <lb/> πολὺ θᾶττον ᾖ· ἀνάγκη γὰρ βραδυτέραν γίνεσθαι τὴν ὦσιν <lb/> τοῦ ἀντωθουμένου. </s> <s id="g0133103"> δεύτερον δέ, ἐὰν ἠρεμῇ· ἀντιτείνει γὰρ καὶ <lb/> τὸ ἠρεμοῦν. </s> <s id="g0133104"> τὸ δὲ κινούμενον ἐπὶ τὸ αὐτὸ τῷ ὠθοῦντι ὅμοιον <lb/> ποιεῖ ὥσπερ ἂν εἰ αὐξήσειέ τις τὴν τοῦ κινοῦντος δύναμιν <lb/> καὶ ταχυτῆτα· ὃ γὰρ ὑπ' ἐκείνου ἂν ἔπασχε, τοῦτο αὐτὸ <lb/> ποιεῖ εἰς τὸ πρὸ ὁδοῦ κινούμενον. </s> </p> <p n="51"> <s id="g0133201prop32"> <lb/> Διὰ τί παύεται φερόμενα τὰ ῥιφέντα; </s> <s id="g0133202"> πότερον ὅταν <lb/> λήγῃ ἡ ἰσχὺς ἡ ἀφεῖσα, ἢ διὰ τὸ ἀντισπᾶσθαι, ἢ διὰ <lb/> τὴν ῥοπήν, ἐὰν κρείττων ᾖ τῆς ἰσχύος τῆς ῥιψάσης; </s> <s id="g0133203"> ἢ ἄτοπον <lb/> τὸ ταῦτ' ἀπορεῖν, ἀφέντα τὴν ἀρχήν. </s> </p> <p n="52"> <s id="g0133301prop33"> <lb/> Διὰ τί φέρεταί τι οὐ τὴν αὑτοῦ φοράν, μὴ ἀκολουθοῦντος <lb/> καὶ ὠθοῦντος τοῦ ἀφέντος; </s> <s id="g0133302"> ἢ δῆλον ὅτι ἐποίησε τοιοῦτον <lb/> τὸ πρῶτον ὡς θάτερον ὠθεῖν, καὶ τοῦθ' ἕτερον· </s> <s id="g0133303"> παύεται δέ, <lb/> ὅταν μηκέτι δύνηται ποιεῖν τὸ προωθοῦν τὸ φερόμενον ὥστε <lb/> ὠθεῖν, καὶ ὅταν τὸ τοῦ φερομένου βάρος ῥέπῃ μᾶλλον τῆς <lb/> εἰς τὸ πρόσθεν δυνάμεως τοῦ ὠθοῦντος. </s> </p> <p n="53"> <s id="g0133401prop34"> <lb/> Διὰ τί οὔτε τὰ ἐλάττονα οὔτε τὰ μεγάλα πόρρω φέρεται <lb/> ῥιπτούμενα, ἀλλὰ δεῖ συμμετρίαν τινὰ ἔχειν πρὸς <lb/> τὸν ῥιπτοῦντα; </s> <s id="g0133402"> πότερον ὅτι ἀνάγκη τὸ ῥιπτούμενον καὶ <lb/> ὠθούμενον ἀντερείδειν ὅθεν ὠθεῖται; </s> <s id="g0133403"> τὸ δὲ μηθὲν ὑπεῖκον διὰ <lb/> μέγεθος ἢ μηδὲν ἀντερεῖσαν δι' ἀσθένειαν οὐ ποιεῖ ῥῖψιν <lb/> οὐδὲ ὦσιν. </s> <s id="g0133404"> τὸ μὲν οὖν πολὺ ὑπερβάλλον τῆς ἰσχύος τῆς <lb/> ὠθούσης οὐθὲν ὑπείκει, τὸ δὲ πολὺ ἀσθενέστερον οὐδὲν ἀνερείδει. </s> <s id="g0133405"> <lb/> ἢ ὅτι τοσοῦτον φέρεται τὸ φερόμενον, ὅσον ἂν <lb/> ἀέρα κινήσῃ εἰς βάθος; τὸ δὲ μηδὲν κινούμενον οὐδ' ἂν <lb/> κινήσειεν οὐδέν. συμβαίνει δὴ ἀμφότερα τούτοις ἔχειν. </s> <s id="g0133406"> <pb xlink:href="080/01/024.jpg" ed="Bekker" n="858b"/> <lb/> τό τε γὰρ σφόδρα μέγα καὶ τὸ σφόδρα μικρὸν ὥσπερ οὐθὲν <lb/> κινούμενά ἐστι· τὸ μὲν γὰρ αὐτὸ καθ' ἓν κινεῖ, τὸ δ' <lb/> οὐθὲν κινεῖται. </s> </p> <p n="54"> <figure id="id.080.01.024.1.jpg" xlink:href="080/01/024/1.jpg"/> <s id="g0133501prop35"> <lb/> Διὰ τί τὰ φερόμενα ἐν τῷ δινουμένῳ ὕδατι εἰς τὸ <lb/> μέσον τελευτῶντα φέρονται; </s> <s id="g0133502"> πότερον ὅτι μέγεθος <lb/> ἔχει τὸ φερόμενον, ὥστε ἐν δυσὶ κύκλοις εἶναι, τῷ μὲν <lb/> ἐλάττονι τῷ δὲ μείζονι, ἑκάτερον αὐτοῦ τῶν ἄκρων. ὥστε <lb/> περισπᾷ ὁ μείζων διὰ τὸ φέρεσθαι θᾶττον, καὶ πλάγιον <lb/> ἀπωθεῖ αὐτὸ εἰς τὸν ἐλάττω. ἐπεὶ δὲ πλάτος ἔχει τὸ <lb/> φερόμενον, καὶ οὗτος πάλιν τὸ αὐτὸ ποιεῖ, καὶ ἀπωθεῖ εἰς <lb/> τὸν ἐντός, ἕως ἂν εἰς τὸ μέσον ἔλθῃ. </s> <s id="g0133503"> καὶ τότε μένει διὰ <lb/> τὸ ὁμοίως ἔχειν πρὸς ἅπαντας τοὺς κύκλους τὸ φερόμενον, <lb/> διὰ τὸ μέσον· καὶ γὰρ τὸ μέσον ἴσον ἀπέχει ἐν ἑκάστῳ <lb/> τῶν κύκλων. </s> <s id="g0133504"> ἢ ὅτι ὅσων μὲν μὴ κρατεῖ ἡ φορὰ τοῦ δινουμένου <lb/> ὕδατος διὰ τὸ μέγεθος, ἀλλ' ὑπερέχει τῇ βαρύτητι <lb/> τῆς τοῦ κύκλου ταχυτῆτος ἀνάγκη ὑπολείπεσθαι καὶ βραδύτερον <lb/> φέρεσθαι. </s> <s id="g0133505"> βραδύτερον δὲ ὁ ἐλάττων κύκλος φέρεται· <lb/> τὸ αὐτὸ γὰρ ἐν ἴσῳ χρόνῳ ὁ μέγας τῷ μικρῷ στρέφεται <lb/> κύκλῳ, ὅταν ὦσι περὶ τὸ αὐτὸ μέσον. </s> <s id="g0133506"> ὥστε εἰς τὸν <lb/> ἐλάττονα κύκλον ἀναγκαῖον ἀπολείπεσθαι, ἕως ἂν ἐπὶ τὸ <lb/> μέσον ἔλθῃ. </s> <s id="g0133507"> ὅσων δὲ πρότερον κρατεῖ ἡ φορά, λήγουσα <lb/> ταὐτὸ ποιήσει. δεῖ γὰρ τὸν μὲν εὐθύ, τὸν δὲ ἕτερον κρατεῖν <lb/> τῇ ταχυτῆτι τοῦ βάρους, ὥστε εἰς τὸν ἐντὸς ἀεὶ κύκλον <lb/> ὑπολείπεσθαι πᾶν. </s> <s id="g0133508"> ἀνάγκη γὰρ αὐτὸ ἐντὸς ἢ ἐκτὸς κινεῖσθαι <lb/> τὸ μὴ κρατούμενον. </s> <s id="g0133509"> ἐν αὐτῷ δὴ τοίνυν ἐν ᾧ ἐστίν, <lb/> ἀδύνατον φέρεσθαι τὸ μὴ κρατούμενον. ἔτι δὲ ἧττον ἐν τῷ <lb/> ἐκτός· θάττων γὰρ ἡ φορὰ τοῦ ἐκτὸς κύκλου. </s> <s id="g0133510"> λείπεται δὲ <lb/> εἰς τὸν ἐντὸς τὸ μὴ κρατούμενον μεθίστασθαι. ἀεὶ δὲ ἕκαστον <lb/> ἐπιδίδωσιν εἰς τὸ μὴ κρατεῖσθαι. </s> <s id="g0133511"> ἐπεὶ δὲ πέρας τοῦ μὴ κινεῖσθαι <lb/> ποιεῖ τὸ εἰς μέσον ἐλθεῖν, μένει δὲ τὸ κέντρον μόνον, <lb/> ἅπαντα ἀνάγκη εἰς τοῦτο δὴ ἀθροίζεσθαι. </s> </p> </chap> </body> <back/> </text> </archimedes>